division of labour and innovation with indivisibilities ... · division of labour and innovation...
TRANSCRIPT
Bank i Kredyt 43 (6) , 2012, 7–28
www.bankandcredit.nbp.plwww.bankikredyt.nbp.pl
Division of labour and innovation with indivisibilities: lessons from A. Smith
Krzysztof Makarski*
Submitted: 28 March 2012. Accepted: 22 May 2012.
AbstractWe study division of labour, innovation, and economic growth in a world with indivisibilities. In order to increase the division of labour, more specialized, labour saving capital varieties must be invented, which can be done only subject to a minimum size requirement. Thus, the division of labour is limited by the size of the market. Furthermore, the division of labour has a major impact on labour productivity. Due to the minimum size requirement, producers want to charge nonlinear prices (two part tariffs). In the model we obtain interesting dynamics. Depending on the parameters our economy can grow unboundedly, can grow up to a certain (even very high) level and then stagnate, or can be stuck in a poverty trap.
Keywords: innovation, indivisibility, division of labour, extent of the market
JEL: J22, O30, O40, D90, D24, D40
* National Bank of Poland; Warsaw School of Economics; e-mail: [email protected].
K. Makarski8
1. Introduction
Most of the economic literature on the division of labour, following A. Smith (1776) and A. Young (1928), emphasizes the importance of the extent of the market as a constraint on the division of labour. A good review of the literature is given by Lavezzi (2001) and Yang and Ng (1998). Since the classic paper of Dixit and Stiglitz (1977), the theory that formalizes economic of scale has been developed. This literature includes, among others, Ethier (1982), Krugman (1979), Judd (1985), Romer (1986; 1987), and Grossman and Helpman (1989). In these papers intermediate goods are aggregated into final goods with a Dixit-Stiglitz technology, thus it is optimal to have as many intermediate goods as possible. Since production of each intermediate good is subject to a fixed cost, the extent of the market limits the number of intermediate goods and therefore specialization.
Houthakker (1956) suggested another way to model the division of labour. In his view there is a trade-off between economies of specialization and the transaction costs associated with specialization. Papers that explore this line of reasoning include Baumgardner (1988), Kim (1989), Yang and Borland (1991), and Becker and Murphy (1992). This literature recognizes “coordination costs”, not the extent of the market, as the major constraint on the division of labour.
Literature on external economies of scale is also related to our topic. In a seminal paper Chipman (1970) argues that external economies of scale could be logically modelled as perfectly competitive. Markusen (1990) provides microfoundations for external economies of scale and shows, when ad hoc specifications used in trade and growth theories, are consistent with those microfoundations. In a more recent paper Grossman and Ross-Hansberg (2010) study external economies of scale at the industry level using the model of Bertrand competition. All those models are based on increasing returns of scale.
We follow the approach proposed by Edwards and Star (1987). They argue that “labour specialization results in scale economies only through indivisibility or other nonconvexity in the use of labour”. But, we introduce indivisibility on capital, rather than labour. In our model the division of labour is limited by the extent of the market without increasing returns on the firm level. Increasing returns arise endogenously and only in an aggregate production function. In our model, the division of labour and capital are related. In fact, the division of labour is embodied in the specialization of capital. The increase in capital diversity leads to more differentiated tasks being performed in the economy and increases the division of labour. As capital becomes more diversified, labour becomes more productive. Thus, technological progress is labour saving, as defined by Acemoglu (2003). The ideas that more advanced technology requires more advanced capital was explored by Boldrin and Levine (2002), but in their model there is no indivisibility and no division of labour.
In our model there are infinitely many technologies, which differ in the level of sophistication. In order to use more sophisticated technology (with more diversified capital) the minimum size requirement must be met. This indivisibility is not at the individual firm level, but rather at the aggregate level. This captures the idea, that in order to use more advanced capital, firms need access to engineers, educated labour force and so on. Thus, action of other firms matter. The key mechanism in our model works as follows. As the extent of the market increases, there is more innovation (i.e. more sophisticated technologies are adopted) and diversity of capital increases, labour productivity increases, which in turn increases the extent of the market. This model is
Division of labour and innovation with indivisibilities... 9
consistent with A. Smith’s claim that the division of labour is limited by the extent of the market, but explores a different mechanism than increasing returns. Moreover, since trade increases the size of the market, it increases the division of labour and labour productivity. Thus it also increases GDP and welfare. It is also worth mentioning that this indivisibility is a source of nonconvexities in the aggregate production function. Perpetual growth requires the economy to adopt ever better technologies, but this process might stop if, at some point, the economy fails to fulfil the minimum size requirement.
We also perform numerical simulations to show dynamic behaviour of the model. We find that the model − depending on parameters or initial conditions − can deliver quite rich dynamics. First, in the model there can be both unbounded growth and no growth. Second, poverty traps are possible. Third, it is possible to grow for some time (even very long) and develop quite sophisticated technologies and then stagnate.
This environment creates problems for the standard notion of equilibrium, so we have to make modifications. The main modification is that the producers are allowed to choose nonlinear prices, in the form of two part tariff pricing structures. With introduction of the two part tariff pricing structures equilibrium exists and is efficient. The two part tariff pricing structures were studied by Brown (1991) in the context of natural monopoly.
The paper is organized as follows. In section 2 we analyse the efficient division of labour. In section 3 we show problems with the standard notion of competitive equilibrium in this environment. In section 4 we propose the notion of equilibrium and establish its properties. Then we conclude the paper.
2. Environment and effi cient division of labour
We consider a finite or infinite period economy,1 with a measure one of homogeneous agents. Each consumer has the constant relative risk aversion (CRRA) utility function )1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
, where ct and 1/θ denote, respectively, consumption and the elasticity of intertemporal substitution consumption. Agents discount future with the discount factor δ < 1 and the total lifetime utility is given by
(1)
There is an infinite number of technologies in the consumption good sector indexed by n ∈ N, where N denotes the set of natural numbers (to avoid confusion we want to clarify that it does not include zero). The production process of technology n ∈ N is divided into n steps. Let
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
, be the set of all possible steps of all possible technologies, the pair ( j, n) denotes step j of technology n. The stock of capital in period t is denoted as ≥ 0,jntk jnk jn where
jntk jnk jn is interpreted as the capital specific to step j of technology n. This capital can be used in the production of the consumption good. The production process that uses technology n is represented by the following production function
1 We denote the time horizon of the economy as
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
. All the results in the paper hold for both fi nite and infi nite version of the model.
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
K. Makarski10
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(2)
where jntjntjnt l jnl jnky ,, represents, respectively, production, capital and labour employment in step j of technology n at period t.
Assume that f is homogeneous of degree 1, and
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
. Note that the Cobb-Douglas production function,
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
satisfies this conditions, therefore we are using it in all the examples throughout the text. The motivation for this production function is that technology with higher n uses more diversified (advanced) capital and thus is more productive. Each technology uses a different type of capital. Therefore, also as technologies become more sophisticated, we get more division of labour. Notice, that there is no minimum size requirement in the production process of consumption good. Capital is produced from the consumption good, once capital is produced it is technology specific. In order to produce capital specific to step j of technology n, denoted as kjnt+1, the minimum size requirement bn /n has to be met. The following production function represents its production process. The quantity of consump tion good used in production of capital specific to step j of technology n is denoted by
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
.
Note, that for simplicity we assume that capital depreciates fully after each period.
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(3)
Assume that
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
, which captures the idea, that the more advanced capital, the bigger the minimum size requirement.2
We assume exogenous supply of labour by household, which is normalized to one. Therefore the sum of labour services across technologies has to satisfy
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(4)
The feasibility condition for the economy is
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(5)
Next, we define an efficient allocation, and show some of its properties.
Definition 1. An efficient allocation is an allocation
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
such that it solves
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
2 This production process modifi es the production function used by Boldrin and Levine (2002) by adding the minimum size requirement and the division of labour.
Division of labour and innovation with indivisibilities... 11
subject to the feasibility condition, the minimum size requirement
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(6)
(7)
labour feasibility (4), and the production function for consumption good (2).
Notice that in order to simplify the definition formulas (5) and (3) are replaced with (6) and (7).Since a technology with a higher index is superior to all technologies with a lower index,
the social planner uses only one technology in each period. Once a more advanced technology is active (after the minimum size requirement is met), there is no point in using inferior technologies. The next proposition states this result formally.
Proposition 1. In any efficient allocation, for any period t, if
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
then for all n' ∈ N,
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
, we have
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
, i.e. there is only one active technology.Proof. By way of contradiction, suppose that there are two technologies active i.e. there exist
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
such that for all
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
and for all
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
. Without loss of generality assume
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
. Then technology n dominates technology n' and by investing in period t in technology n production all that was invested in n' gives higher product with the same resources (investment and labour), thus for all
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
. Contradiction.
Notice, that without full depreciation after one period, this result would not hold, since residual capital from previous periods would have been used in production. Furthermore, in the following proposition we show that steps of the same technology are symmetric.
Proposition 2. In any efficient allocation, for
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
, for all
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
and
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
i.e. all steps within the same technology are symmetric. Proof. Follows from the form of the production functions of consumption and capital goods.
We know from Proposition 2 that allocation in each step of the same technology is symmetric. So we can simplify notation by redefining variables (we do not need to keep track how capital and labour is divided into different steps of the same technology). Denote, respectively, output, labour and capital employed in the production process involving technology n as knt, lnt and ynt. Notice that in any efficient allocation
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
. The production process can be represented by the following production function.
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(8)
and the feasibility conditions (including minimum size requirement)
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(9)
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(10)
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
(11)
K. Makarski12
Next we define an efficient allocation using new notation.
Definition 2. An efficient allocation is an allocation
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
such that it solves
)1)/(1(=
= ≤
≥
∈
)( 1tt ccu
)(0=
tt
T
tcu
,,:),{( NnjnjJ }nj
0,jntk jntk
1=for ,),(1>for ,}),,({min
=
+
1
jlknfjylknf
yjnt
njnt
ntjjntn
jntjnt
jntjntjnt lky ,,
0>kf , 0<kkf , 0>lf , 0<llf
1)(=),( jntn
jntjntn
jnt lklkf
}{NT
1+
+
+
+
+
+
+
+
+ +
+ +_
+
+
+
+
jntk
nbn/
jnt
otherwise,0/,
=:1>
/1)0(=for
for
,=:1=
1
11
nbkn
bkn
njntjntjnt
jntjntjnt
nn bb >1
1=),(
jntJnjl
nntJnn
jntJnj
t yc),(
1),(
=
Ttnjjntjntjntt lkyc 0=),( }),,(,{
)(max0=
0=}),(),,(,{t
tT
tTtJnjjntljntkjntytc
cu
nntJnn
jntJnj
t ykc),(
1),(
≠
nbk njnt /1
t, if 0>1jntk Nn ,
nn 0=1tnjk
Nnn, nj 0>1jntk nj , 0>1tnjk
nn >
nj ,
,
0=1tnjk
0>t Jnj ),( , ,
,
1>j , 111 = ntj
+
++
_ 11nt+1jnt+1jnt j+
++
_ 11ntj
jnt kk
= ll = yy
jntnjnt kk =
nntnt yy =
),(= ntn
ntnt lkfy
ntNn
ntNn
t ykc =1
0=or 11 ntnnt kbk
1=ntNnl
TtNnntntntt lkyc 0=}),,(,{
)(max0=
0=}),,(,{t
tT
tTtntlntkntytc
cu
θ
δΣ
θ– ––
γ
γ
κ
κ
κ
κ
κ
κ
∞
γ αα
γ–
–
∈ ∪
≥
≥
Σ
Σ Σ
Σ
Σ Σ
∈
∈ ∈
J∈
∈
∈ ∈
∈
∈
∈
∈ ∈
∈
Nn∈
∈
δ
≥
≤
≥
≤
≤
≤≤Σ Σ
Σ
Σ
Σ
Σ
γ
δ
jntnjnt ll =
subject to the feasibility conditions (including the minimum size requirement (9)−(11) and the production function for consumption good (8).
Notice that these two specifications are equivalent, in particular Proposition 1 holds for the second specification. Next, using Proposition 1 we can construct the aggregate technology that describes how to trade consumption today for consumption tomorrow. This technology is presented in Figure 1. Next, building on the proof by Jones and Manuelli (1990), we show that the solution to the social planner problem exists.
Proposition 3. There exists an efficient allocation. Proof. See Appendix.
This shows that there exists an efficient allocation and in any efficient allocation only one technology is active at each time. It does not mean that there exists a unique solution. In fact, there might be more than one solution as we show in the following example.
In order to illustrate possible types of solutions to the social planner problem we show an example with two periods and only two technologies, {1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
.3 In this case in the second period consumers consume everything and save nothing, so the optimal choice can be presented graphically. Let y0 be the initial aggregate output
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
, which is determined uniquely
3 Note that this restriction on the set of technologies here and in the following examples is just for the ease of exposition.
Figure 1Aggregate technology
b2 b3 b4
yt+1
yt –ct
Division of labour and innovation with indivisibilities... 13
by the initial
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. Next, we present the types of choice that the social planner makes for different values of y0. There are four types of solutions. First, the planner chooses technology 1 (and there is no indivisibility to operate technology 1), which is shown in Panel A of Figure 2. Second, the planner chooses technology 2, but the indivisibility is binding, which is shown in Panel C of Figure 2. Third, the planner is indifferent between technology 2 and 1, so there are two solutions, which is shown in Panel B of Figure 2. Fourth, the planner finds technology 2 optimal and the indivisibility is not binding, which is shown in Panel D of Figure 2.
In the following sections we propose the concept of equilibrium to be used in this context. First, in the next section we show that the standard competitive equilibrium does not always exists, and then in the following section we propose a concept of equilibrium that assures existence.
3. Nonexistence of standard competitive equilibrium
In the standard definition of competitive equilibrium consumers choose a consumption path, in order to maximize utility, and producers choose production, capital and labour employment to maximize profit.
While solving the following problem, households take period zero prices of consumption goods pt, period zero real wages wt, period zero prices of type n capital qnt and profits π as given.
Figure 2Optimal choice given different initial conditions
Optimal choice
Output
c1
IC
y0 _ c0
b2 y0
Optimal choice
Output
c1
IC
y0_ c0
b2 y0
Output
c1
IC
y0 _ c0
b2 y0
Optimal choice
Output
c1
IC
y0 _ c0
b2 y0
Optimal choice
Panel A Panel B
Panel C Panel D
K. Makarski14
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
(12)
Note that labour supply is normalized to one. Producers, similarly as consumers, take prices as given and solve the following problem
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
(13)
Next, we define a competitive equilibrium.
Definition 3. A competitive equilibrium is an allocation
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
, and prices
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
such that 1. given prices,
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
solves the consumers problem (12). 2. given prices,
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
solves the producers problem (13). 3. markets clear
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
Next we show that given
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
,
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
and b2 = 0.3 competitive equilibrium does not exist. We use Figure 3 to explain it intuitively. When agents face the return to savings given by technology 2, they do not save enough to meet the minimum size requirement. The optimal choice is denoted by
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. On the other hand, when they face the return to savings given by technology 1, they actually save enough to meet the minimum size requirement. Then, the optimal choice is denoted by
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. But, then it is profitable to introduce technology 2.4
Next we show the counterexample when a standard competitive equilibrium does not exist.
Example 1. A competitive equilibrium in this economy (given
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
,
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
, and
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
does not exist.
Proof. By way of contradiction, suppose that a competitive equilibrium exists.Part 1: Suppose that in a competitive equilibrium
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
, then
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. From the households problem and from the producers problem
4 There is also another reason why equilibrium does not exist. There do not exist prices (in a competitive equilibrium) such that it is not optimal to use superior technology, when there are not enough savings to meet the minimum size requirement. And if the producers “think” they can buy as much capital as possible in period 0 they will always want more capital than it is available.
Division of labour and innovation with indivisibilities... 15
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
and from the producers problem
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. Solving
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
, and
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. Contradiction.
Part 2: Suppose that in a competitive equilibrium k21 = 0. In that case, from the producers problem,
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥
. From the households problem and from the producers problem
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥Solving
{1, 2}n
0{1,2}0 = nn yy
{1,2}0 )( nnk
000=
0=0=}{
=][.
)(max
nnNn
ttt
T
t
tt
T
tTttc
kqwcptos
cuδ
γ
π
π
+_
__
_
0=or
),(.
][max=
≤
≥ 11
1
00
00=
0=}),(,{
+
+
+
+
ntnnt
ntntn
ntt
nnntttt
T
ttNnntlntktc
kbk
klkfctos
kqlwcp
tc{ , ntk( , Ttntl 0=})
Ttntt qwp 0=0 })(,,{
Tttc 0=}{
ntk{( , Ttntl 0=})
1=ntl
),(=1 ntn
ntntt lkfkc +
1=T , 2= , 0.99= , 0.5= , 2= ,
(0.5, 0)=)( {1,2}0 nnk
2 00 )( Techcy
1 00 )( Techcy
1=T , 2= , 0.99=
0.5= , 2= , (0.5, 0)=)( {1,2}0 nnk , and 0.3=2b
021k
1=20q
),(==)()(
212
211
0
1
0 lkfpp
cucu
k'
'
221 bk 1=0y 221 <0.263= bk
0>> 112 kb
),(==)()(
111
111
0
1
0 lkfpp
cucu
k'
'
1=0y , 211 0.305= bk
Σ
Σ
Σ
ΣΣ Σ
Σ Σ
Σ
ΣΣ Σ
∈
∈
∈
∈
∈
∞
∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
Nn∈ Nn∈
Nn∈
Nn∈
Nn∈
∈
γ
γ
γ
γ
γ
δ
δ
δ
δ
α
α θ
θ
_
_
≥
≥
≥ . Contradiction. Thus a competitive equilibrium does not exist.5 This counterexample shows that the standard competitive definition is not well suited for
this environment. In the next section we introduce the modifications to this definition so that the problem is well defined.
4. Competitive equilibrium with two part tariffs
In this section we introduce the concept of a competitive equilibrium with two part tariffs (CETPT). In the previous section, we showed that a standard competitive equilibrium does not exist. Here we propose modifications. We allow producers to offer two part tariffs. The introduction of the two part tariffs scheme also requires further changes. In particular, if producers are allowed to sell
5 We want to be clear that we do not show that it never exists, it does not exist for this parametrisation. In fact one can find cases when it exists.
Figure 3Non-existence of standard competitive equilibrium
(y0 – c 0)Tech 1
c1
y0 –c0
IC 2
IC 1
f 1
f 2
y0b2 (y0 – c0 ) Tech 2
K. Makarski16
any quantity they want in the two part tariffs scheme most of the time they would sell as little as possible. Thus, we require that firms must satisfy the demand from consumers at the prices they offer. Furthermore, in our set up, firms own capital. In the spirit of Makowski’s (1980) definition of a competitive equilibrium, a competitive allocation is required to be a no surplus allocation.
4.1. Definition
The producers of consumption goods can either sell it to the consumers or transform it in the technology specific capital. Furthermore, producers offer two part tariffs pricing schemes to the consumers and have to satisfy whatever the demand is. Let Q be the set of prices of the consumption good R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
, where τ denotes an entry price and p a unit price of the consumption good. Let
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
be the sequence of prices,
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
. Let pt be period zero price of period t unit of consumption good and let τt be period t entry price in period t. Thus, pt τt is period 0 entry price in period t. We assume free entry on the producer side which means, that if prices that benefit consumers and generate positive profits exist, somebody will offer them. We assume that wages are determined competitively and we denote the period zero wage rate in period t as wt . We assume that trade takes place in period 0, similarly like in Arrow-Debreu world. Denote the set of initial technologies for which the stock of capital is nonzero as N0. In period 0 the consumers sell the initial capital to the producers at a competitive price, denote the price of capital specific to technology
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
as
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
. We introduce a definition of competitive equilibrium such that if the active technology is given it simplifies to the Arrow-Debreu definition. Additionally, when producers consider whether to introduce a new technology, they take into account an impact that an introduction of the new technology has on prices (therefore in this respect we deviate from Arrow-Debreu world), but nevertheless in equilibrium there is no surplus.
Households take prices as given and solve the following problem
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
(14)
Once the prices are set, given demand, producers solve the following problem
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
(15)
Next we define a competitive equilibrium with two part tariffs (CETPT).6
6 This concept of equilibrium is new and may seem non-standard. Therefore Makarski (2011) provides a game theoretic implementation of this definition.
Division of labour and innovation with indivisibilities... 17
Definition 4. A competitive equilibrium with two part tariffs (CETPT), is an allocation
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
and prices
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
such that
1.
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
are chosen competitively, i.e. for all t, for all n s.t.
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
,
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/ and for all
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/ . 2.
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
solves the households problem (14) given prices.
3.
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
solves the producers problem (15) given prices and demand.
4. markets clear
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
5. free entry(a) profits are zero, π = 0 (b) there does not exist another allocation
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
and prices
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
, such that conditions 1– 4 are satisfied, profits are positive
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
and utility is higher
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
Condition 5 says that there is free entry which in particular means that profits are zero, and there does not exist another set of prices that would benefit producers and consumers. It basically says that producers will switch to better technology if after lowering prices they will find consumers willing to buy enough products so that it is profitable. Furthermore, notice that there is no maximization with respect to consumption in condition 3, because producers do not choose how much consumption to sell, and have to satisfy all the demand.
4.2. Properties of equilibrium
In any CETPT, in any period t, there is only one active technology. This follows from the fact that a technology with a higher index is superior to any technology with a lower index. Therefore when producers start to use more advanced technology, wages in the economy increase and operating less advanced technologies becomes unprofitable. The following proposition states it formally.
Proposition 4. In any equilibrium, for any period t, if
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
, then for all
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
we have
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
, i.e. there is only one active technology. Proof. By way of contradiction, suppose that there exists
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
such that
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
and
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
(recall that all markets with no trade are shut down). Without loss of generality assume n > n' Since technology n is superior and the minimum size is satisfied no producer would use technology n', thus
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
. Contradiction.
K. Makarski18
The following proposition states that any efficient allocation can be decentralized as a CETPT. In the proof of this proposition we construct prices such that an efficient allocation is supported as an equilibrium allocation.
Proposition 5 (second welfare theorem). Any efficient allocation can be decentralized as CETPT.
Proof. Suppose that an allocation
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
solves the social planner problem. Construct prices:
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
, for all t, for all n s.t.
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
and for all
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
,
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
. Pick
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
so that the profits are zero. Then, by construction, the allocation
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
and the prices
R:),{(= pQ , 0}p ,
0=}{ ttQ
QQt
ttp
0Nn
0nq
000}>{0=
0=0=}{
])([.
)(max
nntttctt
T
t
tt
T
tTttc
kqwcptos
cu
+
+
+
+
+_
__
_
0=or
),(.
])([max=
11
1
000}>{0=
0=}),{(
+ ntnnt
ntntn
ntt
nnnttttctt
T
ttntlntk
kbk
klkfctos
kqlwcp
Ttntntt lkc 0=}),(,{ T
tntt qwQ 0=0 })(,,{
10p , tw , 00 )( nq n s.t. 0>ntl ,
),(=
=
ntn
ntltt lkfpw 0Nn , ),(= 000 nn
nkn lkfq
Tttc 0=}{
TtNnntnt lk 0=}),{(
1=ntl
),(=1 ntn
ntntNnt lkfkc +
Σ
Σ
Σ
Σ
ΣΣ Σ
Σ
Σ Σ
Σ
Σ
ΣΣ
Σ
ττ
τ
τ
τ
Δ
Δ
∈
∈
0Nn∈
Nn∈
Nn∈Nn∈
Nn∈ Nn∈ 0
Nn∈Nn∈0
Nn∈
∈
Nn∈
∈ Nn∈
Nn∈
Nn∈
⊂
≥
∞
∞
δ
π
π
≤
≤
≥
γ
γ
γ
π
∈
+
_ _+
tc{ , Ttntnt lk 0=}),(
Tttt wQ 0=},{
0>])([= 00
0
0}>{0=
nnttttctt
T
tkqlwcp
)(>)(0=0=
tt
T
tt
tT
tcucu
0>1+
+
+
+
+
ntk
Nn , nn ,
0=1tnk ,
Nnn,
0>1ntk
0>1tnk
0=1tnk
tc{ , Ttntntnt lky 0=}),,( 1=0p ,
)()/(= 0cucup tt
t n s.t. 0>ntl , ),(= ntn
ntltt lkfpw 0Nn
),(= 000 nn
nkn lkfq Ttt 0=}{
tc{ , Tttntnt lk 0=}),( T
tNnntt qwQ 0=00 )})(,,{
τ Δ
δδ
∈
∈
≠
Nn∈
∈Nn ∈
δ
γ
γ
τ
∈
/
constitute a CETPT. Intuition for the second welfare theorem can be shown on the graph from the example with
two periods and n ∈ {1, 2}. Recall from Figure 2 that there are four possible types of solution to the social planner problem. Next see Figure 4. The slope of the budget line is given by pt / pt+1. In the case of an interior solution, presented in Panel A and D, profits are zero and no entry fee is required. In the case represented in Panel A technology 2 is not available, so there are no firms operating technology 2. In the case of a corner solution represented in Panel C, the budget line is steeper than the line tangent to the production set. Since profits would be negative, in order to break even, they charge a positive entry price. Thus the consumers and the producers do not want to deviate. In the case represented on Panel B there are two equilibria, one like on Panel A and the other like on Panel C.
Slope = p0 / p1
Output
Panel C
Output
c1 c1
c1c1
y0 – c0 y0 – c0
y0 – c0y0 – c0
b2 b2 y0 y0
b2 y0 b2 y0
Panel A
Equilibrium
Slope = p0 / p1
Equilibrium
Slope = p0 / p1
Output
Panel D
Output
IC
IC
IC
IC
Panel B
Equilibrium
Figure 4Optimal choice given different initial conditions
Division of labour and innovation with indivisibilities... 19
From the second welfare theorem and the existence of an efficient allocation, we get the existence of a competitive equilibrium with two part tariffs. Furthermore the second welfare theorem and condition 6 of the Definition 4 imply that the first welfare theorem holds.
Proposition 6 (existence). A CETPT exists. Proof. From Proposition 3 an efficient allocation exists. Using the second welfare theorem this
allocation can be decentralized as a CETPT. Thus, a CETPT exists.
Proposition 7 (first welfare theorem). Any CETPT allocation is efficient. Proof. By way of contradiction, suppose that a competitive equilibrium with two part tariffs
is not efficient. Consider any efficient allocation and prices constructed by the second welfare theorem, that implement it. It is a CETPT. Now for t = 0, increase τt by ε and leave everything else the same. Then these prices still implement the efficient allocation (since profits are reimbursed to consumers) and the producers have positive profits. Thus there exists another allocation, such that a consumer has higher utility, zero profit condition is satisfied, indivisibility is satisfied, consumption sector conditions are satisfied, markets clear and profits of capital good firm are positive. Thus a free entry condition of the Definition 4 is violated. Contradiction.
4.3. Numerical exercises, sensitivity analysis
Next we perform numerical exercises7 to obtain some properties of CETPT equilibrium. This exercise demonstrates possible dynamic behaviour of the model. But, we do not show what are the necessary or sufficient conditions for the described paths. We find the following results.
7 For description of the solution method see Appendix.
Figure 5Unbounded growth (parameters: γ = b = 1.03, δ = 0.99)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
K. Makarski20
First, the economy is very sensitive to changes in the discount factor, δ. On Figure 5 and Figure 6 we show the change of the output path as a result of decrease in the discount factor. In both cases, we assume the following8 θ = 2, bn = 0.3bn, and γ = b = 1.03.9 The only difference is the discount factor, δ, which in case of Figure 5 is equal to 0.99, and in case of Figure 6 is equal to 0.97. Thus the change in the discount factor, changes the output path of the economy from the unbounded growth to stagnation. Furthermore, there is a threshold for δ between 0.97 and 0.99 such that above it, economy grows and below it, stagnates at the level of technology 1.
Second, this economy is sensitive to the initial conditions, and in particular might face poverty traps. On Figure 7 and Figure 8 we show the effect on the path of output of the change in the initial conditions. In both cases we assume the following parameters θ = 2, b0 = 0.3, δ = 97, and γ = b = 1.03. The only difference is the initial output, y0, which is higher in Figure 7, than in Figure 8, thus we obtain the poverty trap. It results from two factors (1) indivisibility and (2) θ > 1. The indivisibility makes the growth path costly in terms of current consumption. In order to sustain growth, consumers must give up a good part of the initial consumption. And they may or may not be willing to do that. Second, as the initial income goes up, with θ > 1, the willingness to save (or to invest) increases, thus it may be the case (like in the examples we show on the figures) that at some point consumers start to prefer the growth path to stagnation path.
8 Recall, θ is the utility function parameter, bn represents the size of indivisibility and γ productivity improvement associated with the new technology adoption.
9 Recall that 1/θ is elasticity of intertemporal substitution, bn is the minimum size requirement and γ is the parameter of the production function describing the improvement between successive technologies. In all our numerical simulations the necessary condition for unbounded growth was γ ≥ b, but we have not checked whether it is true in general.
Figure 6Bounded growth (parameters: γ = b = 1.03, δ = 0.97)
0
0.05
0.10
0.15
0.20
0.25
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Division of labour and innovation with indivisibilities... 21
Third, in case that γ < b in our simulations, we always obtain stagnation at some level. It is possible that the economy grows for a long period of time and then stops at some point. We also notice that the level of stagnation depends on the discount factor. On Figure 9 we show the effect of the increase of the discount factor on the stagnation level. In both cases we assume the following parameters θ = 2, γ = 1.0296 and b0 = 1.03. We find that the lower discount factor, the lower the level at which economy stagnates. Furthermore, notice that economy may stagnate at a very high level of sophistication. For example economies from Figure 9 stagnate at the level of technology 396, with δ = 0.99, and at 408, with δ = 0.991. Notice that γ < b assumption means that the indivisibility
Figure 7Poverty trap (initial conditions: y0 = 0.63, n0 = 0.20)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Figure 8Poverty trap escaped (initial conditions: y0 = 0.73, n0 = 0.25)
0
2
4
6
8
10
12
14
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
K. Makarski22
grows faster than the gain from indivisibility, thus in this economy the unbounded growth may not feasible.
We conclude that CETPT may be sensitive to the change of the discount factor and the initial conditions. Poverty traps are also possible. Furthermore, for certain parameters the unbounded growth is not feasible, but the economy may still grow for a very long time and stop at a very sophisticated level of development. In this case, the level at which economy stops depends on the discount factor.
5. Conclusion
In the model presented in this paper the division of labour is limited by the extent of the market. If the size of the market increases, it leads to more innovation, increase of the labour productivity and further increase of the size of the market. The output paths in this model are sensitive to the discount factor. Slight changes in the discount factor, may lead to big differences in the growth paths. Furthermore, this economy is sensitive to the initial conditions and poverty traps are possible. Moreover, we introduce a concept of a competitive equilibrium with two part tariffs. We show that this equilibrium exists, and both first and second welfare theorems hold.
We also show that the model is able to generate very rich dynamics (depending on parameters). First, we find that while patient enough economies grow unboundedly, the impatient ones stagnate. Second, the model gives rise to poverty traps. Finally, if the consumption improvements in technology are not high enough (comparing to the indivisibility steps) economy will eventually stagnate. Surprisingly, it can appear even after a long period of development at a very high level of technological advancement.
Figure 9Growth followed by stagnation
0
10,000
20,000
30,000
40,000
50,000
60,000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51
Discount factor = 0.99 Discount factor = 0.991
Division of labour and innovation with indivisibilities... 23
References
Acemoglu D. (2003), Labor- and capital-augmentingt change, Journal of the European Economic Association, 1(1), 1−37.Baumgardner J.R. (1988), The division of labor, local markets, and worker organization, Journal of Political Economy, 96(3), 509−527. Becker G., Murphy K. (1992), The division of labor, coordination costs, and knowledge, Quarterly Journal of Economics, 107(4), 1137−1160. Boldrin M., Levine D.K. (2002), Factor saving innovation, Journal of Economic Theory, 105(1), 18−41.Brown D.J. (1991), Equilibrium analysis with non-convex technologies, in: W. Hildebrand, H. Sonnenschein (eds.), Handbook of mathematical economics, Vol. 4, North-Holland, Amsterdam, New York.Chipman J. (1970), External economies of scale and competitive equilibrium, Quarterly Journal of Economics, 84(3), 347−363.Dixit A., Stiglitz J. (1977), Monopolistic competition and optimum product diversity, American Economic Review, 67(3), 297−308. Edwards B., Starr R. (1987), A note on indivisibilities, specialization, and economies of scale, American Economic Review, 77(1), 192−194. Ethier W.J. (1982), National and international returns to scale in the modern theory of international trade, American Economic Review, 72(3), 389−405. Grossman G.M., Helpman E. (1989), Product development and international trade, Journal of Political Economy, 97(3), 1261−1283. Grossman G.M., Rossi-Hansberg E. (2010), External economies and international trade redux, Quarterly Journal of Economics, 125(2), 829−858. Houthakker H.S. (1956), Economics and biology: specialization and speciation, Kyklos, 9(2), 181−189.Jones L.E., Manuelli R. (1990), A convex model of equilibrium growth: theory and policy implications, Journal of Political Economy, 98(5), 1008−1038.Judd K. (1985), On the performance of patents, Econometrica, 53(3), 579−585.Kim S. (1989), Labor specialization and the extent of the market, Journal of Political Economy, 97(3), 692−709.Krugman P. (1979), Increasing returns, monopolistic competition and international trade, Journal of International Economics, 9(4), 469−479. Lavezzi A.M. (2001), Division of labor and economic growth: from Adam Smith to Paul Romer and beyond, mimeo, University of Pisa.Makowski L. (1980), A characterization of perfectly competitive economies with production, Journal of Economic Theory, 22(2), 208−221.Makarski K. (2011), Game theoretic implementation of competitive equilibrium with two part tariffs, mimeo, Szkoła Główna Handlowa w Warszawie. Markusen J.R. (1990), Micro-foundations of external economies, Canadian Journal of Economics, 23(3), 495−508.Romer P.M. (1986), Increasing returns and long-run growth, Journal of Political Economy, 94(5), 1002−1037.
K. Makarski24
Romer P.M. (1987), Growth based on increasing returns due to specialization, American Economic Review, 77(2), 56−62.Smith A. (1776), An inquiry into the nature and causes of the wealth of nations, W. Stahan and T. Cadell, London. Yang X., Borland J. (1991), A microeconomic mechanism for economic growth, Journal of Political Economy, 34(3), 199−22. Yang X., Ng S. (1998), Specialization and division of labor: a survey, in: K. Arrow, Y. Ng, X. Yang (eds.) Increasing returns and economic analysis, Macmillan, London. Young A.A. (1928), Increasing returns and economic progress, The Economic Journal, 38(152), 527−542.
Acknowledgements
I thank Michele Boldrin, V.V. Chari, Lukasz Drozd and Nicolas Figueroa for helpful comments. All errors are mine.
Division of labour and innovation with indivisibilities... 25
Appendix
Proofs from section 2From Proposition 1 we know that the social planner in each period uses only one technology. Thus, to find a solution to the social planner problem it is enough to look among allocations that use only one technology. Before we proceed with the proof, we need to introduce new notation. Denote a technology active at time t + 1 as nt + 1. Denote the stock of capital, labour and production of technology nt + 1 as, respectively,
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
and
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
. It is convenient to define the
exogenous technology social planner problem (given
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
). In this problem technology is given exogenously, technology nt + 1 must be active at time t + 1, and the indivisibility
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
must be satisfied (which means that the option of not investing anything and consuming all the output is not available anymore).
Definition 5. The exogenous technology social planner problem (given
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
) is
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
(16) (17) (18)
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
(19)
(20)
Notice that a solution to this problem does not exist if
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
. Nevertheless if
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
for all t, the solution exists since
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
. To shorten notation, denote
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
as {n}. Define the set of sequences of technologies for which the problem above has solution as:
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
there exists a sequence
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
such that it solves the exogenous technology social planner problem (given {n})}.
Proof of Proposition 3. From Proposition 1 it is enough to show that there exists
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
and there exists an allocation
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
such that (1)
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
solves the exogenous technology social planner (given {n}); (2) there does not exist another sequence
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
and another allocation
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
that solves exogenous technology social planner (given {n'}), and
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
.
The proof is as follows. First, we prove that the set of all feasible technologies is compact. Next, we define an utility function on this set of technologies (it is given by an allocation that solves the exogenous technology social planner problem), and we show that it is continuous. This leads to the conclusion that there exists a solution (since a continuous function on a compact set attains its optimum).
Consider the metric on the set Ξ,
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
, where t + 1 is the smallest t + 1 such that
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
.
K. Makarski26
Nonemptiness. Notice that for the following sequence of technologies, {n}, where
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
for all t , the model becomes the standard one sector growth model and a solution to exogenous technology social planner problem exists. So the set Ξ is nonempty.
Compactness. Notice that given y0 and n0, there exist a sequence {n̑} such that
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
, which by Tychonoff theorem is compact in the product topology. Furthermore η is closed, which implies that it is compact. Since Ξ is compact in the product topology and any set compact in the product topology is compact in the topology induced by the metric d1 (due to the fact that any open set in the product topology is open in the topology induced by d1), then Ξ is compact in topology induced by metric d1.
Continuity. Define the feasible set
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
given y0, n0, there exists {n} such that
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
. Notice that for each t, yt is bounded from above, and by continuity of f it is closed, hence by Tychonoff theorem Ψ is compact in the product topology. Define the correspondence
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
, which assigns the set of feasible allocations to
the set of technologies. Consider the metric on Ψ:
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
. Since Ψ is
compact in the product topology and any set compact in the product topology is compact in the topology induced by the metric d2 (by the similar argument as above), Ψ is compact in the topology induced by d2. Given the metric on Ξ correspondence φ is continuous. Define the function
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
represented as
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
where
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
, then U is continuous. By the Maximum Theorem
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
defined by
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
is continuous.
Well defined? Assume there exists
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
such that for any
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
,
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
then the utility function is well defined.Since
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
is a continuous function defined on a compact set Ξ, it attains its maximum. This concludes the proof.
Solution methodAs we showed earlier a CETPT allocation and a planner allocation are equivalent. So, it is
enough to solve the planner problem. Recall that the planner operates only with one technology, thus we can rewrite the planner problem recursively
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
s.to
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
where the correspondence
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
is defined in the following way
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
Note that the inverse of this correspondence
1t 1tn
1tnk , 1=
≥
≤
≤
≤
≤
≥
∈
+
+
+
+
+ +
+
+
+
+
+
+
+
+
+ +
+
1tnl ),( tn
tnk
γ
γ
γ
Ξ
Ξ
∈
⊂ ×
×
∞
∞
Ξ
Ξ
Ξ
Ξ
Ξ
Ψ
Ψ
Ψ
Ψ
Ψ
Γ
Γ
Γ
Γ
Γ
Γ
φ
φ
β β
β
γ
δ∑
f
Tttn 0=1}{ )
11 tntn bk
Tttn 0=1}{
11
1
0=0=}
1,{
),(.
)(max
+
tntn
tn
tntnt
tt
T
tTttnktc
kb
kfkctos
cu
),(1
tn
tntnt kfkc
11 tntn kb
),(1
tn
tnt
+
+
1
1
t
n kfb
1=n
0=b
Tttn 0=1
+t 1
+t 1
+
+
t 1
+t 1
+t 1
+t 1 +t 1+t 1
+t 1 +t 1 +t 1 +
+
t 1 +t 1 +t 1
+t 1
}{
:}{{= nTtnt kc 0=},{
}{nTtnt kc 0=},{
Ttnt kc 0=},{
}{n
,
,
},{ 0=1
Ttnt t
kc
)(>
≤
→
→
→
→
≥≤
)(0=0= t
tT
tttT
tcucu
11
=
=
≠
}){},ˆ({1 tnnd
ˆ nn
}ˆ{n
}ˆ{1,...,0=t n
:}{{ 0=t
∞
∞
∞
0=t
0=
_
t
0=t
ty
ty and),( nnknfy , nkn }nn bkn
:
),(1),(
=}){},({2tt
ttt
yxdyxd
yxd
RU :
)( tt yu
ty
RU :
})}({}{:}){},({{max=})({ nyynUnU
0>
≥
, 1<1
}{ ty
0< yy t
)(U
)())((max=)( 1
0yVy'
'
yuyVy
yy )(1
++ RR: , )(ky
n
nnn
nn
n
n
bkbbk
bfbfkf
k=for
),(for )],,(),,([
),,(=)(
11
)(1 y
ʹ
ʹ ʹ
ʹ∑ ∑δ δ
+
γ
∑
∑
δ
δ
δ
δ
θ
∈
∈
∈
+
+
+_ _
_
_
_
t 1
≤
∈
∈γγ γ
is a function. With this specification of the state space we have only one state variable which simplifies numerical simulations.
Division of labour and innovation with indivisibilities... 27
The standard way of solving numerically this type of problems for economies that converge to the steady state is to restrict grid to points between zero and the steady state (or a little above the steady state). But here, since this economy in general does not converge to the steady state, we cannot proceed in the standard way. We propose the following solution of this problem. First, we fix the initial grid of y. We find numerically a value function (using the value function iterations method) and an output path given some initial conditions. Next we double the initial state space (in the sense that the maximal value of y is at least twice as big as the initial maximal value) run the simulations again and compare the results before and after enlargement: a value function, a path given the initial conditions. If the new results are close enough, we end the simulations.
