economic growth theory and the dynamics of the russian economy
TRANSCRIPT
Journal of Mathematical Sciences, Vol. 133, No. 4, 2006
ECONOMIC GROWTH THEORY AND THE DYNAMICS OF THE RUSSIAN ECONOMY
V. D. Matveenko∗ UDC 519.863
Using the endogenous growth model earlier proposed by the author, we study the interrelation between the growth rateand the investment rate as well as the connection of these variables with labor market institutions of the economy.The conditions leading to economic growth and recession are found. It is shown that an increase in the investmentrate is, generally speaking, neither necessary nor sufficient for an increase in the growth rate. Using the model, wediscuss the dynamics of the Russian economy, in particular, the transformational recession of the 1990s and theeconomic recovery that followed the 1998 crisis. Bibliography: 20 titles.
In this paper, we study relations between the growth rate and investmant rate and their links to labor marketinstitutions using an endogenous growth model proposed by the author in the early 1980s. This model (calledthe fK model) generalizes the AK model, which became popular in the recent decade. Questions concerningconditions of economic growth and economic decline are considered. It is shown that an increase in investmentrate is, generally speaking, neither necessary nor sufficient condition for economic growth. The dynamics ofthe Russian economy is discussed, in particular, the transformational decline of the 1990s and the rise of theeconomy following the 1998 crisis.
1. In the early 1980s, the author proposed a vintage model of economic growth, which, being a discrete timeversion of the Kantorovich model [1–3], included at the same time a lot of new elements. Simultaneously,an aggregated endogenous growth model was constructed (in what follows, it is called the fK model), whichdemonstrated the same mechanism of growth that acted in the vintage model more clearly. The author reportedboth models at Kantorovich’s seminar in Moscow, and Leonid Kantorovich suggested to publish the result involumes edited by him (see [4, 5]) and in the paper [6], which he communicated to Doklady Akad. Nauk SSSR (aprestigious journal, where a paper had to be communicated by a member of the Russian Academy of Sciences).Later the model was elaborated in [7, 8].
The topicality of studying the fK model not only did not decrease during these twenty years, but evenincreased, as the fK model can be regarded as a generalization of the AK model, now very popular in theworld. Apparently, the AK model was first discussed in [9], but it attracted special attention in the 1990s (see[10]), which can be explained by two reasons. First, the AK model is a simple model of endogenous growth, andthe interest to such models has sharply increased during the last two decades. Second, the AK model belongsto the Harrod–Domar type models. The main conclusion of this kind of models is the presence of a positivedependence between the investment rate and the growth rate. For half a century such models have been usedin the practice of international financial organizations (see [11]) and serve as a basis for decisions concerning thesize of aid to developing countries. The failure of many attempts to accelerate development by the use of foreignaid provoked a new wave of interest to theoretical and empirical studies based on endogeneous growth models.
The difference between the formulation of the fK model and that of other models of economic growth is inthe use of a production function of the form Y = Kf(V ), where Y is the output, K is the capital, V is thewages (consumption) per unit of capital (or, in a more general sense, the circulating capital). This function is ageneralization of the function Y = AK, A = const, used in the AK model. This choice of a production functionis justified on the basis of simple microeconomic models in Secs. 3 and 4. More profound microfoundations wereconsidered in [12, 13].
Note that such a variable as the consumption to capital ratio also plays a key role in some other endogenousgrowth models, in particular, in the Lucas model (see [14–16]).
Despite a rather small difference in formulation, the fK model differs considerably by its properties bothfrom traditional neoclassical models, where all stationary paths have the same long-run growth rate and themaximum production level is achieved under minimum wages, and from the AK model.
Our research shows that the fK model is free of the deficiencies of the AK model pointed out in [10, 11].The convergence takes place. A positive dependence between the investment rate and the growth rate takes
∗St.Petersburg Institute of Economics and Mathematics and European University at St.Petersburg, St.Petersburg, Russia, e-mail:
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 312, 2004, pp. 215–238. Original article submitted April 28,2004.
1072-3374/06/1334-1491 c©2006 Springer Science+Business Media, Inc. 1491
place only in a certain range of changes in the investment rate, while for higher values of the investment ratethe dependence is negative. Thus empirical results that cast doubt on the AK model do not contradict the fKmodel.
We study a whole family of paths of the fK model (among them, in particular, paths with constant investmentrates and optimal paths) and show that growth takes place only under relatively small investment rates, otherwisewe have decline. In other words, paths of decline are those where the wage (or the circulating capital) per unitof productive capital is too small or where it is too high.
These results show in what cases one must expect the failure of an attempt to accelerate development byincreasing investment.
The fK model is also suitable for explaining the dynamics of the Russian economy and, in particular, thetransformational decline of the 1990s and the rise following the 1998 crisis.
The term transformational decline was introduced by Janos Kornai to identify the decline in production afterthe price liberalization in the Central Eastern Europe and the former USSR. The nature of this phenomenondiffers essentially from the cyclic decline arising periodically in market economies. A lot of different causes ofthe transformational decline were suggested (see a survey in [12]).
In the Western literature on the theory of economic transition, the point of view prevails according to whichonly the new sector of the economy can form a basis for economic growth. (The new sector is formed by newlyemerged private firms and self-employment, while the old sector consists of state firms and privatized firms thatdid not pass restructuring; see details in [12].) However, the large share of the state sector in Russia, the veryslow speed of restructuring, and even attempts at renationalization make it relevant to consider a model ofeconomic development on the basis of the old sector.
2. The fK model is formulated in the following way:
Yt = Ktf(Vt), t = 1, 2, . . . ,
Yt = Kt+1Vt+1 + It+1,(1)
Kt+1 = νKt + It+1,
It+1 ≥ 0, t = 0, 1, . . . .(2)
Here Yt is the output in time period t, K is the capital, Vt is the wage per unit of capital, Ct = KtVt is theconsumption (total wages), It is the investment, ν ∈ (0, 1) is the share of capital that remains after depreciation(i.e., 1 − ν is the depreciation rate). It is assumed that the function f possesses the standard properties:
f(0) = 0, f ′(.) > 0, f ′′(.) < 0, f ′(0) > ν, limV →∞
f ′(V ) < ν.
The nonnegativity condition It+1 ≥ 0 is the condition of irreversibility of investment: the capital cannot beused for consumption.
Lemma 1. The condition of irreversibility of investment It ≥ 0 in period t (where t = 1, 2, . . .) is equivalent tothe inequality
Yt−1 ≥ νKt−1Vt.
Proof. SinceYt−1 − KtVt = It = Kt − νKt−1,
either It ≥ 0, Yt−1 ≥ KtVt ≥ νKt−1Vt or It < 0, Yt−1 < KtVt < νKt−1Vt, and the lemma follows. �Corollary 1. The condition of irreversibility of investment It ≥ 0 in period t (where t = 2, 3, . . .) in thenontrivial case Kt−1 ≥ 0 is equivalent to the inequality f(Vt−1) ≥ νVt.
3. Let us formulate two simple microeconomic models leading to a production function of the form Y = Kf(V ).In this section, we consider a model related to the Russian economy, and in the next section, a model related toan industrialized country.
Assume that for a certain period of time (e.g., a month), an individual possesses a time T which he dividesbetween the time of working in a firm of the old sector of the economy, L1, the time of moonlighting, L2, andthe leisure Le:
T = L1 + L3 + Le.
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In a simple model, it turns out that all the workers are secondary employed. In a more detailed model [13], itis shown that the possibility of moonlighting by itself considerably influences employment and wages in the oldsector even if not all individuals use these possibility.
The output of an old sector firm is described by the production function AF (K, NL1), which is assumed tohave constant returns to scale and other standard properties. Here N is the number of workers employed in thefirm.
In the sphere of moonlighting, an individual can get a job for any time with hourly wage rate W2. (The wageincludes the whole labor income in all numerous forms of payment used in Russia.)
The preferences of an individual are described by a utility function U(C, Le) with the standard properties.Here C is the total wage received by the individual during the period of time, both in the old sector and in thesphere of moonlighting.
An old sector firm proposes to an individual a factual time of work Lf (this flexible variable may differ fromthe working time written in the formal contract) and a per period (monthly) wage rate W 1. The individual maychoose one of the two alternatives:(1) to accept the proposal and, if it is profitable for him, to combine the work in the firm with moonlighting,
or(2) to reject the proposal and to work in the sphere of moonlighting only.Thus the individual solves the problem
max{
maxL2+Le=T−Lf
U(W 1 + W2L2, Le), maxL2+Le=T
U(W2L2, Le)}
.
It is easy to see that if W 1 < W2Lf , then the individual will prefer the second alternative, i.e., L1 = 0, but if
W 1 > W2Lf , he will choose the first alternative, i.e., L1 = Lf . In the boundary case, we assume that the first
variant is chosen.It follows that if the firm maximizes the actually received labor NLf for each value of its wage costs KV =
W 1N , then it solves the problemmaxNLf
under the conditionW2L
f ≤ KV
N(= W 1).
Obviously, at a point of maximum,
NLf =KV
W2.
Consequently, the factual (informally defined) hourly wage in the firm W f = KVNLf is equal to the reservation
hourly wage W2, and the output is equal to
F (K, NLf) = F
(K,
KV
W2
)= KF
(1,
V
W2
)= Kf(V ),
where the function f(V ), constructed under fixed parameter W2, possesses the standard properties: f(0) = 0,f ′(.) > 0, and f ′′(.) < 0.
Thus the fK model emphasizes the role played in the Russian economy by the (factual) labor supply, as wellas the role played in the formation of labor supply by the possibility of moonlighting. An emirical research ofthe influence of moonlighting possibilities on the labor supply in Russia is provided in [13].
4. A similar form of production function may be derived by the use of the efficiency wages model (see [17]).Let a production function F (K, e(W )N) depend on the capital K and the effective labor e(W )N . Here e(W )
is the effort of a worker dependent on his wage rate W ; N is the labor (the number of workers or hours). Thefunction F has constant returns to scale and other standard properties. Concerning the function e, it is assumedthat e(0) = 0, e′(.) > 0, and e′′(.) < 0.
The firm determines the wage rate W maximizing the effective labor e(W )N given the value of wage costsWN = KV . This problem can be reduced to finding the unconditional maximum of the function e(W )/W .Thus W is uniquely determined by the condition
e′(W )e(W )
W = 1.
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(It means that the function e(W ) has unit elasticity at the point W .) In spite of that,
F (K, e(W )N) = F
(K, e(W )
KV
W
)= KF
(1,
e(W )
WV
)= Kf(V ).
The function f(V ) possesses the standard properties. We have obtained the same form of production functionas above.
5. The vector St = (Kt, Yt) will be regarded as the state of the model at time period t, and Vt will be regardedas a control variable. Let a path {St} with initial state S0 = (K0, Y0) have a control sequence {Vt}∞t=1.
The numberw(St) = νKt + Yt
will be called the wealth of the economy in the state St. We see that
w(St) = (ν + f(Vt))Kt = Kt+1(1 + Vt+1), t = 1, 2, . . . ,
w(S0) = νK0 + Y0 = K1(1 + V1).
The growth factor of the wealth is
α(Vt+1) =w(St+1)w(St)
=(ν + f(Vt+1))Kt+1
(1 + Vt+1)Kt+1=
ν + f(Vt+1)1 + Vt+1
.
Let us describe the structure of the path with initial state S0 = (K0, Y0) and control sequence {Vt}∞t=1. Itfollows from (1) and (2) that
Ktf(Vt) −Kt+1Vt+1 = Kt+1 − νKt.
From here we find the growth factor of the capital:
Kt+1
Kt=
ν + f(Vt)1 + Vt+1
,
and the growth factor of the output:Yt+1
Yt=
ν + f(Vt)1 + Vt+1
f(Vt+1)f(Vt)
. (3)
Moreover,
K1 =w(S0)1 + V1
, (4)
and, consequently,
KT =w(S0)1 + VT
α(V1)α(V2) . . .α(VT−1), T ≥ 2, (5)
YT = KT f(VT ), T ≥ 1.
We have expressed the path in terms of its initial state and the control sequence.In the case of the AK model, the growth factors of the wealth of the economy, of the capital, and of the
output coincide and are equal to ν+A1+Vt+1
.
6. Let us describe balanced paths of the model. Denote by V ν the control for which
f(V ν) = νV ν .
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Proposition 1. For each control V such that V ≤ V ν, the path {St} with initial state S0 = K0(1, f(V )) thatis determined by the constant control Vt ≡ V is a balanced path with growth factor equal to α(V ); the modelhas no other balanced paths.
Proof. Let us check that the control V is applicable in the state S0. The control V may be used if and onlyif Y0 ≥ νK0V . With Y0 = K0(V ), the inequality is equivalent to f(V ) ≥ νV , which is in turn equivalentto V ≤ V ν .
Since for Vt = V the state St+1 is proportional to the vector (1, f(V )), the described path does exist and isindeed a balanced path.
Let us prove the uniqueness. If {St} is a balanced path with growth factor α that has control sequence {Vt}∞t=1,then it follows from the expressions for the capital and output growth factors that Vt = const, t = 1, 2, . . . .Moreover,
K1
K0=
Y1
Y0=
K1f(V1)Y0
,
whence Y0 = K0f(V1).This means that all balanced paths have the form described above. �
At a balanced path with control V , the growth factors of the wealth, capital, and output are all equal toα(V ). The maximum growth factor at a balanced path is reached under the control V that is a solution of thefollowing equation:
f ′(V ) = α(V ).
Assume that α(V ) > 1. It is not difficult to see that in this case the equation α(V ) = 1 has two roots; denotethem by V11 and V12 and let V11 < V12. We have V12 < V ν .
For V ∈ (V11, V12) we have α(V ) > 1, which means that growth takes place at balanced paths with controlV . For V ∈ (0, V11) ∪ (V12, V
ν), we have α(V ) < 1, and the corresponding balanced paths are paths of decline.The economic meaning of the indicated conditions is that economic decline takes place at a balanced path if
and only if the wage per unit of capital (wage “in a workshop”) V at this path is either too small or too high.For unbalanced paths, the condition of decline in period t + 1 > 1 is
f(Vt+1)1 + Vt+1
<f(Vt)
ν + f(Vt).
This condition also means that the wage per unit of capital Vt+1 is either relatively small or relatively high.In the special case of the AK model, where f(V ) ≡ A, the condition of decline takes the form
Vt+1 > ν − 1 + A,
i.e., decline in the AK model is related to either too high wage per unit of capital or too small productivity ofcapital (A < 1 − ν). However, in the AK model, in contrast to the fK model, decline cannot be caused by atoo small wage per unit of capital.
7. Now we turn our attention to a relation between the growth rate of a balanced path and the accumulation(investment) rate. Paths with constant rates of accumulation s ∈ (0, 1) will be called Solow paths.
In the fK model,
st =It+1
Yt=
Kt+1 − νKt
Yt=
Kt+1Kt
− ν
f(Vt)=
f(Vt) − νVt+1
(1 + Vt+1)f(Vt).
Hence, at a Solow path,
Vt+1 =f(Vt)(1 − s)ν + sf(Vt)
=1 − s
s + νf(Vt)
.
Thus at a Solow path the control sequence {Vt}∞t=1 monotonically converges to a steady control V (s), which isa solution of the equation
V =1 − s
s + νf(V )
.
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It is easy to verify that the function V (s) is decreasing. The inverse function (the accumulation rate at a balancedpath)
s(V ) =f(V ) − νV
(1 + V )f(V )
is also decreasing.It follows from the properties of the function α(V ) that the asymptotic growth factor of a Solow path (or the
growth factor of a balanced path) increases in s when s ∈ (0, s(V )) and decreases when s ∈ (s(V ), 1) (see thefigure).
It is easy to show that the growth rate of output at a Solow path is equal to
Yt+1
Yt= (1 − s)
f(Vt+1)Vt+1
.
Hence the sequence of growth rates of output converges monotonically to the steady state equal to ν +sf(V (S)).In the AK model, at a Solow path,
Vt+1 =1 − s
s + νA
= const .
Hence the growth rate of a Solow path in the AK model is constant and equal to sA+ν. Thus in the AK modelthere is a positive relation between the accumulation rate and the growth factor. It is the assumption of such akind of relation that lies in the basis of the methodology of international financial organizations (see [11]). Theabsence of such a relation in the fK model is its important feature.
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There is no convergence in the AK model. Barro and Sala-i-Martin [10] pointed this out as the most importantempirical shortcoming of the model. The fK model is free of this shortcoming.
To check the conclusion from the AK model empirically, Jones [18] considered 15 OECD countries and came tothe conclusion that though in the post-war period the investment rates increased, the growth rates did not. In hisopinion, this fact witnesses to the inconsistency of the AK model. This conclusion was disputed by McGrattan[19], who considered a wider sample of countries for a longer period of time. Note that neither Jones’s norMcGrattan’s results contradict the fK model, where there is no positive relation between the growth rate andthe accumulation rate.
8. In what follows, an important role will be played by paths of pure consumption, i.e. paths where theinvestment is equal to zero: It = 0, t = 1, 2, . . . .
Proposition 2. Each path of pure consumption {St} = {(Kt, Yt)} converges to a balanced path determined bythe control V ν, or, more precisely,
Kt+1
Kt≡ ν = α(V ν),
Yt
Kt→
t→∞f(V ν).
Proof. Let {St} be a path of pure consumption with control sequence {Vt}∞t=1. Then
Kt+1 = νKt, t = 0, 1, . . . ,
Yt = Ktf(Vt) = Kt+1Vt+1, t = 1, 2, . . . .
HenceVt+1 =
1ν
f(Vt). (6)
It follows from the properties of the function f that νV < f(V ) < νV ν provided that V < V ν .Consequently, if V1 < V ν (which is true for Y0 < νK0V
ν), then, by induction,
Vt < Vt+1 < V ν , t = 1, 2, . . . .
Thus the sequence {Vt}∞t=1 increases monotonically and is bounded from above; hence it has a limit, which, asfollows from (6), is equal to V ν .
Similarly, if V1 > V ν , then the sequence {Vt}∞t=1 converges to V ν decreasing monotonically.This implies the conclusion of the proposition. �The sequence of the growth factors of wealth at a path of pure consumption converges to α(V ν) = ν < 1; this
means that each path of pure consumption converges to the origin.The growth factor of output at a path of pure consumption, as follows from (3) and (6), is equal to
Yt+1
Yt=
f(Vt+1)Vt+1
.
Consequently, if Y0 < νK0Vν, then the growth factor of output at a path of pure consumption decreases and
converges to ν.
9. Let us prove that V ν is the maximum possible level of wages per unit of capital that can be maintained forlong periods of time.
Proposition 3. Let {St} be an arbitrary path, and let {Vt}∞t=1 be its control sequence. If V1 < V ν , thenVt < V ν for all t = 2, 3, . . . . If V1 > V ν , then either there exists a positive integer T such that Vt < V ν for allt = T , T + 1, . . . , or {Vt}∞t=1 decreases monotonically and Vt → V ν as t → +∞.
Proof. Let V1 < V ν . Then f(V1) < f(V ν) = νV and hence Y1 = K1f(V1) < νK1Vν. It follows that V2 < V ν .
The assertion follows by induction.Let V1 > V ν ; then f(V1) < νV1. On the other hand, f(V1) ≥ νV2, hence V2 < V1. By induction, either
there exists a positive integer T such that VT < V ν and we arrive at the case considered above, or the sequence{Vt}∞t=1 decreases monotonically and is bounded from below by the number V ν, in which case it has a limit Vand V ≥ V ν. Since f(Vt) ≥ νVt+1, we also have the inequality f(V ) ≥ νV , which is equivalent to V ≤ V ν .Consequently, V = V ν. �
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Corollary 2. If the initial state S0 = (K0, Y0) of a path {St} is such that Y0 < νK0Vν, then
Yt < νKtVν , t = 1, 2, . . . .
Proof. Since Y0 ≥ νK0V1, the inequality V1 < V ν holds and, by Proposition 3, Vt < V ν for all t = 2, 3, . . . .Hence
f(Vt) < f(V ν) = νV ν , t = 1, 2, . . . ,
which implies the inequality to be proved. �10. Given an arbitrary initial state S0 = (K0, Y0) with Y0 > 0, let us consider optimal paths for severalalternative optimality criteria. In the fK model, as well as in the vintage model considered in [6], it turns outthat different optimal paths can be constructed according to the same simple general rule.
To each of the optimality criteria under consideration, a fixed generating control corresponds, which does notdepend on the initial state. Let V be the generating control, and let the model be in a state S0 = (K0, Y0).Then the current control Vt+1 is constructed as
Vt+1 = min{
V,Y1
νKt
}.
In other words, if the output is high enough (Yt ≥ νKtV ), then we use the generating control (Vt+1 = V ), andif the output is not high enough (Yt < νKtV ), then there is no investment (It+1 = 0), i.e., pure consumptiontakes place in period t + 1. A path constructed in this way will be called a path generated by the control V .
Romer [20] considered a similar way of describing separate optimal paths, whereas we speak about a wholefamily of paths corresponding to different alternative optimality criteria and different initial states. The possibil-ity to consider many alternative optimality criteria simultaneously is useful for studies of economies in transition,where an optimality criterion is not finally defined and may change.
In particular, the following kinds of optimality criteria can be considered.
A. Paths with stepwise maximum output. A path moves from a current state St to the state St+1 with themaximum possible output Yt+1. It is easy to verify that the generating control V corresponding to this optimalitycriterion is the point of maximum of the function
f(V )1 + V
.
B. Paths with stepwise maximum profit. A path moves from a current state St to the state St+1 with themaximum profit Yt+1 − Vt+1Kt+1. The corresponding generating control V π satisfies the condition f ′(V π) = 1.
C. Efficient path. A path {St} is said to be efficient if there is no path {St} with the same initial state S0 = S0
such that Sk > Sk for some period k. The generating control corresponding to the criterion of efficiency is thecontrol V defined in Sec. 6. In Sec. 11 we will show that efficient paths possess a rather strong property: if {St}is an efficient path with initial state S0, then for any other path {St} with the same initial state there exists apositive integer T such that ST ST .
D. Path with maximum total discounted consumption. Given a discount factor β ∈ (0, 1), we seek for themaximum
max∞∑
t=0
βtCt+1 (7)
over all paths with initial state S0. For β ≥ 1/α(V ), the series (7) diverges. For β ∈ (0, 1/α(V )), the generatingcontrol, which is denoted by V (β), can be found as a solution of the equation
1 − β(ν + f(V )) + βV f ′(V ) = 0 (8)
(a proof is given in Sec. 12). Note that V (β) is a decreasing function. For a sufficiently small β, when “thesociety is impatient,” the inequality V (β) ≥ V12 holds and decline takes place at the balanced path determinedby the control V (β).
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A path with initial state S0 that solves problem (7) will be called β-optimal.Let a path be generated by a control V . Three cases are possible: (1) Vt = V for all t; the path becomes
balanced; (2) the path is a path of pure consumption; it is characterized by Proposition 2; (3) there exists aperiod of time t such that for t < t the path is a path of pure consumption and for t ≥ t the control V is beingapplied, and the path becomes balanced.
Depending on the initial state S0 and the generating control V , one or another structure of the optimal pathis possible. Basic cases are enumerated in the following table.
0 < V < V11 V11 < V < V12 V12 < V < V ν V > V ν
Proportional Proportional Proportional Lack of investment
Y0 > νV K0 decline growth decline and declinestarting from t=2
Lack of investment, Lack of investment, Lack of investment, Lack of investmentY0 < νV K0 then proportional then proportional then proportional and decline
decline growth decline
11. Now we will study the structure of efficient paths.
Theorem 1. For any initial state S0 there exists a unique efficient path. It is generated by the control V anddominates any other path with the same initial state S0.
Proof. 1. Let Y0 ≥ vV K0. Let {St} be a path with initial state S0 and control sequence {Vt ≡ V }∞t=1. (Such apath exists because f(V ) > νV .) Let {St} be another path with the same initial state S0, and let {Vt} be itscontrol sequence. According to (5),
KT
KT
=1 + V
1 + VT
α(V1)
α(V )· · · α(VT−1)
α(V ), (9)
YT
YT
=1 + V
f(V )
α(V1)
α(V )· · · α(VT−1)
α(V )
f(VT )1 + VT
. (10)
The following subcases are possible.1.1. The sequence {Vt} contains a subsequence {Vts} diverging to +∞.1.2. The sequence {Vt} contains a subsequence {Vts} converging to a positive number V , where
1.2.1. V = V ;
1.2.2. V = V .In cases 1.1 and 1.2.1, there exists a positive number T such that
α(Vts) < const < α(V ) for ts > T.
It follows from (9) and (10) thatKt
Kt
→ 0,Yt
Yt
→ 0 for t → ∞.
In case 1.2.2, α(Vt)/α(V ) ≤ 1 for all t and this inequality is strict for at least one index t. Consequently,
limt→∞
Kt
Kt
< 1, limt→∞
Yt
Yt
< 1.
In all this cases St � St for all sufficiently large numbers t; this means that the path {St} is efficient.2. Now let Y0 < νV K0. We will use the following auxiliary result.
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Lemma 2. Let a state S0 = (K0, Y0) be such that Y0 ≤ νV K0. Then in period t the maximum consumption C1
and the maximum wealth of the economy w(S1) will be achieved for I1 = 0 (i.e., under the control V 1 = Y0/νK0).
Proof. Obviously, the maximum of C1 will be achieved for I1 = 0.For I1 ≥ 0, the wealth of the economy is equal to
νK1 + Y1 = ν2K0 + νI1 + (νK0 + I1)f(
Y0 − I1
νK0 + I1
).
The maximum of this function is achieved at the point I1 = 0 if and only if at this point the derivative withrespect to I1 is nonpositive, i.e., if
ν + f
(Y0
νK0
)− f ′
(Y0
νK0
)νK0 + Y0
νK0≤ 0,
or, which is the same,f ′(V 1) ≥ α(V 1).
Taking into account the properties of the function f , the latter inequality is equivalent to the inequality V 1 ≤ Vcoinciding with the assumption of the lemma. �
It follows from Lemma 2 that for Y0 ≤ νV K0 the initial part of an efficient path coincides with a path of pureconsumption. Since, by Proposition 2, at a path of pure consumption Y1/Kt →
t→∞νV ν and V ν > V , an efficient
path at some period of time T satisfies the inequality YT ≥ νKT V . Then we have case 1 considered above. �12. Now we turn to studying the structure of β-optimal paths. Let us assume that
β <1
α(V )(11)
(in this case, problem (7) is well-posed).Let V (β) be a solution of Eq. (8).
Lemma 3. Under condition (11), the following inequality holds:
V (β) > V . (12)
Proof. Assume to the contrary that V (β) ≤ V . Then f ′(V (β)) ≥ α(V (β)), and it follows from (8) that
1 − βα(V (β))(1 + V (β)) + βV (β)α(V (β)) ≤ 0,
i.e., βα(V (β)) > 1. But, on the other hand, βα(V (β)) ≤ βα(V ) < 1. The contradiction obtained proves thelemma. �
We will also assume that the discount factor is not too small, namely, the following inequality holds:
f(V ) > νV (β), (13)
which is equivalent to
β >1
ν + f(f(V )
ν
)− f(V )
νf ′
(f(V )ν
) . (14)
(Indeed, Eq. (8) can be written in the form
β =1
ν + f(V (β)) − V (β)f ′(V (β)).
Since ν + f(V ) − V f ′(V ) is an increasing function, inequalities (13) and (14) are equivalent.)It follows from (12) and (13) that
f(V (β)) > νV (β), (15)
i.e.,V (β) < V ν.
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Proposition 4. Under conditions (11) and (13), if an initial state S0 = (K0, Y0) is such that Y0 ≥ νK0V (β),then the β-optimal path with initial state S0 is determined by the control V (β) (i.e., Vt ≡ V (β), t = 1, 2, . . .).
Proof. According to (4) and (5),
C1 = K1V1 = w(S0)V1
1 + V1,
Ct = KtVt = w(S0)α(V1) . . . α(Vt−1)Vt
1 + Vt, t > 1.
Problem (7) reduces to finding
σ(β) = max∞∑
k=2
{V1
1 + V1+
∞∑k=2
Vk
1 + Vk
( k−1∏i=1
βα(Vi))}
under the conditionf(Vt) ≥ νVt+1, t ≥ 1.
It is clear thatσ(β) ≤ max
0≤V ≤V νgβ(V ), (16)
where
gβ(V ) =V
1 + V
∞∑i=0
(βα(V ))i =V
(1 + V )(1 − βα(V )).
With fixed β, the function gβ(V ) is defined for V ∈ [0, +∞) and is strictly concave. It has a unique point ofmaximum V (β); the latter satisfies Eq. (8).
In view of (15), inequality (16) is fulfilled as an equality.If an initial state S0 = (K0, Y0) is such that Y0 ≥ νK0V (β), then the control V (β) is applicable and the
β-optimal path is determined by the control sequence{Vt ≡ V (β)
}∞t=1
. �
Now let an initial state S0 = (K0, Y0) be such that
V ≤ Y0
νK0< V (β). (17)
In this case, the control V (β) cannot be used directly at the first step, but it can be used at the second step,because, taking into account (13),
f
(Y0
νK0
)> νV (β).
Proposition 5. In the case (17), at a β-optimal path, I1 = 0.
Proof. A control V1 < V cannot be applied at a β-optimal path. Indeed, otherwise one can increase bothconsumption and wealth of the economy by reducing I1, because
dw
dI1(S1) = (1 + V1)[α(V1) − f ′(V1)] < 0
for V1 < V .Thus V1 ≥ V . It follows from (13) that f(V1) > νV (β) and hence, according to Proposition 4, V2 = V (β).
Consequently, the value of problem (7) is equal to
Vβ(S0) = C1 + βVβ (S1) = Y0 − I1 + β(νK0 + I1)(
ν + f
(Y0 − I1
νK0 + I1
))1
1 − βf ′(V (β)).
One can verify that the inequality dVβ(S0)/dI1 < 0 holds for V1 < V (β). Consequently, the maximum is reachedfor I1 = 0. �
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Theorem 2. Under conditions (11) and (13), any β-optimal path is generated, and starting from some time isdetermined, by the control V (β).
Proof. If at a β-optimal path in a period t the inequality Yt < νKtV (β) holds, then, as follows from Lemma 2and Proposition 5, It+1 = 0.
Hence, taking into account Proposition 2, for any β-optimal path {St} there exists a period T such thatYT ≥ νKTV (β). Then, according to Proposition 4, the control sequence {Vt} of the path {St} is such thatVt = V (β) for all t ≥ T + 1.
It follows from (8) that V (β) > V , so that no efficient path can be a solution of the well-posed problem (7).To any control V , where V > V , the discount factor
β(V ) =1
ν + f(V ) − V f ′(V )
corresponds, under which V is the generating control for problem (7). The function β(V ) decreases for V ∈(V , +∞).
It is easy to verify that if a sequence of discount factors {βk} is such that βk ∈ (0, 1/α(V )) and βk → 1/α(V )as k → ∞, then V (βk) → V as k → ∞. �13. Let us see what influence changes in the exogenous parameters W2 (see Sec. 3) and β produce on thegenerating control V (β) and on the pattern of β-optimal paths.
Consider an increase in the alternative wage rate W2. In this case, as can be seen from the results of Sec. 10,f(V ) decreases for each value of V and the corresponding growth factor α(V ) also decreases.
In particular, if the generating control V does not change and the growth factor α(V ) was greater than 1before the increase in W2, then the growth factor will become (for a sufficiently high W2) less than 1, i.e., growthwill give place to decline. On the contrary, a decrease in W2 leads to an increase in the growth factor α(V ) foreach value of V .
Now let the discount factor β decrease. As was observed in Sec. 7, the control V (β) generating β-optimalpaths increases. In particular, the inequality V (β) < V12 may be replaced by V (β) > V12; in this case, long-runeconomic growth is replaced by decline.
In Russia, beginning from the 1960s, a growth in the average real wages (slow before 1987 and fast after)was accompanied by a decline in the output to capital ratio, which was a puzzle for Soviet economists. Usingthe fK model, one can explain the decline in the growth factor before 1987 by a slow increase in the parameterW f = W2 (see Sec. 3), and after 1987, by a sharp increase in W f as a result of emergence and development ofthe new sector of the economy (first in the form of cooperatives); since this moment, the parameter W f = W2
became a reservation wage in the proper sense of the word. The model admits different growth factors dependingon the value of the parameter.
The dynamics of the Russian economy of the last 15 years can be explained in the framework of this model inthe following way. The emergence, since the late 1980s, of cooperatives and then private firms and possibilitiesof private enterprise meant an increase in the alternative (reservation) wage W2. This led to a decrease in thegrowth factor α(V ) in the old sector of the economy: growth was replaced by decline. The development of thenew private sector did not compensate for the decline in the old sector, because the new sector turned out tobe, in the whole, low-productive; the goal of many firms in the new sector was only the current welfare of theirowners; besides, the majority of new firms functioned in a very unfavorable environment.
The crisis of 1998 led to a decline in the reservation wage W2, which, from the point of view of the model,was the main cause of an increase in the growth factor.
Translated by V. D. Matveenko.
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