lecture notes in macroeconomic theory econ...
TRANSCRIPT
Lecture Notes in MacroeconomicTheory∗
Econ 8108
March 28, 2013
Contents
1 Introduction 3
2 Neoclasscial Growth Model Without Uncertainty 3
3 Adding Uncertainity 63.1 Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Problem of the Social Planner . . . . . . . . . . . . . . . . . . 73.3 AD Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 A Comment on the Welfare Theorems . . . . . . . . . . . . . . 8
4 Recursive Competitive Equilibrium 84.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Economy with leisure . . . . . . . . . . . . . . . . . . . . . . . 114.3 Economy with uncertainty . . . . . . . . . . . . . . . . . . . . 124.4 Economy with Medals (or Fireworks/Parties/Parks) . . . . . . 12
4.4.1 Income Tax . . . . . . . . . . . . . . . . . . . . . . . . 124.4.2 Lump Sum Tax . . . . . . . . . . . . . . . . . . . . . . 134.4.3 Taxes and Debt . . . . . . . . . . . . . . . . . . . . . . 13
∗ These notes were started by graduate students at Penn and continued by the graduatestudents from Minnesota.
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4.5 Economy with Externalities . . . . . . . . . . . . . . . . . . . 154.6 An Economy with Capital and Land . . . . . . . . . . . . . . 16
5 Adding Heterogeneity 175.1 An International Economy Model . . . . . . . . . . . . . . . . 18
6 Asset Pricing - Lucas Tree model 20
7 Measure Theory 21
8 Industry Equilibrium 248.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 A Simple Dynamic Environment . . . . . . . . . . . . . . . . . 258.3 Introducing Exit Decision . . . . . . . . . . . . . . . . . . . . 278.4 Stationary Equilibrium . . . . . . . . . . . . . . . . . . . . . . 298.5 Adjustment Costs . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Models with Growth 329.1 Exogenous Growth . . . . . . . . . . . . . . . . . . . . . . . . 32
9.1.1 Population Growth . . . . . . . . . . . . . . . . . . . . 329.1.2 Labor Augmenting Productivity Growth . . . . . . . . 34
9.2 Endogenous Growth . . . . . . . . . . . . . . . . . . . . . . . 359.2.1 AK Model . . . . . . . . . . . . . . . . . . . . . . . . . 369.2.2 Human Capital Models . . . . . . . . . . . . . . . . . . 379.2.3 Growth Through Externalities by Romer . . . . . . . . 389.2.4 Monopolistic Competition, Endogenous Growth and
R&D (Romer) . . . . . . . . . . . . . . . . . . . . . . . 38
10 Search Models 4210.1 The Search Problem . . . . . . . . . . . . . . . . . . . . . . . 4210.2 A Continuous Time Formulation . . . . . . . . . . . . . . . . . 4410.3 Generating Transitions . . . . . . . . . . . . . . . . . . . . . . 4510.4 On-the-Job Search . . . . . . . . . . . . . . . . . . . . . . . . 4610.5 Matching and Bargaining . . . . . . . . . . . . . . . . . . . . . 4710.6 Competitive Search . . . . . . . . . . . . . . . . . . . . . . . . 49
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1 Introduction
A model is an artificial economy. Description of a model’s environment mayinclude specifying the agents’preferences and endowment, technology avail-able, information structure as well as property rights. Neoclassical GrowthModel becomes one of the workhorses of moderm macroeconomics becauseit delivers some fundamental properties of moderm economy, summaried by,among others, Kaldor, i.e. annual output growth rate at roughly 2%; wagegrowth rate at roughly 2%; Labor share roughly constant at 0.66, rate ofreturn to capital and labor roughly constant, etc.Equilibrium can be defined as a prediction of what will happen and there-
fore it is a mapping from environments to outcomes (allocations, prices, etc.).One equilibrium concept that we will deal with is Competitive Equilibrium1.Charicterizing the equilibrium, however, usually involves finding solutions toa system of infinite number of equations. There are generally two ways ofgetting around this. First, invoke the welfare theorem to solve for the allo-cation first and then find the equilibrium prices associated with it. The firstway sometimes may not work due to, say, presence of externality. So the sec-ond way is to look at Recursive Competitive equilibrium, where equilibriumobjects are functions instead of variables.
2 Neoclasscial Growth Model Without Uncertainty
The commodity space is
L = (l1, l2, l3) : li = (lit)∞t=0 lit ∈ R, sup
t|lit| <∞, i = 1, 2, 3
The consumption possibility set is
X(k0) = x ∈ L : ∃(ct, kt+1)∞t=0
such that ∀t = 0, 1, . . .
ct, kt+1 ≥ 0
x1t + (1− δ)kt = ct + kt+1
− kt+1 ≤ x2t ≤ 0
− 1 ≤ x3t ≤ 0
k0 = k0 1Arrow-Debreu or Valuation Equilibrium.
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The production possibility set is: Y =∏
t Yt where
Yt = (y1t, y2t, y3t) ∈ R3 : 0 ≤ y1t ≤ F (−y2t,−y3t)
Definition 1 An Arrow-Debreu equilibrium is (x∗, y∗) ∈ X × Y , and a con-tinuous linear functional ν∗ such that
i. x∗ ∈ arg maxx∈X,ν∗(x)≤0
∑∞t=0 β
tu(ct(x),−x3t)
ii. y∗ ∈ arg maxy∈Y ν∗(y)
iii. x∗ = y∗
Now, let’s look at the one-sector growth model’s Social Planner’s Problem:
max∑∞
t=0 βtu(ct,−x3t) (SPP )
s.t.
ct + kt+1 − (1− δ)kt = x1t
0 ≤ x2t ≤ kt
0 ≤ x3t ≤ 1
0 ≤ y1t ≤ F (−y2t,−y3t)
x = y
k0 given.
Suppose we know that a solution in sequence form exists for (SPP) and isunique.
Two important theorems show the relationship between CE allocations andPareto optimal allocations:
Theorem 1 Suppose that for all x ∈ X there exists a sequence (xk)∞k=0,
such that for all k ≥ 0, xk ∈ X and U(xk) > U(x). If (x∗, y∗, ν∗) is anArrow-Debreu equilibrium then (x∗, y∗) is Pareto effi cient allocation.
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Theorem 2 If X is convex, preferences are convex, U is continuous, Y isconvex and has an interior point, then for any Pareto effi cient allocation(x∗, y∗) there exists a continuous linear functional ν such that (x∗, y∗, ν) is aquasiequilibrium, that is (a) for all x ∈ X such that U(x) ≥ U(x∗) it impliesν(x) ≥ ν(x∗) and (b) for all y ∈ Y , ν(y) ≤ ν(y∗).
Note that at the very basis of the CE definition and welfare theorems thereis an implicit assumption of perfect commitment and perfect enforcement.Note also that the FWT implicitly assumes there is no externality or pub-lic goods (achieves this implicit assumption by defining a consumer’s utilityfunction only on his own consumption set but no other points in the com-modity space).
From the First Welfare Theorem, we know that if a Competitive Equilib-rium exits, it is Pareto Optimal. Moreover, if the assumptions of the SecondWelfare Theorem are satisfied and if the SPP has unique a solution then thecompetitive equilibrium allocations is unique and they are the same as thePO allocations. Prices can be constructed using this allocations and firstorder conditions.
One shortcoming of the AD equilibrium is that all trade occurs at the be-ginning of time. This assumption is unrealistic. Modern Economics is basedon sequential markets. Therefore we define another equilibrium concept,Sequence of Markets Equilibrium (SME). We can easily show that SME isequivalent to ADE. Therefore all of our results still hold and SME is the rightproblem to solve.
Note that the (SPP) problem is hard to solve, since we are dealing withinfinite number of choice variables. We have already established the factthat this SPP problem is equivalent to the following dynamic problem:
v(k) = maxc,k′
u(c) + βv(k′) (RSPP )
s.t. c+ k′ = f(k).
We have seen that this problem is easier to solve.
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What happens when the welfare theorems fail? In this case the solutions tothe social planners problem and the CE do not coincide and so we cannotuse the theorems we have developed for dynamic programming to solve theproblem. As we will see in this course, in this case we can work with RecursiveCompetitive Equilibria. In general, we can prove that the solution to the RCEcoincides with a sequential markets problem but not the other way around(for example when we have multiple equilibria). However, in all the modelswe see this course, this equivalence will hold.
3 Adding Uncertainity
3.1 Markov Process
In this part, we want to focus on stochastic economies where there is aproductivity shock affecting the economy. The stochastic process for pro-ductivity that we are assuming is a first order Markov Process that takes onfinite number of values in the set Z = z1 < · · · < znz. A first order Markovprocess implies
Pr(zt+1 = zi|ht) = Γij, zt(ht) = zj
where ht is the history of previous shocks. Γ is a Markov matrix with theproperty that the elements of its columns sum to 1.
Let µ be a probability distribution over initial states, i.e.∑i
µi = 1
and µi ≥ 0 ∀i = 1, ..., nz.
Next periods the probability distribution can be found by the formula:µ′ = ΓTµ.
If Γ is “nice”then ∃ a unique µ∗ s.t. µ∗ = ΓTµ∗ and µ∗ = limm→∞(ΓT )mµ0,∀µ0 ∈ ∆i.
Γ induces the following probability distribution conditional on z0 on ht =z0, z1, ..., zt:
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Π(z0, z1) = Γi for z0 = zi.
Π(z0, z1, z2) = ΓTΓi for z0 = zi.
Then, Π(ht) is the probability of history ht conditional on z0. The ex-pected value of z′ is
∑z′ Γzz′z
′ and∑
z′ Γzz′ = 1.
3.2 Problem of the Social Planner
Let productivity affects the production function in an arbitrary way, F (z,K,N).Problem of the social planner problem (SPP) in sequence form is
maxct(ht),k′(ht)∈X(ht)
∞∑t=0
∑ht
βtπ(ht)u(ct(ht))
s.t ct(ht) + kt+1(ht) = ztF (kt(ht−1), 1).
Therefore, we can formulate the stochastic SPP in a recursive fashion:
V (zi, K) = maxc,a′
u(c) + β∑j
ΓjiV (zj, K ′)
s.t. c+ k′iF (K, 1).
This gives us a policy function K ′ = G(z,K).
3.3 AD Equilibrium
AD equilibrium can be defined by:
maxct(ht),k′(ht),x1t(ht),x2t(ht),x3t(ht)∈X(ht)
∞∑t=0
∑ht
βtπ(ht)u(ct(ht))
s.t∞∑t=0
∑ht
p(ht)x(ht) ≤ 0.
where X(ht) is the consumption feasibility set after history ht occured. Notethat we are assuming the markets are dynamically complete, i.e. there is
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complete set of securities for every possible history that can appear.
By the same procedure as before, SME can be written in following way:
maxct(ht),it(ht),at(ht−1)
∞∑t=0
∑ht
βtπ(ht)u(ct(ht))
s.t ct(ht) + k′(ht) +∑zj
b(ht, zj)q(ht, z
j) = kt(ht−1)R(ht) + w(ht) + b(ht−1).
Here we have introduced Arrow securities to allow agents to trade with eachother against possible future shocks.However, in equilibrium and when there is no heterogeneity, there will beno trade. Moreover, we have two ways of delivering the goods specified inan Arrow security contract: after production and before production. In anafter production setting, the goods will be delivered after production takesplace and can only be consumed or saved for the next period. This is theabove setting. It is also possible to allow the consumer to rent the Arrow se-curity income as capital to firms, which will be the before production setting.
An important condition which must hold true in the before production settingis the no-arbitrage condition,
∑zt+1
q(ht, zt+1) = 1.
3.4 A Comment on the Welfare Theorems
Situations in which the welfare theorems would not hold include externalities,public goods, situations in which agents are not price takers (e.g. monopo-lies), some legal systems or lacking of markets which rule out certain contractswhich appears complete contract or search frictions. In all of these situationfinding equilibrium through SPP is no longer valid. Therefore, in these sit-uations, as mentioned before, it is better to define the problem in recursiveway and find the allocation using the tools of Dynamic Programming.
4 Recursive Competitive Equilibrium
4.1 A Simple Example
What we have so far is that we have established the equivalance betweenallocation of the SPP problem which gives the unique Pareto optima (which
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is same as allocation of AD competitive equilibrium and allocation of SME).Therefore we can solve for the very complicated equilibrium allocation bysolving the relatively easier Dynamic Programming problem of social plan-ner. One handicap of this approach is that in a lot of environments, theequilibrium is not Pareto Optimal and hence, not a solution of a social plan-ner’s problem, e.g. when you have taxes or externalities. Therefore, theabove recursive problem would not be the right problem to solve. In some ofthese situations we can still write the problem in sequence form. However, wewould lose the powerful computational techniques of dynamic programming.in order to resolve this issue we will define Recursive Competitive Equilib-rium equivalent to SME that we can always solve for.
In order to write the decentralized household problem recursively, we need touse some equilibrium conditions so that the household knows what prices areas a function of some economy-wide aggregate state variable. We know thatif capital is Kt and there is 1 unit of labor, then w (K) = Fn (K, 1) , R (K) =Fk (K, 1) . Therefore, for the households to know prices they need to knowaggregate capital. Now, a household who is deciding about how much to con-sume and how much to work has to know the whole sequence of future prices,in order to make his decision. This means that he needs to know the pathof aggregate capital. Therefore, if he believes that aggregate capital changesaccording to K ′ = G(K), knowing aggregate capital today, he would be ableto project aggregate capital path for the future and therefore the path forprices. So, we can write the household problem given functionG(·) as follows:
Ω(K, a;G) = maxc,a′
u(c) + βΩ(K ′, a′;G) (RCE)
s.t. c+ a′ = w(K) +R(K)a
K ′ = G(K),
c ≥ 0
The above problem, is the problem of a household that sees K in the econ-omy, has a belief G, and carries a units of assets from past. The solution ofthis problem yields policy functions c(K, a;G), a′(K, a;G) and a value func-tion Ω(z,K, a;G). The functions w(K), R(K) are obtained from the firm’s
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FOCs (below).
uc[c(K, a;G)] = βΩa[G(K), a′(K, a;G);G]
Ωa[K, a;G] = (1 + r)uc[c(K, a;G)]
Now we can define the Recursive Competitive Equilibrium.
Definition 2 A Recursive Competitive Equilibrium with arbitrary expecta-tions G is a set of functions2 Ω, g : A × K → R, R, w,H : K → R+ suchthat:
1. given G; Ω, g solves the household problem in (RCE)
2. K’=H(K;G)=g(K,K;G) (representative agent condition)
3. w (K) = Fn (K, 1)
4. R (K) = Fk (K, 1)
We define another notion of equilibrium where the expectations of thehouseholds are consistent with what happens in the economy:
Definition 3 (Rational Expectation Equilibrium) A Rational Expecta-tions Equilibrium is a set of functions Ω, g, R, w,G∗ such that
1. Ω(K, a;G∗), g(K, a;G∗) solves HH problem in (RCE).
2. G∗(K) = g(K,K;G∗) = K ′
3. w (K) = Fn (K, 1)
4. R (K) = Fk (K, 1)
2 We could add the policy function for consumption gc(K, a;G).
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What this means is that in a REE, households optimize given what theybelieve is going to happen in the future and what happens in the aggregate isconsistent with the household’s decision. The proof that every REE can beused to construct a SME is left as an exercise. The reverse turns out not to betrue. Notice that in REE, function G projects next period’s capital. In fact,if we construct an equilibrium path based on REE, once a level of capital isreached in some period, next period capital is uniquely pinned down by thetransition function. If we have multiplicity of SME, this would imply that wecannot construct the function G since one value of capital today could implymore than one value for capital tomorrow. We will focus on REE unlessexpressed otherwise.
4.2 Economy with leisure
We may extend the previous framework to the elastic labor supply case.Note that aggregate employment level is not a state variable but is insteadpredicted by aggregate states. The households problem is as follows:
Ω(K, a;G,H) = maxc,a′,l
u(c, n) + βΩ(K ′, a′;G,H) (RCE)
s.t. c+ a′ = w(K,N)n+R(K,N)a
K ′ = G(K),
N = H(K)
with solution a∗(K, a;G,H), n∗(K, a;G,H).We may thus define an RCE with rational expection to be a collection of
functions (Ω, a∗, n∗) , (G,H), (R,w) such that
1. Given (G,H), (R,w) , (Ω, a∗, n∗) solves HH problem
2. G(K) = a∗(K,K;G,H)
3. H(K) = n∗(K,K;G,H)
4. w (K,N) = Fn (K,N)
5. R (K,N) = Fk (K,N)
Note that condition 1 is the optimality condition. Condition 2 and 3are imposed because of rational expection. Condition 4 and 5 are marginalpricing equations.
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4.3 Economy with uncertainty
Go back to our simple framework with inelastic labor supply. But assumethere is productivity shock z that can take on finite number of values, whoseevolution follows a markov process governed by Γzz′ .Then Current periodproductivity shock, denoted by z,should be included into aggregate statevariables that help predicting prices and future aggregate staet variables.Moreover, assume that households can assumulate state contigent capital.
Ω(K, z, a;G) = maxc,a′(z′)
u(c) + β∑z′
Ω(K ′, z′, a′ (z′) ;G)Γzz′ (RCE)
s.t. c+∑z′
q (K, z, z′) a′ (z′) = w(K, z) +R(K, z)a
K ′ = G(K, z),
c ≥ 0
Solving this problem gives policy function g (K, z, a, z′;G)An RCE in this case is a collection of functions (Ω, g) , (q, w,R,G) such
that
1. Given (q, w,R,G) , (Ω, a∗) solves HH problem
2. g (K, z,K, z′;G) = G (K, z) ,∀K, z, z′
3. w (K,N) = Fn (K,N)
4. R (K,N) = Fk (K,N)
5.∑
z′ q (K, z, z′) = 1 (no arbitrage condition)
4.4 Economy with Medals (or Fireworks/Parties/Parks)
4.4.1 Income Tax
We have an economy in which the government levies taxes in order to pur-chase medals. Medals are goods which provide utility to the consumers (for
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this example).
Ω(K, a;G) = maxc,a′
u(c,M) + βΩ(K ′′;G)
s.t. c+ a′ = [w(K) +R(K)a](1− τ)
K ′ = G(K),
c ≥ 0
Now the social planner function method cannot be used: the CE will not bePareto optimal anymore (if τ > 0 there will be a wedge, and the effi ciencyconditions will not be satisfied).
4.4.2 Lump Sum Tax
The government levies each period T units of goods in a lump sum way andspends it in medals. Assume consumers do not care about medals. Thehousehold’s problem becomes:
Ω(K, a;G) = maxc,a′
u(c) + βΩ(K ′, a′;G)
s.t. c+ a′ + T = w(K) +R(K)a
K ′ = G(K;M,T ),
c ≥ 0
A solution of this problem are functions g∗a(K, a;G,M, T ) and Ω(K, a;G) andthe equilibrium can be characterized by G∗(K,M, T ) = g∗a(K,K;G∗,M, T )and M∗ = T (the government budget constraint is balanced period by pe-riod). We will write a complete definition of equilibrium for a version withgovernment debt (below).
4.4.3 Taxes and Debt
Assume that government can issue debt. Note that in a sequence version ofthe household problem in SME, in order for households not to achieve infiniteconsumption, we need a no-Ponzi condition:
limt→∞
at∏ts=0 Rs
<∞
This is the weakest condition that imposes no restrictions on the first or-der conditions of the household’s problem. It is harder to come up with its
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analog for the recursive case. One possibility is to assume that a′ lies in acompact set A or a set that is bounded from below3. We will give a completedefinition of RCE of this economy.
A government policy consists of taxes and spending:
(τ (K,B) ,M (K,B))
In this environment, debt issued is relevant for the household because it per-mits him to correctly infer the amount of taxes. Therefore the householdneeds to form expectations about the future level of debt from the govern-ment. The government budget constraint now satisfies (with taxes on laborincome):
M(K,B) +R (K) ·B = τ (K,B)w(K) +B′ (K,B)
Where also B′ (K,B) = GB(·). Notice that the household does not careabout the composition of his portfolio as long as assets have the same rateof return which is true because of the no arbitrage condition. Therefore, theproblem of a household with assets equal to a is given by:
Ω(a,K,B; τ(K,B),M(K,B), GK , GB
)= max
c,a′u (c) + βΩ
(a′, K ′, B′;M, τ,GK , GB
)s.t. c+ a′ ≤ (1− τ(K,B))w (K) +R (K) a
K ′ = GK (K,B;M, τ)
B′ = GB (K,B;M, τ)
τ = τ(K,B), M = M(K,B)
Definition 4 (Rational Expectation Equilibrium with Government Debt)A Rational Expectations Recursive Competitive Equilibrium given policiesM(K,B) and τ(K,B) is a set of functions Ω, g, R, w,GK , GB such that
1.
2. Ω(K,B, a; ·), g(K,B, a;GK , GB,M(·), τ(·)) solves the HH problem.
3. w(K) = F2(K, 1), R(K) = F1(K, 1).
3We must specify A such that the borrowing constraint implicit in A is never binding.
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4. Representative agent conditiong(K,B,K +B;GK , GB, ·) = GK(K,B; ·) +GB(K,B; ·).
5. Government Budget ConstraintB ·R(K) +M(K,B) = w(K)τ(K,B) +GB(K,B; ·).
6. Government debt is bounded∃ B, B such that ∀B ∈ [B, B] we have GB(K,B) ∈ [B, B] (and ∀K ∈[0, K]).
4.5 Economy with Externalities
Let’s consider an economy where the consumer cares about aggregate con-sumption in additions to his own, in particular u(c, C), where u2(c, C) < 0(derivative w.r.t. second component, “keeping up with the Joneses”):
Ω(K, a;G) = maxc,a′
u(c, C) + βΩ(K ′, a′;G)
s.t. c+ a′ = w(K) +R(K)a
K ′ = G(K),
c ≥ 0
Another possibility could be that the consumer cares about aggregate con-sumption from the previous period, u(c, C−1), or “catching up with the Jone-ses”. In that case we would have an additional state variable C−1, since thisinformation becomes relevant to the consumer when he is solving his prob-lem. Other externalities could appear in how the consumer enjoys leisure, inthe production function, etc.
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4.6 An Economy with Capital and Land
Consider an economy with with capital and land but without labor. Theagent’s problem is
V (K, a) = maxc,a′
u(c) + βV (K′, a′)
s.t.
c+ a′ = (1 +D(K))a
K′ = G(K)
The solution requires a function a′ = h(K, a). We could use yet anothernotation c + q(K)a′ = a, a′ = h(K, a). The problem for the firm is thefollowing:
Ω(K, k) = maxk′,d
d+ q(K) · Ω(K ′, k′)
s.t.
K ′ = G(K)
Where d = f(k)−k′+ (1− δ)k (full depreciation obviously means δ = 1).An RCE consists of functions V,Ω, h, g, q, G,D, d, and the conditions thathave to hold are similar to the case with uncertainty. In particular:
• V and Ω solve the problems of the household and the firm, respectively
• G(K) = g(K, k)
• h[K,Ω(K,K)/(1 + D(K))] = Ω(G(K), G(K)) · q(K) or alternativelyh[K,Ω(K,K)] = Ω(G(K), G(K))
• uc(f(K)−G(K)) = βq(K)uc(f(G(K))−G(G(K)))
• q(K) = 1/(1 +D(K)).
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5 Adding Heterogeneity
In the previous section we looked at situations in which RCE were useful.In particular these were situations in which the welfare theorems failed andso we could not use the standard dynamic programming techniques learnedearlier. In this section we look at another way in which RCE are helpful- inmodels with heterogeneous agents.
First, lets consider a model in which we have two types of householdsthat differ only in the amount of wealth they own. There is a measure µR ofagents that start of poor and a measure µP that start of poor. Agents areidentical other than their initial wealth position and there is no uncertaintyin the model. The agent’s problem is
V (KR, KP , a;GR, GP ) = maxc,a′
u(c) + βV (K ′R, K ′P , a′R, GP )
s.t. c+ a′ = Ra+ w
R = R(KP , KR)
w = w(KP , KR)
K ′i = Gi(KR, KP ) i ∈ R,P
Definition 5 (Rational Expectation Equilibrium with Agents that Differ in Wealth)A Rational Expectations Recursive Competitive Equilibrium is a set of func-tions V, g, R,w,GR, GP such that
1.
2. V, g solves the HH problem.
3. w,R are the marginal products of labor and capital respectively
4. Representative agent conditionsg(KP , KR, KP ;GK , GP ) = GP (KR, KP ).g(KP , KR, KR;GK , GP ) = GR(KR, KP ).
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Now consider a slightly different economy with a unit measure of agents.There are two types of them. Type i has labor skill εi and initial endowmentWi. Measure of agents, µA, µB satisfies µAεA +µBεB = 1 (below we will con-sider the case µA = µB = 1/2). Note here we also assume i′ = i, so neitheragent is changing his type.
An important issue here is to determine what constitutes a suffi cient sta-tistic for describing household problem. In particular, we should determinewhether K is enough for learning current prices and predict K ′. It turns outthat this is only true under particular assumptions that make the savingsfunction of each individual linear in K.
Without further information, the household will need to keep track also of thedistribution of wealth. In this simple context, it is enough to know (AA, AB).Alternatively, one can use as a suffi cient statistic (K, s) where s is the wealthshare of households of type A. The problem of the household i ∈ A,B canbe written as follows.
Ωi (a,AA, AB;GA, GB) = maxc,a′
u (c) + βΩi (a′, A′A, A
′B;GA, GB)
s.t. c+ a′ ≤ w (K) εi +R (K) a
A′j = Gj (AA, AB) for j ∈ A,BK = µAAA + µBAB
Notice that we have indexed the value function by the agent’s type (if theywere the same we would not need this index).Remember that you should define an RCE for this economy as a homeworkproblem (see HW2).
5.1 An International Economy Model
In an international economy model the specifications which determine thedefinition of country is an important one. We can introduce the idea of dif-ferent locations, or geography, countries can be victims of different policies,additionally trade across countries maybe more diffi cult due to different re-strictions.
Here we will see a model with two countries, A and B, such that labor is notmobile between the countries but with perfect capital markets. Two countries
18
may have different technologies FA(KA, 1) and FB(KB, 1). We need one morevariable, we can choose AA (total wealth for country A, alternatively we coulduse the share of total wealth for one of the countries, check last year’s notesif you are curious).So country i’s problem becomes:
Ωi (KA, KB, AA, a;GA, GB) = maxc,a′
u (c) + βΩi (K′A, K
′B, a
′, A′A;GA, GB)
s.t. c+ a′q(KA, KB, AA) ≤ wi(Ki) + a
K ′j = Gj (KA, KB, AA) for j ∈ A,BK = KA +KB
An RCE for this economy is a set of functions gi, Gi, Hi, R, wi, suchthat the following conditions hold:
• gi(KA, KB, AA, a) solves the household’s problem in each country.
• HA(Ka, KB, AA) = gA(KA, KB, AA, AA)
• GA(·)+GB(·) = gA(KA, KB, AA, AA)+gB(KA, KB, AA, KA+KB−AA),total wealth tomorrow A′.
• R(KA, KB, AA) = (1 − δ) + FAK(KA, 1) = (1 − δ) + FB
K (KB, 1), wi =F iN(Ki, 1).
• FAK [GA(·), 1] = FB
K [GB(·, 1)].
What happens if trade is costly? For example suppose that there were trans-portation costs and only 1−η of amount exported from any country reachedthe other. In particular we have
(1− η)XA = MB
(1− η)XB = MA
Note that in this model we will not see a country simultaneously exportingand importing since we have only one good. We could have also introduceda time to ship. You should write down the RCE for these cases.
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6 Asset Pricing - Lucas Tree model
Consider an economy in which the only asset is a tree that gives fruits. Thefruit d is constant over time but the agent is subject to preference shocks forthe fruit each period, θ ∈ Θ. The agents problem is
V (θ, s) = max θu(c) + β∑θ′
Γθθ′V (θ, s′)
s.t. c+ p · s = s(p+ d)
p = p(θ)
Definition 6 (Rational Competitive Equilibrium with a Lucas Tree and Preference Shocks)A Rational Expectations Recursive Competitive Equilibrium is a set of func-tions V, g, p such that
1.
2. V, g solves the HH problem.
3. g(θ, 1) = 1,∀ θ ∈ Θ
The FOC for this problem using Envelope theorem is: −p(θi)θiuc +β∑′
θ Γθθ′θ′u′c = 0. Arranging the terms this gives us:
p (θi) =∑θ′
βθuc (d)
θ′uc (d)Γθθ′ (p (θ′) + d)
for all θ ∈ Θ = θ1, . . . θn.
Notice that this is just a system of n equations with unknowns p (θi)ni=1.We can use the power of matrix algebra to solve it. Let
p =
p (θ1)...
p (θn)
;U =
θ1uc(d)θ1uc(d)
Γθ1θ1θ2uc(d)θ1uc(d)
Γθ1θ2 · · ·θnuc(d)θ1uc(d)
Γθ1θn...
.... . .
...θ1uc(d)θnuc(d)
Γθnθ1θ2uc(d)θnuc(d)
Γθnθ2 · · ·θnuc(d)θnuc(d)
Γθnθn
20
and rewrite the system above as p = βU (p+ d). Hence, the price for theshares is given by
p = (I − βU)−1 βd
What happens if we add state contingent shares into the model? Then theagent’s problem becomes
V (θ, s, b) = max θu(c) + β∑θ′
Γθθ′V (θ, s′, b′)
s.t. c+ p · s+∑θ′
q(θ, θ′)b(θ′) = s(p+ d) + b(θ′)
p = p(θ)
You have to define an equilibrium for this economy as part of HW2. Notethat in this example, the no arbitrage condition will be
∑θ′ q(θ
′, θ) = p.
7 Measure Theory
This section will be a quick review of measure theory to be able to use in thesubsequent sections.
Definition 7 For a set S, S is a set (or family) of subsets of S. B ∈ Simplies B ⊂ S, but not the other way around.
Definition 8 σ-algebra S is a set of subsets of S, with the following proper-ties:
1. S, ∅ ∈ S
2. A ∈ S ⇒ Ac ∈ S (closed in complementarity)
3. for Bii=1,2..., Bi ∈ S ⇒ [∩iBi] ∈ S (closed in countable intersections)
21
A σ-algebra is a structure to organize information. Examples are thefollowing:
1. Everything, aka the power set (all the possible subsets of a set S)
2. ∅, S
3. ∅, S, S1/2, S2/2 where S1/2 means the lower half of S (imagine S as anclosed interval on R).
If S = [0, 1] then the following is NOT a σ− algebra
S =
∅, [0, 1
2),
1
2
,
[1
2, 1
], S
Remark 1 A convention is (i) use small letters for elements, (ii) use capitalletters for sets, (iii) use “fancy” letters for a set of subsets (or family ofsubsets).
Definition 9 A measure is a function x : S → R+ such that
1. x(∅) = 0
2. if B1, B2 ∈ S and B1 ∩B2 = ∅ ⇒ x(B1 ∪B2) = x(B1) + x(B2)
3. if Bi∞i=1 ∈ S and Bi ∩ Bj = ∅ for all i 6= j ⇒ x(∪iBi) =∑
i x(Bi)(countable additivity)
Countable additivity means that measure of the union of countable disjointsets is the sum of the measure of these sets.
Definition 10 Borel-σ-algebra is a σ-algebra generated by the family of allopen sets (generated by a topology).
Since a Borel-σ-algebra contains all the subsets generated by intervals, youcan recognize any subset of a set using Borel-σ-algebra. In other words,Borel-σ-algebra corresponds to complete information.
Definition 11 Probability (measure) is a measure such that x(A) = 1.
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Definition 12 Given a measure space (S,S, x), a function f : S → R ismeasurable if
∀a b; f(b) ≤ a ∈ S
One way to interpret a σ-algebra is that it describes the information availablebased on observations. Suppose that S is comprised of possible outcomes ofa dice throw. If you have no information regarding the outcome of the dice,the only possible sets in your σ-algebra can be ∅ and S. If you know thatthe number is even, then the smallest σ-algebra given that information isS = ∅, 2, 4, 6, 1, 3, 5, S. Measurability has a similar interpretation.A function is measurable with respect to a σ-algebra, if it cannot give usmore information about possible outcomes than the σ-algebra. We can alsogeneralize Markov transition matrix to any measurable space.
Definition 13 A function Q : S × S → [0, 1] is a transition probability if
• Q(·, s) is a probability measure for all s ∈ S.
• Q(B, ·) is a measurable function for all B ∈ S.
In fact Q(B, s) is the probability of being in set B tomorrow, given thatthe state is s today. Consider the following example: a Markov chain withtransition matrix given by
Γ =
0.2 0.2 0.60.1 0.1 0.80.3 0.5 0.2
Where Γij is the probability of j given a present state i. Then
Q(1, 2, 3) = Γ31 + Γ32 = 0.3 + 0.5 = 0.8
Suppose that x1, x2, x3 is the fraction of types 1,2,3 today. We can calculatethe fraction of types tomorrow using the following formulas
x′1 = x1Γ11 + x2Γ21 + x3Γ31
x′2 = x1Γ12 + x2Γ22 + x3Γ32
x′3 = x1Γ13 + x2Γ23 + x3Γ33
In other wordsx′ = ΓT · x
23
where xT = (x1, x2, x3). We can extend this idea to a general case witha general transition function. We define the Updating Operator as T (x,Q)which is a measure on S with respect to the σ-algebra S such that
x′(B) = T (x,Q)(B) =
∫S
Q(B, s)x(ds)
A stationary distribution is a fixed point of T , that is x∗ = T (x∗, Q). Weknow that if Q has nice properties4 then a unique stationary distributionexists (for example, we discard “flipping” from one state to another) andx∗ = limn→∞ T
n(x0, Q) ∀x0.
Example: Consider unemployment in a very simple economy (we have anexogenous transition matrix). There are two states: first one is employed(e)and second one is unemployed (ue). The transition matrix is
Γ =
(0.95 0.050.50 0.50
)As part of your homework you have to compute the stationary distribution.
8 Industry Equilibrium
8.1 Preliminaries
Now we are going to study a type of models initiated by Hopenhayn (1992).We will abandon the general equilibrium framework from the previous sec-tion to study the dynamics of distribution of firms in a partial equilibriumenvironment.
To motivate things let’s start with the problem of a single firm that produces agood using labor input according to a technology described by the productionfunction f . Let asume that this function is increasing, strictly concave andf (0) = 0. A firm that hires n units of labor is able to produce sf (n). wheres is a productivity parameter. Markets are competitive in the sense that afirm takes prices as given and chooses n in order to solve
π (s, p) = maxn≥0
psf (n)− wn
4See SLP, Ch. 11.
24
A solution to this problem is a function n∗ (s, p)5. Given the above assump-tions, n∗ is an increasing function of s (more productive firms have moreworkers) as well as p, the FOC is psf ′∗) = w.
Suppose now there is a mass of firms in the industry, each associated witha productivity parameter s ∈ S ⊂ R+. Let x be a measure defined over thespace (S,BS) that describes the cross sectional distribution of productivityamong firms. We will use this measure to define statistics of the industry.For example, at this point it is convenient to define the aggregate supplyof the industry. Since individual supply is just sf (n∗ (s, p)), the aggregatesupply can be written as
ys (p) =
∫S
sf (n∗ (s, p))x (ds)
Suppose now that the demand of the market is described by some functionyd (p). Then the equilibrium price, p∗ is determined by the market clearingcondition yd (p∗) = ys (p∗).
So far, everything is too simple to be interesting. The ultimate goal here isto understand how the object x is determined by the fundamentals of theindustry. Hence we will be adding tweaks to this basic environment in orderto obtain a theory of firms distribution in a competitive environment. Let’sstart by allowing firms to die.
8.2 A Simple Dynamic Environment
Consider now a dynamic environment. The situation above repeats every pe-riod. Firms discount profits at rate r which is exogenously given. In addition,assume that a single firm faces each period a probability δ of disappearing.We will focus on stationary equilibria, i.e. equilibria in which the price of thefinal output p stays constant through time.
Notice first that firm’s decision problem is still a static problem. We caneasily write the value of an incumbent firm as follows
V (s, p) =
∞∑t=0
(1− δ1 + r
)tπ (s, p) =
(1 + r
r + δ
)π (s, p)
5As we declared in advance, this is a partial equilibrium analysis. Hence, we ignore thedependence of the solution on w to focus on the determination of p.
25
Note that we are considering that p is fixed (therefore we can omit it fromthe expressions above). Observe that every period there is positive mass offirms that die. As before, let x be the measure describing the distribution offirms within the industry. The mass of firms that die is given by δx (S). Wewill allow these firms to be replaced by new entrants. These entrants drawa productivity parameter s from a probability measure γ.
One might ask what keeps these firms out of the market. The answer “noth-ing”gives a flaw to fix by assuming that there is a fixed entry cost that eachfirms must pay in order to operate in the market. Moreover, we will assumethat the entrant has to pay this cost before learning s. Hence the value of anew entrant is given by the following function.
V E (p) =
∫s
V (s, p) γ(ds)− cE
Entrants will continue to enter if this is bigger than 0 and continue to decidenot to enter if this value is less than zero. So stationarity occurs where this isexactly equal to zero (this is the free entry assumption, and we are assumingthat there is an infinite number (mass) of prospective firms).
Let’s analyze how this environment shapes the distribution of firms in themarket. Let xt be the cross sectional distribution of firms in period t. Forany B ⊂ S the number of firms with s ∈ B in period t is given by xt (B)Next period, some of them will die, and that will attract some newcomers.Hence next period measure of firms on set B will be given by:
xt+1 (B) = (1− δ)xt (B) +mγ (B)
That is, m firms enter and γ(B) will belong in the set B. As you mightsuspect, this relationship must hold for every B ∈ BS. Moreover, since weare interested in stationary equilibria, the previous expression tells us thatthe cross sectional distribution of firms will be completely determined by γ.Stationarity on the system we implies:
x∗(B;m) =m
δγ(B)
Now note that demand supply relation has form
yd (p∗(m)) =
∫S
sf (n∗ (s, p)) dx∗ (s;m)
26
whose solution, p∗(m), is continuous function under regularity conditionsstated in SLP. We have two equations and two unknowns p and m.
Definition 14 A stationary distribution for this environment consists offunctions p∗, x∗ and m∗ such that
1. yd (p∗(m)) =∫Ssf (n∗ (s, p)) dx∗ (s;m)
2. V E (p) =∫sV (s, p) γ(ds)− cE
3. x∗(B) = (1− δ)x∗(B) +m∗γ(B),∀B ∈ B
8.3 Introducing Exit Decision
We want to introduce more (economic) content by making the exit of firmsendogenous (a decision of the firm). Let’s introduce now a cost of operation.Suppose firms have to pay cv each period in order to stay in the market andassume S = [s, s]. By adding such a minor change, the solution still hasreservation productivity property under some conditions (to be discussedbelow). In words, there will be a minimum s which will make profitable forthe firm to stay in the market. To see that this will be the case you shouldprove that the profit before variable cost function π (s, p) is increasing in s.Hence the productivity threshold is given by the s∗ that satisfies the followingcondition:
π (s∗, p) = cv
for an equilibrium price p. Now instead of considering γ as the probabilitymeasure describing the distribution of productivities among entrants, youmust consider γ defined as follows
γ (B) =γ (B ∩ [s∗, s])
γ ([s∗, s])
for any B ∈ BS.
One might suspect that this is an ad hoc way to introduce the exit decision.To make the things more concrete and easier to compute, we will assume thats is a Markov process. To facilitate the exposition, let’s make S finite andassume s has transition matrix Γ. Assume further that Γ is regular enough
27
so that it has a stationary distribution γ. For the moment we will not putany additional structure on Γ.
The operation cost cv is such that the exit decision is meaningful. Let’sanalyze the problem from the perspective of the firm’s manager. He hasnow two things to decide. First, he asks himself the question “Should Istay or should I go?”. Second, conditional on staying, he has to decide howmuch labor to hire. Importantly, notice that this second decision is still astatic decision. Later, we will introduce adjustment cost that will make thisdecision a dynamic one.
Let φ (s, p) be the value of the firm before having decided whether to stay orto go. Let V (s, p) be the value of the firm that has already decided to stay.V (s, p) satisfies
V (s, p) = π (s, p)− cv +1
1 + r
∑s′∈S
Γss′φ (s′, p)
Each morning the firm chooses d in order to solve
φ (s, p) = maxd∈0,1
dV (s, p)
Let d∗ (s, p) be the optimal decision to this problem. Then d∗ (s, p) = 1means that the firm stays in the market. One can alternatively write:
φ (s, p) = maxd∈0,1
d
[π (s, p)− cv +
1
1 + r
∑s′∈S
Γss′φ (s′, p)
]or even
φ (s, p) = max
[π (s, p)− cv +
1
1 + r
∑s′∈S
Γss′φ (s′, p) , 0
]
All these are valid. Additionally, one can easily add minor changes to makethe exit decision more interesting. For example, things like scrap value orliquidation costs will affect the second argument of the max operator above,which so far is just zero.
What about d∗ (s, p)? Given a price, this decision rule can take only finitelymany values. Moreover, if we could ensure that this decision is of the form
28
“stay only if the productivity is high enough and go otherwise”then the rulecan be summarized by a unique number s∗ ∈ S. Without doubt, that wouldbe very convenient, but we don’t have enough structure to ensure that suchis the case. Because, although the ordering of s (lower s are ordered beforehigher s) gives us that the value of s today is bigger than value of smaller s′,depending on the Markov chain, on the other hand, the value of productivitylevel s tomorrow may be lower than the value of s′ (note s′ < s) tomorrow.Therefore we need some additional regularity conditions.
In order to get a cutoff rule for the exit decision, we need to add an as-sumption about the transition matrix Γ. Let the notation Γ (s) indicate theprobability distribution over next period state conditional on being on states today. You can think of it as being just a column of the transition matrix.Take s and s. We will say that the matrix Γ displays first order stochasticdominance (FOSD) if s > s implies
∑s′≤b Γ (s′ | s) ≤
∑s′≤b Γ (s′ | s) for any
b ∈ S. It turns out that FOSD is a suffi cient condition for having a cutoffrule. You can prove that by using the same kind of dynamic programmingtricks that we have used in the first semester for obtaining the reservationwage property in search problems. Try it as an exercise. Also note that thisis just a suffi cient condition.
Finally, we need to mention something about potential entrants. Since wewill assume that they have to pay the cost cE before learning their s, theycan leave the industry even before producing anything. That requires us tobe careful when we describe industry dynamics.Now the law of motion becomes;
x′(B) = mγ(B ∩ [s∗, s]) +∑s′
1s′∈B∩[s∗,s]Γ(s, s′)x(ds)
8.4 Stationary Equilibrium
Now that we have all the ingredients in the table, let’s define the equilibriumformally.
Definition 15 A stationary equilibrium for this environment consists of alist of functions (φ, n∗, d∗), a price p∗ and a measure x∗ such that
29
1. Given p∗, the functions φ, n∗, d∗ solve the problem of the incumbentfirm
2. V E (p∗) = 0
3. For any B ∈ BS (assuming we have a cut-off rule with s∗ is cut-off instationary distribution)6
x∗(B) = mγ(B ∩ [s∗, s]) +∑s′
1s′∈B∩[s∗,s]Γ(s, s′)x(ds) (1)
You can think of condition (2) as a “no money left over the table” condi-tion, which ensures additional entrants find unprofitable to participate inthe industry.
We can use this model to compute interesting statistics. For example theaverage output of the firm is given by
Y
N=
∑sf(n∗(s))x∗(ds)∑
x∗(ds)
Next, suppose that we want to compute the share out output produced bythe top 1% of firms. To do this we first need to compute s such that∑s
s x∗(ds)
N= .01
where N is the total measure of firms. Then the share output produced bythese firms is given by ∑s
s sf(n∗(s))x∗(ds)∑ss sf(n∗(s))x∗(ds)
6If we do not have such cut-off rule we have to define
x∗ (B) =
∫S
∑s′∈S
Γss′1s′∈B1d(s′,p∗)=1x∗ (ds) + µ∗
∫S
1s∈B1d(s,p∗)=1γ (ds)
where
µ∗ =
∫S
∑s′∈S
Γss′1d(s′,p∗)=0x∗ (ds)
30
Suppose now that we want to compute the fraction of firms who are in thetop 1% two periods in a row. This is given by∑
s≥s
∑s′≥s
Γss′x∗(ds)
We can use this model to compute a variety of other statistics include theGini coeffi cient.
8.5 Adjustment Costs
To end with this section it is useful to think about environments in whichfirm’s productive decision is no longer static. A simple way of introducingdynamics is by adding adjustment costs.
We will consider labor adjustment costs. First let think of this sequentially,not recursively. These costs work pretty much like capital adjustment costsas you might suspect. Consider a firm that enters period t with nt−1 unitsof labor. We have then three alternatives (these are some particular specifi-cations, there are endless others):
• Convex Adjustment costs: if the firm wants to vary the units of labor,it has to pay α (nt − nt−1)2 units of the numeraire good. The cost heredepends on the size of the adjustment.
• Training costs or hiring costs: if the firm wants to increase labor, ithas to pay α (nt − (1− δ)nt−1)2 units of the numeraire good only ifnt > nt−1, i.e. 1nt>nt−1α (nt − (1− δ)nt−1)2. δ measures attrition.
• Firing costs: Similarly we can also have firing costs.
The recursive formulation of the firm’s problem conditional on stayingwould be
V (s, n−, p) = maxn≥0
psf (n)−wn−α (n− (1− δ)n−)2−cv+1
1 + r
∑s′∈S
Γss′V (s′, n, p)
31
for the case with pure adjustment costs. In class we also saw an examplein which the firm pays a cost κ to post vacancies. In this case the firm’sproblem is
Ω (s, n−, p) = max0,maxv≥0
psf (n)− wn− κv +1
1 + r
∑s′∈S
Γss′V (s′, n, p)
wheren = vφ+ (1− δ)n.
9 Models with Growth
Previously we have seen the Neo-classical Growth Model as our benchmarkmodel and built on it for the analysis of more interesting economic ques-tions. One peculiar characteristic of our benchmark model, unlike its namesuggested, was lack of growth (after reaching steady state). Many interestingquestions in economics are related to the cross-country differences of growthrates and we will cover some models that will allow for growth so that wewill be able to attempt to answer such questions.
9.1 Exogenous Growth
We know that in our standard NGM there are basically two ways of growth,one in which everything grows, which is not necessarily a per-capita growth,and the other is per-capita growth. We will be focusing on per-capita growth.The title exogenous growth refers to the structure of models in which thegrowth rate is determined exogenously, and is not an outcome of the model.The simplest one of these is one in which the determinant of the growth rateis population growth.
9.1.1 Population Growth
Suppose the population of our economy grows at rate γ and we have theclassical CRTS technology in capital and labor inputs.
Yt = AF (Kt, Nt)
32
Nt = N0γt
Note that our economy is no longer stationary but as we will see, withinthe exogenous growth framework we can make these economies look likestationary ones by re-normalizing the variables. Thus, at the end of the dayit will only be a mathematical twist on our standard growth model. Oncewe do that, we will be looking for the counterpart of a steady state thatwe have in our stationary economies, the Balanced Growth Path, in whichall the variables will be growing at constant rates but not necessarily equal.Back to our population growth model, we know
AF (K,N) = A[KFK(K,N) +NFN(K,N)]
Question is, if N is growing at rate γ, can this economy have a balancedgrowth path? Can we construct one? We know that by the CRTS propertyFK and FN are homogenous of degree zero. If we assume capital stockgrows at rate γ as well, then prices stay constant and per-capita variablesare constant and output grow at the same rate. So we get growth on abalanced growth path without per-capita growth. One question is how dowe model population growth in our representative agent model. One way isto assume there is a constant proportion of immigration to our economy fromoutside but this has to assume the immigrants are identical to our existingagents in our economy, which is a bit problematic. The other way could beto assume growing dynasties which preserves the representative agent natureof our economy. If we do so, the problem of the social planner becomes:
max∞∑t=0
βtNtU(CtNt
)
s.t.
Ct +Kt+1 = AF (Kt, Nt) + (1− δ)Kt
nonegativity
To transform the feasibility set to per capita terms, divide all terms by Nt
and to make the environment stationary by dividing all the variables by γt
33
and assume N0 = 1 we get,
max∞∑t=0
(βγ)tN0U(ct)
s.t.
ct + γkt+1 = AF (kt, 1) + (1− δ)ktnonegativity
So how is this transformed model any different than our NGM? By the dis-count factor, the agents in this economy with growth discounts the futureless but everything else is identical to NGM of course with the exception ofthis economy growing at a constant rate.
9.1.2 Labor Augmenting Productivity Growth
Now suppose we have a labor augmenting productivity growth with constantpopulation normalized to one, i.e. have the following CRTS technology:
Yt = AF (Kt, γtNt)
AF (Kt, γtN0) = A[KtFK(Kt, γ
tN0) + γtN0FN(Kt, γtN0)]
max∞∑t=0
βtU(CtNt
)
s.t.
Ct +Kt+1 = AF (Kt, γtNt) + (1− δ)Kt
nonegativity
and since we have a population of one, these variables are already per-capitaterms. For stationarity, we have to normalize the variables to per productivity
34
units, by dividing all by γt. Then the problem becomes:
max
∞∑t=0
βtU(γtct)
s.t.
ct + γkt+1 = AF (kt, 1) + (1− δ)ktnonegativity
Suppose we have a CRRA preferences, then the question is how can werepresent the preferences as a function of ct only. Using CRRA:
∞∑t=0
βt(γtct)
1−σ − 1
1− σ ≈∞∑t=0
(βγ1−σ)tc1−σt − 1
1− σ
Note that we have added some constant to right hand side, but we knowthat adding and subtracting constants to objective function does not changethe solution of the problem. Once again it is the exact same problem as theNGM with a different discount factor. Note that the existence of a solutionto this problem depends on (βγ1−σ). In this set-up we get per-capita growthat rate γ. Also note that CRRA is the only functional form for preferencesthat is compatible with BGP. This is because as per-capita output grows,for consumption to grow at a constant rate, our agent has to face the sametrade-off at each period. Now suppose we have the TFP growing at rate γwith a CRTS Cobb-Douglas technology
Yt = AtF (kt, 1)
At+1
At= γ
What would be the growth rate of this economy? We can show that likethe previous cases the growth rate of the economy is the growth rate for theproductivity of labor, which is γ
σ1−σ in this case.
9.2 Endogenous Growth
So far in the models we covered growth rate has been determined exoge-nously. Next we will look to models in which the growth rate is chosen by
35
the model itself. We do know for a fixed amount of labor, the curvature of ourtechnology limits the growth due to diminishing marginal return on capitaland with depreciation there is an upper limit on capital accumulation. So ifour economy is to experience sustainable growth for a long period of time, weeither give up the curvature assumption on our technology or we have to beable to shift our production function up. Given a fixed amount of labor, thisshift is possible either by an increasing TFP parameter or increasing laborproductivity. The simplest of such models where we can see this is the AKmodel, where the technology is linear in capital stock so that diminishingmarginal return on capital does not set in.
9.2.1 AK Model
Recall the simple one-sector growth model from the first mini. When thereis curvature in the production function as a function of capital, there is nolong-run growth. When the production function is linear in capital there isa balanced growth path but there is no transitional dynamics. From, thissimple observation, we can see that in order to get long-run growth we needa model that behaves similar to the AK growth model. One way to achievethis is to assume there is human capital and production function is CRS w.r.tphysical capital and human capital. When investment in human capital isdone using consumption goods, you have seen that this economy behaves asan AK economy. This is not true when the cost of investment in humancapital is time. In this section, we build another model that behaves similarto an AK model.
max∞∑t=0
βtU(ct)
s.t.
yt = ct + kt+1 = Akt
ct, kt+1 ≥ 0
This problem will give following Euler equation:
Uc(ct) = βUc(ct+1)A
When preferences are CRRA, i.e.
U(c) =c1−σ − 1
1− σ
36
and when we are on a balanced growth path (consumption is growing at arate γc) then Euler equation gives the following growth rate: γc = (βA)
1σ .
9.2.2 Human Capital Models
Now assume production technology is such that we have another productionfactor namely human capital. Then followings are the feasibility conditionand law of motion for the human capital
ct + IKt + [IHt ] = AKθt (Htn
1t )
(1−θ)
Kt+ 1 = (1− δ)Kt + IKt[n1t + n2
t = 1]
ct, kt+1 ≥ 0
Equations and variables in brackets appear when they are needed in the fol-lowing specifications of law of motion for capital. Law of motion for thecapital can be different for different interpretation of human capital forma-tion. If we think human capital is built “with bricks”:
Ht+1 = (1− δh)Ht + IHt
If we think it is formed by learning by doing than we can define individualhuman capital formation in two ways:
ht+1 = g(n2t , ht)
orht+1 = g(n2
t , Ht)
Lucas defined a human capital model with the specification of schooling andinelastic labor supply. Now that there is no limit to the accumulation ofhuman capital and sustainable growth on a BGP is feasible. Furthermore,an analysis of the characterization of the balanced growth path will indicatethat this model indeed has transitional dynamics, so unlike the AK modelif economy starts out of this optimal growth path economy can adjust andconverge to it by responding to prices in a de-centralized setting. If one de-fine a learning by doing model, s/he can see there is a natural limit to thegrowth of human capital and such an economy might not have a BGP. Thekey ingredient of endogenous growth with labor is then the reproducibilityof the human capital without such a limit.
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9.2.3 Growth Through Externalities by Romer
Then let’s write the growth model with externality model of Romer (1986).We have seen in the AK model that the growth rate is determined solelyby model primitives and endogenized but still it is not directly or indirectlydetermined by the agents’choices in our model. In Lucas’human capitalmodel,the growth rate is determined by the choice of agents, specificallyby the optimal ratio of human and physical capital.The source of growthin Lucas’model is reproducibility of human capital. In the next model,Romer introduces the notion of externality generated by the aggregate capitalstock to go through the problem of diminishing marginal returns to aggregatecapital. In this model, the source of growth will be the aggregate capitalaccumulation, which is possible with a linear aggregate technology in capitalas we saw in the AK model.The firms in our model will not be aware of thisexternality and will have the usual CRTS technology and observe the sourceof growth coming from the TFP parameter. As usual with externalities,the equilibrium outcome will not be optimal. Each firm has the followingtechnology:
yt = Atkθt (Ktnt)
1−θ
We can write this technology as follows
yt = Atkθtn
1−θt
where At = AK1−θt . With this technology and assuming CRRA utility Euler
equation for house hold in balanced growth path reduced to
1 = βγ−σ(1 + r)
where r = MPK.
9.2.4 Monopolistic Competition, Endogenous Growth and R&D(Romer)
There are three sectors in the economy: a final good sector, an intermediategoods sector and R&D sector. Final goods are produced using labor (as wewill see there is only one wage since there is only one type of labor) and
38
intermediate goods according to the production function
Nα1t
∫ At
0
xt (i)1−α di
where x (i) denotes the consumption of intermediate good of variety i ∈[0, At]
7. The intermediate goods are produced using capital according to alinear technology
x (i) =k (i)
η
The aggregate demand of capital from this sector is∫ At
0ηx (i) di.
A new good is a new variety of the intermediate good. In every period, aflow of new intermediate goods is created by using labor according to thefollowing production technology
At+1
At= 1 + ξN2t
Notice that with after some manipulation, one can express growth in thestock of intermediate goods as follows
At+1 − At = AtξN2t
Hence, the flow of new intermediate goods is determined by the stock of themin the economy. This type of externality is the key feature of the model8.This assumption provides us with a constant returns to scale technology inthe R&D sector.
Certainly, this is not a model that one can map to the data. Instead it hasbeen carefully crafted to deliver what is desired and it provides an interestinginsight in thinking about endogenous growth.
Now let’s posit a form of competition between sectors. We will assume thatfinal good producers act as price takers. Intermediate good firms act as
7The function that aggregates consumption of intermediate goods is often referred asDixit-Stiglitz aggregator.
8Perhaps, the basic idea of this production function might be traced back to IsaacNewton’s quote “If I have seen further it is only by standing on the shoulders of giants”.
39
monopolistic competitors, which means that they set the price of their varietyalthough they take the rental price of capital as given. Finally, the R&Dsector also acts as a price taker.
Solving the model
Let’s consider first the problem of a final good producer. In every periodthey choose N1t and xt (i) for every i in order to solve
maxNα1t
∫ At
0
xt (i)1−α di− wtN1t −∫ At
0
qt (i)xt (i) di
where qt (i) is the price of variety i in period t. First order conditions for thisproblem are
N1t : αNα−11t
∫ At
0
xt (i)1−α di = wt
xt (i) : (1− α)Nα1txt (i)−α = qt (i) ∀i
From the second condition one obtains
xt (i) =
((1− α)
qt (i)
) 1α
N1t (1)
which is the demand for variety i of the final good producer.
Let’s consider now the problem of an intermediate firm. These firms act asa price setter. They choose qt (i) in order to solve
πt (i) = max qt (i)xt (qt (i))− rtηxt (qt (i))
where xt (qt (i)) is the demand function in (1). Notice that we have pluggedthe functional form for the technology, i.e. xt (qt (i)) = 1
ηkt (i). First order
conditions for this problem are
qt (i) : xt (qt (i))N1t + (qt (i)− rtη)∂xt (qt (i))
∂qt (i)= 0
which can be written as
(1− α)1α
qt (i)1α
N1t =(qt (i)− rtη)
α
(1− α)1α
qt (i)1+αα
N1t
40
Rearranging yields
qt (i) =rtη
(1− α)(2)
Note what happens when α = 0. Plugging this into the demand functionyields
xt (i) =
((1− α)2
rtη
) 1α
N1t (3)
and the demand for capital services is just ηxt (i).Let Yt be the production of the final good and plug (2) and (3) in its
corresponding production function to get
Yt = N1tAt
((1− α)2
rtη
) 1−αα
(4)
Hence the model displays constant returns to scale in At.
Let’s study now the problem of the R&D firm. This firm chooses N2t in orderto solve the following problem
max ptAt−1ξN2t − wtN2t
An equilibriumwith positive production of R&D goods require pt = wt/At−1ξ,which pins down the price of them. It remains to find an equation to pindown the actual production of these goods. Well, in equilibrium it must bethat no more of the R&D goods are produced because nobody finds it prof-itable. That is, the present discounted value of a firm producing it must beequal to the price of introducing an additional variety
pt =∞∑s=t
πs (i)∏sτ=t (1 + rτ )
τ−t (5)
Notice that the stock of new goods will be growing at a rate ξN2t. In fact,in a balanced growth path, aggregate variables will be growing at this ratewhereas labor of course will remain constant.
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So far we have been silent about the consumer side of the model. Let justadd it for the sake of completeness. We will assume the consumer choosespositive streams of consumption, labor services and capital services in orderto solve
max∞∑t=0
βtu (ct)
s.t. ct + kt+1 ≤ rtkt + wt for all t
k0 given
Finally, market clearing condition are the usual ones
N1t +N2t = 1
ct + kt+1 = Yt
for every t. One final comment, in this economy we have GDP = Yt +ptξAtN2t = Ct + [Kt+1 − (1− δ)Kt] + ptξAtN2t.
10 Search Models
The last topic we will study in the course is an attempt to abandon theassumption of frictionless markets. The ultimate goal is to embed the basicsearch problem into a richer environment in which we can study the dynamicsof the labor market from an equilibrium perspective. A good reference forthis part is Pissarides 2000.
10.1 The Search Problem
An infinitely lived agent receives each period an offer to work at wage w.The agent has to decide whether to accept or reject the offer taking intoaccount the fact that next period he will receive a new offer drawn according
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to the distribution function F with support [0, w]. The lifetime utility of arisk neutral agent with discount factor β who accepts the offer w would be
Ω (w) =
∞∑t=0
βtw =w
1− β
If the agent rejects the offer he will remain unemployed for one period beforehe can receive another offer. Let b denote the unemployment compensationfor the agent. The lifetime utility of an agent rejecting the offer would be
U = b+ β
∫ w
0
max (Ω (w) , U) dF (w)
Notice that the value of rejecting the offer does not depend on the currentoffer w9, people working do not receive offers (there is no “on the job search”).Hence, the problem of a worker with an offer w in hand can be written asfollows
V (w) = maxd∈0,1
dΩ (w) + (1− d)U
where d = 1 means that the worker accepts the offer. It should be obviousthat the optimal policy for the player is to accept whenever Ω (w) ≥ Uand reject otherwise10 Hence, as it was the case when we studied IndustryEquilibrium, the optimal policy can be fully characterized by a cutoff valuew∗ which is often called reservation wage. Clearly, w∗ can not be less than b.Actually, w∗ is strictly higher since the value of being unemployed is not justthe one period return b but also the value of having the chance of receivinga higher offer next period. Formally, as long as b ∈ (0, w) we have that
U ≥ b+ βU
U ≥ b
1− β
Since Ω (w∗) = U , the result follows.
9It could depend but we are assuming that draws across periods are independent ofeach other.10Let’s assume for simplicity that a a worker that is indifferent always accepts.
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10.2 A Continuous Time Formulation
Let consider now a the limit of the economy just presented when the intervalof time tends to zero. Let ∆ be the size of the time interval and r be therate at which the agent discounts the future. The lifetime utility of an agentaccepting the offer can be expressed recursively as follows
Ω (w) = ∆w +1
1 + ∆rΩ (w) (1)
Suppose that job offers occur at each time interval with probability α∆.Then, the value of rejecting an offer can be expressed as
U = ∆b+∆α
1 + ∆r
∫ w
0
max (Ω (w) , U) dF (w) +1−∆α
1 + ∆rU (2)
Write (1) as follows
r
1 + ∆rΩ (w) = w
Make ∆ go to zero to get
rΩ (w) = w
This last expression is interpreted as the flow value of accepting a job whichis equal to the wage w. Let proceed in the same way with (2). Rewrite it asfollows:
r + α
1 + ∆rU = b+
α
1 + ∆r
∫ w
0
max (Ω (w) , U) dF (w)
Make ∆ go to zero to get
(r + α)U = b+ α
∫ w
0
max (Ω (w) , U) dF (w)
Or equivalently
rU = b+ α
∫ w
0
max (Ω (w)− U, 0) dF (w)
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This last expression indicates that the flow value of rejecting an offerconsists of two parts. The instantaneous compensation b and the expectedcapital gain from getting a better offer, which happens with probability α.The reservation wage w∗ in this setting satisfies
w∗ = b+ α
∫ w
0
max (Ω (w)− U, 0) dF (w)
Which can be written as follows
w∗ = b+α
r
∫ w
w∗(w − w∗) dF (w)
= b+α
r
((w − w∗)−
∫ w
w∗F (w) dw
)= b+
α
r
(∫ w
w∗dw −
∫ w
w∗F (w) dw
)= b+
α
r
∫ w
w∗(1− F (w)) dw
where the second line follows from integration by parts.
Now suppose the agent can affect the probability of receiving an offer byexerting effort. Let assume that an effort level of e determines a probabilityof receiving an offer while unemployed equal to α (e) and consumes e unitsof the numeraire. Then, the unemployed worker has a non trivial problem.He has to choose e in order to solve
U = maxeb− e+
α (e)
r
∫ w
w∗(1− F (w)) dw
A solution to the agent’s problem consists now of a pair (w∗, e∗) that indicateswhat offers to accept and how much effort to make.
10.3 Generating Transitions
Suppose now that workers are fired with exogenous probability λ. Assumingthere is a unit measure of agents in the economy we can use this information
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to calculate the stationary rate of unemployment. We denote the rate ofunemployment by u and the derivative of this indicator with respect to timeby u. Each unemployed worker transits to employment at rate α [1− F (w∗)]whereas each employed worker becomes unemployed at rate λ. Therefore wecan write
u = (1− u)λ− α [1− F (w∗)]u
Using the fact that in steady state this expression must equal zero we getthe stationary rate of unemployment
u∗ =λ
λ+ α [1− F (w∗)]
Note that, for example, a change in α affects u∗ “directly”, but also affects w∗.
10.4 On-the-Job Search
A more interesting type of transition would be to allow workers to do on thejob search. Suppose that offers arrive at rate α0 for unemployed workers andat rate α1 for employed workers. Following similar steps as above we can getthe value of being unemployed
rU = b+ α
∫ w
w∗max (Ω (w)− U, 0) dF (w)
The value of being employed needs some modification. Now, the agent canget an offer above his current wage. Hence the value of being unemployedbecomes
rΩ (w) = w + α1
∫ w
0
max (Ω (w′)− Ω (w) , 0) dF (w′) + λ [U − Ω (w)]
The expression for the reservation wage w∗ reflects how these arrival ratesaffect the optimal decision of the agent
w∗ = b+ (α− α1)
∫ w
w∗
[1− F (w)
r + λ+ α1 [1− F (w)]
]dw
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10.5 Matching and Bargaining
We now use the methods developed so far to think about vacancies and un-employment and how they can exists in an economy at the same time. Todo so, we need to put forward a General equilibrium model with firms andhouseholds. Assume that unemployment is represented by u and number ofvacancies are v. There exists an exogenous matching function m(u, v) thatis the rate of matching between firms and workers. Matching function willhave the properties:
m(0, v) = m(u, 0) = 0
m1 > 0,m2 > 0,m11 < 0,m22 < 0
tm(u, v) = m(tu, tv)
i.e., m(·, ·) is a CRS, concave and increasing function.
If we assume all unemployed and vacancies are homogenous, then the arrivalrate of a job for a worker, αw and arrival rate of a worker for a firm, αe aregiven by
αw =m(u, v)
u, αe =
m(u, v)
v
Firms and workers match given the described matching technology. Notem(u, v) = u ·m(1, v/u) = u ·αw(q) where q = v/u, also we require m(u, v) ≤min(u, v).
We need a theory of how wages get determined upon matching. Oneway to model this is to use Nash bargaining. Assume that the value ofan unemployed worker is is U and the value of a vacancy for a firm is V .Moreover, suppose that Ω(w) is the value of a worker with wage w andJ(y − w) is the value of a firm paying w to a worker who produce y unit ofoutput. Then, Nash bargaining implies that w is chosen to solve the followingproblem:
maxw
(Ω(w)− U)µ(J(y − w)− V )1−µ
where µ is the bargaining power for the worker. Given the Bellman equations
47
we developed before, we have
rΩ(w) = w + λ(U − Ω(w))
rJ(π) = π + λ(V − J(π))
rU = b+ αw(Ω(w)− U)
rV = −c+ αe(J(π)− V )
where c is the operating cost of a vacancy and π = y −w is the profit of thefirm (notation: αw(q), αe(q)).
The FOC of the Nash Bargaining problem gives:
µ[J(y − w)− V ]Ω′(w) = (1− µ)[Ω(w)− U ]J ′(y − w)
Given the Bellman equations,
J ′(y − w) = Ω′(w) =1
r + λ
which impliesµ[J(y − w)− V ] = (1− µ)[Ω(w)− U ]
The surplus from the match is S = J(y−w)− V + Ω(w)−U and the aboveequation says that workers receive fraction µ of the total surplus and firmsreceive the rest. Surplus can also be determined by
S =y − rV − rU
r + λ
Hence
Ω(w) = U + µS = b+ µy − rV − rU
r + λ
determines the wage rate.
In order to characterize equilibrium, we either need a free entry conditionfor firms or a fixed number of vacancies. One free-entry condition wouldbe V = 0 which say firms can freely create vacancies. This section closelyfollows Chapter 2 of Pissarides (2000). An alternative to V = 0 is to assumethat there is an exogenous number of “licensed”firms and this number isequal to u · q + (1− u) (then it can be that V > 0).
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Definition 16 A stationary equilibrium for this economy is w∗, u∗, v∗, α∗w, α∗e,Ω(.), U, V
and J(.) st:
u = λ(1− u∗) + u∗(1− α∗w) = 0
α∗w =m(u∗, v∗)
u∗
α∗e =m(u∗, v∗)
v∗
rΩ(w∗) = w∗ + λ(U − Ω(w∗))
rJ(y − w∗) = y − w∗ + λ(V − J(y − w∗))rU = b+ α∗w(Ω(w∗)− U)
rV = −c+ α∗e(J(y − w∗)− V )
we need two additional conditions: Nash bargaining and the free entrycondition for firms (count equations and unknowns).
10.6 Competitive Search
Suppose as previous section that at some point in time there are v vacan-cies posted by firms looking for workers and u unemployed workers lookingfor jobs. We assume the number of contacts between firms and workers isgiven by a matching technology m = m(u, v). This function is an exogenousspecification such that:
m(0, v) = m(u, 0) = 0
m1 > 0,m2 > 0,m11 < 0,m22 < 0
tm(u, v) = m(tu, tv)
i.e., m(·, ·) is a CRS, concave and increasing function.
If we assume all unemployed and vacancies are homogenous, then the arrivalrate of a job for a worker, αw and arrival rate of a worker for a firm, αe aregiven by
αw =m(u, v)
u, αe =
m(u, v)
vNote that CRS assumption of the matching function provide the conditionthat that αw and αe depend only on the ratio v/u, market tightness. Note
49
also that αw is an increasing and αw is a decreasing function of v/u.
In the previous section we studied bargaining, and here we discuss anotherpossibility: ex ante wage posting. In addition to posting, the models in thesection also adopt the idea of directed search, which means that agents can,as the name suggests, direct their search efforts towards particular wages.Models which combine the combination of posting and directed search arecalled “competitive search models”.
In a static setting, at the beginning of the period, there are large numbers uand v of unemployed workers and vacancies, and we let q∗ = u/v. Each firmwith a vacancy must pay cost k, and we can either assume free entry (mak-ing v endogenous) or fix the number of vacancies (v exogenous). Any matchwithin the period produces output y. At the end of the period, unmatchedworkers will get b, while unmatched vacancies get 0.
Consider a worker facing a menu of different wages. Let U denote the highestvalue that he can get by applying for a job at some firm. Then a worker iswilling to apply to a particular job offering a wage w ≥ b only if he believes thequeue length q, i.e. the number of workers who apply, is suffi ciently small.Equivalently, he is willing to apply only if the hiring probability αw(q) issuffi ciently large:
U ≤ αw(q)w + [1− αw(q)]b
If this inequality is strict, all workers would want to apply to this firm,reducing the right hand side. Therefore, in equilibrium, this will hold byequality. Then employers problem will be
V = maxw,q−k + αe(q)(y − w)
s.t.
U = αw(q)w + [1− αw(q)]b
Eliminating w using constraint and also using αe(q) = qαw(q):
V = maxq−k + αe(q)(y − b)− q(U − b)
First order condition for this problem gives
α′e(q)(y − b) = U − b
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Since αe is concave this is also suffi cient condition. It implies that all employ-ers choose the same q, which in equilibrium must equal the economy-wideq∗. Then U can be found from
α′e(q∗)(y − b) = U − b
Using this U , wage rate can be found from
U = αw(q∗)w + [1− αw(q∗)]b
which will be
w∗ = b+q∗α′e(q
∗)
αe(q∗)(y − b)
If the number of vacancies v is fixed this gives profit
V = −k + [αe(q∗)− q∗α′e(q∗)](y − b)
or we can use free entry V = 0 to endogenize v and hence q∗.
51