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The Neoclassical Growth Model in Discrete Time

D. Krueger Lecture Notes Chapter 3P. Krusell Lecture Notes Chapter 3, 5

The model will now be enriched by production, which gives rise to the neoclassicalgrowth model.We will present in discrete-time model to discuss the rationale behind the method ofdynamic programming.We will also discuss the connection between Pareto optimal allocations and allocationsarising in a competitive equilibrium.

1 The Neoclassical Growth ModelNeed to describe the environment of the model by specifying technology, preferences,endowments and the information structure.

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Time is discrete and indexed by t = 0, 1, 2, ... There are households and firms.There is a large number of identical, in finitely lived households. Thus, we can, with-out loss of generality, consider the maximization problem of a single representativehousehold. At period 0 the household is endowed with one unit of time and k0 unitsof initial capital. The preference of the representative household is representable by atime-separable utility function:

∞Pt=0

βtU(ct).

The technology of the firm is to employ capital and labor services (kt, nt) to produceoutput goods yt, according to the production function yt = F (kt, nt). Output goodscan be consumed or invested yt = ct + it. Suppose the capital stock depreciates at aconstant rate δ over time. Thus, the stock of capital evolves according to

kt+1 = (1− δ)kt + it,

which implies that the gross investment it equals net investment (kt+1 − kt) plus depre-ciation δkt.

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We assume that kt+1 ≥ 0, but not it ≥ 0, implying that the existing capital stock can bedisinvested to be consumed, i.e., capital is putty-putty (not putty-clay).We will assume that households own the capital stock and make the investment deci-sion. They will rent out capital to the firms.Finally, there is no uncertainty in this economy and households and firms have fullinformation.

1.1 Pareto Optimal AllocationsConsider the problem of a social planner. Given our the environment of our model, thesolution to the social planner’s problem is Pareto efficient.An allocation {ct, kt, nt}∞t=0 is feasible if for all t ≥ 0,

ct + kt+1 ≤ F (kt, nt) + (1− δ) kt,

ct ≥ 0, kt+1 ≥ 0, 0 ≤ nt ≤ 1, k0 ≤ k0.

An allocation {ct, kt, nt}∞t=0 is Pareto efficient if it is feasible and there is no other feasible

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allocation {ect,ekt, ent}∞t=0 such that (recall that this is a representative agent model)∞Pt=0

βtU (ect) > ∞Pt=0

βtU(ct).

Thus, the problem of the social planner is

w¡k0¢= max{ct,kt,nt}∞t=0

∞Pt=0

βtU(ct)

s.t. ct + kt+1 ≤ F (kt, nt) + (1− δ) kt,

ct ≥ 0, kt+1 ≥ 0, 0 ≤ nt ≤ 1, k0 ≤ k0.

We make the following assumptions on preferences and technology.

(A1) U is continuously differentiable, strictly increasing, strictly concave and bounded.It satisfies the Inada conditions, limc→0U

0(c) =∞ and limc→∞U 0(c) = 0.(A2) F is continuously differentiable and homogenous of degree 1, strictly increasingin both arguments and strictly concave. F (0, n) = F (k, 0) = 0 for all k, n > 0. Also Fsatisfies the Inada conditions, limk→0F

0(k, 1) =∞ and limk→∞F 0(k, 1) = 0.

By the Inada conditions, ct = 0 and kt+1 = 0 cannot be optimal, so we can ignore the

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non-negativity constraints.By (A1) and (A2), we have nt = 1 for all t (households do not value leisure) and k0 = k0(production function is strictly increasing in capital). Given the results, we denote

f(k) = F (k, 1) + (1− δ)k, for all k,

which gives the total amount of the final good available for consumption or investment.By (A2), f is continuously differentiable, strictly increasing and strictly concave, f(0) =0, f 0(k) > 0 for all k, limk→0 f

0(k) =∞ and limk→∞ f 0(k) = 1− δ.The feasibility constraint becomes ct ≤ f (kt) − kt+1. By (A1), U is strictly increasing,implying that goods will not actually be thrown away, because they are valuable. Thus,the feasibility constraint will be binding, i.e., ct = f (kt)− kt+1.Then we rewrite the social planner’s problem

w¡k0¢= max{kt+1}∞t=0

∞Pt=0

βtU(f (kt)− kt+1) (1)

s.t. 0 ≤ kt+1 ≤ f (kt) , k0 = k0 given.

Note that by solving this social planner’s problem, we will also have solved for compet-itive equilibrium allocations of our model. But how to solve this problem? This problem

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is an infinite-dimensional optimization problem, and thus the solution to the problem isan infinitely sequential allocation (k1, k2, ...).The idea of dynamic programing (DP) is to simplify the maximization problem by ex-ploiting the stationarity of the environment of the model. We call a problem stationarywhenever the structure of the choice problem that a decision maker faces is identicalat every point in time. To be concrete,

w (k0) = max{kt+1}∞t=0

0≤kt+1≤f(kt),k0 given

∞Pt=0

βtU(f (kt)− kt+1)

= max{kt+1}∞t=0

0≤kt+1≤f(kt),k0 given

½U(f (k0)− k1) + β

∞Pt=1

βt−1U(f (kt)− kt+1)

¾

= maxk1

0≤k1≤f(k0),k0 given

⎧⎪⎨⎪⎩U(f (k0)− k1) + β

⎡⎢⎣ max{kt+2}∞t=0

0≤kt+2≤f(kt+1),k1 given

∞Pt=0

βtU(f (kt+1)− kt+2)

⎤⎥⎦⎫⎪⎬⎪⎭

Note that the maximization problem inside the bracket is also a social planner’s problemthat maximizes lifetime utility of the representative agent from period 1 onwards, given

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initial capital stock k1. Since the technology or the utility functions doesn’t change overtime, the maximization problem from period 1 onwards remains basically the same.If a problem is stationary, we can think of a function that, for every period t, assignsto each possible initial level of capital kt an optimal level for next period’s capital kt+1:kt+1 = g(kt), where the function g(·) has no other argument than current capital, anddoes not vary with time. We will refer to g(·) as the policy function.

Thus, the optimal value of the maximization problem inside the bracket can be rewrittenas w (k1). The social planner’s problem becomes

w (k0) = max0≤k1≤f(k0)k0 given

{U(f (k0)− k1) + βw (k1)} .

(Under which conditions is this re-formulation of the maximization problem correct?)

Again, how to solve this problem? The maximization problem is much easier becausewe now maximize over just one variable, k1. However, the function w (·) shows up onthe right hand side and we do not know the form of this function.

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1.2 Recursive Formulation of the Social Planner’s ProblemWe now study this recursive formulation of the social planner’s problem. Note that thefunction w (·), as formulated in (1), is associated with the sequential formulation. Wedefine v (·) to be the value function for the recursive formulation of the social planner’sproblem:

v (k) = max0≤k0≤f(k)

{U(f (k)− k0) + βv (k0)} . (2)

The capital stock k that the planner brings into the current period is called the “statevariable,” because it completely summarizes the state of the economy today and deter-mines what allocations are feasible from today onwards.

(2) is a functional equation (the Bellman equation), which says that the discountedlifetime utility of the representative agent is given by the utility that this agent receivestoday, U(f (k)−k0), plus the discounted lifetime utility from tomorrow onwards, βw (k0).Thus, the planner faces a trade-off: more consumption and higher utility today, or alarger capital stock to work with and higher discounted future utility from tomorrowonwards.Hence, for a given k this maximization problem is much easier to solve by simply choos-

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ing the next period’s capital stock k0. Solving the functional equation means that we finda value function v solving (2) and an optimal policy function k0 = g(k) as a function of k(for each possible value that k).

1.3 Various Methods to Solve the Value functionLet U(c) = ln(c) and F (k, n) = kαn1−α. Assume full depreciation of capital goods, i.e.δ = 1. Thus, f(k) = kα and the functional equation becomes

v (k) = max0≤k0≤kα

{ln(kα − k0) + βv (k0)} . (3)

Remember that the solution to this functional equation is the function v(·).

1.3.1 Guess and VerifyThis method works well for our example, but may not work well for more general cases.Let us guess

v(k) = A+B ln(k),

where A and B are coefficients to be determined.

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Given the guess, we plug it into (3):

v (k) = max0≤k0≤kα

{ln(kα − k0) + β [A+B ln(k0)]} .

Given the log utility function, the constraints 0 ≤ k0 ≤ kα never bind. The objectivefunction is strictly concave and the constraint set is compact, for any given k. Thus, thefirst order necessary condition is also sufficient for the unique solution:

k0 =βB

1 + βBkα.

Plug the solution, which maximize the value function, back to (3):

A+B ln(k) = − ln(1 + βB) + α ln(k) + βA + βB ln

µβB

1 + βB

¶+ αβB ln(k).

Note that the equality must hold for every capital stock k, i.e., we must have

B = α + αβB,

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i.e.,

B =α

1− αβ.

Then we can solve for A:

A =1

1− β

∙αβ

1− αβln(αβ) + ln(1− αβ)

¸.

Thus, the optimal policy function k0 = g(k) is

g(k) =βB

1 + βBkα = αβkα.

The optimal policy of the social planner for this example is to save a constant fractionαβ of total output k as capital stock for tomorrow and and give the household consumea fraction (1− αβ) of total output today.In general, there may be other solutions to the functional equation. But in this example,the solution is unique.Given our policy function g(k) we can construct a sequence of capital stock {kt+1}∞t=0

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that will solve the sequential problem (1):

k0 = k0, k1 = g(k0) = αβkα0 , k2 = (αβ)1+α kα

2

0 ,

and in general kt = (αβ)Pt−1

j=0 αj

kαt

0 . Since α < 1, we have

limt→∞

kt = (αβ)11−α .

1.3.2 Value Function Iteration: Analytical ApproachConsider the following iterative procedure for our example:(1) Guess an arbitrary function v0(k), say, v0(k) = 0.(2) The solve

v1 (k) = max0≤k0≤kα

{ln(kα − k0) + βv0(k0)} .

Now we can solve the maximization problem because we know the function v0(k) onthe right hand side. Given v0(k

0) = 0, for all k0, we have

k0 = g(k) = 0,∀k.

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Then the date 1 value function will be

v1 (k) = α ln k.

(3) Similarly, we can solve for v2 (k), v3 (k), ...,

vn+1 (k) = max0≤k0≤kα

{ln(kα − k0) + βvn(k0)} .

Thus, we obtain a sequence of value functions {vn}∞n=0 and policy functions {gn}∞n=0.Hopefully these sequences will converge (Recall Stokey & Lucas Chapter 4).

1.4 The Euler Equation Approach and Transversality ConditionsWe now relate our example to the traditional approach of solving optimization problems.Again this approach works only in simple examples.

1.4.1 The Finite Horizon CaseWe consider the social planner problem for a situation in which the representative con-

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sumer lives for T <∞ periods. The social planner problem in (1) becomes

wT¡k0¢= max{kt+1}Tt=0

TPt=0

βtU(f (kt)− kt+1) (4)

s.t. 0 ≤ kt+1 ≤ f (kt) , k0 = k0 given.

Given that the utility function satisfies the Inada conditions the constraint 0 ≤ kt+1 ≤f (kt), for t = 0, 1, ..., T − 1, will never bind. What about the constraint kT+1 ≥ 0? Sincethe world ends after period T , the consumer receives no utility from capital that is leftbeyond date T , thus kT+1 = 0.Since we have a finite-dimensional maximization problem and the constraint set is non-empty, closed and bounded, a solution to the maximization problem exists (Bolzano-Weierstrass theorem), so that wT

¡k0¢

is well-defined.Moreover, due to the assumption of strict concavity in U (·) and f(·), the overall objec-tive function is strictly concave and the constraint set is convex, and thus the solutionto the maximization problem is unique. That is, the first order conditions are not onlynecessary, but also sufficient.

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Combining the FOCs, we have

U 0 (f (kt)− kt+1) = βU 0 (f (kt+1)− kt+2) f0 (kt+1) , t = 0, 1, ...T − 1. (5)

By variational argument, suppose the planner saves one more unit of good today, thehousehold reduces consumption by one unit, at utility cost of U 0 (f (kt)− kt+1). By to-morrow there is one more unit of capital to produce with, yielding additional productionf 0 (kt+1). Each additional unit of goods consumed by the household raises utility levelU 0 (f (kt+1)− kt+2) tomorrow. At the optimum the net benefit of such a variation in allo-cations must be zero. This first order condition is also called the Euler equation.

The Euler equations are a system of T second order difference equations, with theterminal condition kT+1 = 0.Recall the example U(c) = ln(c) and f(k) = kα. The Euler equation is

kαt+1 − kt+2 = αβkα−1t+1 (kαt − kt+1) .

Define zt ≡ kt+1kαt

(saving rate), then we have

zt+1 = 1 + αβ − αβ

zt, (6)

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which becomes a first order difference equation.

Since we have the terminal condition kT+1 = 0, which implies zT = 0. This is theterminal condition for (6). We can solve this equation backwards for the entire sequence{zt}Tt=0, given zT = 0:

zt = αβ1− (αβ)T−t

1− (αβ)T−t+1,

and thus

kt+1 = αβ1− (αβ)T−t

1− (αβ)T−t+1kαt , ct =

1− αβ

1− (αβ)T−t+1kαt .

Then, the discounted utility at time zero is

wT (k0) = α ln k0TPj=0(αβ)j −

TPj=1

βT−j ln

µjP

i=0(αβ)i

¶+αβ

TPj=1

βT−j

(j−1Pi=0(αβ)i

"ln (αβ) + ln

ÃPj−1i=0 (αβ)

iPji=0 (αβ)

i

!#).

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Since

limT→0

kt+1 = limT→0

αβ1− (αβ)T−t

1− (αβ)T−t+1kαt = αβkαt ,

in the limit the discounted utility at time zero is

limT→0

wT (k0) =1

1− β

∙αβ

1− αβln(αβ) + ln(1− αβ)

¸+

α

1− αβln k0.

This says that given our example the optimal policy for the social planner’s problemwith infinite time horizon is the limit of the optimal policies for the T -horizon planningproblem.But note that to obtain this result we have allowed the limit of the finite maximizationproblem to be equal to maximization of the problem in which time goes to infinity. Butin general we can not interchange maximization and limit-taking.In Fig.3.3, start with zT = 0 on the y-axis and going backwards to t = 0, the savingsrate zt approach αβ.Thus, for large T and for small t (close to t = 0) the solution to the finite maximizationproblem comes close to the optimal infinite time horizon policies.

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1.4.2 The Infinite Horizon CaseRecall the infinite horizon social planner’s problem in (1). The Euler equation has thesame form as the finite horizon planner’s problem in (5):

U 0 (f (kt)− kt+1) = βU 0 (f (kt+1)− kt+2) f0 (kt+1) , t = 0, 1, 2.... (7)

Given k0, we have no terminal condition since t goes to infinity, and thus we can notsolve the problem backwards. Still we need something similar to the terminal condition.The missing terminal condition for the difference equation is called the transversalitycondition (TVC):

limt→∞

λtkt+1 = 0,

where λt is the Lagrange multiplier on the feasibility constraint. Using the FOCs and

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the Euler condition, this condition can be expressed as

0 = limt→∞

λtkt+1

= limt→∞

βtU 0 (f (kt)− kt+1) kt+1 = limt→∞

βt−1U 0 (f (kt−1)− kt) kt

= limt→∞

βt−1 [βU 0 (f (kt)− kt+1) f0 (kt)] kt

= limt→∞

βtU 0 (f (kt)− kt+1) f0 (kt) kt, (8)

where βtU 0 (f (kt)− kt+1) f0 (kt) is the value in discounted utility terms of one more unit

of capital.The idea of the condition is as follows: it cannot be optimal for the consumer to choosea capital sequence such that, in discounted utility terms, the shadow value of kt re-mains positive as t goes to infinity. This could not be optimal because it means house-holds save too much, and a reduction in saving and an increase in consumption todaywould not violate the feasibility constraint and would increase the lifetime utility of thehouseholds.Note that this condition does not require that the capital stock itself converges to zeroin the limit, only the shadow value of the capital stock has to converge to zero.

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The following theorem (similar to Theorem 4.15 in Stokey and Lucas, but focus onbounded returns) provides sufficient conditions for the existence of maximum in infinitehorizon maximization problem.Theorem Let U , and F (and hence f ) satisfy assumptions (A1) and (A2). Then anallocation {kt+1}∞t=0 that satisfies the Euler equation (7) and the transversality condition(8) solves the sequential social planner’s problem, for a given k0.

The no-Ponzi scheme and the transversality conditions play very similar roles in dy-namic optimization in a purely mechanical sense. In fact, they can typically be shownto be the same condition.However, the two conditions are conceptually very different. The no-Ponzi schemecondition is a restriction on the choices of the agent (it is usually added to bepart of the budget constraints of a consumer). In contrast, the transversalitycondition prescribes how to behave optimally, given a choice set.

Note that this theorem does not apply for the case in which the utility function is loga-rithmic (due to unboundedness), but Theorem 4.15 in Stokey and Lucas applies alsoto unbounded utility function.On the other hand, is TVC a necessary condition for maximum, i.e. does every solution

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to the sequential planning problem have to satisfy the TVC?Now consider our example economy with U(c) = ln(c), f(k) = kα. Given the definitionzt = kt+1/k

αt , the Euler equation has the same form as in (6):

zt+1 = 1 + αβ − αβ

zt,

and the TVC is

limt→0

βtU 0 (f (kt)− kt+1) f0 (kt) kt

= limt→0

αβt

1− zt.

Now we have to solve for the equilibrium forwards, conditional on a guess of the initialvalue z0:(1) If z0 < αβ, by Fig.3.3, zt will become negative for some finite t, implying that kt+1 isnegative.(2) If z0 > αβ, by Fig.3.3, zt converges to 1. We will argue that all these paths violatethe TVC (Note that for any z0 > 1 it violates the nonnegativity of consumption and cannot be admissible to be a starting point). Here we rely on the result in Ekelund and

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Scheinkman (1985) showing that the TVC is a necessary condition for the optimum.Thus, any path that begins with z0 > αβ cannot be optimal.To see this, note that if zt converges to 1, then both the numerator and the denominatorin TVC go to zero. We linearize zt+1 in (6) around the steady state z = 1:

zt+1 ' 1 +αβ

(zt)2 |zt=1

(zt − 1) = 1 + αβ (zt − 1) .

Thus,

(1− zt+1) ' αβ (1− zt) = (αβ)2 (1− zt−1) = ... = (αβ)t+1 (1− z0) ,

which means that the following holds for all j ≥ 0:(1− zt+1) ' (αβ)t−j+1 (1− zj) .

Thus, the TVC becomes

limt→0

αβt

1− zt= lim

t→0

αβt+1

1− zt+1= lim

t→0

βj

αt−j (1− zj)=∞.

(3) If z0 = αβ, then zt = αβ for all t > 0. Note thatzt = αβ clearly satisfies the Euler

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equation, and the TVC is

limt→0

αβt

1− zt= lim

t→0

αβt

1− αβ= 0.

From the sufficiency of the Euler equation jointly with the TVC, the sequence {zt}∞t=0given by zt = αβ the unique optimal solution for the infinite-dimensional sequentialsocial planner’s problem.The solution zt = αβ implies that the optimal policy kt+1 = αβkαt , with k0 given, whichis exactly the constant saving rate policy that we derived in the recursive problem.

We next show that, by solving the social planner’s problem, we have solved for a (the)competitive equilibrium in this economy.

2 Competitive Equilibrium with Capital AccumulationOnce we have solved the social planner’s problem for a Pareto efficient allocation{c∗t , k∗t+1}∞t=0, we are interested in how the Pareto efficient allocation can be supportedby the allocations and prices that arise when firms and consumers interact in the mar-kets.

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For a competitive equilibrium the ownership structure of the economy must be clearlyspecified. We assume that(1) Households own all factors of production (labor and capital stock) and rent them outto the firms.(2) Households own the firms, i.e. they are residual claimants of the firms.

We also assume that the consumption goods market and the factor markets (for laborand capital services) are perfectly competitive.

2.1 Arrow-Debreu (A-D) Market EquilibriumWe first consider the Arrow-Debreu (A-D) market structure in which markets are openedonly at time zero in which goods from all future periods are traded. All terms of the con-tracts they had agreed upon in period 0 are perfectly enforceable.We assume that there is a single, representative firm that behaves competitively. The

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representative firm’s problem is, given a sequence of prices, {pt, wt, rt}∞t=0,

π = max{kt,nt}∞t=0

∞Pt=0

pt [yt − rtkt − wtnt] (9)

s.t. yt = F (kt, nt),

yt, kt, nt ≥ 0.The firm’s decision problem involves just a one-period choice - it is not of a dynamicalnature (for example, we could imagine that firms live for just one period). All of themodel’s dynamics come from the household’s capital accumulation problem.Sometimes we directly characterize the representative firm’s maximization problem by

rt = FK(kt, 1), wt = Fn(kt, 1). (10)

Note that n∗t = 1 because the households do not value leisure and F (kt, nt) is strictlyincreasing in nt.The representative household’s maximization problem is, given a sequence of prices,{pt, wt, rt}∞t=0,

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max{ct,kt+1,nt}∞t=0

∞Pt=0

βtU (ct) (11)

s.t.∞Pt=0

pt (ct + kt+1 − (1− δ) kt) ≤∞Pt=0

pt (rtkt + wtnt) ,

ct, kt+1 ≥ 0,∀t, k0 given.

(Note that specification in Kreuger’s note, where he distinguishes between the cap-ital stock and capital services that households supply to the firm, and assume thathouseholds cannot provide more capital services than the capital stock at their dis-posal produces. In this case, capital stock is “putty-putty.”)

We assume that U and F satisfy assumptions A1 and A2 outlined above. Thus, thenonnegativity constraints on consumption and capital stock do not bind follows directlyfrom the Inada conditions.

An Arrow-Debreu Competitive Equilibrium consists of prices {pt, wt, rt}∞t=0 and alloca-tions {ct, kt+1, nt}∞t=0 such that(1) Given prices {pt, wt, rt}∞t=0, the allocation {ct, kt+1, nt}∞t=0 solves the representativehousehold’s maximization problem (11).

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(2) Given prices {pt, wt, rt}∞t=0, the allocation {kt, nt}∞t=0 solves the representative firm’smaximization problem (9).(3) Market clearing conditions

nt = 1,

ct + kt+1 − (1− δ) kt = F (kt, 1).

We now characterize the equilibrium.Firstly, the FOCs of the static profit maximization problem for the representative firmare given by (10), which also implies that the profits the firms are zero.Next, since the utility function of the representative household is strictly increasing inconsumption, the Arrow-Debreu budget constraint holds with equality in equilibrium.The Euler equations are

βU 0 (ct+1)

U 0 (ct)=pt+1pt

=1

1 + rt+1 − δ.

Let f(kt) = F (kt, 1) + (1− δ)kt. Then the real rental rate of capital is

rt = Fk(kt, 1) = f 0(kt)− (1− δ).

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Thus, the marginal return on capital is f 0(kt) = 1 + rt − δ to the households. This isbecause households own capital and thus they must pay for the cost of depreciation.The Euler equation becomes

U 0 (ct) = βU 0 (ct+1) f0(kt+1),

which says that in equilibrium the marginal rate of substitution of consumption goodsbetween t and t + 1 equals to the marginal return on capital (or the marginal rate oftechnical transformation between t and t + 1.Note that this is exactly equivalent to the Euler equation in (7) from the social planner’sproblem.Again, since the households’ maximization problem is an infinite-dimensional optimiza-tion problem, we need an extra condition to ensure the optimality of the households’maximization problem. The transversality condition (TVC) is given by:

limt→∞

λtkt+1 = 0,

We can show that this leads to the same expression as in (8):

limt→∞

βtU 0 (f (kt)− kt+1) f0 (kt) kt = 0,

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which means that the value of the capital stock saved for tomorrow must converge tozero as time goes to infinity.Finally, we can solve for the equilibrium prices. By the Euler equation

pt+1pt

=1

1 + rt+1 − δ,

we have

pt+1 =pt

1 + rt+1 − δ= ... =

1

Πts=0 (1 + rs+1 − δ)

, t = 1, 2, ...

Therefore a competitive equilibrium allocation satisfies the optimality conditions for thecentralized economy, i.e., the competitive equilibrium is optimal. This is the First Wel-fare Theorem for the economy we consider here, which holds under very general condi-tions (local nonsatiation). The second welfare theorem, which more restrictive assump-tions, states that any Pareto efficient allocation can be decentralized as a competitiveequilibrium with transfers, i.e. there exist prices and redistributions of initial endow-ments such that the prices, together with the Pareto efficient allocation is a competitiveequilibrium for the economy with redistributed endowments. But the second welfaretheorem is substantially harder to prove, particularly in infinite-dimensional spaces.

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Suppose we have proved the first welfare theorem and we have established that thereexists a unique Pareto efficient allocation (this in general requires representative agenteconomies). Then we have established that, if there is a competitive equilibrium, itsallocation must be Pareto efficient.

2.2 Sequential Market EquilibriumIn sequential markets, we assume that firms manage the capital stock (even thoughhouseholds still make decision regarding capital accumulation). Thus, the marginalreturn on capital to the households (net of depreciation) is ert = rt + 1− δ.The representative firm’s problem is, given a sequence of prices, {wt, ert}∞t=0,

π = max{kt,nt}∞t=0

∞Pt=0[yt − ertkt + (1− δ) kt − wtnt] (12)

s.t. yt = F (kt, nt),

yt, kt, nt ≥ 0.The FOCs are er = FK(kt, 1) + (1− δ) , wt = Fn(kt, 1).

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The representative household’s maximization problem is, given a sequence of prices,{wt, ert}∞t=0,

max{ct,kt+1,nt}∞t=0

∞Pt=0

βtU (ct) (13)

s.t. ct + kt+1 ≤ ertkt + wtnt,

ct, kt+1 ≥ 0,∀t, k0 given.

A Sequential Markets equilibrium is prices {ert, bwt}∞t=0 and allocations {bct,bkt+1, bnt}∞t=0,such that(1) Given prices {ert, bwt}∞t=0, the allocations {bct,bkt+1, bnt}∞t=0, solve (13).(2) Given prices {ert, bwt}∞t=0, the allocation {bkt, bnt}∞t=0 solves the representative firm’smaximization problem (12).(3) Market clearing conditions

nt = 1,

ct + kt+1 − (1− δ) kt = F (kt, 1).