Macroeconomic Theory IIHomework 4 - Solution

Professor Gianluca Violante, TA: Diego Daruich∗

New York University

Spring 2014

Problem 1

Time is discrete and indexed by t ∈ {0, 1, ...,∞} . The economy is populated by a continuum of measure one ofinfinitely-lived households indexed by i on [0, 1]. Preferences for individual i are given by

U(ci0, c

i1, ...

)= E0

∞∑t=0

βtu(cit),

where β ∈ (0, 1) is the discount factor, ct ≥ 0 denotes consumption at time t, u′ > 0 and u′′ < 0. Each householdi in period t receives a stochastic idiosyncratic endowment of the consumption good εit ∈ {εh, εl} which follows aMarkov chain where πεhh = Pr

{εit = εh|εit−1 = εh

}, πεll = Pr

{εit = εl|εit−1 = εl

}, πεhl = 1−πεhh, and πεlh = 1−πεll.

Households can trade a risk-free asset a in zero-net supply with return Rt, and they can accumulate debt up tothe exogenous borrowing limit −b.

Part a

Define a stationary recursive competitive equilibrium (RCE) for this economy, and discuss its existence anduniqueness.

First, let us define some objects. Let S = A×E denote the state space for an individual agent, where A = [−b, a](for some a)1 and E = {εh, εl}. Also, let B(S) be the Borel σ-algebra induced by S. It is helpful to define thetransition function Q : B(S) × S → [0, 1], as the function that gives us, for any arbitrary Borel sets A × E , theprobability that an agent with current states (a, ε) will be ”in” those Borel sets next period. We can define Q as

Q[A× E , (a, ε)] = 1[a′(a, ε) ∈ A]∑ε′∈E

π(ε′, ε) (1)

∗The solution is based on an earlier version by Dan Greenwald and Miguel de Faria e Castro, NYU. Comments anderrors should be addressed to ddaruich@nyu.edu.

1How do we know such a exists? See proof of existence and uniqueness of stationary distribution given prices. Otherwise,we could also write the set A as being non-compact for now.

1

where 1 is the indicator function, a′(a, ε) is the policy function for savings of an agent with states (aε) andπ(ε′, ε) = Pr(εt+1 = ε′|εt = ε).

Also note that there is no aggregate uncertainty in this economy: in a stationary equilibrium, all aggregatevariables will be constant, including the value of aggregate production. Let (π∗, 1 − π∗) denote the stationarydistribution of ε. Then, aggregate output is constant at

Y = π∗εh + (1− π∗)εl

and it is easy to check that

π∗ =1− πll

2− πll − πhhSince there is no aggregate uncertainty, I shall assume that Rt is time-invariant when defining a stationaryequilibrium. That is, Rt = R.

Before defining the equilibrium, it is also useful to write down the problem of an agent with states (a, ε). Wecan write the recursive formulation of that agent’s problem as

V (a, ε) = maxc,a′

{u(c) + β

∑ε′∈E

π(ε′, ε)V (a′, ε′)

}(2)

subject to

c+ a′ = Ra+ ε

c ≥ 0

a′ ≥ −b

A stationary recursive competitive equilibrium consists of a value function V : S → R, policy functions(c, a′) : S → R+ ×A, a price R and a stationary measure λ : B(S)→ [0, 1] such that

1. Given the price R, V is the solution to the Bellman Equation in (2), and (c, a′)(a, ε) are the associatedpolicy functions.

2. Markets clear:

goods :

∫A×E

c(a, ε)dλ(a, ε) =

∫A×E

εdλ(a, ε) = Y

assets :

∫A×E

a′(a, ε)dλ(a, ε) = 0

3. For all (A× E) ∈ B(S), the invariant measure λ satisfies

λ(A× E) =

∫A×E

Q[A× E , (a, ε)]dλ(a, ε)

where Q is defined in (1).

Existence and uniqueness of stationary distribution given prices

2

The major difference between this exchange economy and the production economy discussed in class is thatassets need to be in zero net supply. This is equivalent to defining the demand of assets as 0 ∀r. The supplyof capital behaves in a similar way to the discussion in class. To prove the existence of a stationary equilibriumwe need to verify that there exists a unique invariant distribution λ. To satisfy theorem 12.13 in SLP for theexistence of the equilibrium we need the existence and uniqueness of the invariant distribution, which (usingtheorem 12.12 in SLP) follows from the following assumptions2:

• For compactness of the state space we need to assume DARA preference. Then compactness is guaranteedfor every r that satisfies β (1 + r) < 1.3

• The Feller property is immediate and does not require additional assumptions.

• For the monotonicity of the transition function we need to assume that πhh > πll. Then monotonicityfollows from the fact that policy function a′ is increasing in (a, ε).

• MMC is satisfied if the Markov chain is ergodic.

Existence and uniqueness of the recursive competitive equilibrium

Given aggregate demand and supply are continuous functions of interest rate, we can show existence by checkingthat demand is greater than supply as interest rate R is close to 0 and supply is greater than demand as R isclose to 1

β . Unfortunately as usual, we cannot show uniqueness as there could be more than one interest ratesthat correspond to zero net asset.

There are several reasons why the choice of aggregate asset holdings might be nonmonotonic in the interest rate.First, income effect and substitutional effect of interest rate work in opposite directions. Which one dominatesthe other is determined by the specific set of preference parameters. Second, even we consider only the incomeeffect, the effect interest rate on aggregate asset allocation is still ambiguous, because agents are allowed to holdnegative asset balances (borrow). If a < 0, then higher interest rate will make the household more constrainedand so a′ might decrease with interest rate.

Part b

Suppose now that b = 0, and that u (c) = c1−γ

1−γ .What is so special about this case? Determine the equilibrium risk-free return as a function of the model’s pa-rameters and prove that the return is lower than 1/β. Explain how the risk-free rate changes with the ratio εh/εl,with the persistence parameter πhh, and with risk aversion γ.

Now we have that b = 0 (so that agents cannot borrow) and CRRA utility. Note that if agents cannot borrow,they cannot lend either - since the asset market must clear at zero net supply, whoever wants to lend needs some-one who is willing to borrow in order to undertake the asset trade - but agents cannot borrow in the first place.Therefore, this economy will be autarkic and agents will simply consume their own current income, ct = εt, ∀t ≥ 0.

2See notes at the end of the document.3Using Chamberlain and Wilson (2000) result for βRgeq1, since we have “enough stochasticity.” And that R = 0 would

lead to too much borrowing in an equilibrium (since no one has to pay, they all borrow up to the limit). Then, we usestandard dynamic programming results to argue that savings policy function a′ is increasing (using the concavity of theutility function) and continuos (theorem of the maximum).

3

What would agents want to do? Assume for a moment that (πhh, πll) are both strictly less than one (so thatno state is absorbing). Then agents will have a consumption smoothing motive, as high agents will want tosave for when they get a low income shock, and low agents will want to borrow. The latter are technologicallyconstrained by b. High endowment agents, however, do not face any sort of technological constraint. Therefore,aggregate conditions will have to be such that high endowment agents find it optimal not to save. Since highendowment agents determine the interest rate, what this tells us is that R must be such that, in equilibrium, theEuler Equation for agents with high endowments reads:

u′(εh) ≥ βR[πhhu′(εh) + (1− πhh)u′(εl)]

Applying our specific functional form yields

(εh)−γ ≥ βR[πhh(εh)−γ + (1− πhh)(εl)−γ ]

Rearranging:

R ≤ 1

β

ε−γhπhhε

−γh + (1− πhh)ε−γl

Any interest rate that is lower than the RHS is consistent with an equilibrium in which high endowment agentsfind it optimal not to save and simply consume their entire endowment - this is fairly intuitive: for high agentsnot to save, the interest rate must be extremely low. Since γ > 0, x−γ is a decreasing function of x. Therefore,ε−γh < ε−γl . Thus we have that

ε−γhπhhε

−γh + (1− πhh)ε−γl

< 1

Once again, this holds assuming that πhh < 1. But, then, we immediately obtain that

R <1

β

as we wanted to show.

The comparative statics are as follows:

1. Endowment Ratio - Rewrite the condition on the interest rate to read

R ≤ 1

β

(εh/εl)−γ

πhh(εh/εl)−γ + 1− πhh

where I simply multiply and divide the RHS by ε−γl . Clearly, when this ratio increases, the RHS termdecreases. Therefore, the maximum possible value for the interest rate that is consistent with equilibriumdecreases, R ↓. The intuition is that an increase in this ratio raises the variance of income and thusincreases the incentives to smooth consumption (since the agent is risk-averse), which can only be donethrough saving. Since the incentives to save increase, the interest rate must decrease further so as toensure the autarkic market clearing for assets (i.e. to ensure that rich agents do not save). Note that ifthe consumer were risk-neutral, γ = 0, this effect would not exist.

2. Persistence - Since ε−γh < ε−γl , as πhh ↑ we observe the RHS term increasing, so that the maximum interestrate compatible with equilibrium increases R ↑. An increase in the persistence of high income makes incomeless volatile for rich agents (i.e. the likelihood of remaining rich increases), thereby decreasing the incentivesto save. This allows R to increase slightly while still ensuring autarky in the asset market.

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3. Risk Aversion - This effect is less obvious. The easiest way to think about this is to take the inverse ofthe RHS

πhh + (1− πhh)(εh/εl)γ

When γ ↑, since εl < εh the above expression increases. Thus our expression must decrease, meaning thatR ↓ .This is in fact intuitive: if the agent becomes more risk-averse, he or she will have more incentives tosave, forcing the interest rate down to ensure equilibrium.

Problem 2

Consider a stationary economy populated by a continuum of measure one of infinitely lived, ex-ante equal agentswith preferences over sequences of consumption and leisure given by

E0

∞∑t=0

βt[u (cit) + eψitv (1− hit)

].

Agents face individual shocks to their preference for leisure ψit, which follow the stochastic process ψit = ρψi,t−1+ηit, with ηit ∼ N (0, ση) . Agents can save but cannot borrow. Production takes place through the aggregatetechnology

Ct +Kt+1 − (1− δ)Kt = AKαt H

1−αt

where Ct,Kt and Ht are, respectively, aggregate consumption, aggregate capital, and aggregate hours at time t.Labor and asset markets are competitive and clear, every period, with prices wt and rt, respectively.

The government taxes capital income at a fixed flat rate τ . Tax revenues are returned to agents as tax-exemptlump-sum transfers b. Households can evade taxes by deciding every period t the fraction of capital income φitto declare in their tax return, i.e., the fraction of capital income on which they pay taxes. Let xit be the totalundeclared taxes at time t.

The government, knowing that agents may have evaded taxes at time t− 1, at time t can monitor and perfectlyverify the past period individual tax returns. Let π be the probability that, at time t, the time t− 1 tax returnedof a household is subject to monitoring. The household finds out whether her t− 1 period tax return is monitoredat the beginning of period t, i.e., before consumption decisions are taken. In the event the household is caught, attime t the tax agency gives her a fine equal to z (xi,t−1), where xi,t−1 is the tax amount due from the past period,with z (0) = 0 and z′ (xi,t−1) > 1.

Part a

Write down the problem of the household in recursive form, making explicit the individual and the aggregate statevariables.

There are several ways to set-up this problem. The individual state variables at time t will be (ait, ψit, xit−1,mit),where ait is current asset holdings, ψit is the preference shock (you could had written ψit−1, ηit instead), xit−1 isthe amount of taxes that went undeclared in the previous period and mit ∈ {0, 1} denotes whether the agent isbeing monitored by the government this period or not.

5

The aggregate state for this economy is the distribution of individual states λ(a, ψ, x,m), from which otheraggregate states such as K or b can be derived. I always implicitly assume that the agent takes λ as a givenwhen solving the recursive problem.

I am going to write the problem in a way that it is not needed to include the additional decision variableφ, and the agent simply chooses over x′ (the amount of taxes that go undeclared from this period to the next).We can then set up the recursive problem as

V (a, x, ψ, n) = maxc,a′,h,x′

{u(c) + eψv(1− h) + βEψ′ [πV (a′, x′, ψ′, 1) + (1− π)V (a′, x′, ψ′, 0)]

}subject to

c+ a′ = [1 + r(1− τ)]a+ x′ + wh+ b− nz(x)

x′ ≤ τrax′, c, a′ ≥ 0

h ∈ [0, 1]

A few comments: first, the expectation in the value function is taken with respect to the two independent randomvariables (ψ′, n′). Second, the choice of φ is subsumed in the first and second constraints.

Part b

Write down the individual first-order necessary condition that characterizes the optimal tax evasion choice.

Let (λ1, λ2, µ1, µ2, µ3) denote the Lagrange multipliers for each constraint (and let us just assume the agentchooses interior leisure, for simplicity). Then, we can write the FOC with respect to x′ as

u′(c) + βEψ′ [πV2(a′, x′, ψ′, 1) + (1− π)V2(a

′, x′, ψ′, 0)]− λ2 + µ1 = 0

where I am also implicitly assuming that c > 0. Now, we need to apply the envelope condition to the two valuefunctions: when the agent is monitored and when he or she is not. For the monitoring case, with n = 1, theagent will pay an extra tax z(x) for each unit of x that was not reported. Then

V2(a, x, ψ, 1) = −u′(c)z′(x)

While for the no-monitoring case, n = 0, the previous choice of x does not affect the consumer in any way

V2(a, x, ψ, 0) = 0

Iterating forward and plugging back in the FOC with respect to x′, we see that

u′(c) = βπEψ′ [u′(c′)z′(x′)] + λ2 − µ1

Clearly, we will have that x′geq0, hence µ1 = 0 (since no agent would ever overreport the amount of taxes itneeds to pay). We cannot, however, say anything about λ2 - some agents may prefer to report losses, which isnot allowed -. Thus the FOC for tax evasion is

u′(c) = βπEψ′ [u′(c′)z′(x′)] + λ2.

However, if the solution were interior, λ2 = 0.

6

Part c

Define a stationary recursive competitive equilibrium for this economy.

Let the state space be S = A × X × Ψ × N = R2+ × R × {0, 1}. Let B(S) denote the product σ-algebra

over S. A Stationary Recursive Competitive Equilibrium for this economy is a value function V : S → R, policyfunctions for the household (c, a′, x′, h) : S → R3

+ × [0, 1], policies for the firm (H,K), prices (r, w), governmentpolicies (τ, b), and a stationary measure λ : S → [0, 1] such that

1. Given prices and government policies, V solves the Bellman Equation in Part 1, and (c, a′, x′, h) are theassociated policy functions.

2. Given prices (r, w), the firm chooses (H,K) optimally

FK(K,H) = r + δ

FH(K,H) = w

3. Markets clear

labor :

∫Sh(a, x, ψ, n)dλ = H

assets :

∫Sa′(a, x, ψ, n)dλ = K ′

goods :

∫Sc(a, x, ψ, n)dλ+K ′ = F (K,H) + (1− δ)K

4. The government budget constraint is balanced∫Sτradλ−

∫Sx′(a, x, ψ, n)dλ+ π

∫Sz(x)dλ = b

5. For all Borel sets (A,X ,P,N ) ∈ B(S), the invariant measure satisfies

λ(A,X ,P,N ) =

∫SQ[(A,X ,P,N ), (a, x, ψ, n)]dλ(a, x, ψ, n)

where Q : B(S)× S → [0, 1] is a transition function defined as

Q[(A,X ,P,N ), (a, x, ψ, n)] =1[a′(a, x, ψ, n) ∈ A ∩ x′(a, x, ψ, n) ∈ X ]×

× [π1[{1} ∈ N ] + (1− π)1[{0} ∈ N ]]

∫Pψ′dF (ψ′, ψ)

where F (ψ′, ψ) is the transition function implied by the AR structure ψ′ = ρψ + η, with η ∼ N (0, ση).

Problem 3

Time is discrete and indexed by t ∈ {0, 1, ...,∞} . The economy is populated by a continuum of measure one ofinfinitely-lived households indexed by i on [0, 1]. Preferences for individual i are given by

U(ci0, c

i1, ...;h

i0, h

i1, ...

)= E0

∞∑t=0

βtu(cit, 1− hit

),

7

where β ∈ (0, 1) is the time discount factor, c ≥ 0 denotes consumption and h ≥ 0 hours worked. Household i attime t supplies the optimal amount of hours worked hit to a competitive labor market, taking the hourly wage ωtas given.

Each household owns a single private firm, i.e. firm i is identified with household i. Each firm produces the finalgood y (which can be used for conumption or investment) through

yit = F(zit, k

it, n

it

),

where F is a production technology displaying constant-returns in capital and labor. The variable zit represents afirm-specific productivity shock which follows the Markov process π (z′, z) = Pr

{zit+1 = z′|zit = z

}with zmin = 0.

Households can freely hire labor on the market for their firm, but here is no market to trade claims on physicalcapital, thus households can only invest the capital they own in their firm. Moreover, kit ≥ 0. Capital depreciatesat rate δ. Let πit denote the profits accruing to individual i from operating her firm at time t.

Households can trade a risk-free bond b in zero-net supply with return Rt and they can accumulate debt up to the“natural borrowing constraint”.

Assume that the economy is in a stationary equilibrium.

Part a

What is the natural borrowing constraint faced by each household?

This question highlights some of the difficulties in establishing what a natural borrowing constraint shouldlook like. Basically, the natural borrowing limit will depend on whether we assume that the agent is able tocommit the repayment of its own debts or if the agent is allowed to default. It is first helpful to look at thebudget constraint for an individual agent i. We can define it sequentially as

cit + kit+1 + bit+1 = F (zit, kit, n

it)− ωnit + ωhit +Rbit + (1− δ)kit

on the LHS we say that the agent can spend money in consumption, capital investment or bond investment. Thefirst two terms on the RHS are the net profit of the private firm (production minus wage bill), the third term iswage income from the labor market and the last two terms are bond return and undepreciated capital (that canbe sold in the goods market). Note that I dropped the time indices on prices (R,ω), since we are focusing on astationary equilibrium in which there is no aggregate uncertainty.

The household derives income from two sources: capital and labor. Since we are studying a natural borrowinglimit, we have to look at what can the agent afford to pay in a worst case scenario, when zmin = 0. Imposingcit + kit+1 ≥ 0, we find the following:

bit ≥bit+1 − [F (zit, k

it, n

it)− ωnit]− ωhit − (1− δ)kit

R

Iterate forward to obtain that

bit ≥ −1

R

∞∑j=0

F (zit+j , kit+j , n

it+j)− ωnit+j + ωhit+j + (1− δ)kit+j

Rj+ limT→∞

biTRT

8

Now, the natural debt limit should be such that the agent should be able to repay its debts in every state of theworld. In particular, the agent should be able to repay its debts in a worst-case scenario where zt = zmin = 0forever. Clearly, in such an event, the agent would optimally set nit = 0 (the agent would not hire any worker,since the marginal gain from doing so is zero) and kit+j = 0,∀j ≥ 1 (the logic is the same, the agent would not

purchase any capital).4 This, and imposing a transversality condition on bond holdings, allows us to slightlysimplify the expression to yield

bit ≥ −1

R(1− δ)kt −

1

R

∞∑j=0

ωhit+jRj

Iterating one period ahead leaves us with

bit+1 ≥ −1

R(1− δ)kit+1 −

1

R

∞∑j=0

ωhit+1+j

Rj

That is, the agent can borrow against the capital stock he or she will hold in the next period (adjusted fordepreciation), and future human capital income. There is a fundamental problem with this, however: both kt+1

and {ht+1+j}∞j=0 are endogenous variables, chosen by the agent! Therefore, it is as if the agent can influence itsown natural borrowing limit - so this seems to be a very complicated problem!

There are two approaches to this. The first is just to assume that the agent will do whatever it takes torepay its debts, even if this includes setting ht = 1 forever and not investing in capital today (regardless of thestates). This leaves us with

bt+1 ≥ −1

R

∞∑j=0

ω × 1

Rj= − 1

R− 1ω

The other approach is to assume that the agent will choose optimally, and will do so in order to repay its debts.In this case, the natural debt limit will be a function of the current states, Φ(zt, kt, bt). Since we are dealing withworst case scenarios, we just have to look at the policies h(zt, kt, bt) when zt = 0, kt = 0 and bt = −Φ. Thus weget that

bt+1(zt, kt, bt) ≥ −Φ(zt, kt, bt) = − 1

R(1− δ)k′(zt, kt, bt)−

1

R

∞∑j=0

ωh(0, 0,−Φ)

Rj

Note that finding this alternative debt limit requires computing the optimal policies (as function of the states),and then solving a fixed point problem that determines Φ:

Φ(zt, kt, bt) =1

R(1− δ)k′(zt, kt, bt) + h[0, 0,−Φ(zt, kt, bt)]

1

R− 1

If we have the policy functions, we can compute the function Φ, hence the natural borrowing limit will fluctuateover time. For simplicity, I assume that Φ = 1

R−1ω in the remainder of the exercise.

Part b

List explicitly the individual state variables and write down the problem of the household in recursive formulation.

The discussion in the previous exercise already spoiled some of the work we develop here: the individual statevariables will be the current productivity level zt, the current stock of capital kt and bond holdings bt. The statespace is S = Z ×K × B, where Z is the state space for the Markov process π, K = R+ and B = [−Φ,+∞),

4We are assuming here that F (0, n, k) = 0 for all (n, k). Otherwise, the statement of zmin = 0 makes no difference.

9

where Φ is the natural borrowing limit computed before.

We can express the agent’s problem recursively as

V (z, k, b) = maxc,k′,b′,h,n

{u(c, 1− h) + β

∑z′∈Z

π(z′, z)V (z′, k′, b′)

}(3)

subject to

c+ k′ + b′ = F (z, k, n)− ωn+ ωh+Rb+ (1− δ)kc, n, k′ ≥ 0

b′ ≥ −Φ

h ∈ [0, 1]

Part c

Define a Recursive Competitive Equilibrium (RCE) for this economy.

A SRCE for this economy is a value function V : S → R, policies (c, k′, n, b′, h) → R3+ × [−Φ,+∞) × [0, 1],

prices (R,ω) and a stationary distribution λ : B(S)→ [0, 1] such that:

1. Given prices, V solves the Bellman Equation (3) and (c, k′, n, b′, h) are the associated policy functions.

2. Markets clear:

(labor) :

∫Z×K×B

h(z, k, b)dλ(z, k, b) =

∫Z×K×B

n(z, k, b)dλ(z, k, b)

(bonds) :

∫Z×K×B

b′(z, k, b)dλ(z, k, b) = 0

(goods) :

∫Z×K×B

c(z, k, b)dλ(z, k, b) +

∫Z×K×B

k′(z, k, b)dλ(z, k, b)

=

∫Z×K×B

F [z, k, n(z, k, b)]dλ(z, k, b) + (1− δ)∫Z×K×B

kdλ(z, k, b)

3. For all (Z ×K × B) ∈ B(S), the invariant distribution satisfies

λ(Z ×K × B) =

∫Z×K×B

Q[(Z ×K × B), (z, k, b)]dλ(z, k, b)

where the transition function Q : B(S)× S → [0, 1] is defined as

Q[(Z ×K × B), (z, k, b)] = 1[k′(z, k, b) ∈ K ∩ b′(z, k, b) ∈ B]∑z′∈Z

π(z′, z)

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Part d

Discuss the existence and uniqueness of the RCE for this economy.

The existence and uniqueness of the stationary invariant distribution given prices (w, r) can be shown in asimilar way to problem 1. So is the existence of a stationary equilibrium.

As for uniqueness, the argument in problem 1 still holds here since natural borrowing constraint is negative. Whatis worse, the actual effect of wage increase on labor supply is also ambiguous because of the income affect (lessleisure) and the substitution effect (more leisure). On top of this, is the effect coming from the interconnectedeffects of the wage and the interest rate. Therefore uniqueness cannot be guaranteed.

Part e

Imagine now that a new financial market opens allowing agents to invest their own k in other households’ firms.What is a simple way to represent the new economy with this additional market? In particular, list what theindividual state variables become in this case.

A simple way to model this would be to add an additional state variable κ, corresponding to capital invested inother firms. Correspondingly, there would be a new control κ′. The budget constraint would now look like

c+ k′ + b′ + κ′ = F (z, k, n)− ωn+ ωh+Rb+ (1− δ)k + (1 + r − δ)κ

where r is the equilibrium return on capital in the economy (constant in a stationary equilibrium). Since capitalwould now become freely mobile across firms, it does not matter in which firm capital is invested, since returnswill have to be equalized so as to prevent arbitrage opportunities. In equilibrium, all capital and labor willconcentrate in the firms with the highest productivity level zmax due to constant returns to scale.

Since capital becomes concentrated in those firms, the distribution of capital per se becomes irrelevant. Tounderstand why, note that the single source of heterogeneity in this economy were the productivity shocks.However, agents no longer face these: if zit < zmax, the agent simply sets k′ = 0 and invests everything in κ′,in the highest productivity firm. Thus all agents face the same return on capital in the economy. This meansthat heterogeneity is completely eliminated and the problem now admits aggregation as a representative agentproblem, with resource constraint given by

C +K ′ = F (zmax,K,N) + (1− δ)K

Notes on FP, Monotonicity and MMC Conditions

Feller Property

Alternative 1 For this property we need operator T to preserve continuity and boudedness. Boundedness isstraightforward from compactness of A × E. For continuity argument we need to provide some details of theproof. Lets take an arbitrary sequence {(an, εn)}∞n=1 such that limn→∞ (an, εn) = (a, ε) . We know that

Q ((an, εn) , s)→weakly Q ((a, ε) , s) as n→∞

11

iff for any f ∈ Climn→∞

∫f (s)Q ((an, εn) , ds) =

∫f (s)Q ((a, ε) , ds)

5By Aleksandrov TheoremQ ((an, εn) , s)→weakly Q ((a, ε) , s) as n→∞

iffQ ((an, εn) , S)→ Q ((a, ε) , S) for all S such that Q ((a, ε) , ∂S) = 0

Note that since a′ is continuos then limn→∞ a′ (an, εn) = a′ (a, ε) . Also note that Q ((an, εn) , (A× E)) →

Q ((a, ε) , (A× E)) is violated only if a′ (a, ε) ∈ ∂A. Hence if a′ (a, ε) 6∈ ∂A then Q ((an, εn) , S) → Q ((a, ε) , S) .One can show that Q ((a, ε) , ∂ (A× E)) = 0 iff (a) or (b) hold:

a a′ (a, ε) 6∈ ∂A

b∑

ε′∈E π (ε′, ε) = 0

In case (a) we just have shown that Q ((an, εn) , (A× E))→ Q ((a, ε) , (A× E)) . In case of (b) Q ((an, εn) , S)→Q ((a, ε) , S) obviously holds since both Q ((a, ε) , S) and Q ((an, εn) , S) are zero starting from some n. Combiningthis altogether we get that T preserves continuity.

Alternative 2 Using theorem 9.14 in SLP. Our space is measurable and the transition function satisfies that

1. A is a convex Borel set in R.

2. E is countable and we are using the Borel sigma-algebra.

3. Savings policy function is continuous (using theorem of the maximum).

Hence, all assumptions are satisfied, so the transition function has the Feller property.

Monotonicity

Monotonicity has two components: monotonicity in a and in ε. T preserves monotonicity iff for any f that isincreasing in both arguments Tf is also increasing.

First lets prove monotonicity in a. We need to show that for all a, a : a ≥ a∫f (s)Q ((a, ε) , ds) ≥

∫f (s)Q ((a, ε) , ds) for all increasing f

We know that this is equivalent to Q ((a, ε) , ds) <FSD Q ((a, ε) , ds) . We also know that there is a criterion forFSD relation which is

g <FSD h

iffG (x) ≤ H (x) for all x

5Here convergence is in terms of r.v. that are associated with probability measures Q ((an, εn) , ·) and Q ((a, ε) , ·)

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where G (·) is cdf for g and H (·) is cdf for h. In our case cdf is Q ((a, ε) , S (α, e)) where

S (α, e) = {(a, ε) | (a, ε) ∈ (A× E) and (a, ε) ≤ (α, e)}

Note that since a′ (a, ε) is monotonic then Q ((a, ε) , S (α, e)) looks like a step and threshold for Q ((a, ε) , S (α, e))is higher then for Q ((a, ε) , S (α, e)) hence Q ((a, ε) , S (α, e)) ≤ Q ((a, ε) , S (α, e)) .From this we can concludethat

Q ((a, ε) , S (α, e)) <FSD Q ((a, ε) , S (α, e))

and hence ∫f (s)Q ((a, ε) , ds) ≥

∫f (s)Q ((a, ε) , ds) for all increasing f

Now lets prove monotonicity in ε.We need to show that for all ε, ε : ε ≥ ε∫f (s)Q ((a, ε) , ds) ≥

∫f (s)Q ((a, ε) , ds) for all increasing f

We will show it for the case of E = {εH , εL} . The argument that we need to use is also based on FSD relation.However we need to impose more conditions. Since

Q ((a, ε) , A (α, e)× E (α, e)) = I{a′ (a, ε) ∈ A (α, e)

} ∑ε′∈E(α,e)

π(ε′, ε)

thenI{a′ (a, ε) ∈ A (α, e)

}≥ I

{a′ (a, ε) ∈ A (α, e)

}for all (α, e) (4)

and ∑ε′∈E(α,e)

π(ε′, ε)≥

∑ε′∈E(α,e)

π(ε′, ε)

for all (α, e) (5)

implies thatQ ((a, ε) , A (α, e)× E (α, e)) ≥ Q ((a, ε) , A (α, e)× E (α, e)) for all (α, e)

It is easy to check that increasing a′ is sufficient for 4 to hold and π (εH , εH) ≥ π (εL, εL) implies 5. QED

MMC

Note that necessary condition for MMC to hold is ergodicity of Markov chain.

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