Macroeconomic Theory IIHomework 5 - Solution

Professor Gianluca Violante, TA: Diego Daruich∗

New York University

Spring 2014

Problem 1

Consider an economy populated by a continuum of measure one of infinitely lived, ex-ante equal agents withpreferences over sequences of consumption given by

E0

∞∑t=0

βtu (cit) .

Agents face individual uninsurable endowment shocks εit to labor productivity, where εit ∈ {ε1, ..., εN}. Thisprocess is governed by the Markov chain πε (ε′, ε) = Pr {εi,t+1 = ε′|εit = ε} . Production takes place through theaggregate technology Yt = ztK

αt H

1−αt where Kt and Ht are, respectively, aggregate capital and aggregate efficiency

units of labor at time t. Capital depreciates at rate δ. Aggregate total factor productivity zt ∈ {zb, zg} follows aMarkov process πz (z′, z) = Pr {zt+1 = z′|zt = z} .

Agents can hold two assets: risky capital ait and a risk-free one-period bond bit. Let qt be the price of the bondat time t: buying b units of the risk-free bond at price qt at time t guarantees b units of final good next period.Assume that agents face borrowing limits

(−a,−b

)on the two assets. Labor and asset markets are competitive.

Part a

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

ALTERNATIVE 1: NATURAL STATESThe individual states for this problem are current asset holdings a, current bond holdings b and the current levelof labor productivity ε. The aggregate states are the distribution of individual states λ(a, b, ε) and the aggregateproductivity level z. Note that this problem is not as simple as in the previous problem set, since we no longer

∗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.

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have a stationary steady state. Due to the shock z, there is now aggregate uncertainty. The direct implicationis that, in order to optimize, the agent will be forced to forecast prices (which would otherwise be constant,but that now depend on the current aggregate states). This is done by specifying the laws of motion for theaggregate states: for z, π(z′, z) is given, while for λ we specify λ′ = G(λ, z).

The recursive formulation for the agent’s problem can be written as

V (a, b, ε; z, λ) = maxc,a′,b′

{u(c) + β

∑ε′∈E

∑z′∈Z

π(ε′, ε)π(z′, z)V (a′, b′, ε′; z′, λ′)

}

subject to

c+ a′ + q(z, λ)b′ = (1 + r(z, λ))a+ b+ w(z, λ)ε

a′ ≥ −ab′ ≥ −bλ′ = G(λ, z)

ALTERNATIVE 2: REDUCED STATE SPACEThe state of the agent is labor productivity ε and the total assets at the beginning of the period ω. ω includesthe number of bonds and the capital including return.

The aggregate state includes z and the distribution of wealth and productivity in the economy λ (A, E). Thehousehold problem in recursive form is then:

v (ω, ε; z, λ) = maxc,a′,b′

u (c) + β∑

ε′∈E,z′∈Zv(ω′, ε′; z′, λ′

)π(ε′, ε)π(z′, z)

(1)

s.t.

c+ a′ + q(z, λ)b′ = ω + w (z, λ) ε

ω′ = b′ + a′(1 + r

(z′, λ′

))(2)

a′ ≥ −ab′ ≥ −bλ′ = G

(z, λ, z′

)

Part b

Define a recursive competitive equilibrium for this economy.

I will focus on the second alternative from now on.

Let A = [−a,+∞] and B = [−b,+∞]. Also, let E = {ε1, ..., εN}. Let S = [−a − b,+∞) × E be the individualstates and Q = Z × Λ denote the vector of aggregate states, where Λ is the set of probability measures on S. ARecursive Competitive Equilibrium for this economy is a value function V : S ×Q→ R, policy functions for the

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household (c, a′, b′) : S×Q→ R+×A×B, policies for the firm (H,K), pricing functions (q, r, w) : Q→ R3+ and

a law of motion G such that

1. Given the pricing functions [r(z, λ), w(z, λ), q(z, λ)], V solves the Bellman Equation for the household (asin part 1) and (c, a′, b′) are the associated policy functions.

2. Given price functions r(z, λ), w(z, λ), H and K are the optimal policies for the firm

zFK(K,H) = r(z, λ) + δ

zFH(K,H) = w(z, λ)

3. Markets clear

labor :

∫Sεdλ(ω, ε) = H

assets :

∫Sa′(ω, ε)dλ(ω, ε) = K ′ = K(z′, G(z, λ, z′))

bonds :

∫Sb′(ω, ε)dλ(ω, ε) = 0

goods :

∫Sc(ω, ε)dλ(ω, ε) +K ′ = zF (K,H) + (1− δ)K

4. The law of motion G is defined as follows for any (W, E) ∈ B(S)

λ′(W, E) = G(W,E)(λ, z) =

∫SQz,z′ [(W, E), (ω, ε)]dλ(ω, ε)

where Qz,z′ : B(S)× S → [0, 1] is a transition function specified as

Qz,z′ [(W, E)), (ω, ε)] = 1[ω′(ω, ε;λ, z, z′) ∈ W]∑ε′∈E

π(ε′, ε)

where ω′ is the function defined before that depends on a′ and b′.1

Part c

Outline a numerical algorithm that solves for the equilibrium allocations and prices.

We adopt the algorithm of Krusell and Smith (1998) to solve for the recursive competitive equilibrium in thepresence of aggregate uncertainty. First, note that in order to solve for the policy functions, the agent mustforecast future prices. In the baseline model (that you saw in class), the agent only needs to forecast (r, w), sothat keeping track of aggregate capital K ′ was enough (through the production function and the firm’s FOC)to keep track of prices one period ahead. The same logic applies here, since ε is assumed to be at its stationarydistribution, aggregate labor demand H is constant, hence both wages and interest rates can be obtained fromK. We are therefore going to use the following law of motion

logK ′ = b0z + b1z logK

1Note that ω′ is based, among other things, on the interest rate next period. In general, this complicates things becausethe interest rate depends on the whole distribution tomorrow. However, exogenous labor and the Krussel-Smith guess onaggregate capital will simplify this.

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for z = {zb, zg}. But there is an additional complication now: the agent must also track the price of the bond.Since the bond market is independent of the firm’s optimality condition, this means that an additional law ofmotion is required. Since the bond is in zero net supply, we cannot obviously track its stock, but we can trackits price. Moreover, since the bond and the asset are two possible investment channels over which agents mustbe indifferent in equilibrium, it is natural to assume that the bond price will be related to the main determinantof the asset return: the capital stock. To see this, note first that the bond will be priced by a marginal buyer(i.e., someone who is buying the bond in a positive amount). Therefore, taking the FOC for b′, a′ for some agentthat chooses interior solutions for these assets, and combining these FOC we obtain that

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q(λ, z)=

∑ε′∈E

∑z′∈Z πz(z

′, z)πε(ε′, ε)(1 + r(λ′, z′))u′[c′(a′, b′, ε′;λ′, z′)]∑

ε′∈E∑

z′∈Z πz(z′, z)πε(ε′, ε)u′[c′(a′, b′, ε′;λ′, z′)]

Clearly, forecasting q′ involves forecasting r′, which in turn requires a forecast of K ′. This, in turn, can beobtained with K! I therefore assume that

log q = d0z + d1z logK

for z = {zb, zg}.

The only additional difference with respect to the algorithm seen in class is that this additional law of mo-tion poses a non-trivial problem: while an incorrect guess of K ′ results in inaccuracy for the agent’s policies, anincorrect guess of q′ will result on the bond market not clearing. Therefore, we need to add a loop in which weadjust q at every t so that the bond market clears.

The algorithm to solve for the equilibrium is then the following:

1. Guess values for the coefficients in the laws of motion, {b0z, b1z, d0z, d1z}z∈Z .

2. Given laws of motion for (K ′, q), solve the household problem. That is, find policies (c, a′, b′)(a, b, ε;K, z).Note that aggregate labor is constant, so that the household can back-up current prices as well as forecasts(R′, w′, q′) using the laws of motion.

3. Simulate the economy for N individuals and T periods. For example, N = 10, 000 and T = 20, 000. Drawfirst a sequence of aggregate shocks {zt}Tt=1. Next, draw a sequence of individual productivity shocks

{εit}N,Ti=1,t=1. Use the decision rules to generate sequences of asset and bond holdings {ait, bit}N,Ti=1,t=1. Usingthese, compute, for each period:

Bt =1

N

N∑i=1

bit

If Bt 6= 0, adjust qt until the bond market clears. Do this for every period until the bond market clears forevery period.2 Once it clears, store {qt} and compute

At =1

N

N∑i=1

ait

4. Drop enough periods in the beginning of the simulation in order to eliminate dependence on initial condi-tions. Using the remaining sequences {At, qt}T , regress for each z ∈ {zb, zg}

log At+1 = β0z + β1z log At

log qt = δ0z + δ1z log At

2This is not strictly necessary but some papers have shown it to be more stable than just checking at the very end.

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5. Check if

max

{maxi=g,b

|biz − βiz|,maxi=g,b

|diz − δiz|}< ε

where ε is some tolerance level. If yes, stop. If not, set bz = βz, dz = δz and return to step 1.

6. Recall that this equilibrium is approximate: we still need to verify how good this approximation is to thefully rational-expectation equilibrium. For this purpose, compute a measure to the fit of the regression instep 40, for example by using R2. Next, try augmenting the state space with another moment, for exampleusing m2 = E

(a2i). Repeat steps 1.-5. until convergence. If the R2 of the new regression has improved

significantly, keep adding moments until R2 is large and does not respond to addition of new explanatorymoments. Otherwise, stop: it means that additional moments do not add new useful information inforecasting prices.

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Problem 2

Consider an economy populated by a continuum of measure one of infinitely lived, ex-ante equal agents withpreferences over sequences of consumption and leisure given by

E0

∞∑t=0

βtu (cit, 1− hit) .

Agents face individual uninsurable endowment shocks εit to labor productivity, where εit ∈ {ε1, ..., εN}. Thisprocess is governed by the Markov chain π (ε′, ε) = Pr {εi,t+1 = ε′|εit = ε} . Agents can save and borrow througha unique non-state-contingent asset ait. Assume that they can borrow up to the natural limit (what is it?).

Agents supply hit hours of labor to the market, where each hour has productive efficiency equal to εit. Labor andasset markets are competitive and clear, every period, with prices wt (per efficiency unit) and rt, respectively.

Production takes place through the aggregate technology Yt = AKαt L

1−αt where Kt and Lt are, respectively,

aggregate capital and aggregate units of labor in efficiency units at time t.

The government in this economy sets government expenditures Gt according to the mood of its president, toeither Gh or Gl, with Gh > Gl. The mood of the president follows a publicly known random process φ (G′, G) =Pr {Gt+1 = G′|Gt = G} . Proportional taxes on consumption expenditures τ ct adjust to balance the budget everyperiod.

Part a

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

First, let us derive the natural borrowing limit. The budget constraint for agent i can be written as

(1 + τ ct )cit + ait+1 = (1 + rt)ait + wtεithit

Non-negative consumption requires that

(1 + rt)ait + wtεithit − ait+1 ≥ 0

or

ait ≥ −wtεithit1 + rt

+ait+1

1 + rt

Iterate forward and impose a transversality condition to obtain that

ait+1 ≥ −∞∑i=1

wt+iεt+iht+i∏ij=1(1 + rt+j)

Since both transfers and interest rates are dynamic equilibrium objects, we will need to guess and verify thevalue of this object. This may be a complicated task because the two objects are related. Therefore, we need toinvestigate the worst case either analytically or by simulation (for example investigate how this value is affected

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by different paths of taxes in the economy). Zhang (1997) suggests an endogenous borrowing constraints thatguarantees participation in the assets market because a default would lead to autarky. In other words, derivationof the natural borrowing limit is slightly more complicated due to the fact that there is aggregate uncertainty inthis economy (induced by G), and so prices do not have to be constant (as the economy will not necessarily settlein a stationary equilibrium). In order to simplify the computation, let us assume that creditors have access to atechnology that allows them to fully enforce the debt contracts, thus they can force the agent to work full-timein order to repay the debts hit = 1 for all t and states of the world. Furthermore, the debt limit must allowrepayment in every possible state of the world, in particular in those where εit = min{ε1, . . . , εN} = εmin

ait+1 ≥ −εmin

∞∑i=1

wt+i∏ij=1(1 + rt+j)

Without further assumptions on the aggregate dynamics, there is little we can do to simplify the above expression.A possibility would be the following: compute the stationary equilibrium when Gt = Gh,∀t, and the stationaryequilibrium when Gt = Gl,∀t. Compute all possible transition paths between the two steady states, and obtainall possible values for wages and the interest rate along these paths. Note now that the natural borrowing limitis increasing in the wage and decreasing in the interest rate. Since it must hold for all possible states, take theminimum wage you computed along those paths, wmin as well as the maximum interest rate rmax. We can thenwrite the natural debt limit as

ait+1 ≥ −εminwmin

rmax

Now, onto the main course. The individual states are (a, ε), whereas the aggregate states are G and the distri-bution of individual states λ(a, ε). The problem of the household in recursive form can then be written as

V (a, ε;G,λ) = maxc,a′,h

{u(c, 1− h) + β

∑ε′∈E

∑G′∈G

π(ε′, ε)φ(G′, G)V (a′, ε′;G′, λ′)

}

subject to

[1 + τ(G,λ)]c+ a′ = [1 + r(G,λ)]a+ w(G,λ)εh

a′ ≥ −Φ

c ≥ 0

h ∈ [0, 1]

λ′ = Γ(G,λ)

where Φ is a law of motion for the distribution of individual state, as perceived by the individual agent.

Part b

Define a recursive competitive equilibrium for this economy.

Let S = A × E × G × Λ = [Φ,∞] × {ε1, ..., εN} × {Gh, Gl} × Λ denote the state space for an individ-ual agent where Λ is the set of probability measures on A × E. B(S) denotes the product σ-algebra in-duced by S. A recursive competitive equilibrium for this economy consists of a value function V : S → R,policies (c, a′, h) : S → R+ × [−Φ,+∞) × [0, 1], policies for the firm (H,K) : R2

+ → R2+, price functions

(r, w) : G× Λ→ R2+, government policy τ : G× Λ→ R+ and a law of motion Γ : G× Λ→ Λ such that

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1. Given prices and government policies, V solves the Bellman Equation associated with the household’sproblem and (c, a′, h) are the associated policies.

2. Given prices, the firm optimally chooses H and K

r(G,λ) + δ = FK(K,H)

w(G,λ) = FH(K,H)

3. Markets clear

labor :

∫A×E

εh(a, ε;G,λ)dλ(a, ε) = H(G,λ)

assets :

∫A×E

a(a, ε;G,λ)dλ(a, ε) = K(G,λ)

goods :

∫A×E

c(a, ε;G,λ)dλ(a, ε) +

∫A×E

a′(a, ε;G,λ)dλ(a, ε) +G = F (K,H) + (1− δ)K(G,λ)

4. The government budget constraint is balanced

τ(G,λ)

∫Sc(a, ε;G,λ)dλ(a, ε) = G

5. The law of motion Γ satisfies, for any A× E ∈ B(S)

λ′(A× E) = Γ(G,λ) =

∫SQG[(A× E), (a, ε)]dλ(a, ε)

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

QG[(A× E), (a, ε)] = 1[a′(a, ε;G,λ) ∈ A]∑ε′∈E

π(ε′, ε)

Part c

Outline a numerical algorithm that solves for the equilibrium allocations and prices.

Since there is aggregate uncertainty but no aggregation, we have to use a Krusell-Smith algorithm. Note,however, that since labor is endogenous, and the labor supply is not fixed, capital is not a sufficient forecast forall future prices, next period. In other words, the agent has to forecast r and w in order to optimize. In a settingwith fixed labor supply, a forecast of K gives us a forecast of K/H, which in turn allows the agent to forecastmarginal productivities and both prices. Now, H is no longer fixed hence two forecasts are required. I am goingto assume that agents use the following forecasting rules3

logK ′ = b0G + b1G logK

logH = d0G + d1G logK

The algorithm then goes as follows:

3You could alternatively guess that logH ′ = d0G + d1G logH, but you would then need to expand the state space byadding the previous aggregate labor supply. Most of this guess forms are quite ad-hoc, but you should try to think whyyou find them reasonable (and possibly argue about that) and be consistent with the state space. Also, you should showthat they give the agent enough information to forecast all the elements they are interested in (here, prices and taxes).

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1. Guess coefficients for the laws of motion {b0G, b1G, d0G, d1G}G∈G

2. Solve the household problem and obtain the decision rules (c, a′, h)(a, ε;G,λ). Note that forecasts of(r′, w′, τ ′) are required. For the first two, our laws of motion are enough. How can we obtain the lastone? Note that from the government budget constraint we have that τC = G. Goods market clearing alsoimplies that

τ =G

Y + (1− δ)K −K ′ −Gand everything in the RHS can be forecasted with either the exogenous stochastic processes or the assumedlaws of motion.

3. Simulate the economy for N individuals and T periods. Draw first a sequence of aggregate shocks {Gt}Tt=1.

Then, draw individual productivity shocks {εit}T,Nt=1,i=1 (potentially conditional on the path of the aggregate

shocks, if there were correlation). Given initial conditions for each agent {a0i}Ni=1, compute the path of

policies {(ait, hit}N,Ti=1,t=1. For each period, compute the average capital stock and labor supply

Kt =1

N

N∑i=1

ait

Ht =1

N

N∑i=1

εithit

4. Note that in this model, we need to clear three different markets (capital, labor and the government budgetconstraint) every period (the fourth, goods, clears by Walras’ Law). In order to avoid numerical instability,we should make sure that all markets clear at each t. Therefore, an additional inner loop is required: if Ht

does not coincide with the forecast, update wt until the labor market clears. At the same time, since H ischanging, so is Y , so we need to update the tax rate so that the government budget constraint is satisfied.This can be done by using the previous expression

τ =G

Y + (1− δ)K −K ′

Once both the labor market and the government budget constraint clear at each t, store the sequences{Kt, Ht}∞t=0.

5. Discard the first S periods in order to avoid dependence on initial conditions. Using the remaining se-quences, regress

log Kt+1 = β0G + β1G log Kt

log Ht+1 = δ0G + δ1G log Ht

and estimate {β0G, β1G, δ0G, δ1G}G∈G

6. For a given tolerance level ε, check if

maxG∈G

{maxi=1,2

|biG − βiG|,maxi=1,2

|diG − δiG|}< ε

If yes, then stop. Otherwise, substitute b = β, d = δ and return to step 2.

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7. Recall that this equilibrium is approximate: we still need to verify how good this approximation is to thefully rational-expectation equilibrium. For this purpose, compute a measure to the fit of the regression,for example by using R2. Next, try augmenting the state space with another moment, for example using

m2 ={E(a2i), E(

(hiεi)2)}

. Repeat steps 1-6) until convergence. If the R2 of the new regression

has improved significantly, keep adding moments until R2 is large and does not respond to addition ofnew explanatory moments. Otherwise, stop: it means that additional moments do not add new usefulinformation in forecasting prices.

Part d

Suppose you want to compute the welfare costs of living under a moody government, compared to facing a gov-ernment with a stable mood (and stable expenditures G, for all t). Define your preferred welfare criterion andexplain how you would calculate this welfare loss, in practice.

First, one needs to define a reasonable benchmark against which to compare the moody economy. We couldeither use one of the constant mood steady states G = {Gl, Gh}, or define some sort of weighted average (i.e.the unconditional mean of G if a stationary distribution exists).

Once the benchmark is defined, the problem becomes relatively straightforward: let (c∗t , h∗t ) denote the allo-

cations that would be chosen under a ”stable government”, and let (ct, ht) denote those obtained under a moodyone. We could compute ω(a, ε) such that

E0

∞∑t=0

u[(1 + ω(a, ε)ct, 1− ht] = E0

∞∑t=0

u[c∗t , 1− h∗t ]

where (a, ε) is some pair of initial individual states. This would tell us, for each possible starting point, how muchwould we have to change consumption on average so as for the agent to attain the same ex-ante expected utilityin both economies. This would induce a distribution for ω(a, ε) that would allow us to formulate statements onwelfare comparisons.4

4We could also do a welfare cost evaluation using the change in initial asset position that would make the agents equallyhappy. This would allow them to optimally allocate it, by changing both their consumption and labor allocations.

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Problem 3

Consider the Krusell-Smith model that we studied in class (the one described in the lecture notes, i.e. the neo-classical growth model with idiosyncratic uninsurable risk, aggregate risk and inelastic labor supply).

Part a

Describe how you would calculate the welfare gains from eliminating business cycles (i.e. eliminating fluctuationsin z) for an agent in a generic state (a, ε). Think, in particularly, about how to handle the fact that the evolutionof idiosyncratic uncertainty ε [the risk you do not eliminate] depends on aggregate uncertainty z [the risk you toeliminate] through the joint Markov chain π (z′, ε′, z, ε)

The idea here is that we want to preserve in the economy without business cycles an idiosyncratic processthat will reflect the one in the economy with aggregate shocks. The interesting part of the problem is thatalthough the average household may not be affected by the cycles5, some consumers may suffer significantly (e.g.poor or unemployed). In order to evaluate this, we first need understand how to remove the aggregate produc-tivity. Lucas’ approach was to replace it by its mean, which we will follow here. However, what implicationsdoes this have on the individual’s shocks? I will discuss two options here.

Note: Before moving to procedures to remove the cycles, it is important to notice that the question is asking forthe welfare gains of “eliminating business cycles” which is different from “how does an economy with businesscycles compare to one in which the cycles have been removed (in welfare terms).” The first also includes thisissue of the transition from one economy with cycles to one without (while the second just compares long runwelfare). Having determined a methodology to remove the cycles, the transition experiment can be approached(for example, as suggest by Krussel and Smith [1999])by replacing the stochastic aggregate process with itsconditional expectation as of the initial date, leaving a deterministic movement which disappears in the longrun. I will avoid this in this problem though and focus (as required) on how to remove the aggregate uncertaintyproperly.

Alternative 1The evolution of the idiosyncratic risk in the economy with aggregate shocks depends on the productivity shockz as follows:

π zb, eg zb, eb zg, eg zg, ebzb, eg x x

zb, ebzg, eg x x

zg, eb

5See Lucas (1987 or 2003)

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Imrohoroglu (1989) suggests that agents only observe the idiosyncratic risk and can use only this to forecast theshock in the next period. This amounts to simply collapsing each set of four probabilities as the one markedabove into one probability as follows:

π′ eg ebeg x

eb

where π′ is the evolution of the idiosyncratic risk in the economy without business cycles. This is done byreplacing the probabilities by the expectations (each set of four probabilities like the one marked above will berepresented by one) of the probabilities as follows:

π′(e′|e)

=∑z∈Z

θe,z(π(zg, e

′|z, e)

+ π(zb, e

′|z, e))

where θe,z is the stationary distribution of (e, z) in the economy with aggregate shocks. Therefore, conditionalon the idiosyncratic risk, the expected probability (and value) of the exogenous idiosyncratic risk is the same inboth economies. Note that this imposes that the new process remains first-order Markov, which will not be thecase in the next alternative.

Alternative 2Krusell and Smith (1999) started a literature on what is referred to as the ”Integration Principle”.6 However,these authors later on noticed a “mistake”7 in their procedure which led to a new paper: Krussel, Mukoyama,Sahin and Smith (2009) [KMSS]. I base the following explanation on that then. Integrating the aggregate riskout of an income process refers to averaging over the aggregate states conditional on each idiosyncratic state.However, since individual shock are correlated with the aggregate state, the individual variable is not fullyidiosyncratic, which makes the integration non-trivial. It is first necessary to construct a pure process for id-iosyncratic luck that is uncorrelated across individuals. Conditional on every realization of this new idiosyncraticvariable, one can integrate out the influence of aggregate risk on individual employment.Following the Krussel-Smith model seen in class (exogenous labor), how should the cyclical components be re-moved?

1. Aggregates:For aggregate shocks (productivity zt ∈ {zg, zb} and associated unemployment levels ut ∈ {ug, ub}) wereplace them by their unconditional means

z = πgzg + πbzb

u = πgug + πbub

where πg and πb are the unconditional probabilities of good and bad states.

2. Individual Variables:One could remove the aggregate component of the individual shock in several ways. KMSS integrate overthe aggregate shock.Suppose the individual variable of interest y is a function h of two independent(!) random variables,

6The basic idea is also well explained by Krebs (2006).7The implementation assumed that individual shocks were unaffected by the removal of cycles.

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y = h(i, z) where i is the idiosyncratic shock and the z is the aggregate shock. Let their densities be fi(i)and f(z), respectively. We can then identify the idiosyncratic shock process in the absence of aggregaterisk, y(i) as:

y(i) =

∫zh(i, z)fz(z)dz

for each i with density f(i).8 But how do we identify this independent variable i in our economy? KMSSshow that this can be done by:

• Define i to be uniform on [0, 1]. Let π1|z be the probability of being employed when the aggregatestate is z. Assume here that π1|g > π1|b.

• Define:(1) ε(i, zg) equal to 1 if i ≤ π1|g and 0 otherwise;(2) ε(i, zb) equal to 1 if i ≤ π1|b and 0 otherwise.

• Integration:(1) if i ≤ π1|b the individual is employed no matter the aggregate, so ε(i) = 1;(2) if i ≥ π1|g the individual is unemployed no matter the aggregate, so ε(i) = 0;(3) if π1|b < i < π1|g the individual is employed only if the aggregate state is good, which occurs withprobability πg so ε(i) = 1× πg + 0× πb = πg.

• Hence, our new employment variable ε has a 3-state distribution (with different support) and can beshown to be less risky.9

In a dynamic economy this gets even more complicated but KMSS show that the new process will increaseits support and will not remain first-order Markov, but will settle down to a stationary process.

8Note that i is not the individual employment state. In our model, employment is not independent of z. This was themistake in Krussel and Smith (1999). As a result the same exact employment process was applied for the models with andwithout aggregate shocks.

9Some probability mass has moved from the extremes to the center.

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References

[1] Imrohoroglu, A., (1989) ”The Cost of Business Cycles with Indivisibilities and Liquidity Constraints,” Journalof Political Economy, 97: 1364-1383.

[2] Krebs, T (2006) ”Multi-dimensional risk and the cost of business cycles,” Review of Economic Dynamics, 9:640-658.

[3] Krusell, P., and A. Smith (1999) ”On the Welfare Effects of Eliminating Business Cycles,” Review of EconomicDynamics, 2: 247-272.

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