Macroeconomics 2Lecture 12 - Idiosyncratic Risk and Incomplete Markets

Equilibrium

Zsofia L. Barany

Sciences Po

2014 April

Last week

I two benchmarks: autarky and complete markets

I non-state contingent bonds: the PIH and certainty equivalence

I precautionary savings: due to prudence and borrowingconstraints

I a natural borrowing constraint

I initial look at the existence of an equilibrium

This week

models that feature a non-trivial endogenous distribution of wealth

need these models to analyze:

I what fraction of the observed wealth inequality is due touninsurable earnings variation across agents?

I how much of aggregate savings is due to the precautionarymotive?

I how do various policies affect the distribution? effects oninequality and welfare?equilibrium model important, since changes in policy affectequilibrium prices

I can we generate a reasonable equity premium (i.e. excessreturn of stocks over a risk-free bond), once we introduce arisky asset?

I how large are the welfare losses of a rise in labor market risk?

look at models with a continuum of agents facing individualincome shocks and trade a risk-free asset:

I Bewley (1986) introduced them, with agents holding moneyas the risk-free asset

I Laitner (1992) added altruism

I Huggett (1993) endowment economy and the asset in zeronet supply

I Aiyigari (1994) production economy and aggregate productionfunction

I Huggett (1996) OLG version of the Aiyigari model

Key elements

1. consumption choice under the fluctuation of income

2. aggregate neoclassical production function

3. asset market equilibrium

⇒ look for stationary equilibria

1. the income fluctuation problem

I idiosyncratic risk

I only non-state contingent asset (risk-free bond) and anexogenous borrowing constraint

I ⇒ two reasons to save: inter-temporal substitution andpre-cautionary motive (shield against future negative incomeshocks)

I agents who have a long sequence of bad shocks will have lowwealth and will be close to the borrowing constraint

I agents who have a long sequence of good shocks will havehigh wealth

I ⇒ endogenous distribution of wealth

I integrating wealth over all agents → aggregate supply ofcapital

2. aggregate production function

I competitive representative firm

I profit maximization

I CRS production technology

I ⇒ aggregate demand for capital

3. equilibrium on the asset market

I aggregate supply of capital = aggregate demand for capital

I ⇒ equilibrium interest rate

note: in an AD equilibrium, the economy is observationallyequivalent to a representative agent model with a stationaryamount of savings, and the steady state is described byβ(1 + r) = 1here the agents have more reasons to save ⇒ more capital ⇒lower interest rate ⇒ β(1 + r) < 1

Consumers - workers

I continuum (measure 1) of infinitely lived agents

I time-separable utility function: E0∑∞

t=0 βtu(ct)

u′ > 0, u′′ < 0, β ∈ (0, 1)

I the budget constraint

ct + at+1 = (1 + rt)at + wt lt

* lt is the efficiency units of labor the agent supplies* at+1 is the amount of risk free bond the agent buys, theprice is one; it pays 1 + rt+1 in the next period* the interest rate is independent of the individual state

I the exogenous liquidity constraint

at+1 ≥ −φ

I inelastic labor supply, with an exogenous process for laborproductivity:lt drawn from density g(l) with support [lmin, lmax ]

I the shocks are iid across individuals and over time

I ⇒ the aggregate supply of efficiency units of labor is:

Lt =

∫ lmax

lmin

lg(l)dl for all t ≥ 0

I ⇒ no aggregate uncertainty, Lt exogenous

Financial position of households

I at the beginning of period t, an individual takes assets, at andlabor income, wt lt as given

I as the productivity process is iid, future draws, lτ areunpredictable from current draw lt

I assume for now that the real wage, w and the real interestrate, r are expected to remain constant

I let at denote accumulated assets plus the borrowing limit:at ≡ at + φ

I the borrowing constraint is then equivalent to: at+1 ≥ 0

I let zt denote the maximum amount of consumption that canbe obtained in period t:zt = wlt + (1 + r)at + φ = wlt + (1 + r)at − rφ

Consumer’s problem in a recursive form

v(z ;w , r) = maxc,a′

(u(c) + β

∫ lmax

lmin

v(z ′;w , r)g(l)dl

)s.t. c + a′ = z

z ′ = wl ′ + (1 + r)a′ − rφ

a′ ≥ 0

I where r ,w are exogenous to the individual, but endogenous tothe economy

I z is the only state variable

I we assumed that w , r are constant, which is true in astationary environment r ,w

solution: policy function A(z ;w , r) which determines the choicevariable a′

I if individual resources are very low in the current period, theindividual will consume everything, i.e. A(z ; r ,w) = 0

I higher levels of z induce some saving, so resources are dividedbetween consumption and saving

I the policy function, A(z ; r ,w), and the random draw of ldetermine the evolution of each individuals’s availableresources over time:

zt+1 = wlt+1 + (1 + r)A(zt ;w , r)− rφ

the economy is described by the distribution of z across individuals

the law of motion for this is:

λt+1(z) =

∫ ∞0

g

(z − (1 + r)A(z ;w , r) + rφ

w

)λt(z)dz

under some conditions there exists a unique, stable, stationarydistribution for z : λ∗

the long-run total amount of assets in the economy is then:∫ ∞0

(A(z ; r ,w)− φ)λ∗(z)dz

Technology, markets, feasibility

I representative firm has a CRS technology: Yt = F (Kt , Lt)

I the depreciation rate of capital is δ ∈ (0, 1)

I all markets (final good, labor, capital) are competitive

I the feasibility constraint of the economy is

F (Kt , Lt) = Ct + It = Ct + Kt+1 − (1− δ)Kt

Stationary Recursive Competitive Equilibrium

consists of a value function v : z → R, policy functions for thehousehold A : z → R+, a stationary probability distribution λ∗, realnumbers (K , L,w , r) given g(·) and φ such that

I the policy function A solves the household’s problem and v isthe associated value function, given w and r

I prices satisfy r + δ = FK (K , L) and w = FN(K , L)

I the labor market clears,∫ lmax

lminlg(l)dl = L

I the capital market clears,∫∞0 (A(z ; r ,w)− φ)λ∗(z)dz = K

I given A and g , λ∗ satisfies

λ∗(z) =

∫ ∞0

g

(z − (1 + r)A(z ;w , r) + rφ

w

)λ∗(z)dz

Existence and uniqueness of equilibrium

I equilibrium in the labor market exists, and is unique

I demand for capital: K (r) = F−1K (r + δ), continuous,decreasing

I supply of capital: A(r) =∫∞0 (A(z ; r ,w)− φ)λ∗r (z)dz

* if we could show that this crosses the demand, then weprove existence→ possible, but complicated, we skip it here* if in addition we could show that A(r) is increasing, then wewould also have uniqueness→ no results on monotonicityincome and substitution effects of r (could rule out turning byspecific utility function)hard to assess how r affects the distribution of assets

Computation of the equilibrium

1. make an initial guess, r0 ∈ (−δ, 1− 1β )

2. given r0, obtain the wage rate w(r0)

3. given r0 and w(r0), solve the agent’s problem and obtainA(z ; r0,w0)

4. given these and g(l) by iteration we can obtain the stationarydistribution λ(r0)

I simulate a large number of households (say 10,000): initializeeach individual in the sample with a pair (a0, l0) and hence z0,using the decision rule A(z0; r0,w(r0)) and a random numbergenerator that replicates g(l), update the hhs individual state(z) in every period t

I for every t, compute a set of cross-sectional moments Jt whichsummarize the distribution of assets (mean, variance, variouspercentiles); stop when Jt and Jt−1 are close enough → thecross-sectional distribution has converged(we know that for any given r , a unique invariant distributionwill be reached for sure)

5. compute the supply of capital

A(r0) =

∫ zmax

0(A(z ; r0,w(r0))− φ)λr0(z)dz

this can be done by using the model generated data in step 4

6. compare K (r0) and A(r0)* if K (r0) > A(r0), then the new guess for r has to be higher* if K (r0) < A(r0), then the new guess for r has to be lowerfor example by bi-section:

r1 =1

2(r0 + FK (A(r0), L)− δ)

7. update your guess to r1 and go back to step 1; keep iteratinguntil reaching convergence of the interest rate

Chapter 4 by Rios-Rull in Cooley (1995) describes how to computeequilibria in these types of models

Calibration

this is already needed for the numerical solution method describedbefore

I technology: Cobb-Douglas, capital share α = 1/3

I depreciation: δ = 0.06

I utility function: CRRA, the coefficient of relative risk aversionγ ∈ [1, 5], typically 1 or 2 is most commonly used

I β could be chosen to match the aggregate wealth-incomeratio for the US (around 3 without residential capital)this requires internal parameter calibration, which is quite hardan alternative is to match the wealth-income ratio of 3 undercomplete markets, implies to achieve 3 in the calibratedincomplete markets model we need a β slightly below 0.951

I g to match the distribution of labor income

I φ - natural debt limit, or match the fraction of people in thedata that hold negative wealth (15%) - internal calib

Precautionary savings in the Aiyigari model

Figure IIB shows the amount of savings that would occur ifmarkets were complete:

I under complete markets we can use a representative agent,who receives a constant earning in each period

I if r < λ, then the interest rate does not compensate enoughfor the impatience, so the agent would be up against theborrowing constraint, at −φ

I if r > λ, then the agent would accumulate an infinite amountof assets

I the steady state would be at ef , as the desired asset holdingsare represented by the bold line FI (λ = 1/β − 1)

⇒ the distance between the two capital levels is the amountaccumulated for self-insurance

can use the model to predict the amount of aggregateprecautionary savings in the US:

I log utility and iid shocks ⇒ r∗ ≈ 1/β − 1, no precautionarysavingsintuition: low risk aversion and low persistence shocks → lowself-insurance motive

I CRRA utility, γ = 5, 0.9 autocorrelation of income shocks ⇒precautionary savings is 14% of aggregate outputintuition: high risk aversion → consumption fluctuations arevery costly, high persistence → income can stay low for a longtime

I these are the two extremes, a more realistically calibratedmodel (γ ≈ 2) predicts that the precautionary saving ratewould be around 5% of GDP, about 25% of total savings

note: equilibrium considerations put some discipline on the model -given demand for capital only one interest rate works(amount of savings if r gets closer to 1/β − 1 behaves differentlyin the equilibrium model)

Comparative statics

I borrowing limit: suppose we increase φ, i.e. make theborrowing limit more generous ⇒ A(r) shifts up, theequilibrium amount of capital decreases and the interest rateincreases

I risk aversion: increase risk aversion ⇒ individuals are moreconcerned about consumption smoothing, they accumulatemore buffer-stock savings, for any r A(r) is larger ⇒ A(r)shifts down, more capital, lower interest rate

I income process: increase the variance of income ⇒ again,people accumulate more, A(r) shifts downincreasing the persistence has a similar, but quantitativelybigger effect

Efficiency and constrained efficiency

I is the competitive equilibrium efficient, does is achieve thefirst best?

I answer: it clearly is not, as the first best allocation is the oneachieved under complete markets, where βR = 1where agents fully insure themselves against the idiosyncraticlabor risk

I in the Aiyigari model there is an over-accumulation ofcapital compared to the first-best

I first best can be achieved by public insurance: tax away all ofthe income, and redistribute it equally across agentsnote: this would not work if the households also had alabor-leisure choice → this would introduce a trade-offbetween insurance and efficiency ⇒ next week we will look atthese questions more in detail

I a different question: do the markets perform efficientlyrelative to the set of allocations achievable with the samestructure? → constrained efficiency

I the planner tells each agent how much to consume and howmuch to save

I facing the same technology and asset structure

I there is an externality in the competitive equilibrium: eachagent’s decision has an impact on prices, which he does nottake into account when making his choices

I ⇒ the competitive equilibrium is constrained inefficient

I the planner takes this effect into account when telling theagents how much to consume and save

the planner maximizes

Ω(λ) = maxf (a,l)

∫A×L

u(aR(λ) + lw(λ)− f (a, l))dλ+ βΩ(λ′)

* subject to what? * where a′ = f (a, l) is the amount of assetsthat the planner asks an individual with assets a and laborproductivity l to have next period* when the planner assigns f (a, l) to an individual, he changes thetotal amount of saving (capital) next period, thus influencing thetotal amount of resources of everyone in the next period* the Euler equation for the optimal saving of an agent with statea, l is:

uc ≥ βR(λ′)

∫ lmax

lmin

u′cg(l ′)dl + β

∫A×L

(a′F ′KK + l ′F ′LK )u′cdλ′

the extra term compared to the competitive EE is:

β

∫A×L

(a′F ′KK + l ′F ′LK )u′cdλ′

→ the planner internalizes the effects of individual savings onprices

I the blue term captures the effect of an extra unit of saving onnext period capital income and labor income

I more savings ⇒ more capital ⇒ more labor income and lesscapital income

I this effect is averaged over all agents, weighted by themarginal utility from consumption

I term can be positive or negative, as FKK < 0 and FKL > 0why isn’t there such a term in case of complete markets?

I if the income of the poor is labor-intensive, then theexpression is positive

I ⇒ the planner wants the agents to save more

I the competitive equilibrium features under-accumulation ofcapital

I intuition: the planner wants to redistribute from the rich tothe poor

I if the poor have mostly labor income, then the way toredistribute is to increase equilibrium wages by inducingagents to save more than in equilibriumlarger individual savings increase the aggregate capital stockand increase wages

I Davila, Hong, Krusell and Rios-Rull calibrate this model tothe US economy and find that the constrained efficient capitalstock is 3.5 (!!) times higher than the competitive equilibriumcapital stock

Wealth inequality

I in this model agents are ex-ante identical, they only differ dueto the variation in their income realizations⇒ differences are only driven by luck

I the path of shocks lead to an endogenous consumption andwealth distribution

I natural questions: if idiosyncratic earnings shocks are the onlysource of heterogeneity how much can the model explain ofthe observed wealth inequality?

mean mean/median Gini share of top 1%

earnings 21.1 1.57 0.61 7.5%wealth 47.4 4.03 0.80 31%

Source: Budria, Diaz-Gimenez,Quadrini, Rios-Rull (2002)

data: both wealth and earnings are skewed, but wealth much moreso, the top 1% of the wealth distribution holds more than 30% oftotal wealth

model generates too much asset holding at the bottom and notenough at the top, the Gini generated by the model is 0.4 asopposed to 0.8 in the data

1. reduce the incentives for the poor to save for self-insurance* modeling the welfare state properly helps a lot ∼ publicinsurance schemessee for example Hubbard, Skinner and Zeldes (1995)

2. increase the incentives for the rich to accumulate capital* entrepeneurship (Quadrini (2000), Kitao (2008))* heterogeneity in the discount factor (Krusell, Smith (1997))* bequests (De Nardi (2003))* very high income realizations with very low probability(Castaneda, Diaz-Gimenez, Rios-Rull (2003))

Aggregate uncertainty

so far we assumed that there is no aggregate uncertainty ⇒ cannotstudy the effects of incomplete markets on macro dynamics

Krusell and Smith (1998) introduce aggregate uncertainty, whereaggregate shocks have two effects on the economy:

1. change total factor productivity

2. change employment

extra difficulty in solving these models: the entire distribution ofwealth becomes a state variable (this is infinite dimensionalobject), which we have to keep track of as it is not stationary ⇒impossible to solve explicitly for the equilibrium allocations

good news: we can approximate the exact equilibrium

I aggregate consumption and saving resemble those of arepresentative agent

I little is lost by considering only the mean of the assetdistribution

I it is enough to keep track of the mean wealth rather than theentire wealth distribution in order to forecast future prices

I ⇔ we also do not gain much from looking at theheterogeneous agent economy when thinking about businesscycles

Intuition:

I under complete markets there is a linear relationship betweensaving and wealth

I this is not true in the Aiyigari and in the Krusell and Smithmodel: precautionary saving should be higher for agents whohold fewer assets ⇒ non-linear saving function

I for agents who are rich enough, the policy function is roughlylinearthey achieve good self-insurance with a small amount ofsaving, most of their saving is due to the inter-temporalmotive rather than the insurance motive

I non-linearity is created by those who cannot self-insure bysaving and borrowing → those who are very poor and areclose to the borrowing constraint

I but these agents are a relatively unimportant part of theoverall distribution of assets