Aiyagari Rao (EER) - Frictions in Asset Pricing and Macroeconomics

8/7/2019 Aiyagari Rao (EER) - Frictions in Asset Pricing and Macroeconomics

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ELSEVIER European Economic Review 3X (1994) 932-939

EUROPEANECONOMICREVIEW

Frictions in asset pricing and macroeconomics *S. Rao Aiyagari

Federul Reserw Bad oj Minneapolis. Research Depurtment. P.O. Boz 31, Minneupolis.MN 554RO-O-791. USA

AbstractThis paper is about a useful way of taking account of frictions in asset pricing andmacroeconomics. I start by noting that complete frictionless markets models have anumber of empirical deficiencies. Then I suggest an alternative class of models withincomplete markets and heterogeneous agents which can also accommodate a varietyof other frictions. These models are quantitatively attractive and computationallyfeasible and have the potential to overcome many or all of the empirical deficienciesof complete frictionless markets models. The incomplete markets model can also differsignificantly from the complete frictionless markets model on some important policyquestions.Kerr words: Complete markets; Incomplete markets: Precautionary motive; BorrowingconstraintsJEL c.lussificution: D52; D58; E2; E62

1. IntroductionThis paper is about a useful way of taking account of frictions in asset

pricing and macroeconomics. I start by noting that complete frictionlessmarkets models have a number of empirical deficiencies. Then I suggest analternative class of models with incomplete markets and heterogeneousagents which can also accommodate a variety of other frictions. Thesemodels are quantitatively attractive and computationally feasible and havethe potential to overcome many or all of the empirical deficiencies ofcomplete frictionless markets models. By quantitatively attractive I mean thatthese models can be calibrated using macro and micro data in the same way

*The views expressed herein are those of the author and not necessarily those of the FederalReserve Bank of Minneapolis or the Federal Reserve System.0014.2921/94/$07.00 (_I1994 Elsevier Science B.V. All rights reservedSSDI 0014-2921(93)E0073-T


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as the standard representative agent growth model of Brock-Mirman is usedfor studying growth, business cycles with and without money, asset pricing,etc. The incomplete markets model can also differ significantly from thecomplete frictionless markets model on some important policy questions.

The models are of the Bewley (undated, 1984), Lucas (1980), and Lucasand Prescott (1974) type and contain a large number of infinitely lived agentssubject to idiosyncratic shocks to earnings and/or tastes which cannot beinsured against. This is the market incompleteness. Even if all agents areidentical ex-ante, they will become heterogeneous ex-post. Because themodels have heterogeneity, they can also accommodate other frictions, likeborrowing/liquidity constraints. Because of the large number of agents andindependent risks, there may be little or no aggregate uncertainty. Thesemodels capture the notion that uncertainty at the individual level is muchmore important than uncertainty at the aggregate level.

Because of incomplete insurance markets, there is a self-insurance motivefor accumulating and trading assets which has important implications for avariety of empirical issues, including the following: (i) variability and comove-ment of individual consumptions and aggregate consumption, (ii) wealth andincome distributions, (iii) portfolio compositions at different wealth levels, (iv)asset returns, (v) importance of precautionary saving in capital accumulation,(vi) taxation.

2. Empirical deficiencies of the complete frictionless markets modelThe following are well known implications of the complete frictionless

markets model which are quite counterfactual.(i) Individual consumptions should be perfectly correlated with each other

and with per capita consumption, (ii) each individuals consumption shouldvary as much as anyone elses and as much as per capita consumption (ifindividuals risk aversion coefficients are not too different), (iii) an individ-uals position in wealth distribution should not vary much over time oracross states (if individuals risk aversion coefficients are not too different),(iv) every individual should hold some amount of risky assets with favorablereturns. If individuals risk aversion coefficients are not too different, then allindividuals should hold roughly similar portfolios, (v) there is no role forasset trading, and there are no predictions regarding transactions volumesand transactions velocities of different assets, (vi) the risk-free rate is toohigh, and the equity premium is too low.

Aiyagari (1993~) describes in some detail the derivation of the aboveimplications and the wealth of empirical evidence against these.


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934 S. R. Aiyuxari /European Economic Review 38 (I994 i 932-9393. Growth model with uninsured idiosyncratic shocks

Consider an economy with a continuum of infinitely lived agents ofmeasure unity who receive idiosyncratic shocks to their labor endowments.Let I, denote an individuals labor endowment, and suppose that it is i.i.d.across agents and follows some Markov process over time. We normalizeper capita labor endowment to unity so that E(I,) = 1. There are no aggregateshocks. We describe the steady state of such an economy without insurancemarkets but with trading in risk-free assets-capital, government debt andprivate debt. In this steady state there will be fluctuations in an individualsconsumption. income, and wealth, but per capita variables and cross-sectiondistributions will be constant over time.

Let c,, u,, w, r, and r denote an individuals consumption in period t, anindividuals assets at the beginning of period t, the wage rate, the return onassets, and the lump sum tax, respectively. The typical consumer maximizesthe expected discounted sum of utilities of consumption E,~Y,( 1 +p))U(c,)subject to

ct + a ,+,dwl,+(l+r)a,-t, a,>-d. (3.1)In (3.1) d is a borrowing limit.Let K be the per capita capital stock, N the per capita labor input, and

J(K,N) a standard neoclassical aggregate production function. Note thatN equals unity in equilibrium. Then, from producer profit maximization, wehave

r=f;(K,l)-6. (3.2a)w=f;(K,f), (3.2b)

where S is the depreciation rate of capital.Let G denote per capita government consumption. and let B denote percapita government debt. Then the government budget constraint is given by

G+rB=z. (3.3)Let A denote per capita assets held by consumers. In a steady state

equilibrium of this economy we must haveA=K+B. (3.4)

The steady state of this economy is characterized by an interest rate r*which solves


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-(d+B) L;_yLcr*(r;G,d+ B) =x(r), IASSETS,CAPITALFig. 1 (3.5)where Cc*(r,. ) is the per capita assets desired by consumers (net of govern-ment debt) as a function of the interest rate and K(T) is the per capita capitalexpressed as a function of the interest rate.

The steady state determination under incomplete markets (IM) is shown inFig. 1 and is marked IM. The crucial feature of this picture is thatcc*(r; G, d +B) tends to infinity as r approaches the time preference rate pfrom below. The intuition behind this is that when r equals p the consumerwould like to maintain a smooth marginal utility of consumption profile.However, since there is some probability of receiving a long string of lowlabor endowment shocks, the only way for the consumer to maintain asmooth marginal utility of consumption profile would be to have aninfinitely large amount of assets. Note also that under incomplete markets,r Eq. (3.5) is obtained in the following way. We can use (32a) to define K as a function of rwhich is the right-hand side of (3.5). Also, (3.2a and b) can be used to write w as a function of r.Let this be denoted w(r). Then rewrite the consumers budget constraint by substituting fortaxes r from (3.3) into (3.1) and defining a: =a,-& This leads to the following formulation ofthe consumers budget constraint: c,ta :+r < c~(r)l,+(l +~)a:-G. a:> -d-B. The steady stateequilibrium condition in the asset market is now given by A*=K. The solution to theconsumers problem yields a decision rule for asset accumulation: a,*+, = a(a:, I,, r, G. d + E). Thisdecision rule can be used together with the Markov process for the labor endowment shock tocalculate the stationary distribution of assets, denoted by H(a *: r, G. d+B). This stationarydistribution then implies an expression for per capita assets (net of government debt).A* = Ja*dH = a*(r; G, d + B). This is the left-hand side of (3.5).


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936 S. R. Aiyquri /Europw~ Ecorromic~ ReGw 3X ( 1994 i 932-939due to the precautionary motive and borrowing constraints, the consumerwill hold assets over and above the credit limit to buffer earnings shockseven when r is less than p. This would not be the case under completemarkets.

Under complete markets (CM) the idiosyncratic labor endowment shockwould be fully insured against and effectively there would be no uncertaintyin individual earnings. The asset demand function in this case is described bythe dashed line in Fig. 1, leading to the steady state marked CM. This is theusual result that the capital stock satisfies the modified golden rule.Calibration. The above model can be calibrated in the same way asrepresentative agent models are. One can use data on factor shares tocalibrate the production function, use data on G, B, and t, and use estimatesfor 6 and preference parameters. A key feature of this model is that inaddition to the usual macro data one also needs to use micro data on, say,earnings, in order to calibrate the I, process.Implications of the incomplete markets model(1) Individuul und per cupitu consumption. The following quantitativeexample, taken from Aiyagari (1993a), indicates the variabilities of individualconsumption, income, and assets. The example assumes a constant relativerisk aversion utility function with risk aversion coefficient denoted p and afirst-order autoregressive process for the logarithm of the labor endowmentshock (equivalently, for the logarithm of earnings) with serial correlationdenoted /je and coefficient of variation (c.v.) denoted oe. In this examplewe assume that ( cre, pe, ,L() (0.2,0.6,3):

Consumption Net income Gross income AssetsC.. 0.17 0.21 0.27 0.62

The key points to note are the following. Individual consumption variesabout ten times as much as U.S. per capita consumption. Since the modelonly has idiosyncratic shocks, individual consumptions are uncorrelated witheach other and with per capita consumption (which is constant). If one wereto introduce some aggregate shocks into the model, then it would be possibleto generate the prediction that individual consumptions would be positivelybut less than perfectly correlated with each other and with per capitaconsumption. Since the cross-section distributions coincide with the long-rundistributions, the relative variabilities of consumption, income, and wealthhave obvious implications for wealth and income distributions.(2) Wealth and income distributions. The model naturally generatesgreater dispersion in wealth than in income and skewness in the wealthdistribution. Both of these features are qualitatively consistent with the data.Aiyagari (1993a) contains some quantitative illustrations of these features.


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S.R. Aiyagari /European Economic Review 38 (1994) 932-939 937(3) Precautionary saving and capital accumulation. As can be seen in Fig. 1,the steady state with incomplete markets is characterized by a lower interestrate and higher capital as compared to the complete markets case. Theadditional capital accumulation implies a higher saving rate. This incrementin the saving rate may be attributed to precautionary saving. The followingquantitative example, taken from Aiyagari (1993a), shows how importantprecautionary saving may be:

(a,, P.&l Reduction in net Increase in aggregatereturn to capital saving rate(0.2,O.l) 0 0(0.2,0.6,3) 0.3,, 0.6:j0(0.4.0.6,3) 1.4, 3(0.4.0.9.5) 4.59/, 14:

As can be seen from the above table, persistence in earnings and riskaversion can have a strong impact on the saving rate. Persistence isimportant because what matters to the individual is variability in permanentincome, and more persistence implies greater variability in permanentincome.(4) Asset returns and asset market transactions. In Aiyagari and Gertler(1991) a version of this model is used to study asset returns and assetmarket transactions. If we assume that there is no capital and interpret d + Bas representing liquid assets, then the risk-free rate will equal RF (see Fig. 1)and will be lower than what it would be under complete markets. If we nowintroduce another asset into the model which is costly to trade, then thereturn on this asset must be higher due to a transaction/liquidity premium.The typical individual uses the low return liquid asset to buffer earningsshocks and only occasionally buys or sells the high return illiquid asset. As aconsequence, the model naturally generates a higher transactions velocity forthe liquid asset relative to the illiquid asset. The quantitative exercises inAiyagari and Gertler (1991) indicate that the transactions velocity of theliquid asset can be from 10 to 20 times that of the illiquid asset, which isconsistent with the data on the relative turnover rates of bank money marketfunds and stocks. When there are fixed costs of trading the illiquid asset, thismodel also generates portfolio compositions such that people at the high endof the wealth distribution have relatively more illiquid assets compared topeople at the low end of the wealth distribution. This is qualitativelyconsistent with the data.(5) Capital taxation. An interesting policy implication of the completemarkets model is that under Ramsey taxation the optimal capital income taxrate is zero in the long run (Charnley, 1986). Lucas (1990a) argues that forthe U.S. the welfare gains of switching to zero capital income taxation arequite large. The incomplete markets model suggests otherwise. The optimalcapital income tax rate is always positive (see Aiyagari, 1993b). Conse-


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quently. switching to zero capital income taxation may well involve welfarelosses instead of welfare gains.

The intuition for this is that even under incomplete markets, long-runoptimality under Ramsey taxation requires that the pre-tax return on capitalequal the time preference rate. Because of incomplete markets, the only wayto support this is by a tax on capital income since, in the absence of such atax, the return to capital will always be less than the time preference rate.The situation is depicted in Fig. 1 in which i is the after-tax return and p isthe pre-tax return. As can be seen, the amount of the tax will depend on thestrength of the precautionary saving motive, which, in turn, depends on therisk aversion and the variability and persistence of earnings.

The following quantitative example, taken from Aiyagari (1993b), suggeststhat it is not too difficult to generate capital income tax rates close to theobserved value.

Capitaltax rate

(0.4,0.6.3) (0.4,0.6,5) US. data0.14 0.25 0.36

4. Future developmentsThere are two important ways in which the model described in this paper

needs to be extended. One is by introducing aggregate uncertainty inaddition to the idiosyncratic shocks, and the other is to properly takeaccount of private information and the resulting incentive compatibilityrestrictions. These are important for several issues, including the following: (i)asset returns, (ii) financial propagation mechanisms, (iii) transmission ofmonetary shocks.

For a complete resolution of the risk-free rate/equity premium puzzle weneed to have aggregate uncertainty so as to simultaneously account for therisk premium and the transactions/liquidity premium. Recent models offinancial propagation mechanisms (Williamson. 1987; Bernanke and Gertler,1989) are based on optimal contracting in environments with privateinformation. Recent models of the transmission of monetary shocks empha-size the uneven distribution of monetary shocks across households andmarkets (Grossman and Weiss, 1983; Rotemberg, 1984; Lucas, 1990b).

Obviously, problems of private information are at the root of incompleterisk sharing. In the incomplete markets model described in this paper theabsence of risk sharing was simply taken as given, but it is obviously moredesirable to explain this on the basis of some informational frictions. Recentwork by Green (1987), Phelan and Townsend (1991). and Atkeson and Lucas(1992) has pursued this approach.

Introducing aggregate uncertainty involves a considerable computational


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S.R. Aiwgari /European Economic Review 38 (1994) 932-939 939

burden. This is because the cross-section wealth distribution is an endoge-nous state variable which evolves stochastically over time in response toaggregate shocks. There is a considerable amount of difficult but excitingwork ahead.

ReferencesAiyagari, S. Rao, 1993a. Uninsured idiosyncratic risk and aggregate saving, Working paper 502

(Research Department, Federal Reserve Bank of Minneapolis, Minneapolis. MN).Aiyagari. S. Rao, 1993b, Optimal capital income taxation with incomplete markets, borrowing

constraints, and constant discounting, Working paper 508 (Research Department, FederalReserve Bank of Minneapolis, Minneapolis, MN).

Aiyagari, S. Rao, 1993c, Explaining financial market facts: The importance of incompletemarkets and transaction costs, Federal Reserve Bank of Minneapolis Quarterly Review 17,Winter, 17-31.

Aiyagari, S. Rao and Mark Gertler, 1991, Asset returns with transactions costs and uninsuredindividual risk. Journal of Monetary Economics 27, 311-331.

Atkeson. Andrew and Robert E. Lucas. Jr., 199 2, On efficient distribution with privateinformation, Review of Economic Studies 59. 427453.

Bernanke, Ben and Mark Gertler. 1989. Agency costs, net worth. and business fluctuations,American Economic Review 79, 14-31.

Bewley. Truman, 1984, Notes on stationary equilibrium with a continuum of independentlyfluctuating consumers. Manuscript (Yale University, New Haven, CT).

Bewley, Truman, undated, Interest bearing money and the equilibrium stock of capital,Manuscript (Yale University, New Haven, CT).

Charnley. Christophe, 1986, Optimal taxation of capital income in general equilibrium withinfinite lives, Econometrica 54. no. 3. 607-622.

Green, Edward, 1987, Lending and the smoothing of uninsurable income, in: Edward C. Prescottand Neil Wallace, eds.. Contractual arrangements for intertemporal trade (University ofMinnesota Press, Minneapolis. MN) 3-25.

Grossman. Sanford and Laurence Weiss, 1983, A transactions-based model of the monetarytransmission mechanism, American Economic Review 73. 871-880.

Lucas, Robert E. Jr., 1980, Equilibrium in a pure currency economy, in: John Kareken and NeilWallace. eds.. Models of Monetary Economies (Federal Reserve Bank of Minneapolis,Minneapolis, MN) 131-145.

Lucas, Robert E. Jr., 1990a, Supply side economics: An analytical review, Oxford EconomicPapers 42,293-3 16.

Lucas, Robert E. Jr.. 1990b. Liquidity and interest rates, Journal of Economic Theory 50,237-264.

Lucas, Robert E. Jr. and Edward C. Prescott. 1974, Equilibrium search and unemployment.Journal of Economic Theory 7, 188-209.

Phelan Christopher and Robert M. Townsend, 1991, Computing multi-period, informationconstrained optima. Review of Economic Studies 58. 8533881.

Rotemberg, Julio, 1984, A monetary equilibrium model with transactions costs, Journal ofPolitical Economy 92, [email protected], Stephen, 1987, Financial intermediation. business failures, and real business cycles,Journal of Political Economy 95, 11961216.

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Aiyagari Rao (EER) - Frictions in Asset Pricing and Macroeconomics