STOCHASTIC CONSUMPTION-SAVINGS MODEL: CANONICAL APPLICATIONS SEPTEMBER 13, 2010 BASICS. Introduction

Transcription

1 STOCASTIC CONSUMPTION-SAVINGS MODE: CANONICA APPICATIONS SEPTEMBER 3, 00 Introduction BASICS Consumption-Savings Framework So far only a deterministic analysis now introduce uncertainty Still an application of basic consumer theory The cornerstone of modern macro theory Starting point: two time periods Important: all analysis conducted from the perspective of the very beginning of period so a future (period for which to save But uncertainty exists about (some period- primitives Soon will extend to infinite number of periods Dynamic stochastic analysis the foundation of modern macroeconomic theory An explicit accounting of time An explicit accounting of risk Two-period stochastic model illustrates many central ideas, results, and methods September 3, 00

2 Macro/Finance Fundamentals ASSET MARKETS Risk about the future (period requires adopting a view about the nature of future (period- returns on assets Asset return State-contingent asset returns Suppose period- realized return on asset depends on realized period- endowment r = r probability q r probability p r probability -p-q Corresponding to y = y probability q y probability p y probability -p-q Schedule of returns known in period Equivalent to complete set of Arrow-Debreu assets A-D security: asset that pays one unit of numeraire in a particular realized state, zero otherwise Complete markets: A-D security exists for each of the possible realizations of uncertainty Complete asset markets span the uncertainty space Will later consider incomplete asset markets September 3, 00 3 Model Structure EVENT TREE Timeline of events More useful to think of as event tree Probability q: Realization y a0 Economic outcomes during period : income, consumption, savings a Probability p: Realization ybar a Beginning of planning horizon Period Probability -p-q: Realization y End of planning horizon Economic outcomes during period : stochastic income, state-contingent consumption, savings September 3, 00 4

3 Model Structure EXPECTED UTIITY Preferences v(c, c with all the usual properties ifetime expected utility function Assume separable across time periods: v(c,c = u(c + u(c (deterministic case Strictly increasing in each of c and c Diminishing marginal utility in each of c and c v(c,c v(c,c But realized c cannot be known at time decisions are made in period, due to period- income risk ow to incorporate risk into utility metric? Expected lifetime utility Assume consumers maximize c M ( Evc (, c = uc ( + Euc ( = uc ( + qu( c + pu( c + ( p q u( c A decision-theoretic (not experiential utility metric von-neumann-morgenstern (944 foundations / Econ 603 September 3, 00 5 c Stochastic Consumption-Savings Model: Solution CONSUMPTION DYNAMICS 5 equations, 5 unknowns In principle, can solve Solution to consumer problem is an asset position and state-contingent consumption profile c, c, c M, c ; a that satisfies ( State-by-state period- budget constraint M M c + a = y + ( + r a c + a = y + ( + r a c + a = y + ( + r a = 0 = 0 = 0 Euler equation M qu '( c pu'( c ( p q u' ( c u ( c = ( + r + ( + r + ( + r E ( r u' ( c u'( c u'( c = + u ( c Period- budget constraint c+ a = y + ( + r0 a0 ( taking as given r, r, r ; y, a0, r0 and the stochastic distribution G(. of y Could express solution in alternative ways e.g., using lifetime budget constraints September 3,

4 s APPICATIONS Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming Quadratic period-utility uc ( Risk aversion Risk-free asset returns Risky period- income (with arbitrary distribution Precautionary savings Introduction to asset pricing αc = γ c September 3, 00 7 Stochastic Consumption-Savings Model: Application CERTAINTY EQUIVAENCE Optimal period- (current consumption ( ( c = A + B y + + r a + C E y 0 0 Depends only on the mean of risky future income, E y Independent of second- and higher-moments of risky future income Distribution function G(. of period- income y = y probability q y probability p y probability -p-q E y = y ( ( Var y = q y y + ( p q y y Certainty Equivalence Mean-preserving spreads of G(. do not affect optimal choice of c E.g., (p =, q = 0 Period- income has no risk But c is identical s (period- savings is identical September 3,

5 CERTAINTY EQUIVAENCE A benchmark result in intertemporal consumption theory Result depends on Quadratic utility Riskless (aka non-state-contingent asset returns Only source of risk is income risk Only version of the intertemporal consumption model with analytical solution Strong implication: risk about future (income does not affect current consumption and savings decisions Intuitively plausible? Empirically relevant? Probably not but why not? Model does feature both Income risk (Var y > 0 Risk averse utility with respect to consumption need to define formally September 3, 00 9 Stochastic Consumption-Savings Model: Applications APPICATIONS Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing September 3,

6 Macro/Finance Fundamentals RISK AVERSION Illustrate with simple static example Utility function u(c Two possible consumption outcomes c with probability η c with probability -η Expected consumption is c = ηc + (-ηc Definition: an individual is risk averse (with respect to consumption risk if uc ( > ηuc ( + ( η uc ( JENSEN S INEQUAITY Risk aversion A preference for certain (deterministic outcomes to risky (stochastic outcomes Embodied in strictly concave utility ow to measure risk aversion? Need to capture something about concavity of utility September 3, 00 Macro/Finance Fundamentals RISK AVERSION ow to measure? A candidate measure: -u (c But not invariant to positive linear transformations of u(. September 3, 00 6

7 Macro/Finance Fundamentals RISK AVERSION Arrow-Pratt coefficient of absolute risk aversion (ARA u ( c Controls for linear ARA( c u transformations of u(. ( c ARA(c gets at idea of risk aversion in level gains or losses of c from E(c Increasing ARA: ARA (c > 0 Decreasing ARA: ARA (c < 0 Most empirically-relevant case Richer people can afford to take a chance Perhaps also useful to have measure of risk aversion in percentage gains or losses of c from E(c Relative risk aversion (RRA cu ( c RRA( c ( = c ARA( c Adjusts for level of u ( c consumption/wealth September 3, 00 3 Macro/Finance Fundamentals RISK AVERSION CRRA σ σ c c vc (, c = + σ σ σ > 0 uc ( uc ( Continuing to assume utility is additively-separable over time Attitude of consumers toward smoothing consumption between time periods IES = /σ Attitude of consumers toward risky outcomes within a given time period September 3,

8 Macro/Finance Fundamentals RISK AVERSION CRRA σ σ c c vc (, c = + σ σ σ > 0 uc ( uc ( Continuing to assume utility is additively-separable over time Attitude of consumers toward smoothing consumption between time periods IES = /σ Attitude of consumers toward risky outcomes within a given time period cu ''( c RRA( c = =σ u'( c u''( c σ ARA( c = = u'( c c CRRA utility: σ governs both intertemporal attitudes and intratemporal (relative risk attitudes! Inverses of each other!! Must/should IES and RRA be so directly related in reality? Not at all Epstein-Zin (EZ utility function disentangles the two concepts September 3, 00 5 Stochastic Consumption-Savings Model: Applications APPICATIONS Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing September 3,

9 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most critical assumption: quadratic utility αc αc Risk aversion (within period with vc (, c = uc ( + uc ( = γc + γc? Obviously = 0! (whether RRA or ARA So why certainty equivalence? September 3, 00 7 Stochastic Consumption-Savings Model: Application PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most critical assumption: quadratic utility αc αc Risk aversion (within period with vc (, c = uc ( + uc ( = γc + γc? Obviously = 0! (whether RRA or ARA So why certainty equivalence? i.e., why does future income risk not matter for current choices? Euler eqn often the key = E [ u c + r ] γ αc = E [( γ αc ( + r ] u'( c '( ( γ αc = q( γ αc ( + r + p( γ αc ( + r + ( p q( γ αc ( + r M M γ αc = ( + r q( γ αc + p( γ αc + ( p q( γ αc M ( c + ( p q c = ( + r γ α qc + p [ ] γ α γ α c = ( + r Ec c = r ( r Ec α + + September 3, 00 8 γ 9

10 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most critical assumption: quadratic utility αc αc Risk aversion (within period with vc (, c = uc ( + uc ( = γc + γc? Obviously = 0! (whether RRA or ARA So why certainty equivalence? Marginal utility function of order one (or lower implies risk on future income doesn t matter for current consumption Contrapositve Risk on future income matters for current consumption implies marginal utility function must be strictly convex u (c > 0 necessary for breaking certainty-equivalence result (Given u (. > 0 and u (. < 0 u (. > 0 u (. increasing in c u (. decreasing less quickly as c Not satisfied by quadratic utility September 3, 00 9 Stochastic Consumption-Savings Model: Application PRECAUTIONARY SAVINGS Assume utility with u (c > 0 vc (, c = uc ( + uc ( Assume interest rate is not state contingent r = r = r = r risk-free interest rate Insert in definition of solution to intertemporal problem M M c + a = y + ( + r a c + a = y + ( + r a c + a = y + ( + r a = 0 = 0 = 0 [ u c + r ] Euler eqn often the key u'( c = E '( ( [ '( ] u' ( c = (+ r E u c M u'( c = ( + r qu'( c + pu'( c + ( p q u'( c c + a = y + ( + r a 0 0 = E c, so none of the subsequent steps with quadratic u(. follow u (c > 0 current consumption depends on distribution G(. of future risk i.e., on first- and (in principle all higher-order moments of G(. September 3,

11 PRECAUTIONARY SAVINGS u (c > 0 current consumption depends on distribution G(. of future risk Optimal c is smaller than certainty-equivalent c Proof: Implication: optimal s is larger than certainty-equivalent s Precautionary Savings Risk about the future induces prudent (cautious choices in the present Desire to build up a buffer stock of assets to ensure c does not fall too low in future Risk aversion a necessary, but not sufficient, feature of preferences Strictly convex marginal utility the key feature of preferences Classic papers: Kimball (990 Econometrica, Sandmo (970 Review of Economic Studies ow to measure precautionary savings motive? Need to capture something about convexity of marginal utility Kimball (990 provides clever insight September 3, 00 Stochastic Consumption-Savings Model: Application PRECAUTIONARY SAVINGS ow to measure? c c avg = 0.5(c + c c c c avg = 0.5(c + c c A candidate measure: u (c Analogy with measures of risk aversion September 3, 00

12 PRECAUTIONARY SAVINGS ow to measure? Kimball (990: Define v(c = -u (c. Then can apply standard theory of risk aversion to v(c! September 3, 00 3 Stochastic Consumption-Savings Model: Application PRECAUTIONARY SAVINGS Coefficient of absolute prudence: Coefficient of relative prudence: u'''( c u''( c cu '''( c u''( c Measures of the sensitivity of optimal choice to risk Governed by marginal utility function ARA and RRA measure the sensitivity of welfare to risk Governed by the utility function CRRA utility u'''( c σ + = σ u''( c c c uc ( = σ cu '''( c = σ + u''( c Displays constant relative prudence Displays constant relative risk aversion Absolute prudence Relative prudence September 3, 00 4

13 s APPICATIONS Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing September 3, 00 5 Macro/Finance Fundamentals ASSET MARKETS Risk about the future (period requires adopting a view about the nature of asset markets Continue with example of risky period- income y = y y probability q probability p y probability -p-q But now three distinct assets available for purchase in period Asset a : purchase price R in period, pays off one unit in period if y, zero else M Asset a : purchase price R in period, pays off one unit in period if y, zero else Asset a : purchase price R in period, pays off one unit in period if y, zero else Arrow-Debreu securities, aka contingent claims Equivalent to state-contingent asset returns on a single asset September 3,

14 BASICS OF ASSET PRICING Consumer problem M M ma x uc ( + qu( c + p( c + ( p q u( c + λ y+ a0 c R a Ra Ra qλ y a c pλ y a c ( p q λ y a c M M M FOCs Asset prices qλ qu ( c R = = λ u ( c R pλ pu ( c ( p q λ ( p q u ( c M M = = R = = λ u ( c λ u ( c j u'( c / u ( c is willingness to intertemporally substitute consumption between period and state j in period intertemporal MRS (IMRS Contingent claims prices (aka Arrow-Debreu prices, aka state prices reflect IMRS (if markets functioning well In principle, allow for inferences about Risk aversion Prudence Market participants assessment of probabilities of event j occurring September 3, 00 7 Stochastic Consumption-Savings Model: Application BASICS OF ASSET PRICING Generalize the period- risk structure S: number of possible realizations of y (in richer models, risk in other primitives R j : period- price of AD security that pays off one unit in state j, zero otherwise p j : probability of state j occurring in period, with j j ifetime expected utility uc ( + Eu ( c = u( c + puc ( Period- budget constraint State-j period- budget constraint AD price for state j (compute FOCs j j j j j p λ p u ( c R = = λ u'( c S S j f j j u ( c u ( c Define R R = p = E j= j= u ( c u ( c S j= j j + = + 0 j= j j j j c R a y a Is the price of a one-period riskless bond S S j= j p = { } c + a,,,3,..., = y + a j S = 0 September 3,

15 BASICS OF ASSET PRICING One-period riskless bond Purchase price R f in period Pays off one unit ( face value in all states of the world in period (Can scale to any arbitrary face value: $00 bonds, $000 bonds, etc. Introduce in model Period- budget constraint State-j period- budget constraint S f j j + Rb+ = + 0 j= c R a y a { S} c j + a j = y j j +b + a, j,,3,..., = 0 b : bond holdings carried from period to period September 3, 00 9 Stochastic Consumption-Savings Model: Application BASICS OF ASSET PRICING One-period riskless bond Purchase price R f in period Pays off one unit ( face value in all states of the world in period (Can scale to any arbitrary face value: $00 bonds, $000 bonds, etc. Introduce in model Period- budget constraint State-j period- budget constraint FOC on b f λ u ( c R = E = E λ u '( c S j S j u ( c j p R j= u ( c j= = = S f j j + Rb+ = + 0 j= c R a y a { S} c j + a j = y j j +b + a, j,,3,..., Result: risk-free bond price can be decomposed into state prices A complete set of AD securities spans the risk space which makes b a redundant asset; consumer can synthesize b himself ow do these asset structures affect consumer s intertemporal life? September 3, = 0 Price of riskless bond reflects expected IMRS and by no-arbitrage equals sum of state prices. b : bond holdings carried from period to period 5

16 CONSUMPTION, SAVINGS, AND ASSET PRICES Consumption smoothing a primitive feature of preferences (u (.>0, u (.<0 Nature of asset markets affects ability to achieve consumption smoothing Two dimensions of consumption smoothing Intertemporal consumption smoothing: concavity of u(. implies preference for low time-series-variance of consumption R f u ( c = E u '( c Expected IMRS = price of riskfree bond f R u'( c = Eu'( c September 3, 00 3 Stochastic Consumption-Savings Model: Application CONSUMPTION, SAVINGS, AND ASSET PRICES Consumption smoothing a primitive feature of preferences (u (.>0, u (.<0 Nature of asset markets affects ability to achieve consumption smoothing Two dimensions of consumption smoothing Intratemporal consumption smoothing: concavity of u(. implies preference for low cross-state variance of consumption within any period that has risk j j j j j p λ pu ( c R = = λ u'( c A high state price R j reflects igh probability of state j igh u (. in state j i.e., low consumption in state j Or both View as intratemporal optimality condition across future state-contingent c j j j R / p u ( c MRS across states j, k = (riskadjusted relative state price =, jk, {,,3,..., S k k k } R / p u ( c September 3,

17 CONSUMPTION, SAVINGS, AND ASSET PRICES Define m j = R j /p j as discount factor for state j Intratemporal optimality condition m m j k u = u c j ( c, jk, k,,3,..., ( { S} Intertemporal optimality between period and state j in period j j u ( c m =, j,,3,... S u'( c Expected IMRS between period and period S j j Em p m j= R u ( c f = = E u' ( c { } September 3,