STOCHASTIC CONSUMPTION-SAVINGS MODEL: CANONICAL APPLICATIONS FEBRUARY 19, 2013

Transcription

1 STOCHASTIC CONSUMPTION-SAVINGS MODEL: CANONICAL APPLICATIONS FEBRUARY 19, 2013

2 Model Structure EXPECTED UTILITY Preferences v(c 1, c 2 ) with all the usual properties Lifetime expected utility function Assume separable across time periods: v(c 1,c 2 ) = u(c 1 ) + u(c 2 ) (deterministic case) Strictly increasing in each of c 1 and c 2 Diminishing marginal utility in each of c 1 and c 2 v(c 1,c 2 ) v(c 1,c 2 ) But realized c 2 cannot be known at time decisions are made in period 1, due to period-2 income risk How to incorporate risk into utility metric? Expected lifetime utility Assume consumers maximize A decision-theoretic (not experiential) utility metric von-neumann-morgenstern (1944) foundations c 1 H M L ( ) E vc (, c) = uc ( ) + Euc ( ) = uc ( ) + qu( c ) + pu( c ) + (1 p q) u( c ) c 2 February 19,

3 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 H M L that satisfies ( c1, c2, c2, c2; a1) State-by-state period-2 budget constraint H H H H M M c2 + a2 = y2 + (1 + r1 ) a1 c2 + a2 = y2 + (1 + r1) a c L a L 2 = y L 2 + (1 + r L 1 ) a1 = 0 = 0 = 0 Euler equation H M L qu '( c2 ) H pu'( c2 ) (1 p q) u' ( c2) L u ( c2) 1 = (1 + r1 ) + (1 + r1) + (1 + r1 ) E1 (1 r1 ) u' ( c1) u'( c1) u'( c1) = + u ( c1 ) Period-1 budget constraint ( ) c + a1 = y + (1 + r ) a taking as given H L r1, r1, r1 ; y1, a0, r0 and the stochastic distribution G(.) of y 2 Could express solution in alternative ways e.g., using lifetime budget constraints February 19,

4 s APPLICATIONS Use (solution to) stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming Risk aversion Quadratic period-utility Risk-free asset returns Risky period-2 income (with arbitrary distribution) In static form (Arrow (1971) and Pratt (1964 EC)) In dynamic form (Swanson (2012 AER)) Precautionary savings αc uc () = γ c 2 2 February 19,

5 CERTAINTY EQUIVALENCE Assume quadratic utility αc αc vc ( 1, c2) = uc ( 1) + uc ( 2) = γc1 + γc2 2 2 Assume interest rate is not state contingent r H L = r = r = r risk-free interest rate Insert in definition of solution to intertemporal problem H H H M M L L L c2 + a2 = y2 + (1 + r1 ) a1 c2 + a2 = y2 + (1 + r1) a c + a 1 2 = y + (1 + r1 ) a = 0 = 0 = 0 Euler eqn often the key [ u c + r ] u'( c ) = E '( )(1 ) [( )(1 )] γ αc = E γ αc + r γ αc = q( γ αc )(1 + r) + p( γ αc )(1 + r) + (1 p q)( γ αc )(1+ r) H M L H M L γ αc1 = (1 + r1) q( γ αc2 ) + p( γ αc2 ) + (1 p q)( γ αc2) H M L ( c + (1 p q c ) = ( 1 + r) γ α qc + p ) = Ec 1 2 c + a1 = y + (1 + r ) a February 19,

6 CERTAINTY EQUIVALENCE Assume quadratic utility αc αc vc ( 1, c2) = uc ( 1) + uc ( 2) = γc1 + γc2 2 2 Assume interest rate is not state contingent r H L = r = r = r risk-free interest rate Insert in definition of solution to intertemporal problem M M L L L c + a = y + (1 + r) a c + a = y + (1 + r) a c + a2 = y + (1 + r1 ) a H H H = 0 = 0 = 0 Euler eqn often the key [ u c + r ] u'( c ) = E '( )(1 ) [ ] γ α γ α [( )(1 )] γ αc = E γ αc + r γ αc = q( γ αc )(1 + r) + p( γ αc )(1 + r) + (1 p q)( γ αc )(1+ r) H M L H M L γ αc1 = (1 + r1) q( γ αc2 ) + p( γ αc2 ) + (1 p q)( γ αc2) H M L ( c + (1 p q c ) = ( 1 + r) γ α qc + p ) c1 = (1 + r1) Ec 1 2 c = 1 r1 (1 r1) Ec 1 2 α + + c + a1 = y + (1 + r ) a February 19, γ

7 CERTAINTY EQUIVALENCE If not concerned with state-contingent solutions for c 2 solution to consumer problem is an asset position and expected j consumption profile that satisfies ( c, 1 { c2} ; a1) State-by-state period-2 budget constraints c = (1 + r ) a + y j S j j Euler equation c = γ r (1 r) Ec α Period-1 budget constraint c + a1 = y + (1 + r ) a ( ) taking as given r1 ; y1, a0, r0 and the stochastic distribution G(.) of y 2 Optimal period-1 consumption c γ r (1 + r) 1+ r 2 = ( y1 (1 + r0) a0) + 2 E1y 2 α 1 + (1 + r1) 1 + (1 + r1) (1 + r1 ) A B C February 19,

8 CERTAINTY EQUIVALENCE Optimal period-1 (current) consumption ( (1 ) ) c = A + B y + + r a + C E y Depends only on the mean of risky future income, E 1 y 2 Independent of second- and higher-moments of risky future income Distribution function G(.) of period-2 income y 2 = H y2 probability q y probability p 2 y L 2 probability 1-p-q Ey = y H L ( ) ( ) 2 2 Var y = q y y + (1 p q) y y Certainty Equivalence Mean-preserving spreads of G(.) do not affect optimal choice of c 1 E.g., (p = 1, q = 0) Period-2 income has no risk But c 1 is identical s 1 (period-1 savings) is identical February 19,

9 CERTAINTY EQUIVALENCE 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 2 > 0) Risk averse utility with respect to consumption need to define formally February 19,

10 s APPLICATIONS Use (solution to) stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming Risk aversion Quadratic period-utility Risk-free asset returns Risky period-2 income (with arbitrary distribution) In static form (Arrow (1971) and Pratt (1964 EC)) In dynamic form (Swanson (2012 AER)) Precautionary savings uc () = γ c αc 2 2 February 19,

11 Macro/Finance Fundamentals RISK AVERSION Illustrate with simple static example Utility function u(c), with u (.) > 0 and u (.) < 0 Two possible consumption outcomes c H with probability η c L with probability 1-η Expected consumption is c = ηc H + (1-η)c L February 19,

12 Macro/Finance Fundamentals RISK AVERSION How to measure? u(c H ) u(c avg ) u(c H ) u(c avg ) 0.5*( u(c L ) + u(c H ) ) u(c L ) u(c L ) c L c avg = 0.5*(c L +c H ) c H c L c avg = 0.5*(c L +c H ) c H February 19,

13 Macro/Finance Fundamentals RISK AVERSION Illustrate with simple static example Utility function u(c), with u (.) > 0 and u (.) < 0 Two possible consumption outcomes c H with probability η c L with probability 1-η Expected consumption is c = ηc H + (1-η)c L Definition: an individual is risk averse (with respect to consumption risk) if H L uc ( ) > ηuc ( ) + (1 η) uc ( ) JENSEN S INEQUALITY Risk aversion A preference for certain (deterministic) outcomes to risky (stochastic) outcomes Embodied in strictly concave utility How to measure risk aversion? Need to capture something about concavity of utility February 19,

14 Macro/Finance Fundamentals RISK AVERSION How to measure? u(c H ) u(c avg ) u(c H ) u(c avg ) 0.5*( u(c L ) + u(c H ) ) u(c L ) u(c L ) c L c avg = 0.5*(c L +c H ) c H c L c avg = 0.5*(c L +c H ) c H A candidate measure: -u (c) But not invariant to positive linear transformations of u(.) even though implied choices are invariant to any monotonically increasing transformation of u(.) February 19,

15 Macro/Finance Fundamentals RISK AVERSION Arrow-Pratt coefficient of absolute risk aversion (ARA) 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) u () c Controls for linear ARA() c u () c transformations of u(.) cu () c Adjusts for level of RRA() c ( = c ARA() c ) u () c consumption/wealth Controlling for income/consumption, richer people cannot afford to take a chance anymore than anyone else February 19,

16 Macro/Finance Fundamentals RISK AVERSION CRRA 1 σ 1 σ c1 1 c2 1 vc ( 1, c2) = + 1 σ 1 σ σ > 0 uc ( uc ( ) 2 1 ) Continuing to assume utility is additively-separable over time Attitude of consumers toward smoothing consumption between time periods IES = 1/σ Attitude of consumers toward risky outcomes within a given time period RRA(c) = cu''(c) u'(c) =...?... ARA(c) = u''(c) u'(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 February 19,

17 s APPLICATIONS Use (solution to) stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming Risk aversion Quadratic period-utility Risk-free asset returns Risky period-2 income (with arbitrary distribution) In static form (Arrow (1971) and Pratt (1964 EC)) In dynamic form (Swanson (2012 AER)) Precautionary savings uc () = γ c αc 2 2 February 19,

18 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility Other necessary assumptions Non-state-contingent asset returns Future income the only source of risk February 19,

19 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility 2 2 αc1 αc2 Risk aversion (within period) with vc ( 1, c2) = uc ( 1) + uc ( 2) = γc1 + γc2? 2 2 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 )(1 + r) ] u'( c ) '( )(1 ) γ αc = q( γ αc )(1 + r) + p( γ αc )(1 + r) + (1 p q)( γ αc )(1+ r) H M L H M L γ αc1 = (1 + r1) q( γ αc2 ) + p( γ αc2 ) + (1 p q)( γ αc2) H M L ( c + (1 p q c ) = ( 1 + r) γ α qc + p ) [ ] γ α γ α c1 = (1 + r1) Ec 1 2 c = 1 r1 (1 r1) Ec 1 2 α + + February 19, γ

20 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility 2 2 αc1 αc2 Risk aversion (within period) with vc ( 1, c2) = uc ( 1) + uc ( 2) = γc1 + γc2? 2 2 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 à u (.) increasing in c à u (.) decreasing less quickly as c Not satisfied by quadratic utility February 19,

21 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 H L Insert in definition of solution to intertemporal problem H H H M M L L L c2 + a2 = y2 + (1 + r1 ) a1 c2 + a2 = y2 + (1 + r1) a c + a 1 2 = y + (1 + r1 ) a = 0 = 0 = 0 Euler eqn often the key [ u c + r ] u'( c ) = E '( )(1 ) [ '( ] u' ( c ) = (1+ r ) E u c ) c + a1 = y + (1 + r ) a H M L u'( c1) = (1 + r1) qu'( c2 ) + pu'( c2 ) + (1 p q) u'( c2) = E 1 c 2, so none of the subsequent steps with quadratic u(.) follow u (c) > 0 à current consumption depends on distribution G(.) of future risk i.e., on first- and (in principle) all higher-order moments of G(.) February 19,

22 PRECAUTIONARY SAVINGS u (c) > 0 à current consumption depends on distribution G(.) of future risk Optimal c 1 is smaller than certainty-equivalent c 1 Proof: Implication: optimal s 1 is larger than certainty-equivalent s 1 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 (1990 Econometrica), Sandmo (1970 Review of Economic Studies) How to measure precautionary savings motive? Need to capture something about convexity of marginal utility Kimball (1990) provides clever insight February 19,

23 PRECAUTIONARY SAVINGS How to measure? u'(c L ) u'(c L ) 0.5*( u (c L ) + u (c H ) ) u'(c avg ) u'(c H ) u'(c avg ) u'(c H ) c L c avg = 0.5(c L + c H ) c H c L c avg = 0.5(c L + c H ) c H A candidate measure: u (c) Analogy with measures of risk aversion February 19,

24 PRECAUTIONARY SAVINGS How to measure? u'(c L ) u'(c L ) 0.5*( u (c L ) + u (c H ) ) u'(c avg ) u'(c H ) u'(c avg ) u'(c H ) Kimball (1990): Define v(c) = -u (c). Then can apply standard theory of risk aversion to v(c)! -u'(c H ) -u'(c avg ) -u'(c L ) -u'(c H ) -u'(c avg ) -0.5*( u (c L ) + u (c H ) ) -u'(c L ) February 19,

25 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 February 19,

26 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 1 σ c 1 uc () = 1 σ Displays constant relative prudence Displays constant relative risk aversion u'''( c) σ + 1 = u''( c) c cu '''( c) u''( c) = σ + 1 Absolute prudence Relative prudence February 19,