CONSUMPTION-SAVINGS MODEL JANUARY 19, 2018

Transcription

1 CONSUMPTION-SAVINGS MODEL JANUARY 19, 018

2 Stochastic Consumption-Savings Model 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- income (with arbitrary distribution) Precautionary savings Appendix: Asset pricing c uc () c January 19, 018

3 CERTAINTY EQUIVALENCE Assume quadratic utility c c v( c1, c) u( c1 ) u( c) c1 c Assume interest rate is not state contingent 1 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 c a y (1 r1 ) a1 c a y (1 r1 ) a c a 1 y (1 r1 ) a = 0 = 0 = 0 Euler eqn the key u'( c ) E u '( c )(1 r ) c a1 y (1 r ) a January 19, 018 3

4 CERTAINTY EQUIVALENCE Assume quadratic utility c c v( c1, c) u( c1 ) u( c) c1 c Assume interest rate is not state contingent 1 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 c a y (1 r1 ) a1 c a y (1 r1 ) a c a 1 y (1 r1 ) a = 0 = 0 = 0 Euler eqn the key u'( c ) E u '( c )(1 r ) c a1 y (1 r ) a January 19, c E ( c )(1 r) c q( c )(1 r ) p( c )(1 r ) (1 p q)( c )(1 r ) H M L H M L c1 (1 r1 ) q( c ) p( c ) (1 p q)( c ) ( 1 r ) qc pc (1 p q) c H M L 1 Ec 1

5 CERTAINTY EQUIVALENCE Assume quadratic utility c c v( c1, c) u( c1 ) u( c) c1 c Assume interest rate is not state contingent 1 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 c a y (1 r1 ) a1 c a y (1 r ) a c a y (1 r1 ) a = 0 = 0 = 0 Euler eqn the key 1 1 u'( c ) E u '( c )(1 r ) c a1 y (1 r ) a January 19, c E ( c )(1 r) c q( c )(1 r ) p( c )(1 r ) (1 p q)( c )(1 r ) H M L H M L c1 (1 r1 ) q( c ) p( c ) (1 p q)( c ) ( 1 r ) qc pc (1 p q) c H M L 1 c1 (1 r1 ) E1c c 1 r1 (1 r1 ) E1c

6 CERTAINTY EQUIVALENCE If not concerned with state-contingent solutions for c solution to consumer problem is an asset position and expected consumption profile c, E c ; a that satisfies Period- budget constraint in expectation Ec (1 r ) a E y Euler equation c r (1 r ) E c Period-1 budget constraint c a1 y (1 r ) a taking as given r1 ; y1, a0, r0 and the stochastic distribution G(.) of y Optimal period-1 consumption c r (1 r ) 1r y1 (1 r0 ) a0 E1 y 1 (1 r1) 1 (1 r1) 1 (1 r1 ) A B C January 19, 018 6

7 CERTAINTY EQUIVALENCE Optimal period-1 (current) consumption c A B y (1 r ) a C E y Depends only on the mean of risky future income, E 1 y Independent of second- and higher-moments of risky future income Distribution function G(.) of period- income y H y probability q y probability p L y probability 1-p-q E y y 1 H L Var y q y y (1 p q) y y January 19, 018 7

8 CERTAINTY EQUIVALENCE Optimal period-1 (current) consumption c A B y (1 r ) a C E y Depends only on the mean of risky future income, E 1 y Independent of second- and higher-moments of risky future income Distribution function G(.) of period- income y H y probability q y probability p y L probability 1-p-q E y y 1 H L 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- income has no risk But c 1 is identical s 1 (period-1 savings) is identical January 19, 018 8

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 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 January 19, 018 9

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- income (with arbitrary distribution) Precautionary savings Appendix: Asset pricing u() c c c January 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 January 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 January 19, 018 1

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 u( c ) u c (1 ) u c JENSEN S INEQUALITY January 19,

14 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 Risk aversion A preference for certain (deterministic) outcomes to risky (stochastic) outcomes Embodied in strictly concave utility How to measure risk aversion? H L u( c ) u c (1 ) u c Need to capture something about concavity of utility JENSEN S INEQUALITY January 19,

15 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(.) January 19,

16 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 u() c Controls for linear ARA() c u () c Decreasing ARA: ARA (c) < 0 Most empirically-relevant case Intuition Richer people can afford to take a chance transformations of u(.) January 19,

17 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 Intuition 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 RRA() c cara() c Adjusts for level of u () c consumption/wealth Intuition Controlling for income/consumption, richer people cannot afford to take a chance anymore than anyone else January 19,

18 Macro/Finance Fundamentals RISK AVERSION CRRA vc (, c ) c1 1 c uc ( uc ( ) 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 cu ''( c) u''( c) RRA( c)... ARA( c)... u'( c) u'( c) January 19,

19 Macro/Finance Fundamentals RISK AVERSION CRRA vc (, c ) c1 1 c uc ( uc ( ) 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 CRRA utility: σ governs both intertemporal attitudes and intratemporal (relative) risk attitudes! cu ''( c) RRA( c)... u'( c) Inverses of each other! u''( c) ARA( c)... u'( c) Must/should IES and RRA be so directly related in reality? Not at all Epstein-Zin (EZ) utility function disentangles the two concepts January 19,

20 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- income (with arbitrary distribution) Precautionary savings Appendix: Asset pricing u() c c c January 19, 018 0

21 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 January 19, 018 1

22 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility c1 c Risk aversion (within period) with v( c1, c) u( c1 ) u( c) c1 c? Obviously = 0! (whether RRA or ARA) So why certainty equivalence? i.e., why does future income risk not matter for current choices? January 19, 018

23 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility c1 c Risk aversion (within period) with v( c1, c) u( c1 ) u( c) c1 c? Euler eqn the key Obviously = 0! (whether RRA or ARA) So why certainty equivalence? i.e., why does future income risk not matter for current choices? E u c r c E ( c )(1 r) u'( c ) '( )(1 ) c q( c )(1 r ) p( c )(1 r ) (1 p q)( c )(1 r ) H M L H M L c1 (1 r1 ) q( c ) p( c ) (1 p q)( c ) ( 1 r ) qc pc (1 p q) c H M L 1 Because of linear marginal utility! c1 (1 r1 ) E1c c 1 r1 (1 r1 ) E1c January 19, 018 3

24 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility c1 c Risk aversion (within period) with v( c1, c) u( c1 ) u( c) c1 c? Obviously = 0! (whether RRA or ARA) So why certainty equivalence? Marginal utility function of order one (or lower) implies risky future income doesn t matter for current consumption Contrapositve Risky future income matters for current consumption implies marginal utility function must be strictly convex January 19, 018 4

25 PRECAUTIONARY SAVINGS Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility c1 c Risk aversion (within period) with v( c1, c) u( c1 ) u( c) c1 c? Obviously = 0! (whether RRA or ARA) So why certainty equivalence? Marginal utility function of order one (or lower) implies risky future income doesn t matter for current consumption Contrapositve Risky 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 January 19, 018 5

26 PRECAUTIONARY SAVINGS Assume utility with u (c) > 0 v( c, c ) u( c ) u( c ) 1 1 Assume interest rate is not state contingent r r r r risk-free interest rate H L January 19, 018 6

27 PRECAUTIONARY SAVINGS Assume utility with u (c) > 0 v( c, c ) u( c ) u( c ) 1 1 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 c a y (1 r1 ) a1 c a y (1 r1 ) a c a 1 y (1 r1 ) a = 0 = 0 = 0 Euler eqn the key u'( c ) E u '( c )(1 r ) u '( c ) (1 r ) E u '( c ) c a1 y (1 r ) a H M L u '( c1 ) (1 r1 ) qu '( c ) pu '( c ) (1 p q) u'( c ) = E 1 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(.) January 19, 018 7

28 PRECAUTIONARY SAVINGS u (c) > 0 current consumption depends on distribution G(.) of future risk Optimal c 1 is smaller than certainty-equivalent c 1 Proof: January 19, 018 8

29 PRECAUTIONARY SAVINGS u (c) > 0 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 January 19, 018 9

30 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 ) -u'(c H ) -u'(c avg ) c L c avg c H = 0.5(c L + c H ) A candidate measure: u (c) -u'(c H ) -u'(c avg ) -0.5*( u (c L ) + u (c H ) ) -u'(c Analogy L ) with measures of risk aversion -u'(c L ) c L c avg = 0.5(c L + c H ) c H January 19,

31 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 ) January 19,

32 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 January 19, 018 3

33 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 uc () 1 c 1 1 Displays constant relative prudence Displays constant relative risk aversion u'''( c) 1 u''( c) c cu '''( c) u''( c) 1 Absolute prudence Relative prudence January 19,

34 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- income (with arbitrary distribution) Precautionary savings Appendix: Asset pricing u() c c c January 19,

35 APPENDIX: ASSET PRICING JANUARY 19, 018

36 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 H y probability q y probability p y L probability 1-p-q But now three distinct assets available for purchase in period 1 a Asset : purchase price in period 1, pays off one unit in period if, zero else a Asset : purchase price in period 1, pays off one unit in period if, zero else a H M L R R R H L Asset : purchase price in period 1, pays off one unit in period if, zero else y H y y L Arrow-Debreu securities, aka contingent claims NOT EQUIVALENT to state-contingent asset returns on a single asset January 19,

37 STATE-CONTINGENT CHOICES Consumer problem max u( c ) qu( c ) pu( c ) (1 p q) u( c ) y a c R a Ra R H M L H H M L L H H H H M M M L L L L q y a1 c p y a1 c (1 p q) y a1 c a FOCs January 19,

38 BASICS OF ASSET PRICING Consumer problem max u( c ) qu( c ) pu( c ) (1 p q) u( c ) y a c R a Ra R H M L H H M L L H H H H M M M L L L L q y a1 c p y a1 c (1 p q) y a1 c a FOCs Asset prices H H H q qu( c ) R u( c ) 1 1 R p pu( c ) u( c ) M M L L R 1 1 L (1 p q) (1 p q) u( c ) u( c ) 1 1 j u '( c) / u( c1) is willingness to intertemporally substitute consumption between period 1 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 But which asset prices to empirically identify as which state prices?... Market participants assessment of probabilities of event j occurring January 19,

39 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-1 price of AD security that pays off one unit in state j, zero otherwise p j : probability of state j occurring in period, with Lifetime expected utility Period-1 budget constraint State-j period- budget constraint S j j 1) E1u ( c) u( c1 p u c S j1 j j c1 R a1 y1 a0 j1 u( c ) ( ) S j1 p j 1 c j a j j j, 1,,3,..., y a1 j S = 0 January 19,

40 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-1 price of AD security that pays off one unit in state j, zero otherwise p j : probability of state j occurring in period, with Lifetime expected utility Period-1 budget constraint State-j period- budget constraint AD price for state j (compute FOCs) j j j j j p p u( c) R u'( c ) S j j 1) E1u ( c) u( c1 p u c S j1 j j c1 R a1 y1 a0 j1 u( c ) ( ) 1 1 S j1 p j 1 c j a j j j, 1,,3,..., y a1 j S = 0 Define S S j f j j u( c) u( c) R R p E1 j1 j1 u ( c1) u( c1) Is the price of a one-period riskless bond January 19,

41 BASICS OF ASSET PRICING One-period riskless bond Purchase price R f in period 1 Pays off one unit ( face value ) in all states of the world in period (Can scale to any arbitrary face value: $100 bonds, $1000 bonds, etc.) January 19,

42 BASICS OF ASSET PRICING One-period riskless bond Purchase price R f in period 1 Pays off one unit ( face value ) in all states of the world in period (Can scale to any arbitrary face value: $100 bonds, $1000 bonds, etc.) Introduce in model Period-1 budget constraint State-j period- budget constraint FOC on b 1 S f j j 1 Rb j1 c R a y a c j b j a j j j y b1 a1, j 1,,3,..., = 0 = 0 b 1 : bond holdings carried from period 1 to S period January 19, 018 4

43 BASICS OF ASSET PRICING One-period riskless bond Purchase price R f in period 1 Pays off one unit ( face value ) in all states of the world in period (Can scale to any arbitrary face value: $100 bonds, $1000 bonds, etc.) Introduce in model Period-1 budget constraint State-j period- budget constraint FOC on b 1 R f u( c) E1 E1 1 u '( c 1) S j S j u( c ) j p R u ( c ) j1 1 j1 S f j j 1 Rb j1 c R a y a c j b j a j j j y b1 a1, j 1,,3,..., S = 0 = 0 Price of riskless bond reflects expected IMRS and by no-arbitrage equals sum of state prices. Result: risk-free bond price can be decomposed into state prices A complete set of AD securities spans the risk space which makes b 1 a redundant asset; consumer can synthesize b 1 himself Any asset can be decomposed into state prices (Cochrane, Chapter 3.1) January 19, b 1 : bond holdings carried from period 1 to period

44 BASICS OF ASSET PRICING One-period riskless bond Purchase price R f in period 1 Pays off one unit ( face value ) in all states of the world in period (Can scale to any arbitrary face value: $100 bonds, $1000 bonds, etc.) Introduce in model Period-1 budget constraint State-j period- budget constraint FOC on b 1 R f u( c) E1 E1 1 u '( c 1) S j S j u( c ) j p R u ( c ) j1 1 j1 S f j j 1 Rb j1 c R a y a c j b j a j j j y b1 a1, j 1,,3,..., S = 0 = 0 Price of riskless bond reflects expected IMRS and by no-arbitrage equals sum of state prices. Result: risk-free bond price can be decomposed into state prices A complete set of AD securities spans the risk space which makes b 1 a redundant asset; consumer can synthesize b 1 himself b 1 : bond holdings carried from period 1 to period How do these asset structures affect consumer s intertemporal life? January 19,

45 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 E 1 u( c u '( c 1 ) ) Expected IMRS = price of riskfree bond f R u '( c ) Eu '( c ) 1 1 January 19,

46 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 p u( c) R u'( c ) 1 1 January 19,

47 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 A high state price R j reflects High probability of state j High u (.) in state j i.e., low consumption in state j Or both View as intratemporal optimality condition across future state-contingent c R R j j j j j p p u( c) R u'( c ) / p u( c ) MRS across states j, k = (riskadjusted) relative state price, j, k 1,,3,..., S / ( ) j j j k k k p u c 1 1 January 19,

48 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 ), j, k k 1,,3,..., ( ) S Intertemporal optimality between period 1 and state j in period Expected IMRS between period 1 and period Terminology: Stochastic discount factor (SDF) j j u( c ) m, j 1,,3,... S u'( c ) Em 1 1 S j1 p j m u( c f R E1 u '( c1 ) j January 19, )