Lecture 1: Introduction

Transcription

1 Lecture 1: Introduction Fatih Guvenen University of Minnesota November 1, 2013 Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

2 What Kind of Paper to Write? Empirical analysis to: I I provide motivation for the paper test the implications of your model Your paper can answer a question, or explain: I I an existing empirical fact a new fact that you document in your paper Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

3 Model Specification: Partial Equilibrium Choices, choices: V (a, w) =max c,k 0 u(c,`)+ E(V (a 0, w 0 ) w) c + a 0 =(1 + r)a + w(1 `) w 0 f ( w) What functional form to choose for u(c,`)? How about if we also want to model home production? or household preferences? How to specify f ( w)? There is an entire literature on the choice of f (). How about if we have other shocks (health, rate-of-return, etc.)? Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

4 Model Specification: General Equilibrium max 1X t=0 t " C 1 t 1 + (1 N t) 1 1 # s.t. C t + K t+1 (1 ) K t apple F t (K t, N t ) ( t ) What changes between the two formulations? Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

5 Model Specification: General Equilibrium max 1X t=0 t apple (C t (1 N t ) 1 ) 1 1 s.t. C t + K t+1 (1 ) K t apple F t (K t, N t ) ( t ) What changes between the two formulations? Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

6 Model Specification: General Equilibrium max 1X t=0 t apple (C t (1 N t ) 1 ) 1 1 s.t. C t + (K t+1 (1 ) K t ) apple F t (K t, N t ) ( t ) What changes between the two formulations? Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

7 Model Specification: General ( ) X max E t u (c t,`t) t s.t. c t + x zt + x ht + x kt apple F (k t, z t, s t ) (1) z t apple M (n zt, h t, x zt ) (2) k t+1 apple (1 k) k t + x kt (3) h t+1 apple (1 h) h t + G (n ht, h t, x ht ) (4) `t + n ht + n zt apple 1 (5) h 0 and k 0 given Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

8 Risk Aversion What is the risk aversion when preferences are of the form: U(C) = C1 1 Risk aversion is not the curvature of some utility function. It is the answer to a specific question. Depending on what question we ask, the risk aversion we measure will be different. Sometimes it will have a simple relationship to the curvature, and sometimes it will not. Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

9 What is Risk Aversion? Start with a static gamble as studied by Pratt(1964, ECMA). Because the problem is static, there is no saving, so Pratt assumed the outcome of the gamble would be consumed immediately: I bet pays off c + i dollars in state i, realized w.p. p i. If the bet is declined, consumption is c minus the risk premium,. So: u(c ) = nx p i u(c + i ). i=1 Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

10 What is Risk Aversion? When the risk is small, use the Arrow-Pratt approximation. Basically, take the first-order Taylor approximation of the LHS, and the second-order approximation to the RHS (why?) to get: nx u(c) u 0 (c) = p i u(c)+ i u 0 (c)+ 1 2 i u 00 (c) 2 i=1 = u(c) nx i=1 p i {z } =1 + u 0 (c) nx i=1 u 0 (c) = 1 2 u00 (c) var( i ) ) p i i {z } = u00 (c) nx i=1 p i {z } =var( i ) 2 i = u 00 (c) u 0 (c) {z } Absolute risk aversion 1 2 var( i). {z } Amount of risk (6) Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

11 What is Risk Aversion? If the gamble is in fixed monetary units, we are talking about absolute risk aversion. If it is indexed to the average level of the bet, then we are talking about relative risk aversion: u(c(1 r )) = nx p i u(c(1 + i )). i=1 The coefficient of relative risk aversion: RRA(c) = c u00 (c) u 0 (c) (7) Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

12 What is Risk Aversion? In a dynamic model, risk aversion can be as simple as what we have seen so far or it can be as complex as you can imagine. Why? Because in a dynamic context it does not usually make sense to assume that you have to consume the outcome of the bet immediately. For example, a worker who loses his job will usually have the option to borrow to smooth consumption. Or somebody who has a windfall gain from an inheritance, does not have to spend all of it in the current period. And so on. So, in general, risk aversion will depend on the market structure and the type of gamble that is offered, so it can mean different things. Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

13 What is Risk Aversion? In a dynamic model, individuals can typically use financial markets to smooth consumption relative to income, so we should think about wealth/income bets: V (!(1 r )) = nx p i V (!(1 + i )). i=1 r =! V 00 (!) V 0 (!) {z } Absolute risk aversion 1 2 var( i). {z } Amount of risk Result: If (i) preferences are separable over time, and (ii) the market structure is such that (i.e., markets are complete) the envelope condition is V 0 (!) =u then:! V 00 (!) V 0 (!) = c u00 (c) u 0 (c), where we used Euler s = c. Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16 (8)

14 Risk Aversion One can show that this is true even when the individual derives utility from leisure. So, for example, for preferences given by the specifications: or U(c,`)= c1 1 ` (9) c `1 1 U(c,`)=. (10) 1 relative risk aversion is even though preferences also include leisure. Homework: Prove this claim. Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

15 Risk Aversion This explanation also makes it clear that this result is more special and limited than it looks. Because we know that in many models the marginal utility of consumption is not equated across dates and states, most notably when markets are incomplete which is most of the models this book intends to cover! In such cases, immediately consuming the outcome of the bet cannot be any greater than finding the state with the highest marginal utility and consuming in that state. So wealth will have (weakly) higher marginal utility than current consumption yielding an inequality: w V 00 (w) V 0 (w) c u00 (c) u 0 (c) =. (11) Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16

16 Risk Aversion A second case of interest is when preferences are time-non-separable, e.g., Epstein-Zin preferences or habit formation. In this case, even if markets are complete, risk aversion may differ (sometimes substantially) from the curvature of the utility function. With incomplete markets it is not clear what w should be. Wealth gambles are not too meaningful if most of your cash on hand comes from labor income. If it is literally financial wealth, risk aversion may be zero or negative as measured by (11), since w could be zero or negative. If we think that it should include wealth labor income, so it is cash-on-hand, then how do we discount future earnings? In general, the formula above is not very useful in incomplete markets models as a measure because of these difficulties.) Fatih Guvenen (2013) Lecture 1: Introduction November 1, / 16