Endogenous Risk Aversion Marjorie Flavin UCSD and NBER October, - PowerPoint PPT Presentation

Housing, Adjustment Costs, and Endogenous Risk Aversion Marjorie Flavin UCSD and NBER October, 2009 Prepared for the Bank of Spain conference on Household Finance and the Macroeconomy 1

Most macro models use the “CRRA” utility function: 1 c u ( c ) 1 is the parameter governing the curvature of the utility function 2

The household’s problem: The household’s expected lifetime utility is given by: ~ t U E e U ( H , C ) dt (1) t t 0 H = stock of housing t C nondurable consumption (numeraire) t a C H a 1 U ( C , H ) a 3

 W H X B t t t t X (1xn) vector of amounts held of the risky assets t financial assets  (nx1) vector of ones. B amount held of riskless asset t 4

In other papers, I have imposed a borrowing, or collateral constraint that says that the household cannot borrow more than the value of the house: H B t t In this paper, however, there is no constraint on the holding of the riskless asset; B can be either positive or negative. 5

Wealth evolves according to: dW H X C dt X d H d t 0 H t t t Ft 0 Ht when the house is not sold. At the instant the house is sold, the H Household pays an adjustment cost equal to . The adjustment costs is “lumpy” in the sense that it is nonconvex in the size of the adjustment. 6

The Bellman equation is: s V ( H , W ) sup E e u H , C ds e V ( H , W ) 0 0 0 s X , C , s s 0 where is the next stopping time (that is, the next time that the house is sold. 7

Problem can be stated in terms of one state variable, instead of two, with a change of variables. The state variable becomes W y H Asset holdings and nondurable consumption are also stated as a ratio to H X x H C c H 8

a C H U ( C , H ) a a H u ( c ) a c u ( c ) a 9

The Bellman equation is: s V ( H , W ) sup E e U H , C ds e V ( H , W ) 0 0 0 s X , C , s s 0 With the change of variables, the Bellman equation can be written as: s h ( y ) sup E e u c ds e h ( y ) 0 s t x , c , s s 0 a h ( y ) H V ( H , W ) where 10

The first order conditions imply: the marginal utility of consumption is equal to the marginal value of wealth: u nondurable consumption h ' ( y ) c the vector of risky asset holdings is: h ' ( y ) 1 x portfolio allocation h ' ' ( y ) Relative risk h ' ( y ) risk aversion aversion h " ( y ) y 11

The solution to the problem consists of and a M sup ( y ) h ( y ) y satisfies the differential equation . for 12

parameter values 2 a 1 1 8 0 . 059 . 22 r . 01 f . 01 . 05 13

Plot of value function: h(y) y 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 h(y) -200 -400 h(y) Mya -600 -800 -1000 -1200 14

Figure 2: plot of h'(y) 7000 h’(y) 6000 5000 4000 3000 2000 1000 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 y 15

Plot of value function: h(y) y 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 h(y) -200 -400 h(y) Mya -600 -800 -1000 -1200 RRA 1.24 2.34 7.44 16