SLIDE 1
1
Most macro models use the “CRRA” utility function:
2
1 c ) c ( u
1
is the parameter governing the curvature
- f the utility function
Endogenous Risk Aversion Marjorie Flavin UCSD and NBER October, - - PowerPoint PPT Presentation
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:
SLIDE 2
SLIDE 3
3
dt ) C , H ( U e E U ~
t t t
The household’s expected lifetime utility is given by:
(1)
t
H
= stock of housing
t
C
nondurable consumption (numeraire) The household’s problem:
a H C ) H , C ( U
a
1 a
SLIDE 4
4
t t t t
B X H W
t
X
(1xn) vector of amounts held of the risky assets financial assets (nx1) vector of ones.
t
B
amount held of riskless asset
SLIDE 5
5
t t
B H
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: In this paper, however, there is no constraint on the holding of the riskless asset; B can be either positive
- r negative.
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6
Wealth evolves according to: At the instant the house is sold, the 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.
Ht Ft t t t H t
d H d X dt C X H dW
when the house is not sold.
H
SLIDE 7
7
The Bellman equation is:
s s , C , X
) W , H ( V e ds C , H u e E sup ) W , H ( V
s s
where is the next stopping time (that is, the next time that the house is sold.
SLIDE 8
8
Problem can be stated in terms of one state variable, instead of two, with a change of variables. The state variable becomes Asset holdings and nondurable consumption are also stated as a ratio to H
H W y H C c H X x
SLIDE 9
9
a H C ) H , C ( U
a
a c ) c ( u ) c ( u H
a a
SLIDE 10
10
The Bellman equation is:
s s , C , X
) W , H ( V e ds C , H U e E sup ) W , H ( V
s s
t s s , c , x
) y ( h e ds c u e E sup ) y ( h
s s
With the change of variables, the Bellman equation can be written as:
) W , H ( V H ) y ( h
a
where
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11
The first order conditions imply: the marginal utility of consumption is equal to the marginal value of wealth: the vector of risky asset holdings is:
) y ( ' h c u
1
) y ( ' ' h ) y ( ' h x
y ) y ( " h ) y ( ' h Relative risk aversion nondurable consumption portfolio allocation risk aversion
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12
The solution to the problem consists of
.
satisfies the differential equation and for
) y ( h ) y ( sup M
a y
SLIDE 13
13
parameter values
05 . 01 . 01 . r 22 . 059 . 8 1 1 a 2
f
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14
- 1200
- 1000
- 800
- 600
- 400
- 200
0,1 0,2 0,3 0,4 0,5 0,6 0,7
h(y)
y h(y) Mya
Plot of value function: h(y)
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15
1000 2000 3000 4000 5000 6000 7000 0,1 0,2 0,3 0,4 0,5 0,6 0,7
y Figure 2: plot of h'(y) h’(y)
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16
- 1200
- 1000
- 800
- 600
- 400
- 200
0,1 0,2 0,3 0,4 0,5 0,6 0,7
h(y)
y h(y) Mya