Download Macro II Homework 2- Some Useful Mathematical Tools 1

Macro II
Homework 2- Some Useful Mathematical Tools
1. [Consumer Theory]
In consumer theory V (p; y) and x(p; y) denote the indirect utility function
and the demand function.
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V (p; y) ´ maxx2¡(p;y) u(x) where ¡(p; y) = fx 2 R+
: px · yg
x(p; y) 2 argmaxx2¡(p;y) u(x)
If u is strictly concave and continuous, then a standard result that follows
from the Theorem of the Maximum is that V (p; y) and x(p; y) are continuous in
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(p; y) on R++
£ R++ . This relies on showing that the budget set is a continuous
correspondence. In this problem you can assume that V (p; y) and x(p; y) are
well de¯ned over their domain. Answer the following questions:
(i) [Monotonicity of the Value function]
Prove that V (p; y) is monotone increasing in y provided that the utility
function u(x) is increasing in the commodity vector x.
(ii) [Concavity of the Value function]
Prove that V (p; y) is concave in y provided that the utility function u(x)
is increasing and concave in the commodity vector x.
(iii) [Di®erentiability of the Value Function]
Prove that V (p; y) is di®erentiable in y. Be clear on exactly what assumptions you use to prove this result. Also state what the derivative is equal
to.
2. [An Application to Search Models] P
t
An agent maximizes expected utility E[ 1
t=0 ¯ u(ct )]. The agent receives a
wage o®er w this period. IfP
the agent accepts this o®er, the agent works forever
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at wage w receiving utility t=0 ¯ t u(w). If the agent rejects this o®er, the agent
receives utility u(0) = 0 this period and gets an o®er next period drawn from
a distribution F (w). The decision problem repeats each period as long as the
agent rejects the o®er.
(i) Formulate this problem as a dynamic programming problem.
(ii) Does Bellman's equation de¯ne a contraction mapping? If so, then prove
why this is so. Otherwise explain why not.
(iii) Graph the optimal value function against the value of the state variable.
Over what region of o®ers does the agent choose to accept the o®er? Explain.
(iv) Now assume that agents that reject o®ers are required to perform a
service that period that leads to some personal disutility. Speci¯cally, agents
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who reject an o®er now receive period utility equal to ¡1 which is below the
level u(0) = 0 previously associated with rejected o®ers. Compared to your
answer in part (iii), what e®ect does this have on (a) the optimal value function
and (b) the region of o®ers that are accepted? Explain CAREFULLY!!
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