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( ) ( ) PV Intertemporal Budget Constraint The solution below is in terms of FV (answers in PV terms is OK). The UMP is:
( ) ( ) ( ) s.t. ( ) s.t. ( ( ) ) Setting up the Lagrangian gives:
[( ) The first order conditions (FOCs) are:
(
( [( ) ) ) And the KuhnTucker conditions are: According KT conditions we have four cases:
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ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. CaseI:
From first and second FOC we have Solving the system of two equations gives ( ) Substituting this into the third FOC:
( ) ( )
( [ ) ]( ) Hence caseI is a solution only if
( ( ) ) And
( [
Therefore, caseI satisfies when ( ) ) ]( ) ( ) ⁄. CaseII:
From FOCs one and three we have And substituting For into FOC three to be true { ⁄( )} {⁄} must satisfy; therefore, caseII is a solution only if
( ) CaseIII:
From FOC two and three we have Solving the system of two equations and substituting for
(
For to be true {[( )( )⁄ ] FOC one gives )( )
} must satisfy; therefore, caseIII is a solution only if CaseIV:
From FOC three we can see that caseIV cannot be a solution, since
the same time. cannot be equal to zero and greater than zero at 15
ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. For the rest of solutions we assume that there exists an interior solution. That is
(
(
(b) When will this consumer borrow in ? As )( ) ) will she continue to be a borrower? Show all calculations. Answer:
In
the consumer eats: If she is a borrower in , it must be that: That is, she will borrow in
if her income tomorrow is below a threshold level. Observe that as
the right hand
side of this equation becomes smaller and may eventually reverse so that she switches from being a borrower to a
saver.
(c) Assume the consumer is a borrower in
Answer:
Assuming she is a borrower in . If inflation rises, will she borrow more or less? she will borrow: Now, either from the exact formula for real interest rates: Or the Fisher approximation: We see that as inflation goes up, falls so that goes up, implying that she will borrow more in .
16 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (d) If the nominal interest rate equals inflation, when will this consumer exhibit consumption smoothing? Show all
calculations.
Answer:
From above:
[ If then so that:
[ For consumption smoothing: [ [ (6.5) Consider a 3 period economy (
) where at the beginning of
each consumer receives (individualspecific) real incomes
(in units of corn) respectively. In the first two periods, consumers can “borrow” or
“save” corn at the prevailing real interest rate.
Suppos...
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This note was uploaded on 03/20/2014 for the course ECON 204 taught by Professor Ajazhussain during the Fall '09 term at University of Toronto.
 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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