Unformatted text preview: e a consumer has the following utility function over consumption at Assume and that the nominal interest rate and inflation rate are and : . (a) Assume a uniform rate of inflation and nominal interest rate between periods
consumer’s intertemporal UMP for
. Show all calculations.
Hints: ● For
we must have
⏟ and . Solve the Answer
Take the hint and transform the utility function:
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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. ( ) Notice, as we have repeatedly discussed in lectures and HWs, that we do not transform the “utility” but rather the utility
function. Carrying on: Use the hint The agent solves the UMP: Now use the hint that or ⏟ we must have : as such, the UMP becomes Now, the “budget constraint” is: With a uniform real interest rate this becomes: (
For simplicity let ) ( ) ( ) ( ) ( ) ( ( ) ) so that the budget constraint becomes:
( ) The UMP becomes: ( ) ( ) Setup the Lagrangian function:
[ ( ) ( ) The FOCs and KT conditions are: 18
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. (
( ( [ The KT condition ) ) ( ) gives rise to two possibilities:
Possibility #1 Now, to “sign” we need to know
constraint we see that: which will happen when for which we need either ( )
( Let’s solve for ) we need to express (
) in terms of or )
( . Now, if then from the budget ⏟
) . From: ( ) ( ) ( ) And from:
( ( Equating the ) ) ’s we have:
( ) ( ) ( ) Substitute in budget constraint: (
( )
) ( )( ) 19
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. ( )[ ⏟( ) ( ) ( ) ( ) From this we have:
(
Now, we are ready to see when Thus Thus, if ) ) From: whenever: then: ( ) Possibility #2 To “sign” ( we need to know ( ) ( ) ( ) which will happen when and in turn for this we need to know . From: ⏟
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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. From:
( ) ( ⏟( ( ) ⏟( ) ) ) ( ) Now, from the “budget constraint” we have: (
( ) ( )
) ( ) ( ) ( ) (
Now ) whenever:
( ) This is the exact opposite condition for possibility #2.
Thus, if then: ( ) ( )
( (b) Under what conditions will the consumer save corn at
increases? Show all calculations. ) ? What will happen to savings at if the real interest Answer
The consumer will save corn at when: We need to consider the two possibilities.
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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 Huss...
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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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