Show all calculations answer from above if then so

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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 inter-temporal UMP for . Show all calculations. Hints: ● For we must have ⏟ and . Solve the Answer Take the hint and transform the utility function: 17 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Inter-temporal Consumption in ECO 204 (this version 2012-2013) 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 Inter-temporal Consumption in ECO 204 (this version 2012-2013) 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 Inter-temporal Consumption in ECO 204 (this version 2012-2013) 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: ⏟ 20 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Inter-temporal Consumption in ECO 204 (this version 2012-2013) 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. 21 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Inter-temporal Consumption in ECO 204 (this version 2012-2013) 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.

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