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Ch 6 Practice

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Unformatted text preview: ( The constraint says that the versa). Now: ) ( ( ( ( ) ) )( ) ) ( ( ) ) ) ) ( ) of lifetime savings equals zero (so that borrowing is cancelled out by savings and vice ( )( ( ) ( ) ( ( ) ( ) ( ) ) [ ) ( ) The FOCs are: ( [( ) ) 38 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. Re-arrange and equate the first two FOCs: )( ( ) ( )( )( ( )( ) )( ) ( )( )( ) )( ) We can substitute this in the budget constraint: ( ( ) ) ( )( ( ) ( )( ( ) ( )( ) ( )( ) ( )( ) ( ( ) ( )( ) ( )( ) ( )( ( ) ) ) )( ) ( ( [ )( )( ( ) )( ) [ This is the exact same expression for in question (6.6). Substituting { into [ [ ) gives} Next, we know that: 39 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. ( ) ( ( [ Substitute ){ ( ) ( [ ( ) ) [ ( ) ){ } [ [ ( into ) to get: ( Substituting ) } gives {( ){ [ [( }} ) (6.8) Consider a 3 period economy ( ) with a single good (say, corn). Each consumer receives (individualspecific) real income (in units of corn) at the beginning of respectively. The price level at is (i.e. is the base period) and the price levels at and are and , respectively. In each period, the consumer can “borrow” or “save” corn at nominal interest rate . In this model, “saving” in a period means that the agent is consuming less than or equal to her income in that period (i.e. ) while “borrowing” in a period means that the agent is consuming more than her income in that period (i.e. ). The of total lifetime income is (“future period” is ( The of total lifetime income is (“present period” is ): ) ( ) ): ( ) Suppose a consumer has the following utility function over consumption at ( Assume all pecuniary variables and and : ) . (a) Solve the consumer’s inter-temporal UMP for (i) the parameters. What happens if ? , (ii) the of , and (iii) the of in terms of 40 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. Answer: (i) Using the log transformation of the Cobb-Douglas utility function, the UMP can be stated as (In terms of L): ( ) L [( ) FOCs: 1. L ( ( ) ) 2. L ( ( ) ) 3. L 4. L ( ( ) ) From 1 and 3: ( ) ( ) From 1 and 2: ( ) ( ( ) ( ) ) From 4: ( ( ) ) ( )( ) ( ) 41 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 5: ( ) ( ) From 6: ( ) Note: this can also be expressed as: (ii) To compute of and of , we can use the e...
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