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Ch 6 Practice

# Eco 204 s ajaz hussain do not distribute at the

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Unformatted text preview: ) ( )} ( ) {( ) )} ( so that: One can substitute these in the budget constraint to get the actual values for ( ) ( ( ) ) ( ) Thus, when: ( On the other hand, if then is decreasing in ) so that: One can substitute these in the budget constraint to get the actual values for ( ) ( ( ) ( ) ( ) ) ( ) Thus, when: 11 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. ( ) (6.3) In this 1 period problem, a consumer allocates her income function is: ) between consumption ( Assume ( and savings . Her utility ) . (a) Solve the consumer’s UMP. Show all calculations and clearly state any assumptions. Answer: By definition, income must be allocated to consumption and savings. The consumer’s UMP is: ( ) s.t. Being an equality constrained optimization problem, set up the Lagrangian: ( ) ( ) The first order conditions (FOCs) are: ( ) From the 1st FOC: Substitute in second FOC: ( ) ( ) 12 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 in 3rd FOC: ( ) Thus, the complete solution to this UMP is: ( ) Observe that her savings are always independent of income. This, of course, is due to her quasi-linear preferences (recall ( ) then that is always independent of income). (b) Is it possible that this consumer “goes hungry” (i.e. consume nothing)? Answer: It is possible that this consumer consumes nothing. From above, note that she always saves a constant dollar amount ( ) and consumes the remainder of her income. Thus, if her income is less than ( ) she will consume nothing: ( ) (6.4) An individual lives for two periods ( and ) in an “e-corn-omy” (i.e. the only good is corn). Her real income (measured in corn) is in and in time . is the base period so that and allow for the possibility of inflation (i.e. it may be that the nominal interest rate by , inflation by and the real interest by . Consider a consumer with the following utility function for consumption in ( ) Assume and consumption in ). Denote : . In this question you do not have to derive the inter-temporal budget constraint from 1st principles (i.e. you can state the constraint). 13 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. (a) Calculate the consumer’s optimal consumption in . Show all calculations and clearly state any assumptions. Answer: The consumer maximizes utility subject to the constraint that inter-temporal consumption equals inter-temporal income. The inter-temporal budget constraint can be expressed in PV or FV terms: FV Inter-temporal Budget Constrai...
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