A solve the consumers inter temporal ump for i the

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Unformatted text preview: nvelope theorem: [( L of ( ) : L (iii) To compute of | | , we can rewrite the lagrangian in terms of [ L of ) ( ) ( ) ( )] : L Therefore, we need to solve for | | ( ) . Using 3: 42 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. Therefore of Therefore of ( ) (b) What happens to the optimal ( ) ⁄( ) due to, all else equal, a small increase in a small increase in and ? ) Answer: We can use the envelope theorem to answer this question: ( [ L L | ) ( | ( L ) ( ) ( ) ( ) ( ) ( | ( ) ) | ) ) ( | ⁄( L ( ) | (c) Derive the conditions under which an agent will save in increase in ? . What happens to due to, all else equal, a small Answer: The agent will save in [ ( ) [ [ ( ) ( [ ) ( ) 43 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. [ [ [ ( [ ) ( ) Therefore, [ ( ) Now, [ ( ( ( ( ) ( )( ) ) ) ) ( ( ) ( ( )( ) ) ) Since (d) Derive the conditions under which an agent will save in increase in . What happens to due to, all else equal, a small Answer: The agent will save in if ( [ ( [( ( [ ) ) ( ) ( ) ) ) [( ) 44 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. [ [( [ ) [( ) Therefore, [( ) Now, ( ) ( ( ( ( ) )( ) ( ( ) ) ( )( ) [ ) ) ( ) Therefore, ( ) ( ) (6.9) Consider a 2 period economy ( ) with a single good (say, corn). Each consumer receives 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 level at is . 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: ( The ) of total lifetime income is: Suppose a consumer has the following utility function over consumption at ( ) ( and : ) 45 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. Assume all pecuniary variables and . (a) Solve the consumer’s inter-temporal UMP for the of , and the of in terms of the parameters. Answer: The inter-temporal UMP is to choose consumption in each period subject to the constraint that income: ( ) ( ( ) ) ( ( consumption equals ) ) This UMP cannot be solved by calculus because the utility function is not differentiable everywher...
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