Each consumer receives real income in units of corn

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Unformatted text preview: e. Instead, we exploit the fact that at the optimum: Solve this equation and the inter-temporal budget constraint simultaneously. From above: Substitute in: ( ) ( ) [( ) [( ) [( ) Substitute in: [( ) [( ) [( ) Thus: 46 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 these we can get the MU of “income” (the “ ” in Lagrangian UMPs) by using direct calculation: the optimal utility is: ( ) ( [( ) ( [( ) [( [( [( The of ) ) ) : [( To compute ) ) of , express ) in terms of [( [( ) ( ) and ) ) [( (b) What happens to the optimal ( ) due to, all else equal, a small increase in and ? Answer: [( ) [( ( ) ) [( ) [( ) [ ( ( ) ( ) ) 47 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. To answer this, we need to expand more: [ ( ) ( ) )[ ( ) ) ( if and [( [( ) ) ( ) [ [( ) ) [( and [ ( [( Now if [ ( ( [ ) ) ) , then , then [( ) [( ) ( ) ) [ ( more: [( ( ) [( [ ) ) [( To answer this, we need to expand [( ) ( [ ( [( )[ ( ) ) ) ) [( [( ) ) 48 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. ( [( ) ) [ [( and Now if if and ) ) , then , then (c) Derive the conditions under which an agent will save in condition, will the agent continue to save at as ? Answer: To be a saver at ( (and therefore “borrow” at ). According to this , this consumer would have to consume less than her income: [( ) ) [( [( ) There are many ways to express the borrowing condition. Here is one way, useful for graphing saving/borrowing diagrams. The agent saves in when: [( ) [( ) ( ) [( ) [( ) ( ) 49 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 right hand side of this equation is the slope of the “L shaped indifference curves corners’ line” while the left hand side is the slope of the ray from the origin to the endowment point. That is, the agent will save in (the optimal bundle will be to the left of the endowment point) when the corners line is steeper than the endowment point ray: Agent saving in t = 0, “borrowing” in t = 1 C1 FVY C1 S1 < 0 Y0 > Y1 E Y1 α/β C0 Y0 PVY C0 S0 > 0 Observe how the condition for saving is independent of the real interest rate which means that once a saver at the agent will always be a saver regardless of the real interest rate (albeit the level of savings will change). (d) Derive the conditions under which an agent will...
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