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Ch 6 Practice

# E what happens to a solve the consumers inter temporal

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Unformatted text preview: ons under which an agent will save in condition, will the agent continue to save at as ? (e) What happens to : . (a) Solve the consumer’s inter-temporal UMP for (b) What happens to the optimal ) and ). According to this (and therefore borrow at ). According to this due to, all else equal, a small increase in ? Graph vs. and vs. . (6.10) In this question, you will practice the intertemporal consumption/savings model. Consider a 3 period economy with a single good (say, corn). Each consumer receives real income (in corn) at the beginning of respectively. The price level at is (i.e. is the base period) and the price level at and is respectively. In each period, the consumer can borrow or lend corn at nominal interest rate . Assume that the real interest rate (i.e. the nominal interest rate is always greater than or equal to inflation). Suppose the consumer has the utility function: ( ) Solve for the optimal consumption in and interpret your result for the case . Hint: to write down the 3 period inter-temporal budget constraint, observe that the 2 period inter-temporal budget constraint says: FV of consumption = FV of income. 5 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. Solutions (6.1) Consider a 2 period economy ( units of corn) at the beginning of ). ) with a single good (say, corn). Each consumer receives real income respectively. Denote the real interest rate by (assume a common at Suppose a consumer has the following utility function defined over the consumption set ( Assume all pecuniary variables ) ( {( (in }: ) ) . (a) Derive an expression for the of (Future Value of Total Lifetime Income), and the Value of Total Lifetime Income). State all assumptions and show all calculations. Answer: The intertemporal UMP is to choose consumption in each period subject to the constraint that income: ( ) ( ( ) ) ( ( of (Present consumption equals ) ) This UMP cannot be solved by calculus because the utility function is not differentiable everywhere. Instead, we exploit the fact that at the optimum: Solve this equation and the intertemporal budget constraint simultaneously. From above: Substitute in: ( ) ( ) [( ) [( [( ) ) 6 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: [ [ Thus: [ [ From these we can get the of and of by using direct calculation: the optimal utility is: ( [ [ ( [ ( ) [ [ The of ) : [ To compute ) of , express in terms of [ [ ( ) 7 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. ( ) [ b)...
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