In each period the consumer can borrow or save corn

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Unformatted text preview: straint (in “real” terms) on a plot and interpret the and axis intercepts and slope. inter-temporal budget Answer: Let’s use the monotonic transformation The agent considers and . to be imperfect substitutes. | | | | ⁄| | ⁄ | | This implies that if the agent consumes one more unit of corn today she just forego units of corns tomorrow to remain equally happy. FV Budget Constraint (in “real” terms): ( ) ( ( ) ) ( ) Y-axis intercept is: Set 24 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. ( ) X-axis intercept: Set ( ( Slope: ( ) ) ( ( ) ) ). This implies that if the agent consumes one more unit of corn today, she must give up ( ) units of corn tomorrow to remain on the budget line. The |slope| can also be interpreted as the ratio of “Price of 1 unit corn today” to “Price of 1 unit of corn tomorrow” either in PV terms or FV terms. In FV Terms: | | | | In PV Terms: FVY Demand curve for good 1 PVY (b) Solve the consumer’s inter-temporal UMP for What happens if the of ? If it’s easier, feel free to make up values for , and the of (for example: in terms of the parameters. or ). Answer: 25 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 inter-temporal UMP is to choose consumption in each period subject to the constraint that consumption equals income: ( ) ( ) ( )( Use the definition of real interest rate: ( Where )( to transform the UMP into: ( Set (since )( ) ) is the base period) to get: ( )( ( ) ( )( ) )( ( ( )( ( ) ( ) ) ( The ) ) ) ) budget constraint is in terms of the real interest rate. Use the hint and work with the log Cobb-Douglas utility function: ( ) ( ) Setup the Lagrangian: 26 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 FOCs are: ( [ ( ) ) ( ) Re-arrange and equate the first two FOCs: ( ) () We can substitute this in the ( ) budget constraint: ( ( ) ) ( ( () ( ) ) )[ ( ( [ ( ) ( [ ) ) ( ) ) 27 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. [ That is, consumption in (in the present period) is a constant fraction ( () () ) ( [ [( That is, consumption in of the PV income. Now for ) ) (in the future) is a constant fraction of the FV income (that’s neat!). Finally, we can calculate , which by the envelope theorem, is the marginal utility of “income”: [ ( ) ( [ Applying the envelope theorem the MU of a small increase in We can solve for Recall that ( ) ) income is: using any of the first two FOCS. For example, the second FOC implies: [( ) so that: [( [( ) ) 28 ECO 204 Chapter 6: Practice Problems & Solutions for Modeling In...
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