Ch 6 Intertemporal Consumption

# Note that varies from bundle to bundle ie imperfect

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Unformatted text preview: n is defined on the consumption set ( at any bundle in the consumption set, the slope of the indifference for an arbitrary level of utility Notice that the ) is: . Note that varies from bundle to bundle (i.e. imperfect substitutes). Since any bundle on the boundary of the consumption set has inter-temporal UMP as: and since at the optimal solution ( requires that ) ⏟( ( ) we can state the ) ⏟( ) As with consumer theory, it’s easier to work with the log-linear positive monotonic transformation: [( ) We note that by the envelope theorem, the impact on optimal utility due to a change in in (why?) is: other than due to a change The FOCs are: 14 ECO 204 Chapter 6: Inter-temporal Consumption (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ( [( ) ( ) ( ) ) Equating the expressions for yields a familiar result from consumer theory: at the optimal solution, the indifference curve is tangent to the budget line: ( ) E 0 Solving and ( ( ) ) simultaneously with the budget constraint yields the optimal amounts of corn consumption at : 15 ECO 204 Chapter 6: Inter-temporal Consumption (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Consumption at is a constant fraction if the present value of total lifetime income while consumption at is a constant fraction if the future value of total lifetime income. Of course, these can be re-expressed as: ( Substituting and into any FOC and solving for due to a change in either and/or (but not ): ) yields the marginal utility due to a small change in ( arising ) _________________________________________________________________________________________________ It’s instructive to compare these expressions with what we had in consumer theory: [ In the basic and the inter-temporal UMPs, consumption is a fraction of income and the marginal utility due to a small change in income has the form _________________________________________________________________________________________________ Given that: We can analyze when the agent will save/borrow when she’s young. By definition, if she is saving at then: 16 ECO 204 Chapter 6: Inter-temporal Consumption (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. This occurs whenever: [ There are many ways to express the “save at ” condition and here’s one way to do this which gives us a “graphical” condition for the agent to save when she’s young: This is shown below: ( ) * E Similarly, if she is borrowing at then: This occurs whenever: 17 ECO 204 Chapter 6: Inter-temporal Consumption (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. [ There are many ways to express the “borrow at ” condition and here’s one way to do this which gives us a “graphical” condition for the agent to borrow when she’s young: This is shown below: E * ( Now, the conditions for saving and borrowing at ) were:...
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## This note was uploaded on 03/20/2014 for the course ECON 204 taught by Professor Ajazhussain during the Fall '09 term at University of Toronto- Toronto.

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