Ch 6 Intertemporal Consumption

# That means you have savings of 10 the positive

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Unformatted text preview: saves more while the borrower continues to be a borrower but borrows less: 4 Think of it this way: Say you borrow \$10. That means you have “savings” of -\$10. The positive derivative ⏞ ( ) means that the negative saving is getting larger as a number so that it might go from -\$10 to -\$5. That is, you borrow less. 20 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. Save in Agent A & “live off savings” in ( Agent B & “payback loan” in Borrow in ) ( ( 1 ) ) E ( 1 ) 0 0 E 3. Complements Inter-temporal UMP Model Consider an agent who perceives consumption at consumed in and as complements where for each unit of corn she wants to consume (say) units of corn. We can represent her preferences by the complements utility model: ( ) ) Where are utility parameters and the utility function is defined on the consumption set ( . Since any bundle on the boundary of the consumption set has and since at the optimal solution requires that we can state the inter-temporal UMP as: ( ) ( ( ) ) ⏟( ( ) ) ⏟( ) As with consumer theory, this UMP cannot be solved by calculus and we use intuition to see that the optimal bundle is at the intersection of the “corners” line and the inter-temporal budget constraint: 21 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. ( ) * E Along the corners line: This implies that at the optimal solution (why?): Solving together with the inter-temporal budget constraint: ( ) Yields: ( ) [( in terms of and [( We have expressed respectively: ) ) in terms of [( If we want, we can express this in terms of ) and 22 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. ( [( ) ) Let’s work with the expressions: ( [( To find the marginal utility due to a small change in exploit the fact that: ) ) [( ) arising due to a change in either [( (but not ) we ) [( and/or ) Recall that whether an agent with Cobb-Douglas preferences saves or borrowers depends on her preferences, incomes and the real interest rate5. What about an agent with complements preferences – does their choice of being a saver or a borrower depend on (amongst things) the real interest rate? By definition, if an agent is saving at then: This occurs whenever: [( ) ( [( ) ) The choice of an agent with complements preferences being a saver depends on the agent’s preferences and incomes but, unlike an agent with Cobb-Douglas preferences, it does not depend on the real interest rate. This is a startling result: regardless of the real interest rate, “once a saver, always a saver” (and as you will see below, regardless of the real interest rate, “once a borrower, always a borrower”). 5 Cobb-Douglas agents will save whenever and borrow whenever 23 ECO 204 Chapter 6: Inter-temporal Consumption (this version 201...
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