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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: Intertemporal Consumption (this version 20122013) 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 Intertemporal 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 intertemporal 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 intertemporal budget constraint: 21
ECO 204 Chapter 6: Intertemporal Consumption (this version 20122013) 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 intertemporal 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: Intertemporal Consumption (this version 20122013) 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas agents will save whenever and borrow whenever 23
ECO 204 Chapter 6: Intertemporal Consumption (this version 201...
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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.
 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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