Unformatted text preview: e. Instead, we exploit
the fact that at the optimum: Solve this equation and the intertemporal budget constraint simultaneously. From above: Substitute in: ( ) ( ) [( ) [( ) [( ) Substitute in: [( ) [( ) [( ) Thus: 46
ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. [( ) From these we can get the MU of “income” (the “ ” in Lagrangian UMPs) by using direct calculation: the optimal utility
is:
(
)
( [( ) (
[( ) [( [( [(
The of ) ) ) : [(
To compute ) ) of , express ) in terms of
[( [( ) ( ) and ) ) [(
(b) What happens to the optimal ( ) due to, all else equal, a small increase in and ? Answer:
[( ) [(
( ) ) [( ) [( ) [ ( ( ) ( ) ) 47
ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. To answer this, we need to expand more: [ ( )
( ) )[ ( ) ) ( if and [( [( ) ) ( )
[
[( ) ) [( and [
(
[( Now if
[ ( ( [ ) ) ) , then , then [( ) [( ) ( ) )
[ ( more:
[( ( ) [( [ ) ) [( To answer this, we need to expand [( )
(
[
(
[( )[ ( ) ) ) ) [( [( ) ) 48
ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (
[( ) )
[
[( and Now if if and ) ) , then , then (c) Derive the conditions under which an agent will save in
condition, will the agent continue to save at
as ?
Answer:
To be a saver at ( (and therefore “borrow” at ). According to this , this consumer would have to consume less than her income: [( ) ) [( [( ) There are many ways to express the borrowing condition. Here is one way, useful for graphing saving/borrowing
diagrams. The agent saves in
when: [( ) [( ) ( ) [( ) [( ) ( ) 49
ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. The right hand side of this equation is the slope of the “L shaped indifference curves corners’ line” while the left hand
side is the slope of the ray from the origin to the endowment point. That is, the agent will save in
(the optimal
bundle will be to the left of the endowment point) when the corners line is steeper than the endowment point ray: Agent saving in t = 0, “borrowing” in t = 1
C1
FVY C1
S1 < 0 Y0 > Y1
E Y1
α/β C0 Y0 PVY C0 S0 > 0
Observe how the condition for saving is independent of the real interest rate which means that once a saver at
the agent will always be a saver regardless of the real interest rate (albeit the level of savings will change). (d) Derive the conditions under which an agent will...
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 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics, Inflation, S. Ajaz Hussain, Sayed Ajaz Hussain

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