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The constraint says that the
versa). Now: ) (
( ( ( ) ) )( ) ) ( ( ) )
) ) ( ) of lifetime savings equals zero (so that borrowing is cancelled out by savings and vice ( )( ( )
( ) ( (
) ( )
( ) ) [ )
( ) The FOCs are:
( [( ) ) 38 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. Rearrange and equate the first two FOCs:
)( ( ) ( )( )( ( )( ) )( ) ( )( )( ) )( ) We can substitute this in the budget constraint:
(
( ) ) ( )( ( ) ( )( ( ) ( )( ) ( )( ) ( )( ) (
( ) ( )( ) ( )( ) ( )(
( )
) )
)( ) (
( [ )(
)( ( )
)( ) [
This is the exact same expression for in question (6.6). Substituting
{ into [
[ ) gives} Next, we know that:
39
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. ( )
(
( [ Substitute ){ ( ) ( [
( ) ) [ ( ) ){
}
[ [ (
into ) to get: ( Substituting ) } gives
{( ){ [ [( }} ) (6.8) Consider a 3 period economy (
) with a single good (say, corn). Each consumer receives (individualspecific) real income
(in units of corn) at the beginning of
respectively. The price level at
is
(i.e.
is the base period) and the price levels at
and
are and , respectively.
In each period, the consumer can “borrow” or “save” corn at nominal interest rate
. In this model, “saving” in a
period means that the agent is consuming less than or equal to her income in that period (i.e.
) while “borrowing”
in a period means that the agent is consuming more than her income in that period (i.e.
).
The of total lifetime income is (“future period” is
( The of total lifetime income is (“present period” is ):
) ( ) ):
( ) Suppose a consumer has the following utility function over consumption at
(
Assume all pecuniary variables and and : ) . (a) Solve the consumer’s intertemporal UMP for (i)
the parameters. What happens if
? , (ii) the of , and (iii) the of in terms of 40
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. Answer:
(i)
Using the log transformation of the CobbDouglas utility function, the UMP can be stated as (In terms of L):
(
)
L
[(
) FOCs:
1.
L ( ( ) ) 2.
L (
( )
) 3.
L 4.
L ( ( ) ) From 1 and 3:
( ) ( ) From 1 and 2:
( ) ( ( ) ( ) ) From 4:
(
( ) )
( )( ) ( ) 41
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 5:
( ) ( ) From 6: ( ) Note: this can also be expressed as: (ii) To compute of and of , we can use the e...
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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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