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Unformatted text preview: 4, S. Ajaz Hussain. Do not distribute. [(
Because ) the graph has a positive slope since:
(
⏟ [( ) ) Take the 2nd derivative to see if the graph is linear, strictly concave/convex:
(
⏟
Because and [( ) ) , the second derivative is always negative so that the graph is strictly concave. When [( As ) then:
[( And as then because ) :
[( ) Agent will borrow at t = 0 if α/β < Y1/Y0
S0 1 0 r Borrow in t = 0
Save in t = 1 54
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. [(
( ) ) (
[( ) ) ( [( )
( )( )( [(
)[ ( )
) ) ) Now,
( ( )[ )[( (
[( )[ ( )( )
) ( ) ( ) [( (
[( )[ ( ( )] )
[( (
) ) ) ) then Now if Again,
)[( (
( ) )[ ( )( ) (
[(
If )
) then Concave
As , ( )(
( As ) ) , 55
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. ( )(
( ) ) S1
1 If on the other hand r 0 1 then
(
[( ) )
( [( ) ) Convex
As As , ,
( )(
( ) ) S1 0 1 r 1 56
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. (6.10) In this question, you will practice the intertemporal consumption/savings model. Consider a 3 period economy
with a single good (say, corn). Each consumer receives real income
(in corn) at the beginning of
respectively. The price level at
is
(i.e.
is the base period) and the price level at
and
is
respectively.
In each period, the consumer can borrow or lend corn at nominal interest rate . Assume that the real interest rate
(i.e. the nominal interest rate is always greater than or equal to inflation).
Suppose the consumer has the utility function:
( ) Solve for the optimal consumption in
and interpret your result for the case
. Hint: to write down the 3
period intertemporal budget constraint, observe that the 2 period intertemporal budget constraint says: FV of
consumption = FV of income.
Answer:
We write down the 3 period intertemporal budget constraint by setting the FV of consumption = FV of income: ( ) ( ) Optional: You can also derive this from 1st principles:
( ) ( )( ( ) ) ( )( ) I’ve used the real interest rate because in an inflationary economy that the real return on savings:
( )( ) ( )( ) ( ) If you rearrange this, you will get: ( ) Now for the UMP: the utility function is:
( ) This implies that the optimal choices must be: Plugging this into the PV budget constraint yields:
57
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. (
[ )
( (
[ )
( ) ( ) ) (
( ) )
( ) ( [ ) ) ( )
( )
( )
( ( ( ) [ ) Thus:
(
(
If ) )
( ) [ then the expression:
(
( Becomes: ) )
( [ )
[ ( ) That is, the consumer will have consumption smoothing, consuming the average life time income in each period. 58
ECO 204 Chapter 6: Practice Problems & Solutions for Modeling Intertemporal Consumption in ECO 204 (this version 20122013)...
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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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