In the Fisher model, we showed that current consumption depends on

the present discounted value (PV) of lifetime income. This implies that an increase in the interest

rate

R

t

will decrease the PV of lifetime income and will therefore reduce current consumption.

To incorporate this notion into our short run model, we will slightly modify the consumption

equation. Specifically, consider the following model:

Y

t

=

C

t

+

I

t

(1)

C

t

Y

t

=

a

c

b

c

(

R

t

r

)

(2)

I

t

Y

t

=

a

i

b

i

(

R

t

r

)

(3)

where

a

c

,

b

c

,

a

i

,

b

1

, and

r

are positive constants. To simplify things, we are also assuming govern-

ment spending is zero and international trade is balanced (so that net exports are zero).

(a)

[4 Points]

Derive the IS curve in this model and explain how it differs from the IS curve in

the baseline model we've used in class? Illustrate your result(s) in a graph with

R

t

on the

y-axis and

Y

t

on the x-axis.

(b)

[4 Points]

Suppose the economy is in long run steady state, and suddenly, the parameter

a

c

decreases. What will happen to the IS curve in the short run? Illustrate the effect in your

graph from part (a).

(c)

[4 Points]

Assume that the interest rate is equal to the long run return on capital,

R

t

=

r

.

What happens to equilibrium short run output in response to the change in

a

c

? Illustrate the

effect in your graph from part (a).

(d)

[4 Points]

Suppose the Federal Reserve controls

R

t

and can freely choose any value for this

variable. If the Fed would like to immediately offset the effect on short run output from part

(a), what value of

R

t

should it choose? Compute an exact formula for the value you would

suggest. Illustrate your result in your graph from part (a).

(e)

[4 Points]

Does this modified model recommend that the Fed react more aggressively, less

aggressively, or the same as in the baseline model from class (in which

C

t

does not depend

on the interest rate

R

t

)? Justify your answer using the model formulas but also give an

intuitive explanation for your answer.