3. Suppose that the investment function is given by

I =400 – 2,000R +0.1Y

rather than by the investment function given in Chapter 8. Add this investment

function to the other four equations of the IS-LM model:

Y =C + I +G +X

C =220 +0.63Y

X =525 -0.1Y -500R

M =(0.1583Y -1,000R)P

Treat the price level as predetermined at 1.0, and let government spending

be 1,200 and the money supply be 900.

a. Derive an algebraic expression for the IS curve for this model and plot

it to scale. Compare it with the IS curve in the examples of Chapter 8.

Which is steeper? Why?

b. Derive the aggregate demand curve and plot it to scale. How does it

compare with the aggregate demand curve in the example of Chapter 8?

c. Calculate the effect of an increase in government spending on GDP. Is

the effect larger or smaller than in the case where investment does not

depend on output Y? Describe what is going on.

d. Calculate the effect of an increase in the money supply on GDP. How

does the impact compare with the situation where investment does

not depend on output?

4. Multiplier-accelerator interaction: Consider a macro model that has both a

consumption function that depends on lagged income (like Friedman’s

permanent-income equation) and an investment equation that depends,

with a lag, on changes in income. Ignore interest-rate effects. In particular,

assume that the following equations describe the economy:

Y =C + I + G

C =220 + 0.63Yp, with Yp =0.5(Y +Y_1)

I = 900 +0.2(Y_1 -Y_2)

G= 1,200

a. By algebraic substitution of C and I into the income identity, obtain a

single expression for output Y in terms of output in the previous years

(Y_1 and Y_2).

b. Calculate the constant level of output Y that satisfies all the relationships

in the model. (Hint: Set Y_1 =Y and Y_2 =Y in the equation

from part a and solve for Y using algebra.)

c. Suppose that Y is equal to the value you calculated in part b for the

past two years (years 1 and 2). Now, suppose that government spending

increases by $50 billion (in year 3). Calculate the effect on output

in year 3. Calculate the effect on output in years 4 through 10. Be

sure to use the relationship you derived in part a and substitute the

values for Y_1 and Y_2 you calculated in the previous two steps.

d. Plot the values of Y on a diagram with the years on the horizontal axis.

Do you notice any cyclical behavior in Y? Explain what is going on.

(This algebraic model was originally developed by Paul Samuelson of

M.I.T. while he was a student at Harvard in the 1930s.)

5. Paradox of thrift: Assume a closed-economy model with Y = C + I + G.

Suppose that investment demand depends on the level of income but not

on the interest rate, according to the formula

I = e +dY

and that consumption also depends on income according to the consumption

function

C = a +b(1 – t)Y.

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a. Sketch the spending line for the economy that shows how total spending

increases with income Y. (Put spending on the vertical axis and income

on the horizontal axis.) Draw a 45-degree line and indicate

where spending balance is.

b. Suppose that consumers decide to be more thrifty, to save more. They

do this by reducing a once and for all. Show the new point of balance

in the diagram.

c. What happens to investment as a result of consumers’ attempts to

save more? Explain.

d. Explain the paradox of thrift, that the attempt to save more may result

in a reduction in private saving. What happens to total saving?

e. Explain why the paradox of thrift is a short-run phenomenon. Introduce

interest rates into the investment function, and add a money demand

function and price adjustment equation to the model. If the

economy is operating at potential GDP before the reduction in a, will

it eventually return to potential after the reduction in a? What happens

to saving and investment when prices have fully adjusted?

6. Suppose the desired capital stock is given by the expression

K* = vY/RK,

where v is a constant and RK is the rental cost of capital.

a. Assuming that output in the economy is fixed at Y*, will a permanent

increase in the interest rate have a permanent or a temporary effect on

the level of investment?

b. Is your answer to part a consistent with the investment function incorporated

in the IS curve?

c. Suppose now that output in the economy is growing each period so

that ? Y = g. Assuming that the actual capital stock adjusts immediately

to its desired level, answer part a.

7. Sketch an IS-LM diagram. Compare two cases, one in which the investment

demand function depends on income and the other in which it is independent

of income. In which case are both monetary and fiscal policy

more effective? Explain.

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