9. This problem asks you to analyze the IS–LM model algebraically. Suppose consumption is a linear function of disposable income:

C(Y – T) = a + b(Y −T), where a > 0 and 0 < b < 1. Suppose also that investment is a linear function of the interest rate:

I(r) = c − dr, where c > 0 and d > 0.

a. Solve for Y as a function of r, the exogenous variables G and T, and the model’s parameters

a,

b, c, and d.

b. How does the slope of the IS curve depend on the parameter

d, the interest rate sensitivity of investment? Refer to your answer to part (a), and explain the intuition.

c. Which will cause a bigger horizontal shift in the IS curve, a $100 tax cut or a $100 increase in government spending? Refer to your answer to part (a), and explain the intuition.
Now suppose demand for real money balances is a linear function of income and the interest rate:
L(r, Y) = eY − fr, where e > 0 and f > 0.

d. Solve for r as a function of Y, M, and P and the parameters e and f.

e. Using your answer to part (d), determine whether the LM curve is steeper for large or small values of

f, and explain the intuition.

f. How does the size of the shift in the LM curve resulting from a $100 increase in M depend on i. the value of the parameter

e, the income sensitivity of money demand?
ii. the value of the parameter

f, the interest sensitivity of money demand?
g. Use your answers to parts

(a) and

(d) to derive an expression for the aggregate demand curve. Your expression should show Y as a function of P; of exogenous policy variables M, G, and T; and of the model’s parameters. This expression should not contain r.
h. Use your answer to part (g) to prove that the aggregate demand curve has a negative slope.
i. Use your answer to part (g) to prove that increases in G and M, and decreases in T, shift the aggregate demand curve to the right.
How does this result change if the parameter

f, the interest sensitivity of money demand, equals zero?