## PROBLEMS AND APPLICATIONS

1. According to the IS–LM model, what happens in the short run to the interest rate, income, consumption, and investment under the following circumstances?

1. The central bank increases the money supply.

2. The government increases government purchases.

3. The government increases taxes.

4. The government increases government purchases and taxes by equal amounts.

2. Use the IS–LM model to predict the effects of each of the following shocks on income, the interest rate, consumption, and investment. In each case, explain what the central bank should do to keep income at its initial level.

1. After the invention of a new high-speed computer chip, many firms decide to upgrade their computer systems.

2. A wave of credit-card fraud increases the frequency with which people make transactions in cash.

3. A best-seller entitled Retire Rich convinces the public to increase the percentage of their income devoted to saving.

3. Consider the economy of Hicksonia.

1. The consumption function is given by

C = 200 + 0.75(YT).

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The investment function is

I = 200 – 25r.

Government purchases and taxes are both 100. For this economy, graph the IS curve for r ranging from 0 to 8.

2. The money demand function in Hicksonia is

(M/P)d = Y – 100r.

The money supply M is 1,000 and the price level P is 2. For this economy, graph the LM curve for r ranging from 0 to 8.

3. Find the equilibrium interest rate r and the equilibrium level of income Y.

4. Suppose that government purchases are raised from 100 to 150. How much does the IS curve shift? What are the new equilibrium interest rate and level of income?

5. Suppose instead that the money supply is raised from 1,000 to 1,200. How much does the LM curve shift? What are the new equilibrium interest rate and level of income?

6. With the initial values for monetary and fiscal policy, suppose that the price level rises from 2 to 4. What happens? What are the new equilibrium interest rate and level of income?

7. Derive and graph an equation for the aggregate demand curve. What happens to this aggregate demand curve if fiscal or monetary policy changes, as in parts (d) and (e)?

4. Explain why each of the following statements is true. Discuss the impact of monetary and fiscal policy in each of these special cases.

1. If investment does not depend on the interest rate, the IS curve is vertical.

2. If money demand does not depend on the interest rate, the LM curve is vertical.

3. If money demand does not depend on income, the LM curve is horizontal.

4. If money demand is extremely sensitive to the interest rate, the LM curve is horizontal.

5. Suppose that the government wants to raise investment but keep output constant. In the IS–LM model, what mix of monetary and fiscal policy will achieve this goal? In the 1980s, the Canadian government ran budget deficits while the Bank of Canada pursued a tight monetary policy. What effect should this policy mix have?

6. Use the IS–LM diagram to describe the short-run and long-run effects of the following changes on national income, the interest rate, the price level, consumption, investment, and real money balances.

1. An increase in the money supply.

2. An increase in government purchases.

3. An increase in taxes.

7. The central bank is considering two alternative monetary policies:

• holding the money supply constant and letting the interest rate adjust, or

• adjusting the money supply to hold the interest rate constant.

In the IS–LM model, which policy will better stabilize output under the following conditions?

1. All shocks to the economy arise from exogenous changes in the demand for goods and services.

2. All shocks to the economy arise from exogenous changes in the demand for money.

8. Suppose that the demand for real money balances depends on disposable income. That is, the money demand function is

M/P = L(r, Y – T).

Using the IS–LM model, discuss whether this change in the money demand function alters the following:

1. The analysis of changes in government purchases.

2. The analysis of changes in taxes.

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

C(YT) = a + b(YT),

where a > 0 and 0 < b < 1. Suppose also that investment is a linear function of the interest rate:

I(r) = cdr,

where c > 0 and d > 0.

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

2. 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.

3. 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) = eYfr,

where e > 0 and f > 0.

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

5. Using your answer to part (d), determine whether the LM curve is steeper for large or small values of f, and explain the intuition.

6. How does the size of the shift in the LM curve resulting from a \$100 increase in M depend on

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

2. the value of the parameter f, the interest rate sensitivity of money demand?

7. 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. Your expression should not contain r.

8. Use your answer to part (g) to prove that the aggregate demand curve has a negative slope.

9. 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?