PE = 200 + 0.75(Y – 100) + 100 + 100 = 0.75Y + 325.
This equation is graphed in Figure 10–8.
b. To find the equilibrium level of income, combine the planned-expenditure equation derived in part (a) with the equilibrium condition Y = PE:
Y = 0.75Y + 325 Y = 1,300.
The equilibrium level of income is 1,300, as indicated in Figure 10–8.
c. If government purchases increase to 125, then planned expenditure changes to PE = 0.75Y + 350. Equilibrium income increases to Y = 1,400. Therefore, an increase in government purchases of 25 (i.e., 125 – 100 = 25) increases income by 100. This is what we expect to find, because the formula for the government-purchases multiplier is 1/(1 – MPC), the MPC is 0.75, and the government-purchases multiplier therefore has a numerical value of 4.
d. An income level of 1,600 represents an increase of 300 over the original level of income. The government-purchases multiplier is 1/(1 – MPC): the MPC in this example equals 0.75, so the government-purchases multiplier is 4. This means that government purchases must increase by 75 (to a level of 175) for income to increase by 300.
5.Suppose that the money demand function is (M/P)d = 1,000 – 100r, where r is the interest rate in percent. The money supply M is 1,000 and the price level P is 2. a. Graph the supply and demand for real money balances. b. What is the equilibrium interest rate?
c. Assume that the price level is fixed. What happens to the equilibrium interest rate if the supply of money is raised from 1,000 to 1,200?
d. If the Fed wishes to raise the interest rate to 7 percent, what money supply should it set?
a. The downward sloping line in Figure 10–11 represents the money demand function
(M/P)= 1,000 – 100r. With M = 1,000 and P = 2, the real money supply (M/P)= 500. The real
d
s
money supply is independent of the interest rate and is, therefore, represented by the vertical line in Figure 10–11.
b. We can solve for the equilibrium interest rate by setting the supply and demand for real balances equal to each other: 500= 1,000 – 100r r = 5.
Therefore, the equilibrium real interest rate equals 5 percent.
c. If the price level remains fixed at 2 and the supply of money is raised from 1,000 to 1,200, then the new supply of real balances (M/P)s equals 600. We can solve for the new equilibrium interest rate by setting the new (M/P) equal to (M/P): 600 = 1,000 – 100r 100r = 400 r = 4. Thus, increasing the money supply from 1,000 to 1,200 causes the equilibrium interest rate to fall from 5 percent to 4 percent.
d. To determine at what level the Fed should set the money supply to raise the interest rate to 7 percent, set (M/P) equal to (M/P): M/P = 1,000 – 100r. Setting the price level at 2 and substituting r = 7, we find:
s
ds
d
M/2 = 1,000 – 100 ??7 M = 600.
For the Fed to raise the interest rate from 5 percent to 7 percent, it must reduce the nominal money supply from 1,000 to 600.
Chapter 11
PROBLEMS AND APPLICATION:3 3.Consider the economy of Hicksonia.
a. The consumption function is given by C = 200 + 0.75(Y ? T ). 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.
b. 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.
c. Find the equilibrium interest rate r and the equilibrium level of income Y.
d. 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?
e. 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? f. 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?
g. 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)?
a. The IS curve is given by: Y = C(Y – T) + I(r) + G.
We can plug in the consumption and investment functions and values for G and T as given in the question and then rearrange to solve for the IS curve for this economy: Y = 200 + 0.75(Y – 100) + 200 – 25r + 100 Y – 0.75Y = 425 – 25r (1 – 0.75)Y = 425 – 25r Y = (1/0.25) (425 – 25r) Y = 1,700 – 100r. This IS equation is graphed in Figure 11–11 for r ranging from 0 to 8.
b. The LM curve is determined by equating the demand for and supply of real money balances. The supply of real balances is 1,000/2 = 500. Setting this equal to money demand, we find: 500 = Y – 100r.
Y = 500 + 100r.
This LM curve is graphed in Figure 11–11 for r ranging from 0 to 8.
c. If we take the price level as given, then the IS and the LM equations give us two equations in two unknowns, Y and r. We found the following equations in parts (a) and (b):
IS : Y = 1,700 – 100r. LM : Y = 500 + 100r.
Equating these, we can solve for r: 1,700 – 100r = 500 + 100r 1,200 = 200r r = 6.
Now that we know r, we can solve for Y by substituting it into either the IS or the LM equation. We find
Y = 1,100.
Therefore, the equilibrium interest rate is 6 percent and the equilibrium level of output is 1,100, as depicted in Figure 11–11.
d. If government purchases increase from 100 to 150, then the IS equation becomes:
Y = 200 + 0.75(Y – 100) + 200 – 25r + 150.
Simplifying, we find:
Y = 1,900 – 100r.
This IS curve is graphed as IS2 in Figure 11–12. We see that the IS curve shifts to the right by 200.
By equating the new IS curve with the LM curve derived in part (b), we can solve for the new equilibrium interest rate: 1,900 – 100r = 500 + 100r 1,400 = 200r 7 = r.
We can now substitute r into either the IS or the LM equation to find the new level of output. We find
Y = 1,200.
Therefore, the increase in government purchases causes the equilibrium interest rate to rise from 6 percent to 7 percent, while output increases from 1,100 to 1,200. This is depicted in Figure 11–12.
e. If the money supply increases from 1,000 to 1,200, then the LM equation becomes: (1,200/2) = Y – 100r, or Y = 600 + 100r.
This LM curve is graphed as LM2 in Figure 11–13. We see that the LM curve shifts to the right by 100 because of the increase in real money balances.
To determine the new equilibrium interest rate and level of output, equate the IS curve from part (a) with the new LM curve derived above: 1,700 – 100r = 600 + 100r 1,100 = 200r 5.5 = r.
Substituting this into either the IS or the LM equation, we find Y = 1,150.
Therefore, the increase in the money supply causes the interest rate to fall from 6 percent to 5.5 percent, while output increases from 1,100 to 1,150. This is depicted in Figure 11–13.
f. If the price level rises from 2 to 4, then real money balances fall from 500 to 1,000/4 = 250. The LM equation becomes:
Y = 250 + 100r.
As shown in Figure 11–14, the LM curve shifts to the left by 250 because the increase in the price level reduces real money balances.