ECON 304
HOMEWORK # 6
due: June 25 in class
1. Table 1 shows business cycle statistics for the United States for the period 1947-1998.
Table 1
Consumption
Investment
Relative standard
deviation with
respect to output
First-order
Autocorrelation
Correlation with
Output
0.69
2.82
0.88
0.82
0.68
0.90
a. Is investment procyclical, counter-cyclycal or acylical?
Investment is procyclical because its correlation with output is positive (in fact, is very close to one,
therefore, investment is strongly procyclical)
b. Which variable is more volatile, consumption or investment?
Investment is more volatile than consumption because its relative standard deviation (2.82) is higher than
that for consumption (0.69)
In parts [a.] and [b.] be sure to tell which number(s) in Table 1 are you using to support your answer.
c. This question is not based on the information on Table 1. Explain why in a Real Business Cycle (RBC)
model, a 1% increase in productivity A, increases output Y by more than 1%
When A increases, it increases the M P N shifting the labor demand curve out, increasing equilibrium
employment, so N goes up as well. Since Y = AF (K, N ) we also know that:
∆Y
∆A
∆K
∆N
=
+α
+ (1 − α)
Y
A
K
N
Since
∆A
A
= 1% and
∆N
N
> 0 we can infer that
1
∆Y
Y
> 1%
d. This question is not based on the information on Table 1. Explain why in a RBC model, individuals’ desire
to smooth consumption causes investment to be more volatile than output.
As we saw in part (c), a 1% increase in A increases Y by more than 1%. Since in this economy output can
only be consumed or invested, a strong desire to smooth consumption will cause that the extra output
that results from an increase in productivity will mostly be devoted to investment, making it more volatile
than consumption
2. Consider the following economy:
Desired Consumption: C d = 1, 275 + 0.5(Y − T ) − 200r
Desired Investment: I d = 900 − 200r
Real Money Demand: L = 0.5Y − 200i
Full Employment Output: Y ∗ = 4, 600
Expected Inflation: π e = 0
a. Suppose that T = G = 450 and that M = 9, 000. Find an equation describing the IS curve. Hint: Set
desired national saving and desired investment equal, and solve for the relationship between r and Y
The IS curve is found by setting desired saving equal to desired investment. Desired saving is S d =
Y −C d −G = Y −[1, 275+0.5(Y −T )−200r]−G. Setting S d = I d gives Y −[1, 275+0.5(Y −T )−200r]−G =
900 − 200r, or Y = 4, 350 − 800r + 2G − T . Plugging in the values of T and G
IS curve: Y = 4, 800 − 800r
b. Find an equation describing the LM curve (again, assume that T = G = 450 and that M = 9, 000). Hint:
Set real money demand and supply equal and solve for the relationship between r and Y
The LM curve is
M
P
= L = 0.5Y − 200i = 0.5Y − 200(r + π e ) = 0.5Y − 200r. Plugging the value of M
LM curve:
9,000
P
= 0.5Y − 200r
3. Numerical problem # 4, Chapter 9, page 345, Abel & Bernanke
a. What are the general equilibrium levels of the real wage, employment and output?
First, look at labor market equilibrium: Labor supply is N s = 55 + 10(1 − t)w. Labor demand (N d ) comes
from the equation w = 5A − 0.005AN d . Substituting the latter equation into the former, and equating
labor supply and labor demand gives N = 100. Using this in either the labor supply or labor demand
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equation then gives w = 9. Using N in the production function gives Y = 950.
b. For any level of output Y , find an equation that gives the real interest r that clears the goods market (the
IS curve)). What are the general equilibrium values of the real interest rate, consumption and investment?
Next, look at goods market equilibrium and the IS curve:
Sd = Y − Cd − G
S d = Y − [300 + 0.8(Y − T ) − 200r] − G
S d = Y − [300 + 0.8(Y − 20 − 0.5Y ) − 200r] − G
S d = Y − 300 − 0.4Y + 16 + 200r − 50
S d = −334 + 0.6Y + 200r
Setting S d = I d gives:
Sd = I d
−334 + 0.6Y + 200r = 258.5 − 250r
450r = 592.5 − 0.6Y
r = 1.317 − 0.00133Y IS curve
Plugging back the full-employment level of output Y = 950 gives an equilibrium real interest rate of 5%.
Plugging these results into the consumption and investment equations gives C = 654 and I = 246.
c. For any level of output Y , find an equation that gives the real interest rate that clears the asset market (the
LM curve). What is the general equilibrium value of the price level?
Setting money demand equal to money supply gives:
9150/P = 0.5Y − 250(r + 0.02)
r=
[0.5Y − (5 + 9150/P )]
250
With Y = 950 and r = 0.05, solving for P gives P = 20.
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d. Suppose that government purchases increase to G = 72.5. Now what are the general equilibrium values of
the real wage, employment, output, the real interest rate, consumption, investment and the price level?
With G = 72.5, the IS curve becomes r = 1.367 − 0.00133Y . With Y = 950, the IS curve gives r = 10%,
the LM curve gives P = 20.56, the consumption equation gives C = 644, and the investment equation
gives I = 233.5. The real wage, employment, and output are unaffected by the change.
4. Analytical problem # 1, parts (a.) and (b.) Chapter 9, page 346, Abel & Bernanke
a. Reduction in the effective tax rate on capital increases desired investment
The increase in desired investment shifts the IS curve up and to the right, as shown in Figure. Since
the short-run level of output Ye is greater than the full-employment level of output Y , producers start
increasing prices. The price level rises, shifting the LM curve up and to the left to restore equilibrium.
Since the real interest rate rises, consumption declines. In summary, there is no change in the real wage,
employment, or output (since nothing happens in the labor market, the F E line doesn’t change); there is
a rise in the real interest rate, the price level, and investment; and there is a decline in consumption.
New general equilibrium
2.) Prices increase, and this
reduces real money supply,
M/P. This shifts the LM curve
up and to the left, restoring
general equilibrium
r2
r1
1) When investment increases the IS
curve shifts up and to the right
Initial general equilibrium
b. Expected inflation rises
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The rise in expected inflation shifts the LM curve down and to the right, as shown in Figure 9.22. The
price level rises, shifting the LM curve up and to the left to restore equilibrium. Since the real interest
rate is unchanged, consumption and investment are unchanged. In summary, there is no change in the
real wage, employment, output, the real interest rate, consumption, or investment; and there is a rise in
the price level.
1) Increase in πe shifts the LM
curve down and to the right
Initial general equilibrium
=
New general equilibrium
2.) Prices increase, and this
reduces real money supply,
M/P. This shifts the LM curve
up and to the left, restoring
general equilibrium
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