APEC 3006 - APPLIED MACROECONOMICS
Answer Key Problem Set 3
Mankiw 8E, Chapter 5, pp.128-129, Problems and Applications, #1, #3, #5 Mankiw 8E, Chapter 7, pp.201-202, Problems and Applications, #1, #5
CHAPTER 5
QUESTION 1
In the country of Wiknam, the velocity of money is constant. Real GDP grows by 5% per year, the money stock grows by 14% per year, and the nominal interest rate is 11%. What is the real interest rate?
ANSWER: The real interest rate is 2%.
First, we start with the quantity theory of money: . Expressing this in percentage-change form1, we have
% Change in M + % Change in V = % Change in P + % Change in Y.
Rearranging this,
% Change in P = % Change in M + % Change in V – % Change in Y.
Substituting the numbers given in the problem, we thus find:
% Change in P = 14% + 0% – 5%
= 9%.
% Change in P is the inflation rate π. From the Fisher equation, we have
i = r + π,
where i is the nominal interest rate and r is the real interest rate. i is given as 11% in the problem, and we just calculated above that π is 9%. Hence, r is 2%.
QUESTION 3
Suppose a country has a money demand function (M/P)d = kY, where k is a constant parameter. The money supply grows by 12% per year, and real income grows by 4% per year.
a. What is the average inflation rate?
b. How would inflation be different if real income growth were higher? Explain.
c. How do you interpret the parameter k? What is its relationship to the velocity of money?
d. Suppose, instead of a constant money demand function, the velocity of money in this economy
was growing steadily because of financial innovation. How would that affect the inflation rate? Explain.
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How can we transform the quantity theory of money, , to the percentage-change form? One way to do it is using a natural log. We first take natural log in both sides: ln ). Remember the property of log: . Therefore, we have Natural log of a variable is approximately equal to the percentage change of the variable. Hence, we have %Change in M + %Change in V = %Change in P + %Change in Y, which is the expression we want. ApEc 3006 (McCullough)
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ANSWER
a. To find the average inflation rate the money demand function can be expressed in terms of
growth rates: % growth Md – % growth P = % growth Y.
The parameter k is a constant, so it can be ignored. The percentage change in nominal money demand Md is the same as the growth in the money supply because nominal money demand has to equal nominal money supply. If nominal money demand grows 12 percent and real income (Y) grows 4 percent then the growth of the price level is 8 percent.
b. From the answer to part (a), it follows that an increase in real income growth will result in a lower
average inflation rate. For example, if real income grows at 6 percent and money supply growth remains at 12 percent, then inflation falls to 6 percent. In this case, a larger money supply is required to support a higher level of GDP, resulting in lower inflation.
c. The parameter k defines how much money people want to hold for every dollar of income. The
parameter k is inversely related to the velocity of money. All else the same, if people are holding fewer dollars, then each dollar must be used more times to purchase the same quantity of goods and services, which means higher velocity of money.
To see this in our formulas, let’s compare the money demand function:
(M/P)d = kY (1)
and the quantity theory of money:
. (2)
Rearranging the expression (2), we get:
(M/P) = (1/V)×Y. (3)
Since the demand of money and supply of money (in a real term) should be equal, the left-had-sides of expressions (1) and (3) should be equal. Hence, the right-hand-sides should be equal too. In other words,
kY = (1/V)Y.
Cancelling-out Y, we have k = 1/V. Hence, the parameter k has an inverse relationship with V.
d. If velocity growth is positive, then if growth of M and Y stays the same, inflation will be higher.
From the quantity equation we know that:
% growth M + % growth V = % growth P + % growth Y.
Suppose that the money supply grows by 12 percent and real income grows by 4 percent. When velocity growth is zero, inflation is 8 percent. Suppose now that velocity grows 2 percent: this will cause prices to grow by 10 percent. Inflation increases because the same quantity of money is being used more often to chase the same amount of goods.
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QUESTION 5
Suppose that the money demand function takes the form
(M/P)d = L(i, Y) = Y/(5i)
a. If output grows at rate g, at what rate will the demand for real balances grow (assuming constant
nominal interest rates)?
b. What is the velocity of money in this economy?
c. If inflation and nominal interest rates are constant, at what rate, if any, will velocity grow?
d. How will a permanent (once-and-for-all) increase in the level of interest rates affect the level of
velocity? How will it affect the subsequent growth rate of velocity? ANSWER
a. If output Y grows at rate g, then real money balances (M/P)d must also grow at rate g, given that
the nominal interest rate i is a constant.
Expressing the money demand function in percentage change form, we have:
% Change (M/P)d = % Change in Y - % Change in (5i).
When the nominal interest rate does not change, we have
% Change (M/P)d = % Change in Y – 0.
g = % Change in Y.
b. To find the velocity of money, start with the quantity equation MV = PY and rewrite the equation
as (M/P) = (1/V)Y. Since money supply and money demand should be the same, we have (M/P) = (M/P)d. Hence, (1/V)Y = Y/(5i). Therefore, the velocity of money is V = 5i.
c. From part b. we have V = 5i ,where i is the nominal interest rate. If the nominal interest rate is
constant, then the velocity of money must be constant.
d. Again, from part b, we have V = 5i, which means that there is a one-to-one relationship between
the level of velocity and the level of interest rate. Hence, a one-time increase in the level of nominal interest rate will cause a one-time increase in the level of velocity of money. There will be no further changes in the velocity of money, because the increase in interest rate is just a one-time event (The interest rate does not grow over time in this problem).
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CHAPTER 7
QUESTION 1
Answer the following questions about your own experience in the labor force.
a. When you or one of your friends is looking for a part time job, how many weeks does it typically
take? After you find a job, how many weeks does it typically last?
b. From your estimates, calculate (in a rate per week) your rate of job finding f and your rate of job
separation s. (Hint: If f is the rate of job finding, then the average spell of unemployment is 1/f.) c. What is the natural rate of unemployment for the population you represent?
ANSWER
a. In the example that follows, we assume that during the school year you look for a part-time job,
and that on average it takes 2 weeks to find one. We also assume that the typical job lasts 1 semester, or 12 weeks. (Answers to parts (b) and (c) will differ according to what numbers you assume, which does not necessarily have to be 2 weeks and 12 weeks. More important thing is that you show consistency, and understand each concept.)
b. If it takes 2 weeks to find a job, then the rate of job finding in weeks is:
f = (1 job/2 weeks) = 0.5 jobs/week.
How could we get this? From the hint, 1/f is the average spell of unemployment, which we
assumed to be 2 weeks. Hence, f = ½ =0.5. This number can be interpreted as follows: Each week, a half of unemployed people looking for part-time jobs will find jobs. Or, if you are unemployed and looking for a part-time job, the chances that you will find a job during a week of search are a half.
If the job lasts for 12 weeks, then the rate of job separation in weeks is:
s = (1 job/12 weeks) = 0.083 jobs/week.
Just as in the case of f, we can think of 1/s as the average spell of employment, which we assumed to be 12 weeks. Hence, s = 1/12 = 0.083. Each week, about 8.3% of part-time employees lose or leave jobs. Or, if you are employed as a part-time worker, there is 8.3% chance of losing a job in a week.
c. From the text, we know that the formula for the natural rate of unemployment is:
(U/L) = (s/(s+ f )),
where U is the number of people unemployed and L is the number of people in the labor force. Plugging in the values for f and s that were calculated in part (b), we find:
(U/L) = (0.083/(0.083 + 0.5)) = 0.14.
Thus, if on average it takes 2 weeks to find a job that lasts 12 weeks, the natural rate of
unemployment for this population of college students seeking part-time employment is 14%.
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QUESTION 5
Consider an economy with the following Cobb-Douglas production function:
Y = K1/3L2/3.
The economy has 1,000 units of capital and a labor force of 1,000 workers
a. Derive the equation describing labor demand in this economy as a function of the real wage and
the capital stock (Hint: review Chapter 3).
b. If the real wage can adjust to equilibrate labor supply and labor demand, what is the real wage? In
this equilibrium, what are employment, output, and the total output earned by workers? c. Now suppose that Congress, concerned about the welfare of the working class, passes a law
requiring firms to pay workers a real wage of one unit of output. How does this wage compare to the equilibrium wage?
d. Congress cannot dictate how many workers firms hire at the mandated wage. Given this fact,
what are the effects of this law? Specifically, what happens to employment, output, and the total amount earned by workers?
e. Will Congress succeed in its goal of helping the working class? Explain.
f. Do you think that this analysis provides a good way of thinking about a minimum-wage law?
Why or why not?
ANSWER
a. The demand for labor is determined by the amount of labor that a profit-maximizing firm wants
to hire at a given real wage. The profit-maximizing condition is that the firm hire labor until the marginal product of labor (marginal revenue) equals the real wage (marginal cost),
MPL = W/P.
The marginal product of labor is found by differentiating the production function with respect to labor (see Chapter 3 for more discussion).
Hence,
.
Now, to solve for the labor demand, we solve this for L:
Note that the increase in the real wage (W/P) will reduce the demand of labor.
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b. In part (a), we have solved for the real wage (W/P) as a function of K and L:
L above is the labor demand. In equilibrium, we have supply of labor = demand of labor. Since we have labor supply of L=1,000 in the economy, we plug-in L=1,000 into the above equation. Similarly, we plug-in K=1,000. Then we can solve for W/P.
As we have said, in equilibrium, employment is 1,000 (since labor supply = labor demand = 1,000).
To get the output, we plug-in equilibrium values of capital and labor K=1,000 and L=1,000 in the production function:
Y = K1/3L2/3 = 1,0001/31,0002/3 = 1,000.
The total output earned by workers is equal to marginal contribution of labor to output (MPL) times the amount of labor: MPL × L = (W/P) × L = (2/3) × 1,000 = 667 units of output. Note that the labor gets two-thirds (2/3) of total output (Y = 1,000). In Cobb-Douglas production function of the form Y = KαL1-α, α and 1- α stands for the capital’s and labor’s share of output. In this problem α=1/3.
c. The Congress mandated a real wage of 1 unit of output, which is above the equilibrium real wage
of 2/3 of output.
d. Firms will use the labor demand function to decide how many workers to hire at the given real
wage of 1 and capital stock of 1,000. Using the labor demand function we have derived in part (a),
Note that, without this mandate (W/P being the equilibrium wage rate 2/3), L would be 1,000. With the mandate, the employment is only 296.
Plugging-in L=296 and K=1,000 in the production function, the output is: Y = K1/3L2/3 = 1,0001/32962/3 = 444.
The amount of output earned by workers is: MPL × L = (W/P) × L = 1 × 296 = 296 units of output. We can get the same number also by multiplying the share of output the labor gets (the
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exponent to the labor in the production function) and the total output: 2/3 × 444 = 296.
e. The policy redistributes output from the 704 workers who became involuntarily unemployed to
the 296 workers who get paid more than before. The lucky workers benefit less than the losers lose as the total compensation to the working class falls from 667 to 296 units of output.
f. The analysis of minimum wage in this problem focuses on the two effects of these laws: They
raise the wage for some workers while downward-sloping labor demand reduces the total number of jobs. Note, however, that if labor demand is less elastic than in this example, then the loss of employment may be smaller, and the change in worker income might be positive.
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