# Solutions to Problems in Chapter 5

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```					Chapter 5: Aggregate Demand and the Classical Theory of the Price Level            Instructor’s Manual

Chapter 5: Aggregate Demand and the Classical Theory
of the Price Level

Problem 1

a.       The value of the propensity to hold money, k, is calculated according to the relationship
k = M/PY and is shown in the table below.

1980     1981       1982        1983      1984    1985
0.152     0.147     0.152       0.153     0.145   0.155

b.       Since k = M/PY, nominal GDP (PY) is a flow and M is a nominal stock, k has units of
time.

c.       Figure 1 below illustrates the time-path of the propensity to hold money, k. Although k
fluctuated a lot over the period, there is no discernible time trend. In fact, by eye-balling
the graph we can think of k as being approximately constant at 0.149.

0.156

k
0.154

0.152

0.150

0.148

0.146

0.144
1980        1981   1982    1983   1984   1985

Figure 1: Propensity to hold money

Problem 2

The quantity theory of money says that MD = kPYD, where PYD is the nominal demand for
commodities and k is the marginal propensity to hold money. The intuition behind this equation
is that people hold a constant fraction of their nominal demand for commodities as money. The
quantity theory of money is a theory of aggregate demand because if the money market is in
equilibrium, i.e. MD = MS, then the quantity theory equation can be rewritten as P = MS/(kYD).
From this equation, we can see that there is a negative relationship between P and YD. The
intuition behind the downward sloping aggregate demand curve is that an increase in the price
level reduces the real value of the money supply. As real money balances fall, the real demand
for commodities also decrease.

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Chapter 5: Aggregate Demand and the Classical Theory of the Price Level              Instructor’s Manual

Problem 3

There should be a strong positive relationship between the money supply and the price level, and
as a result a positive relationship between money growth and inflation. However, looking at the
quantity equation you see that this relationship is not necessarily a perfect relationship if there is a
change in one of the two other variables in the quantity equation. First, changes in k, the
marginal propensity to hold money, affect the relationship between the money supply and the
price level. An increase in k lowers the price level if the money supply is held constant. Second,
a change in Y, or real GDP, changes the relationship between the money supply and the price
level. An increase in Y also reduces the price level holding the money supply constant.

Problem 4

Nominal variables are variables that are measured in monetary units and, as a result, have not
been adjusted for changes in the price level over time. Real variables are variables that are
measured in units of commodities and as a result are not distorted by changes in the price level.
The proposition of money neutrality says that increases in the money supply will lead to
proportional increases in all nominal variables but will not affect real variables. In other words,
changes in the money supply only lead to proportional changes in the price level and those
variables measured in prices.

Problem 5

One of the implications of perfectly competitive markets is that prices will be perfectly flexible in
these markets. If markets are not perfectly competitive and individuals have some control over
setting prices, then prices most likely will not be perfectly flexible. As a result, in a response to
an increase in the money supply it may not be the case that the price level will increase
proportionally. Looking at the quantity theory you see that if the money supply rises without a
proportional increase in the price level, one of two real variables must adjust to maintain
equilibrium, either k or Y. This would violate the proposition of money neutrality in that changes
in the money supply would affect real variables.

Problem 6

Seigniorage refers to the revenue that a government generates by printing and issuing money.
Many governments find themselves with severe budget problems. For example, many less
developed countries have accumulated large amounts of foreign debt and loan payments. At the
same time, many of these same countries have a very poor tax base from which to raise revenue.
As a result, countries that have experienced a hyperinflation are typically countries that have had
no other alternative but to meet their obligations through printing up large amounts of money,
which produces high levels of inflation.

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Chapter 5: Aggregate Demand and the Classical Theory of the Price Level          Instructor’s Manual

Problem 7

The price level is countercyclical in the classical model. Given that changes in output can only be
created by changes in aggregate supply, when aggregate supply and output increase the price
level must fall. While prices have been countercyclical over certain periods of U.S. history,
prices were strongly procyclical during the Great Depression when both prices and output fell
substantially. The classical model is unable to explain this procyclical movement in prices.

Problem 8

a.      An increase in natural resources would increase labor demand and aggregate supply. As
a result of the increase in labor demand the quantity of labor would rise and real wages
would rise. As a result of the increase in aggregate supply the price level would fall and
aggregate output would rise.

b.      A reduction in the money supply would only affect nominal variables and not real
variables according to the proposition of money neutrality. There would be no change in
the real wage, but nominal wages and the price level would both fall by 20%. Aggregate
demand falls but aggregate supply remains unchanged so that aggregate output remains
unchanged.

c.      Same answer as in part a.

Problem 9

A large increase in labor force participation created by more women entering the labor force
would result in a large increase in labor supply. Real wages would fall and the quantity of labor
would rise. As a result, aggregate supply would rise, which would reduce the price level and
increase output. If both the price level and the real wage fall, it must be the case that nominal
wages fell as well and by more than the decrease in the price level.

Problem 10

If we impose market clearing on the classical aggregate demand equation by equating aggregate
demand YD to aggregate supply YE, we obtain the Quantity Equation of Money:
MS
P        .
kY E
This equation tells us what factors influence the average price level. Preferences affect the price
level through the propensity to hold money, k. Although the classical theory assumes that the
propensity to hold money is constant over time, as Figure 5-9 in the text shows, it has changed a
lot over the years. Technology affects the price level through the full-employment output level
YE, with a higher output leading to lower prices. Finally, as an example of endowments, we could
think of the number of people in the economy. A shrinking labor pool would imply a shrinking
real economy, and this would raise the price level.

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Chapter 5: Aggregate Demand and the Classical Theory of the Price Level          Instructor’s Manual

None of these factors can be considered as having had any important consequence for prices
in the U.S. history. In fact, as Figure 5-7 in the text demonstrates, the inflation rate in the U.S.
has followed the rate of money creation quite closely.

Problem 11

In the classical model, aggregate output is completely determined by changes in aggregate
supply. As a result, the aggregate demand for output is determined solely by the amount of
output that is supplied. Surpluses and shortages of output are not possible in the classical model
because prices are perfectly flexible. As a result, supply does determine, or "create," its own
demand and the classical model is consistent with Say’s Law.

Problem 12

a.      The marginal product of labor is MPL = 3L-1/2. Setting the real wage equal to the
marginal product, the labor demand curve is
w / P = 3L-1/2,
which can be rewritten as LD = 9/((w / P)2).

b.      Using equation 5.4 in the text, the aggregate demand curve is P = 12/YD.

c.      Setting LS = LD and solving for the equilibrium wage, w / P = 1.65 and L = 3.31.

d.      Equilibrium output is YE = 6(3.31)1/2 = 10.9.

e.      From the aggregate demand curve P = 12/10.9 = 1.1.

Problem 13

a.      The marginal product of labor is MPL = AL-2/3. Setting the real wage equal to the
marginal product of labor,
w / P = AL-2/3,
w / P which can be rewritten as LD = (A/ (w / P))3/2.

b.      Setting LS = LD and solving for the equilibrium wage, w / P = 1 and L = 1. Aggregate
output is Y = 3.

c.      Using equation 5.4 in the text, the aggregate demand curve is P = 15/YD. Given Y = 3
then P = 5.

d.      When A = 4 then setting LS = LD, the equilibrium wage is now w / P = 2.30 and
L = 2.30. Aggregate output increases to Y = 12(2.30)1/3 = 15.84. The aggregate price
level falls to P = 15/(15.84) = .95.

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Chapter 5: Aggregate Demand and the Classical Theory of the Price Level                  Instructor’s Manual

Problem 14

a.       Set the real wage equal to the marginal product to obtain LD = 16/((w / P)2). Set labor
supply equal to labor demand to obtain w / P = 2 and L = 4.
Y = 8(4)1/2 = 2. P = MS/kY = 16.

b.       The real wage, labor, and output all remain unchanged. When the money supply rises to
MS = 120, the price level also rises so that P = MS/kY = 20, which is also a 25% increase.

c.       The 25% increase in the money supply lead to a proportional increase in the price level
but did not affect real variables such as Y or L. As a result, these results are completely
consistent with the proposition of money neutrality.

Problem 15

The production function for this economy is Y = L, while the representative agent's utility
function is U = log(Y) + log(1-L).

a.       The simplest way to solve this problem is to maximize the representative agent's utility
subject to the economy's production function, i.e.,
Max U = log Y + log (1-L)
subject to Y = L
This can be rewritten as a problem of maximizing the function U = log Y + log (1-Y),
by substituting for L from the production function into the utility function. The first order
condition for the optimal choice of Y is
U 1   1             1
        0  Y*  .
Y Y 1  Y           2
Thus 1/2 units of output are produced in equilibrium.
b.       Equilibrium employment is obtained from the production function; it is 1/2 unit.

c.       If the labor market is perfectly competitive, the equilibrium real wage paid to workers
will be equal to the marginal product of labor. Since the marginal product of labor is
always one, the equilibrium real wage (w /P)* is also equal to 11. Hence, if the price level
is P=\$6, the equilibrium nominal wage is W* = P (w / P)* =\$6.

d.       If money supply is \$20 and the propensity to hold money k = 1, from the quantity
equation of money, the price level is P = M / (k Y*) = \$40.

Problem 16

A class discussion is helpful here.

1
Recall from Chapter 4, that the labor demand curve is perfectly elastic at the real wage w / P = 1. Hence
the labor supply curve must intersect the labor demand curve at w / P = 1.

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