macro homework answers

National Income

Reference : Mankiw, Macroeconomics, Chapter 2, Problem 3.

• Suppose a woman marries her butler. After they are married, her husband contin-

ues to wait on her as before, and she continues to support him as before (but as a

husband rather than as an employee). How does the marriage affect GDP ? What

is your reaction to this ?

GDP is a measure of recorded transactions only, not of informal work. The services

offered by the husband were recorded in GDP before the marriage, but not after the

marriage. Marriage induces the spouse to withdraw from the labor force and reduces

GDP.

???


• Place each of the following transactions in one of the four components of expen-

diture.

a. Boeing sells an airplane to the Air Force.

b. Boeing sells an airplane to American Airlines.

c. Boeing sells an airplane to Air France.

d. Boeing sells an airplane to Amelia Earhart.

e. Boeing sells an airplane to Jackie Chan.

f. Boeing builds an airplane to be sold next year.

a. Boeing sells an airplane to the Air Force.

Government Spending

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b. Boeing sells an airplane to American Airlines.

Investment

c. Boeing sells an airplane to Air France.

Export (From the United States to France)

d. Boeing sells an airplane to Amelia Earhart.

Consumption. Because Amelia Earhart is an American pilot resident in the US –

Amelia Earhart, born July 24, 1897; missing July 2, 1937 – Only if she purchases

the aircraft under her name; if she purchases on behalf of her company, then it is

accounted as investment.

e. Boeing sells an airplane to Jackie Chan.

Export. Because Jackie Chan is a Hong Kong resident (as far as I know). An

airplane is a big item requiring registration. If Jackie Chan buys a model airplane

worth a few dollars while vacationing in the U.S., it would be classified as U.S.

Consumption.

f. Boeing builds an airplane to be sold next year.

Investment, in the Inventory account.

???


• Consider how each of the following events is likely to affect real GDP. Do you

think the change in real GDP reflects a similar change in economic well-being ?

a. A hurricane in Florida forces Disney World to shut down for a month.

b. The discovery of a new, easy-to-grow strain of wheat increases farm har-

vests.

2

c. Increased hostility between unions and management sparks a rash of strikes.

d. Firms throughout the economy experience falling demand, causing them to

lay off workers.

e. Congress passes new environmental laws that prohibit firms from using pro-

duction methods that emit large quantities of pollution.

f. More high-school students drop out of school to take jobs mowing lawns.

g. Fathers around the country reduce their work-weeks to spend more time with

their children.

a. Potential visitors to the Park are unable to spend as planned on this consump-

tion/service, while the Park does not earn the corresponding income it could have

earned. Real GDP falls and so does welfare.

b. The new technology makes production more efficient, so that the farmers may

produce more output per (man- and machine-) hour; this increased productivity is

likely to translate into increased output; for instance, the farmers may choose to

work and invest as hard as before and to produce more (the fact that the increased

farm output is likely to lower the price of wheat is irrelevant since the question is

concerned with Real GDP, not Nominal GDP); or the farmers may choose to work

less at the farm and to expend more labor and investment in another production

activity, in which case Real GDP rises because of that other activity; or the farmers

may choose to enjoy more leisure, in the latter case Real GDP does not rise; in all

cases, well-being rises.

c. Strikes reduce man-hours and machine-hours, reduce Real GDP and (most likely)

welfare.

3

d. The new law imposes a constraint on the firms and as such may reduce production

and/or raise the costs of production; thus Real GDP is reduced. The reduction in

Real GDP reduces well-being, while the reduction in pollution increases it – the

net effect of the policy on well-being is therefore ambiguous.

e. The immediate effect is to raise Real GDP, since students add to the labor force;

and consequently well-being rises. However, education raises the future produc-

tivity of the workers and, therefore, their future well-being (the greater produc-

tivity may translate into higher production or higher leisure, depending on each

person’s preferences). By taking unskilled lawn-mowing jobs, students forgo the

opportunity to raise their future skills. If they freely choose to do so, the princi-

ple of revealed preference suggests that they must be better off doing so; either

because they have weighed future benefits against present sacrifices or because

they do not care about the future at all – if the drop-outs don’t care about their

future, why should economists care about it for them ? However, if markets fail

to channel loanable funds to the most productive uses, students may be unable to

borrow to finance their education; this market failure would commend some de-

gree of intervention. It may also be that society as a whole benefits from having

educated citizens – fewer graffitis in the subway, fewer elected politicians hurling

bananas at each other.

f. Real GDP falls since the fathers work less, well-being rises since the fathers

choose to forgo labor hours for leisure hours.

???

Reference : Mankiw, Principles, Chapter 10/23, Problem 5.

• Below are some data from the land of milk and honey.

4

Price of Quantity of Price of Quantity ofyear Milk Milk Honey Honey2008 $1 100 $2 502009 $1 200 $2 1002010 $2 200 $4 100

a. Compute nominal GDP, real GDP, and the GDP deflator for each year, using

2008 as the base year.

b. Compute the percentage change in nominal GDP, real GDP, and the GDP

deflator in 2009 and 2010 from the preceding year. For each year, identify

the variable that does not change. Explain in words why your answer makes

sense.

c. Did economic well-being rise more in 2009 or 2010 ? Explain.

a. Refer to in-class discussion.

b. At a glance, you can establish that the CPI inflation rate is 0% in 2008-2009 (since

no prices change), and 100% in 2009-2010 (since both prices double), while the

Real GDP growth rate is 100% in 2008-2009 (since the production of both goods

doubles) and 0% in 2009-2010 (since the production of neither good changes).

c. Well-being, in this problem, is approximated by Real GDP, which grew in 2008-

2009 but not in 2009-2010.

???

5

Cost of Living

Reference : Mankiw, Principles, Chapter 11/24, Problem 7.

• The New York Times cost $0.15 in 1970 and $0.75 in 2000. The average wage in

manufacturing was $3.23 per hour in 1970 and $14.32 in 2000.

a. By what percentage did the price of a newspaper rise ?

b. By what percentage did the wage rise ?

c. In each year, how many minutes does a worker have to work to earn enough

to buy a newspaper ?

d. Did workers’ purchasing power in terms of newspapers rise or fall ?

a. By what percentage did the price of a newspaper rise ?

($0.75−$0.15)/$0.15×100% = 400%

b. By what percentage did the wage rise ?

($14.32−$3.23)/$3.23×100% = 343%

c. In each year, how many minutes does a worker have to work to earn enough to

buy a newspaper ?

1970 : $0.15/($3.23/60) = 2.8 minutes

2000 : $0.75/($14.32/60) = 3.1 minutes

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d. Did workers’ purchasing power in terms of newspapers rise or fall ?

Workers’ purchasing power fell in terms of newspapers.

???

7

Income Distribution

Mankiw, Macroeconomics, Chapter 3, Question 1.

• What determines the amount of output an economy produces ? Describe a simple

model to account for the economy’s total production.

The question can be interpreted in various ways. One interpretation is that the first part

of the question asks for a model of the supply side of output, while the second part asks

for a model of the demand side. I favor this interpretation because it will make us review

several topics from the lesson.

To account for the supply of output, we simplify and focus on essentials. Output very

much depends on three things, the quantity of labor services supplied (best weighted by

an index of labor quality, “skill”, which can be imperfectly measured by wage rates or

labor compensation per hour supplied), the quantity of capital available (a result of past

investments and depreciation), and the available technology (ideas, methods, processes

— patented or not). Other things matter too, but may be more difficult to measure,

such as the infrastructure (public capital), property rights (the rule of law), social capital

(trust, cooperation, labor relations, inequalities). Remark : scientists are not interested

in ill-defined and immeasurable concepts such as “culture.”

A simple model would relate total output to the flow of labor services L, the stock

of capital K, and other factors A, according to a simple function with the following

properties : (1) the replication principle holds (constant returns to scale with respect to

K and L taken together), (2) the law of diminishing returns holds (decreasing marginal

products with respect to K and L taken separately), (3) the factor income shares are

constant. It turns out that a model of perfectly competitive factor markets with a Cobb-

8

Douglas production function has these properties.

Y = A×Kα ×L1−α , α = 0.3

This production function works quite well as a description of the production process,

but it’s not perfect and, more seriously, the role of A is large, suggesting that K and L

together typically account for between half and three-quarters of the output, with the

rest depending on the “other” factors captured in the stock A.

Remark 1. There are several ways to model technology. Generally speaking, tech-

nology refers to the shape of the production function Y = F(K,L). However, sometimes

we may wish to be more specific and we write instead Y = F(K,L,A), where A is re-

ferred to as a “stock of technology”. If we formulate production as : Y = F(K,L,A) =

AKαL1−α , technology is a “catch-all” factor usually called “total factor productivity”

(TFP) – in this case, we may also write Y = AF(K,L) (these are different functions F

of course). If we formulate production as : Y = F(K,L,A) = Kα(AL)1−α , technology

is a “labor-augmenting” factor sometimes called “skills” – in this case, we may also

write Y = F(K,AL). It turns out that with the Cobb-Douglas production function these

representations are equivalent, since Y = Kα(AL)1−α = (A)1−αKα(L)1−α : therefore

we can either refer to A as labor-augmenting technology or to (A)1−α as total factor

productivity.

Remark 2. Typically, we have data for output Y (GDP), for capital K (the sum of

past gross investments net of depreciation and possibly adjusted for quality), and for

labor L (the sum of labor hours weighted by an index of labor quality derived from

observed wages and/or level of education). We may also have some data pertaining

to technology, e.g. the number of patents delivered, the number of scientists engaged

in scientific research, both private and public, the amount spent in research and devel-

opment. However, data on technology, skills, education and even hours of work may

9

be imprecise (the data on hours of work is not available in some countries); moreover,

some improvements in technology occur fortuitously (learning-by-doing, learning-by-

copying); consequently A is likely subject to some stochastic time trend. Furthermore,

the precise shape of the aggregate production function can only be estimated indirectly;

one way is to assume a Cobb-Douglas production function together with perfect factor

markets and use observed values on the factor income shares to infer the capital and

labor exponents α and 1−α . The presence of a residual unaccounted for by measured

inputs is, then, a “measure of our ignorance.” In the simplest model in which only Y , K,

and L are measured, the size of the “residual” A can be thought of as a measure of “total

factor productivity.”

To account for the demand for output, we simplify and focus on essentials. In a

closed economy, output is simply divided between consumption, investment, and gov-

ernment purchases. The simplest model is one where consumption is proportional to

disposable income, where investment is inversely related to the (expected) real rate of

interest, and where government spending and taxes are exogenous.

In the short run, capital, labor, and technology are constant, K = K, L = L, A = A. In

the long run, things are more complicated (see the lesson on economic growth). Putting

it all together,

Y supply = F(K,L,A)

Y demand =C+ I +G

Y supply = Y demand

A× Kα × L1−α = mpc× (Y − T )+ I(r)+ G

As discussed in the lesson, the real rate of interest r adjusts endogenously to ensure that

both the goods market and the loanable funds markets clear.

10

Figure 1: Figure 3-8, Mankiw, Macroeconomics, Worth, Seventh Edition, 2009.

At this point, you are expected to illustrate the workings of the model with a graph,

see e.g. Figure 1.

???

Mankiw, Macroeconomics, Chapter 3, Problem 4.

• Figure 3-5 [2] shows that in U.S. data, the labor share (of total income) is approx-

imately constant over time. Table 3-1 [3] shows that the trend in the real wage

closely tracks the trend in labor productivity. How are these facts related ? Could

the first empirical fact be true while at the same time the second one not be ?

If production is Cobb-Douglas, say Y = AK0.3L0.7, then the labor income share is con-

stant and equal to 0.7 = 70%, while the capital income share is constant and equal to

0.3 = 30%. [See the note I have written on this topic] This fits the empirical facts, ex-

cept that the measured income shares are not exactly constant. The fluctuations in the

11

Figure 2: Figure 3-5, Mankiw, Macroeconomics, Worth, Seventh Edition, 2009.

Figure 3: Table 3-1, Mankiw, Macroeconomics, Worth Publishers, Seventh Edition, 2009.

12

income shares could be interpreted as small departures from the Cobb-Douglas produc-

tion function, together with the usual uncertainties and errors in measurement.

In a growing economy, typically K and A are growing, possibly also L, and conse-

quently Y is growing, but the ratio of total labor compensation to total output tends to be

constant, while the ratio of total capital income to total output tends to be constant too.

If the income shares were not constant, it would imply that over time one of the factors

would tend to dominate and overcome the other. This was Karl Marx’s famous (incor-

rect) prediction that the capital income share (the rate of profit) would decline over time

as more and more capital is accumulated. But the facts have shown that no factor has

a tendency to overcome the other, both are needed in production and the growth of one

factor also benefits the other factor.

If the labor income share is constant, it must be (by definition of the labor income

share) that (real) labor compensation is growing in line with (real) output, and this means

(as you can see from the definition of labor productivity given in Table 3-1 [3]) that real

labor compensation grows in line with labor productivity. So the two facts are closely

related.

This can be seen with a few equations. With constant returns to scale, the distribution

of income satisfies :

Y = MPK×K +MPL×L

With perfect competition, factors are paid their marginal products :

R/P = MPK, W/P = MPL

Putting both assumptions together :

Y = (R/P)×K +(W/P)×L

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The labor income share and capital income share are defined as :

(R/P)KY

,(W/P)L

Y

It is easy to see that they sum to 1 (by definition), so that if we prove that one is constant,

then the other one must be constant automatically. Let’s focus on the labor income share,

labor income share =W/PY/L

=MPLAPL

The labor income share is simply the ratio of the marginal product of labor W/P to the

average product of labor Y/L.

Take Y = F(K,L,A) = AKαL1−α . Let’s calculate the MPL and APL in terms of the

production function and its inputs,

MPL =∂F(K,L,A)

∂L=

∂AKαL1−α

∂L= (1−α) A

(KL

)α

APL =F(K,L,A)

L=

AKαL1−α

L= A

(KL

)α

We may now calculate the labor income share :

labor income share =MPLAPL

=(1−α) A

(KL

)α

A(K

L

)α = 1−α

It follows that the labor income share is constant (and therefore so is the capital income

share), even if A is growing over time. Thus, it is possible for the APL to rise over

time without affecting the distribution of income – because the MPL will also be rising

over time. Any increase in labor productivity (in this simple model) translates into a

corresponding increase in marginal productivity and, therefore, in wages.

14

If you have problems following the derivations above, try with a simple case : A= 1,

α = 1/2,

F(K,L,A) = K1/2×L1/2 =√

K×√

L

In this simple case, the distribution of income is fifty-fifty between capitalists and work-

ers.

A more realistic example is :

F(K,L,A) = e0.02t×K1/3×L2/3

In this case, the distribution of income is one third for capitalists and two thirds for

labor, with the average productivity of labor rising at (the exponential, annual rate of)

two percent a year (2% = 0.02).

???


• Consider a Cobb-Douglas production function with three inputs. K is capital (the

number of machines), L is labor (the number of workers), and H is human capital

(the number of college degrees among the workers). The production function is :

Y = K1/3L1/3H1/3

a. Derive an expression for the marginal product of labor. How does an in-

crease in the amount of human capital affect the marginal product of labor ?

b. Derive an expression for the marginal product of human capital. How does

an increase in the amount of human capital affect the marginal product of

human capital ?

15

c. What is the income share paid to labor ? What is the income share paid

to human capital ? In the national income accounts of this economy, what

share of total income would workers appear to receive ? (Hint : Consider

where the return to human capital shows up.)

d. An unskilled worker earns the marginal product of labor, whereas a skilled

worker earns the marginal product of labor plus the marginal product of

human capital. Using your answers to parts (a) and (b), find the ratio of the

skilled wage to the unskilled wage. How does an increase in the amount of

human capital affect this ratio ? Explain.

e. Some people advocate government funding of college scholarships as a way

of creating a more egalitarian society. Others argue that scholarships help

only those who are able to go to college. Do your answers to the preceding

questions shed light on this debate ?

a. The marginal product of labor is the change in output obtained by employing one

additional unit of the labor input (e.g. one more worker or one more hour of work,

depending on the unit used), keeping the quantity of all the other inputs constant

(keeping both human capital and physical capital constant). The marginal product

of labor is equal to the partial derivative of the production function Y =F(K,L,H)

with respect to the labor input L,

MPL =∂F(K,L,H)

∂L= K1/3H1/3 dL1/3

dL= K1/3H1/3(1/3)L1/3−1 =

K1/3H1/3

3L2/3

The marginal product of labor MPL is increasing in human capital H. Note that

MPL is increasing in physical capital K and decreasing in raw labor L.

16

b. The marginal product of human capital is the change in output obtained by em-

ploying one additional unit of the human capital input (e.g. one more year of

study, one more year of labor market experience, depending on the human capital

concept used), keeping the quantity of all the other inputs constant (keeping both

physical capital and raw labor constant). The marginal product of human capital

is equal to the partial derivative of the production function Y = F(K,L,H) with

respect to the human capital input H,

MPH =∂F(K,L,H)

∂H= K1/3L1/3 dH1/3

dH= K1/3L1/3(1/3)H1/3−1 =

K1/3L1/3

3H2/3

The marginal product of human capital MPH is decreasing in human capital H.

Note that MPH is increasing in physical capital K and increasing in raw labor L.

c. The income shares follow from the above calculations,

MPL×LY

=(1/3)K1/3L−2/3H1/3×L

K1/3L1/3H1/3 = 1/3

MPH×HY

=(1/3)K1/3L1/3H−2/3×H

K1/3L1/3H1/3 = 1/3

MPK×KY

=(1/3)K−2/3L1/3H1/3×K

K1/3L1/3H1/3 = 1/3

The third line is not part of the question but is included for completeness. Note

that since the production function is perfectly symmetric with respect to the three

inputs, the calculations only need to be performed once, symmetry dictates the

other two answers.

Both the return to raw labor L and the return to human capital H are paid to

17

and received by workers. Thus the share of income received by the workers is

worker share = raw labor share +human capital share

=MPL×L

Y+

MPH×HY

=13+

13

=23.

It is a strange economy one in which it is possible to separate out the contri-

butions of L and H — at the level of a single individual both labor hours and skill

are supplied simultaneously and receive a “global” wage payment. The model

makes more sense if there are two types of workers, skilled workers who supply

(N−L)H and unskilled workers who supply L, where N denotes the number of

hours of skilled labor supplied and L denotes the number of hours of unskilled

labor supplied. In terms of the notation used in the original problem, L should

not be equated with total hours supplied, but instead it should be equated with

total unskilled hours supplied, while H should be equated with a concept of hu-

man capital that incorporates hours worked — to see why the distinction matters,

imagine equating H with the number of years of study in an economy where many

workers are employed part-time or, worse, unemployed.

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d. The ratio of the skilled wage to the unskilled wage is :

skilled wageunskilled wage

=MPL+MPH

MPL

=(MPL+MPH)×L

MPL×L

=MPL×L+MPH×H× (L/H)

MPL×L

=MPL×L+MPL×L× (L/H)

MPL×L

= 1+L/H.

where the line before last uses the previous result :

MPH×H = MPL×L

An increase in the amount of human capital H reduces the ratio of skilled wage

to unskilled wage.

e. Government scholarships will induce a greater investment in human capital. An

increase in H reduces gap between the skilled and the unskilled and creates a

more “egalitarian” society even though the scholarship directly helps only those

students who are able to go to college — the indirect effect of the scholarship is

to raise the marginal product of raw labor while lowering the marginal product

of human capital, these general equilibrium adjustment in factor inputs induce a

reduction in the wage gap.

???

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The Closed Economy


• Consider an economy described by the following equations :

Y =C+ I +G

Y = 5,000

G = 1,000

T = 1,000

C = 250+0.75× (Y −T )

I = 1,000−50× r

a. In this economy, compute private saving, public saving, and national saving.

b. Find the equilibrium interest rate.

c. Now suppose that G rises to 1,250. Compute private saving, public saving,

and national saving.

d. Find the new equilibrium interest rate.

a. In this economy, compute private saving, public saving, and national saving.

Private Saving = Y −T −C = 5,000−1,000− (250+0.75× (5,000−1,000)) = 750

Public Saving = T −G = 1,000−1,000 = 0

National Saving = Y −C−G = (Y −T −C)+(T −G) = 750+0 = 750

20

Note that public saving is exactly zero – the government is running a balanced

budget.

b. Find the equilibrium interest rate.

The equilibrium interest rate is the interest rate that simultaneously clears the

market for loanable funds and the market for goods. The market for loanable

funds is in equilibrium when the demand for funds I (private investment) is equal

to the supply of funds S = Y −C−G (national saving) :

I = Y −C−G

1,000−50× r = 5,000− (250+0.75× (5,000−1,000))−1,000

250 = 50× r

r = 5%

c. Now suppose that G rises to 1,250. Compute private saving, public saving, and

national saving.

Private Saving = Y −T −C = 750 (as before)

Public Saving = T −G = 1,000−1,250 =−250

National Saving = Y −C−G = (Y −T −C)+(T −G) = 750−250 = 500

Public saving is now negative – the government is running a budget deficit.

d. Find the new equilibrium interest rate.

21

I = Y −C−G

1,000−50× r = 5,000− (250+0.75× (5,000−1,000))−1,250

500 = 50× r

r = 10%

The new equilibrium interest rate rises from 5% to 10%.

???

22

Capital Accumulation and Growth

• “Devoting a larger share of national output to investment would help restore rapid

productivity growth and rising living standards.” Do you agree with this claim ?

Explain.

You are expected to discuss the difference between “level effects” and “growth effects”

of changes in the investment rate (in the closed economy, the investment rate must be

“identitally” equal to the saving rate).

???

• In the Solow model, how does the saving rate affect the steady-state level of in-

come ? How does it affect the steady-state rate of growth ?

Assume we are referring to a Solow model in which population/workforce L grows at

rate n and in which efficiency E (whatever that is) is growing at rate g. It is useful to

refer to the standard diagram of the Solow model. Get your class notes ready.

In steady state, some variable called little k will be constant. I indicate that a variable

is at a steady state by a star, k∗. Understanding what determines this little k is crucial

to our understanding of what happens to big K, to big Y and to labor productivity Y/L.

But the Solow model diagram only shows you what happens to little k, and little y, it

does not show big K and big Y . So you need to supplement your understanding of the

diagram with some equations, following from definitions,

K = k×EL

Y = y×EL

23

So if we multiply little k and little y by EL, we have big K and big Y .

So what is EL anyway ? It’s Efficient Labor, and its growing (for some exogenous

reason, not explained here) at rate g+ n. (Recall that the growth rate of the product of

two variables is equal to the sum of the growth rates of each of the variables, yes ?)

In steady state, little k∗ and little y∗ are constant, and therefore big K∗ and big Y ∗

are growing at the exogenous rate g+n. Why the star on big K∗ and big Y ∗ ? To remind

you that it’s a steady-state result. By my notational convention the star means “steady

state” – NOT “constant”.

The steady-state level of income Y ∗ is growing at rate g+ n, implying that steady-

state level of income per worker (Y/L)∗ is growing at rate g.

How does the saving rate affect income ? Clearly, s has no effect on the steady-state

growth rate of income(

∆YY

)∗, since

(∆YY

)∗= g+n. That was easy. The effect on Y ∗ is a

little harder. It shows up through the effect on y∗. From Y ∗ = y∗EL, it is clear that s has

no effect on E and L. The only effect of s on Y ∗ is through y∗. The easiest way to see

how s affects y∗ is the Solow diagram. Get your class notes out. A rise in s shifts the

saving-investment line up, raising the steady-state value of little k∗, and consequently

the steady-state value of little y∗. Therefore, a higher saving rate raises steady-state

income, and raises steady-state income per capita (i.e. labor productivity).

???

• In the Solow model, how does the rate of population growth affect the steady-state

level of income ? How does it affect the steady-state rate of growth ?

Assume we are referring to a Solow model in which population/workforce L grows at

rate n and in which efficiency E (whatever that is) is growing at rate g. It is useful to refer

to the standard diagram of the Solow model. Refer to your class notes. I summarize the

main conclusions below.

24

In steady state, little k∗ and little y∗ are constant, and therefore big K∗ and big Y ∗

are growing at the exogenous rate g+n. The steady-state level of income Y ∗ is growing

at rate g+ n, implying that steady-state level of income per worker (Y/L)∗ is growing

at rate g.

How does the population growth rate affect income ? Clearly, n has a direct effect on

the steady-state growth rate of income(

∆YY

)∗, since

(∆YY

)∗= g+n. That was easy. The

effect on Y ∗ is a little harder. It shows up both through the effect on y∗ and through the

effect on EL. From Y ∗ = y∗EL, it appears that n has an effect L, since L(t) = L(0)ent ,

where L(0) is the population size in the reference period t = 0 (It is natural to choose

the reference period to be the date at which n rises). This effect of n on Y ∗ operating

via L is of course the steady-state growth effect we mentioned earlier. As Y ∗ grows at a

higher rate, clearly its level will be getting higher at a rate greater than before. It is best,

however, to think of this effect as a “growth effect” on(

∆YY

)∗and to think of the effect

operating via y∗ as a “level effect”. The easiest way to see how n affects y∗ is the Solow

diagram. Get your class notes out. A rise in n shifts the depreciation-dilution line up,

thus lowering the steady-state value of little k∗, and consequently the steady-state value

of little y∗. Therefore, a higher rate of population growth reduces the steady-state level

of income, but raises the growth rate of income.

The more interesting question, of course, is : how does an increase in population

growth affect steady-state income per capita ? The answer is that it reduces the steady-

state level of income per capita (Y/L)∗ = y∗E, since it reduces y∗, and has no effect on

the steady-state growth rate of income per capita, since it has no effect on (Y/L)∗ = g.

???

• In the Solow model, how does an increase in the saving rate affect the dynamics

of income, when income is below the steady state ?

25

Refer to the class note on the dynamics of the Solow model.

The central equation of the Solow model, in growth-rate form :

∆kk

=s f (k)

k− (δ +n+g)

Suppose the economy is initially in a steady state, denoted k∗1 and y∗1. Consider what

happens immediately after a rise in the saving rate s. The capital-effective-labor ratio k∗1

is pre-determined by past accumulation. The effect on the growth rate of k, following

the rise in s, can be analyzed by looking at the above equation. The rise in s raises ∆k/k.

To be precise, the “impact effect” of a rise in s on the predetermined level k∗1 is :

∆k∗1k∗1

=s f (k∗1)

k∗1− (δ +n+g)

The impact effect is the highest effect. In subsquent periods, the effect gets smaller

and smaller. As the economy converges towards the (new, higher) steady state, the k

ratio rises, at a falling rate, and in the limit (∆k/k)∗ → 0. At the same time, since

y = f (k) = kα , (∆y/y)∗ → 0. In the new steady state, k∗2, the saving rate no longer

affects k and y, since they are constant.

???

26

• Fill in the gaps, assuming y =√

k, s = 0.50, δ = 0.10.

Year k y c i δk ∆k

1 10.000 3.162 1.581 1.581 1.000 0.581

2 10.581 3.253 1.626 1.626 1.058 0.568

3 ..... ..... ..... ..... ..... .....

4 ..... ..... ..... ..... ..... .....

5 ..... ..... ..... ..... ..... .....

6 ..... ..... ..... ..... ..... .....

7 ..... ..... ..... ..... ..... .....

8 ..... ..... ..... ..... ..... .....

∞ ..... ..... ..... ..... ..... .....

[Hint : Use computer power to assist you !]

???


• Consider how unemployment would affect the Solow growth model. Suppose that

output is produced according to the production function Y = Kα [(1− u)L]1−α ,

where K is capital, L is the labor force, and u is the natural rate of unemployment.

27

The national saving rate is s, the labor force grows at rate n, and capital depreciates

at rate d.

a. Express output per worker (y = Y/L) as a function of capital per worker

(k = K/L) and the natural rate of unemployment. Describe the steady state

of this economy.

b. Suppose that some change in government policy reduces the natural rate of

unemployment. Describe how this change affects output both immediately

and over time. Is the steady-state effect on output larger or smaller than the

immediate effect ? Explain.

a. Express output per worker (y=Y/L) as a function of capital per worker (k =K/L)

and the natural rate of unemployment. Describe the steady state of this economy.

y = Y/L = Kα [(1−u)L]1−α/L = (1−u)1−α(K/L)α

Suppose that labor efficiency is constant, g = 0. The steady state occurs where

capital and labor grow at the same rate n so that the capital-labor ratio is constant,

∆K = sKα [(1−u)L]1−α −δK

∆K/L = s(K/L)α(1−u)1−α −δK/L

28

In steady state, ∆K/K = ∆L/L = n, so that the last line simplifies to :

n(K/L)∗ = s((K/L)∗)α(1−u)1−α −δ (K/L)∗

s((K/L)∗)α(1−u)1−α = (δ +n)(K/L)∗

The steady-state capital-labor ratio follows :

(K/L)∗ =(

s(1−u)1−α

δ +n

)1/α

The effect of unemployment is to effectively reduce the economy’s ability to save,

as if reducing the saving rate to s(1−u)1−α .

b. Suppose that some change in government policy reduces the natural rate of un-

employment. Describe how this change affects output both immediately and over

time. Is the steady-state effect on output larger or smaller than the immediate

effect ? Explain.

The long-run effect of a reduction in the natural rate of unemployment is sim-

ilar to the long-run effect of an increase in the saving rate – it raises the long-run

value of the capital-labor ratio, and therefore raises the long-run value of output

per worker. The immediate effect is to raise the marginal product of capital, by

shifting the investment-saving curve, and therefore to stimulate further investment

and development. All good things.

???

Technological Growth

• How does endogenous growth theory explain persistent growth without the as-

sumption of exogenous technological progress ? How does this differ from the

29

Solow model ?

???

• It is often said that saving is a virtue, yet the Solow model shows that saving does

not generate sustained growth : Is there a paradox ?

???

• Governments sometimes implement policies such as tax cuts and government

spending programs to stimulate consumption, and therefore reduce saving : Is

there a paradox ?

???

• The amount of education the typical person receives varies substantially among

countries. Suppose you were to compare a country with a highly educated labor

force and a country with a less educated labor force. Assume that education

affects only the level of the efficiency of labor. Also assume that the countries

are otherwise the same : they have the same saving rate, the same depreciation

rate, the same population growth rate, and the same rate of technological progress.

Both countries are described by the Solow model and are in their steady states.

What would you predict for the following variables ?

a. The rate of growth of total income.

b. The level of income per worker.

c. The real rental price of capital.

d. The real wage.

???

30


• In the United States, the capital share of GDP is about 30 percent; the average

growth in output is about 3 percent per year; the depreciation rate is about 4

percent per year; and the capital-output ratio is about 2.5. Suppose that the pro-

duction function is Cobb-Douglas, so that the capital share in output is constant,

and that the United States has been in a steady state. Hint : Let Y = F(K,EL) =

AKα(EL)1−α , where A > 0 and 0 < α < 1 are fixed parameters.

a. What must the saving rate be in the initial steady state ? [Hint : Use the

steady-state relationship, sy = (d +n+g)k.]

b. What is the marginal product of capital in the initial steady state ?

c. Suppose that public policy raises the saving rate so that the economy reaches

the Golden Rule level of capital. What will the marginal product of capital

be at the Golden Rule steady state ? Compare the marginal product at the

Golden Rule steady state to the marginal product in the initial steady state.

Explain.

d. What will the capital-output ratio be at the Golden Rule steady state ? (Hint :

For the Cobb-Douglas production function, the capital-output ratio is related

to the marginal product of capital.)

e. What must the saving rate be for the economy to reach the Golden Rule

steady state ?

a. What must the saving rate be in the initial steady state ? [Hint : Use the steady-

state relationship, sy = (d +n+g)k.]

31

Note. In the calculations below I am assuming y = kα . The same results can

be derived from y = Akα . Check it as an exercise.

With a Cobb-Douglas production function and perfectly competitive markets,

the capital income share is equal to the exponent on capital,

r×KY

=MPK×K/L

Y/L

=α(K/L)α−1×K/L

Y/L

=αkα

y

= α,

so α = rK/Y = 0.3.

In the steady state of the Solow model, the growth of output is equal to n+g,

so n+g = 0.03, and so d +n+g = 0.04+0.03 = 0.07.

In the Solow model with a Cobb-Douglas production function, the capital-

to-output ratio is k/y = k/kα = k1−α , so k∗ = (k/y)1/(1−α). So, in steady state,

k∗ = (2.5)1/0.7 ' 3.702.

In steady-state,

s(k∗)α = (d +n+g)k∗

s = (d +n+g)(k∗)1−α ' 0.07(3.702)0.7 = 17.5%

b. What is the marginal product of capital in the initial steady state ?

32

From the relation derived above,

MPK× ky

= α

MPK = αy/k = αkα−1

And so, in steady-state,

MPK = α(k∗)α−1 ' 0.3(3.702)−0.7 = 12%

c. Suppose that public policy raises the saving rate so that the economy reaches the

Golden Rule level of capital. What will the marginal product of capital be at the

Golden Rule steady state ? Compare the marginal product at the Golden Rule

steady state to the marginal product in the initial steady state. Explain.

The Golden rule saving rate is the saving rate that maximizes the steady-state

consumption level.

sGR = maxc∗

Steady-state consumption depends on the saving rate in two way :

c∗ = (1− s)y∗

= (1− s)(k∗)α

= (1− s)(

sd +n+g

)α/(1−α)

To maximize c∗, there are two methods you may use. Method 1. Graph the

Solow model and observe that the maximum arises where the slope of the pro-

duction function kα is equal to the slope of the depreciation-population-growth

33

line (d +n+g)k, that is where

MPKGR = αkα−1GR = d +n+g = 7%

It follows that

kGR =

(α

d +n+g

)1/(1−α)

Method 2. Differentiate (1− s)(

sd+n+g

)α/(1−α)with respect to s and set the

result equal to zero (that is look for the turning point of the expression). This will

give you the same value kGR. From there, apply MPKGR = αkα−1GR to derive the

expression MPKGR = d +n+g.

The Golden rule value of the marginal product of capital is equal to 7%. The

initial value of the marginal product of capital is equal to 12%. The initial value

is 5% above the Golden rule value.

d. What will the capital-output ratio be at the Golden Rule steady state ? (Hint: For

the Cobb-Douglas production function, the capital-output ratio is related to the

marginal product of capital.)

From rK/Y = α ,

(K/Y )GR = α/MPKGR = 0.3/0.07' 4.29

e. What must the saving rate be to reach the Golden Rule steady state ?

Since the Golden rule is a steady-state value of the capital-labor ratio, it fol-

34

lows that

kGR =

(α

d +n+g

)1/(1−α)

=

(s

d +n+g

)1/(1−α)

It is clear from the above that sGR = α .

Note. Make sure that you understand that the Golden rule is a rule about the

saving rate (from which the MPKGR and kGR, etc. follow).

???

35

Unemployment


Reference : Mankiw, Principles, Chapter 15, Problem 2.

• Is most unemployment long-term or short-term ? Explain.

Unemployment is typically short term. Most people who become unemployed are able

to find new jobs fairly quickly. But some unemployment is attributable to the relatively

few workers who are jobless for long periods of time.

The following table summarizes some relevant data. Make sure you know how to

interpret the data in words.

Average from January 1990 to June 2007

# of weeks % of total u duration as %1–4 38% 6.1%5–14 31% 18.8%15+ 31% 75.1%

Table 1: Average Duration of Unemployment in the U.S., Source : Bureau of Labor Statistics, http://www.bls.gov/

???

Reference : Mankiw, Macroeconomics, Chapter 6, Question 5.

• How do economists explain the high natural rate of unemployment in the 1970s

and 1980s ? How do they explain the fall in the natural rate in the 1990s and early

2000s ?

36

http://www.bls.gov/

http://www.bls.gov/

This one is for you to research. And I mean it. Hint : there is no single explanation.

???


• Suppose that Congress passes a law requiring employers to provide employees

some benefit (such as health care) that raises the cost of an employee by $4 per

hour.

a. What effect does this employer mandate have on the demand for labor? (In

answering this and the following questions, be quantitative when you can.)

If a firm was not providing such benefits prior to the legislation, the curve

showing the demand for labor would shift down by exactly $4 at each quan-

tity of labor, because the firm would not be willing to pay as high a wage

given the increased cost of the benefits.

b. If employees place a value on this benefit exactly equal to its cost, what

effect does this employer mandate have on the supply of labor ?

If employees value the benefit by exactly $4 per hour, they would be willing

to work the same amount for a wage that’s $4 less per hour, so the supply

curve of labor shifts down by exactly $4.

c. If the wage is free to balance supply and demand, how does this law affect

the wage and the level of employment? Are employers better or worse off ?

Are employees better or worse off ?

37

Figure 4 shows the equilibrium in the labor market. Because the demand

and supply curves of labor both shift down by $4, the equilibrium quantity

of labor is unchanged and the wage rate declines by $4. Both employees and

employers are just as well off as before.

d. Suppose that, before the mandate, the wage in this market was $3 above

the minimum wage. In this case, how does the employer mandate affect

the wage, the level of employment, and the level of unemployment ? Are

employers better or worse off ? Are employees better or worse off ?

If the minimum wage prevents the wage from falling, the result will be in-

creased unemployment, as Figure 5 shows. Initially, the equilibrium quan-

tity of labor is L1 and the equilibrium wage is w1, which is $3 lower than

the minimum wage wm. After the law is passed, demand falls to D2 and

supply rises to S2. Because of the minimum wage, the quantity of labor de-

manded LD2 will be smaller than the quantity supplied LS

2. Thus, there will

be unemployment equal to LS2−LD

2 .

e. Now suppose that workers do not value the mandated benefit at all. How

does this alternative assumption change your answers to parts (b), (c), and (d)

above ?

If the workers do not value the mandated benefit at all, the supply curve of

labor does not shift down. As a result, the wage rate will decline by less than

$4 and the equilibrium quantity of labor will decline, as shown in Figure 6.

Employers are worse off, because they now pay a greater total wage plus

benefits for fewer workers. Employees are worse off, because they get a

lower wage and fewer are employed.

38

Quantity of Labor

Wage

S1

D1

w1

L1

S2

D2

w1 −$4

Figure 4: Chapter 15, Problem 10.

???


• Consider an economy with two labor markets — one for manufacturing workers

and one for service workers. Suppose initially that neither is unionized.

a. If manufacturing workers formed a union, what impact on the wages and

employment in manufacturing would you predict ?

39

Quantity of Labor

Wage

S1

D1

w1

L1

S2

D2

w1 −$4wm = w1 −$3

LS2LD

2


b. How would these changes in the manufacturing labor market affect the sup-

ply of labor in the market for service workers ? What would happen to the

equilibrium wage and employment in this labor market ?

a. Figure 7 illustrates the effect of a union being established in the manufacturing

labor market. In the figure on the left, the wage rises from wU1 to wU

2 and the

quantity of labor demanded declines from U1 to UD2 . Because the wage is higher,

the quantity supplied of labor increases to US2 , so there are US

2 −UD2 unemployed

workers in the unionized manufacturing sector.

40

Quantity of Labor

Wage

D1

w1

L1

S

D2

w1 −$4

w2

L2


b. When those workers who become unemployed in the manufacturing sector seek

employment in the service labor market, shown in the figure on the right, the

supply of labor shifts to the right from S1 to S2. The result is a decline in the wage

in the non-unionized service sector from wN1 to wN

2 and an increase in employment

in the non-unionized service sector from N1 to N2.

???

41

Figure 7: Unionized and Non-Unionized Labor Markets.

42

The Monetary System

Reference : Mankiw, Principles, Chapter 16, Question 10.

• Suppose that the reserve requirement for checking deposits is 10 percent and that

banks do not hold any excess reserves.

a. If the Fed sells $1 million of government bonds, what is the effect on the

economy’s reserves and money supply ?

b. Now suppose the Fed lowers the reserve requirement to 5 percent, but banks

choose to hold another 5 percent of deposits as excess reserves. Why might

banks do so ? What is the overall change in the money multiplier and the

money supply as a result of these actions ?

???

Reference : Mankiw, Macroeconomics, Chapter 4, Question 8.

• List all the costs of inflation you can think of, and rank them according to how

important you think they are.

Illustrate with specific examples.

???


• Suppose that people expect inflation to equal 3 percent but, in fact, prices rise by

5 percent. Describe how this unexpectedly high inflation rate would help or hurt

the following :

a. the government

43

b. a homeowner with a fixed-rate mortgage

c. a union worker in the second year of a labor contract

d. a college that has invested some of its endowment in government bonds

???

44

Money Growth and Inflation


• Calvin Coolidge once said that “inflation is repudiation.” What might he have

meant by this ? Do you agree ? Why or why not ? Does it matter whether the

inflation is expected or unexpected ?

???


• During World War II, both Germany and England had plans for a paper weapon :

they each printed each other’s currency, with the intention of dropping large quan-

tities by airplane. Why might this have been an effective weapon ?

???


• In the country of Wiknam, the velocity of money is constant. Real GDP grows by

5 percent per year, the money stock grows by 14 percent per year, and the nominal

interest rate is 11 percent. What is the real interest rate ?

???


• Some economic historians have noted that during the period of the gold standard,

gold discoveries were most likely to occur after a long deflation. (The discoveries

of 1896 are an example.) Why might this be true ?

???

45

The Open Economy


• What are the net capital outflow and the trade balance ? Explain how they are

related.

Net capital outflow are, by definition, the difference between domestic saving and do-

mestic investment, S− I. Part of domestic saving is invested domestically, part of it

is invested abroad. Part of domestic investment comes from domestic saving, part of it

comes from foreign saving. Typically, as some domestic residents invest abroad, foreign

residents may invest domestically. These cross-border flows of (financial) capital may

reflect a desire to diversify and different preferences. Whenever one dollar of foreign

saving coming into the domestic economy can be matched to one dollar of domestic

saving going into foreign economies, these flows cancel each other out at the macro

level. Net capital flows summarize the big picture. If the nation’s domestic saving ex-

ceeds domestic investment, as a matter of accounting identity, it must be reflected by

positive net capital outflows. Another name for “Net Capital Outflow” is “Net Foreign

Investment.”

Net Foreign Investment is equal to the trade balance as a matter of accounting iden-

tity, S− I = EX − IM. I strongly recommend you work through the manipulations of

the national accounts that yield this identity. See the notes on that.

???


a. For each country, compute the predicted exchange rate of the local currency per

U.S. dollar. (Recall that the U.S. price of a Big Mac was $3.41)

46

b. According to purchasing-power parity, what is the predicted exchange rate be-

tween the Hungarian forint and the Canadian dollar ? What is the actual exchange

rate ?

c. How well does the theory of purchasing-power parity explain exchange rates ?

Country Price of Big Mac Actual Exchange Rate Predicted Exchange Rate

Indonesia 15,900 rupiah 9,015 rupiah/$Hungary 600 forints 180 forint/$Czech Republic 52.9 korunas 21.1 koruna/$Brazil 6.90 real 1.91 real/$Canada 3.88 Can. dollars 1.05 Can. dollar/$

a. Take x units of foreign currency per Big Mac divided by 3.41 dollars per Big

Mac, and get x/3.06 units of the foreign currency per dollar – that is the predicted

exchange rate.

• Indonesia : 15,900/3.41 = 4,663 rupiah/$

• Hungary : 600/3.41 = 176 forint/$

• Czech Republic : 52.9/3.41 = 15.5 koruna/$

• Brazil : 6.9/3.41 = 2.02 real/$

• Canada : 3.88/3.41 = 1.14 Can. dollar/$

b. Under purchasing-power parity, the exchange rate of the Hungarian forint to the

Canadian dollar is 600 forints per Big Mac divided by 3.88 Canadian dollars per

Big Mac equals 155 forints per Canadian dollar. The actual exchange rate is 180

forints per dollar divided by 1.05 Canadian dollars per dollar equals 171 forints

per Canadian dollar.

47

c. The exchange rate predicted by the Big Mac index (155 forints per Canadian

dollar) is somewhat close to the actual exchange rate of 171 forints per Canadian

dollar.

???

• Describe, discuss, and interpret Figure 5-13 [8].

From the definition of the real exchange rate, ε = e×P/P∗, we have :

∆ε

ε=

∆ee+

∆PP− ∆P∗

P∗

or, in other words,

∆ee

=∆ε

ε+π

∗−π

If the real exchange rate is exactly constant, i.e. ∆ε

ε= 0, then the percentage change

in the nominal exchange rate is exactly equal to the inflation differential. If the real

exchange rate is approximately constant, then the relation will hold as an approximation.

The figure shows that the approximation is pretty good, suggesting that for the countries

under consideration and over the period under consideration, the real exchange rate did

not change much. Countries with relatively high inflation tend to have depreciating

currencies. Countries with relatively low inflation tend to have appreciating currencies.

???


• If a small open economy cuts defense spending, what happens to saving, invest-

ment, the trade balance, the interest rate, and the exchange rate ?

48

Figure 8: Figure 5-13 Mankiw, Macroeconomics, Worth Publishers.

49

Domestic saving is defined as saving by private households (the excess of disposable

income over consumption) plus saving by the government (the excess of tax revenue

over public spending). Therefore, a cut in defense spending, keeping tax revenue un-

changed, raises public saving, and therefore domestic saving. Defense spending is part

of public expenditures and has therefore no effect on investment. From the accounting

identity, S− I = EX − IM, thus as S rises, the left-hand side rises, and thus the right-

hand side must rise, i.e. the trade balance must improve. In a small open economy, there

is no effect on the interest rate r∗, since the interest rate is determined on world mar-

kets and not affected by domestic conditions in a small economy. A rise in net exports

results in a depreciation of the real exchange rate, i.e. a fall in the real exchange rate

ε (by Mankiw’s definition), and this in turn will lead to a depreciation of the nominal

exchange rate e (keeping relative prices P∗/P as given).

???

50

Economic Fluctuations


• When real GDP declines during a recession, what typically happens to consump-

tion, investment, and the unemployment rate ?

When real GDP declines during a recession, both real consumption and real invest-

ment usually decline, and the unemployment rate usually rises. Since consumption is

such a large component of GDP it is not too surprising that it declines as GDP declines

(one component of GDP, at least, must decline). What is surprising, actually, is how

little it usually declines. Consumption is much less volatile than GDP. Quite surprising,

perhaps, is how much investment declines as GDP declines. Investment is a process of

creating capacity for future production, so it would be natural to imagine that investment

decisions would be based on expectations about the future rather than current fluctua-

tions in GDP. Consider an analogy : Education is a form of investment. You would

not expect families to significantly reduce their expenditures on their children’s educa-

tion because their current revenue is unusually low. You would expect families to cut

down other expenditures, such as holidays, expensive restaurants, new cars. By analogy,

therefore, you would expect companies to cut down CEO compensation or some other

items like that, rather than cut down on investment. Well, quite the contrary. Shocking,

but true. Now you know.

As GDP declines, jobs are destroyed faster than new jobs are created, resulting in a

rise in the unemployment rate. This is not mathematics. The unemployment rate u is

defined as u = U/L, where U is total unemployment and L is the labor force. As net

job creation falls below zero, clearly U rises. Whether the unemployment rate u rises

depends on whether U rises faster than L rises. In a downturn, however, L typically

51

falls, reflecting the fact that many workers leave the workforce (housewives and house-

husbands choosing willy-nilly to spend more time at home, early retirements), while

many workers do not enter the workforce (students extending their studies or joining

the army). It is therefore typically the case that the unemployment rate rises during a

downturn, a combination of the rise in U and the fall in L.

???


• Suppose the Fed reduces the money supply by 5 percent.

1. What happens to the aggregate demand curve ?

2. What happens to the level of output and the price level in the short run and

in the long run ?

3. What happens to unemployment in the short run and in the long run ?

It is not possible without a numerical model to answer specifically about a 5% cut in

money supply. Suppose instead that money supply is cut by some unspecified amount.

Refer to your notes for an aggregate demand / aggregate supply diagram and shifts

thereof. The reduction in the money supply shifts the aggregate demand curve in (mean-

ing “to the left”). In the short run, aggregate supply is fixed, so that the equilibrium price

level must fall. In the short run, the price level is fixed, so that the sticky-price will re-

main above the equilibrium-price. The short side of the market is therefore the demand

side, implying that aggregate output falls. In the long run, the price level adjusts down-

wards. As the price level falls, output rises. Eventually, at the new equilibrium, the price

level is lower and the output has returned to its original level. The unemployment rate

initially rises, reflecting the fall in output (the recession), and eventually gradually falls

52

until it returns to its natural rate. The movements of the unemployment rate around its

natural rate as output fluctuates around its long-run trend are captured by Okun’s law,

a remarkably stable and predictable relatonship (quite unlike the Phillips curve which

moves around a lot).

???

53

IS-LM


• Consider the economy of Hicksonia. The consumption function is :

C = 200+0.75(Y −T )

The investment function is :

I = 200−25r

Government purchases and taxes are both 100. The money demand function is :

(M/P)d = Y −100r

The money supply M is 1,000 and the price level P is 2.

a. Graph the IS curve for r ranging from 0 to 8.

b. 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

54

interest rate and level of income ?

g. Derive and graph an equation for the aggregate demand curve. What hap-

pens to this aggregate demand curve if fiscal or monetary policy changes, as

in parts (d) and (e) ?

a. The national income identity for a closed economy may be combined with the

consumption function, investment function, and government spending,

Y =C+ I +G

⇒ Y = 200+0.75(Y −100)+200−25r+100

⇒ Y =425−25r

0.25

⇒ Y = 1,700−100r

This is the equation of a straight line in the (Y,r) plane. Two points suffice to plot

the graph, e.g. (0,17) and (1700,0). The IS curve is a downward-sloping straight

line going through both points.

b. The money market clears at all times, so that the demand for money is equal to its

supply,

MS/P = Y −100r

⇒ 1,0002

= Y −100r

⇒ Y = 500+100r

55

This is the equation of a straight line in the (Y,r) plane. Two points suffice to plot

the graph, e.g. (0,−5) and (500,0). The LM curve is an upward-sloping straight


c. The IS and LM curves intersect once, and only once. The intersection of both

curves characterizes the equilibrium. The equilibrium may be computed by solv-

ing the linear system of two equations in two unknowns Y and r,

Y = 1,700−100r (IS)

Y = 500+100r (LM)

The system may be solved for r by subtracting the two equations, and Y may then

be deduced from either equation by substituting the solved value of r,

200r = 1,700−500 = 1,200 (1)

⇒ r = 6 (2)

Y = 500+100×6 = 1,100. (3)

Thus the equilibrium is at (1100,6).

d. It is convenient to write a more general formula, so that all subsequent questions

may be treated together. Introducing obvious notation,

Y =C0 +α(Y −T )+ I0−β r+G

where C0 is “autonomous consumption” and I0 is “autonomous investment.”

MS/P = Y − γi

56

where α is the marginal propensity to consume out of disposable income, and β ,

and γ are the other relevant elasticities of the model.

Remark : While the demand for money depends on the nominal interest rate i, the

demand for investment depends on the expected real interest rate re. The nominal

and real interest rate are related by Fisher’s equation,

re = i−πe

where re denotes the expected real rate of interest and πe denotes the expected

rate of inflation. Hence, a more general formula for the LM curve, which may be

used to analyze changes in prices, is

MS/P = Y − γ(re +πe)

However, as discussed below, we will simply assume that i = re. Rewriting these

equations, therefore,

Y =C0 + I0 +G−αT

1−α− β r

1−α(IS)

Y = MS/P+ γr (LM)

The fiscal-policy multiplier is 11−α

= 4. The tax-policy multiplier is −α

1−α= −3.

These multipliers are derived from the IS curve alone (equivalently, the Keyne-

sian cross) and assume a constant rate of interest. Introducing the LM curve re-

duces the value of these multipliers, by incorporating a feedback channel whereby

57

higher money demand induces the interest rate to rise. Solving the system yields :

Y =γ(C0 + I0 +G−αT )+βMS/P

β +(1−α)γ

r =C0 + I0 +G−αT − (1−α)MS/P

β +(1−α)γ

These equations may be used to answer the next few questions.

Consider the effect of fiscal policy. Only the IS curve is affected by the

change. The IS curve shifts to the right in the direction of greater Y . The ef-

fects of fiscal policy are given by :

∆Y∆G

=γ

β +(1−α)γ= 2

∆r∆G

=1

β +(1−α)γ= 0.02

The first effect is an adjusted fiscal multiplier (treating the rate of interest as en-

dogenous rather than constant). The effect of raising government purchases by 50

units from 100 to 150 is to raise output by 2×50= 100 units from 1,100 to 1,200,

and to induce the the rate of interest to rise by an amount equal to 0.02×50 = 1,

from 6% to 7%. The new equilibrium is therefore (1200,7).

e. Consider now the effect of monetary policy. Only the LM curve is affected by the

change. The LM curve shifts to the right in the direction of greater Y . The effects

of monetary policy are given by :

∆Y∆MS =

β

(β +(1−α)γ)P= 0.25

∆r∆MS =

−(1−α)

(β +(1−α)γ)P=−0.0025

The effect of raising the money supply by 200 units from 1,000 to 1,200 is to

58

raise real output by 200/4 = 50 units from 1,100 to 1,150, and to induce the rate

of interest to fall by an amount equal to 200/400 = 0.5, from 6% to 5.5%.

f. The effect of a rise in the price level can be analyzed with some care. Two aspects

of the question are open to interpretation. First, it is the nominal interest rate that

determines money demand and the real interest rate that determines investment.

To keep things simple, we ignore expected inflation and suppose the real and nom-

inal interest rates as either equal or differ by a fixed constant — in other words,

we assume that inflationary expectations remain unchanged throughout. This is

appropriate for a short-run analysis of an economy exempt from inflationary or

deflationary pressures. Thus, we assume that the price rise is a one-time jump

with no impact on the underlying rate of inflation, so that r = i.

Remark : Considering the effect of changes in inflationary expectations is an

important extension : for instance, the great depression of the 1930s and Japan’s

lost decades are associated with deflationary pressures. During the great reces-

sion of 2008 suggestions were made (by the IMF research department headed by

MIT economist Olivier Blanchard, by Nobel laureates Paul Krugman and Joseph

Stiglitz, and by several other prominent economists) that central banks around the

world, especially the ECB and the Fed, should commit to higher inflation targets

(higher than currently expected, by credibly promising not to contract money sup-

ply as much as currently expected for the foreseeable future), in order to reduce

real rates of interest and stimulate investment. These considerations are left as an

exercise for you.

Secondly, it is not clear whether government purchases are kept constant in

nominal or real terms. While prices remain unchanged, nominal and real values

are equal and it therefore makes no difference. But if prices rise, the real value

of government purchases falls as a result, and the distinction therefore matters.

59

Ultimately we are interested in real output, so we will assume that Y denotes real

output and that real government purchases remain unchanged — in other words,

we assume that nominal government purchases double as the price level doubles.

With these assumptions we have reduced the problem to its simplest possible

expression : the rise in the price affects real money balances M/P but does not

affect anything else in the economy. Consequently, doubling the price level is

“equivalent” to halving the money supply.

Under these assumptions, real output falls by 500/4 = 125 units, from the

initial value of 1,100 to 975. The (real) rate of interest rises by 500/400 = 1.25,

from the initial value of 6% to 7.25%.

These assumptions may not be the most natural, so please do explore alter-

natives. If nominal government purchases are kept constant, instead, the effects

of inflation combined with a fiscal contraction produce a greater contraction of

output and a smaller rise in the rate of interest.

g. The aggregate demand curve is a graph of the output-price relation on the de-

mand side. To derive the aggregate demand curve from the IS-LM equations, first

substitute the interest rate out and treat the price level as an endogenous variable

(rather than as a parameter, as the IS-LM model does), and secondly solve for

output as a function of the price,

i = r =Y −MS/P

γ(LM)

Y =C0 + I0 +G−αT

1−α− β

1−α

Y −MS/Pγ

(IS-LM)

⇒[

1+β

(1−α)γ

]Y =

C0 + I0 +G−αT1−α

+βMS/P(1−α)γ

⇒ Y =γ(C0 + I0 +G−αT )+βMS/P

β +(1−α)γ(AD)

60

The equation for aggregate demand Y as a function of the price level P is a non-

linear equation of the hyperbola type. Changes in fiscal policy G affect the inter-

cept term (the part that does not depend on P, as can be seen by setting P→ ∞),

while changes in monetary policy MS affect the slope term (the coefficient on

1/P). The aggregate demand curve behaves like a normal demand curve : A rise

in either G or MS shift the entire demand curve to the right, in the direction of

greater output Y , at any given price of the output.

???

Consider a standard IS-LM model.

The consumption function is : C = 100+0.5(Y −T )

The investment function is : I = 50−5r

Government purchases are : G = 40

Taxes are : T = 40

The money demand function is : (M/P)d = Y −20 i

The money supply is : M = 100

The price level is : P = 1

Fisher’s equation is : r = i−π where i is the nominal rate of interest, r is the real rate

of interest, and π is the expected rate of inflation.

1. Suppose that markets expect the price level to remain unchanged. Find the equi-

librium interest rate i and the equilibrium level of income Y .

2. Suppose that government purchases are raised from 40 to 70. What are the new

equilibrium interest rate and level of income ?

61

3. Suppose instead that taxes are cut from 40 to 10. What are the new equilibrium


4. Suppose instead that government purchases and taxes are simultaneously raised

from 40 to 70, so as to balance the budget. What are the new equilibrium interest

rate and level of income ?

5. Suppose instead that the money supply is raised from 100 to 160. Suppose, in

addition, that markets expect no future inflation and the price level remains un-

changed. What are the new equilibrium interest rate and level of income ?

6. Discuss the relative merits of these different policies.

7. Suppose that before any policy is actually implemented, markets come to expect

imminent deflation (negative inflation) at a rate of 3%. Suppose that, contrary to

these expectations, the price level actually remains unchanged. Compare the ef-

fect of monetary policy (increasing the money supply from 100 to 160) and fiscal

policy (increasing government purchases from 40 to 70 without changing taxes).

Compare the effects on the interest rate, on the level of income, on consumption,

on investment. Illustrate with a diagram.

Hint : Exact numbers are expected for questions 1–4. A detailed discussion with or

without exact numbers is expected for question 6.



The national income identity for a closed economy may be combined with the

62


Y =C+ I +G

⇒ Y = 100+0.5(Y −40)+50−5(i−π)+40

⇒ Y =170−5i

0.5

⇒ Y = 340−10i

This is the equation of a straight line in the (Y, i) plane. Two points suffice to plot



That’s not quite right though, because investment cannot be negative, so the

correct IS curve must be interrupted for those values of (Y, i) for which investment

is negative; since I = 50− 5r it follows that the IS curve is vertical for r > 10,

or equivalently i−π > 10. Refer to the IS-LM diagram below. This subtletly is

important only for question 6.

The money market clears at all times, so that the demand for money is equal

to its supply,

MS/P = Y −20i

⇒ 100/1 = Y −20i

⇒ Y = 100+20i




The IS and LM curves intersect once, and only once. The intersection of

63

both curves characterizes the equilibrium. The equilibrium may be computed by

solving the linear system of two equations in two unknowns Y and i,

Y = 340−10i (IS)

Y = 100+20i (LM)

Strictly speaking, this is right only if i− π ≤ 10, which may be verified after

computation. The system may be solved for r by subtracting the two equations,

and Y may then be deduced from either equation by substituting the solved value

of i,

30i = 340−100 = 240 (4)

⇒ i = 8 (5)

Y = 100+20×8 = 260. (6)

Since i = 8 = r < 10, the calculations are valid. From this, consumption and

investment may be deduced. Thus the equilibrium is :

Y = 260, i = 8%,C = 210, I = 10.



Y = 300, i = 10%,C = 230, I = 0.



64

Y = 280, i = 9%,C = 235, I = 5.




Y = 280, i = 9%,C = 205, I = 5.

5. Suppose instead that the money supply is raised from 100 to 160. Suppose, in

addition, that markets expect no future inflation and the price level remains un-

changed. What are the new equilibrium interest rate and level of income ?

Y = 280, i = 6%,C = 220, I = 20.


The effect of government purchases is larger than the effect of tax cuts be-

cause tax cuts affect output demand only via the consumers’ marginal propen-

sity to consume out of disposable income, while fiscal policy also affects output

demand directly. Balancing the budget is counterproductive for the purpose of

stimulating an economy operating below capacity, because tax increases are con-

tractionary. Because fiscal policy is more effective than tax policy, the overall

effect is nevertheless expansionary.

Raising government purchases by 30 units or cutting taxes by 30 units raise

65

the budget deficit by the same amount. While cutting taxes is less effective than

raising purchases when measured in terms of output, it is more effective when

measured in terms of consumption, because expansionary fiscal policy raises in-

terest rates more and crowds out private investment more.

Balancing the budget operates a redistribution from private consumption to

the government budget — consumption is 205 with a balanced budget as opposed

to 230 with a budget deficit. Balancing the budget also reduces the crowding out

of fiscal policy — investment is 5 with a balanced budget as opposed to 0 with a

budget deficit.








There is a difficulty here : since investment cannot be negative, the IS curve

must be vertical for r > 10. With expectations of deflation π =−3%, this means

the IS curve must be vertical for i > 7% (this follows from r = i−π = i+3 > 10).

Investment is, in effect, given by :

i≤ 7 I = 50−5r = 5(7− i)

i > 7 I = 0

If the IS-LM equations yield a rate of interest greater than 7%, and therefore

negative investment, this solution cannot be the answer to the question. To find the

correct answer, set I = 0 into the IS equation and obtain output Y as the residual

66

value. Consumption is then obtained by subtracting government purchases from

output, since C = Y −G.

The IS-LM diagrams below illustrate the different situations.

Monetary policy with expectations of deflation π =−3% :

Y = 260, i = 5%,C = 210, I = 10.

Fiscal policy with expectations of deflation π =−3% :

Y = 300, i = 10%,C = 230, I = 0.

Note that if monetary policy could turn expectations around, it would become

more effective, say π =+3% :

Y = 300, i = 1%,C = 230, I = 30.

However, the effect of rising prices would, eventually, reduce the effectiveness of

the policy. For expectations of greater inflation to be an effective part of monetary

policy, it is best if they precede actual inflation for a time. You may, on rare

occasions, have expectations of inflation without some inflation down the road,

but it cannot be the norm for otherwise expectations would be irrational.

???

Consider a standard IS-LM model.

The consumption function is : C = 40+0.5(Y −T )

The investment function is : I = 20−5r

67

IS

LM

Y

r

Figure 9: Equilibrium in the IS-LM Model with Baseline Parameters

Government purchases are : G = 40

Taxes are : T = 40

The money demand function is : (M/P)d = Y −20 i

The money supply is : M = 100

68

IS

LM

Y

r

Figure 10: Equilibrium in the IS-LM Model with Expected Deflation

The price level is : P = 1

Fisher’s equation is : r = i−π where i is the nominal rate of interest, r is the real rate

of interest, and π is the expected rate of inflation.



69

IS

LM

Y

r

Figure 11: Fiscal Policy in the IS-LM Model with Expected Deflation






70

IS

LM

Y

r

Figure 12: Monetary Policy in the IS-LM Model with Expected Deflation




71







on investment. How different would your answers be if markets came to expect

imminent inflation ? Illustrate with a diagram.

Hint : Exact numbers are expected for questions 1–4. A detailed discussion with or

without exact numbers is expected for question 6.



The national income identity for a closed economy may be combined with the


Y =C+ I +G

⇒ Y = 40+0.5(Y −40)+20−5(i−π)+40

⇒ Y =80−5(i−0)

1/2

⇒ Y = 160−10i




72

That’s not quite right though, because investment cannot be negative, so the

correct IS curve must be interrupted for those values of (Y, i) for which investment

is negative; since I = 20− 5r it follows that the IS curve is vertical for r > 4, or

equivalently, i− π > 4. Refer to the IS-LM diagram below. This subtletly is

important only for question 6.

The money market clears at all times, so that the demand for money is equal

to its supply,

MS/P = Y −20i

⇒ 100/1 = Y −20i

⇒ Y = 100+20i




The IS and LM curves intersect once, and only once. The intersection of

both curves characterizes the equilibrium. The equilibrium may be computed by

solving the linear system of two equations in two unknowns Y and i,

Y = 160−10i (IS)

Y = 100+20i (LM)

Strictly speaking, this is right only if i−π ≤ 4, which may be verified after com-

putation. The system may be solved for i by subtracting the two equations, and Y

73

may then be deduced from either equation by substituting the solved value of i,

30i = 160−100 = 60 (7)

⇒ i = 2 (8)

Y = 100+20×2 = 140. (9)

Since i = 2 = r < 4, the calculations are valid. From this, consumption and in-

vestment may be deduced. Thus the equilibrium is :

Y = 140, i = 2%,C = 90, I = 10.



Y = 180, i = 4%,C = 110, I = 0.



Y = 160, i = 3%,C = 115, I = 5.




74

Y = 160, i = 3%,C = 85, I = 5.


The effect of government purchases is larger than the effect of tax cuts be-

cause tax cuts affect output demand only via the consumers’ marginal propen-

sity to consume out of disposable income, while fiscal policy also affects output

demand directly. Balancing the budget is counterproductive for the purpose of

stimulating an economy operating below capacity, because tax increases are con-

tractionary. Because fiscal policy is more effective than tax policy, the overall

effect is nevertheless expansionary.

Raising government purchases by 30 units or cutting taxes by 30 units raise

the budget deficit by the same amount. While cutting taxes is less effective than

raising purchases when measured in terms of output, it is more effective when

measured in terms of consumption, because expansionary fiscal policy raises in-

terest rates more and crowds out private investment more.

Balancing the budget operates a redistribution from private consumption to

the government budget — consumption is 85 with a balanced budget as opposed

to 110 with a budget deficit. Balancing the budget also reduces the crowding out

of fiscal policy — investment is 5 with a balanced budget as opposed to 0 with a

budget deficit.





75




There are two difficulties with this question, one involving the IS curve and

one involving the LM curve.

First, there is a difficulty with the IS curve. Since investment cannot be neg-

ative, the IS curve must be vertical for r > 4. With expectations of deflation

π =−3%, this means the IS curve must be vertical for i > 1% (this follows from

r = i−π = i+3 > 4). Investment is, in effect, given by :

i≤ 1 I = 20−5(i−π) = 5(1− i)

i > 1 I = 0

If the IS-LM equations yield a rate of interest greater than 1%, and therefore

negative investment, this solution cannot be the answer to the question. To find the

correct answer, set I = 0 into the IS equation and obtain output Y as the residual

value. Consumption is then obtained by subtracting government purchases from

output, since C = Y −G.

Secondly, there is a difficulty with the LM curve. Since the nominal inter-

est rate cannot be negative, the equilibrium of the economy could be off the LM

curve. This is a “liquidity trap” situation. In this case, the equilibrium is found as

the intersection between the IS curve and the horizontal axis i = 0. (by contrast,

the real interest rate could be negative if inflations of expectations were inconsis-

tent with inflation, which happens on rare occasions)

The IS-LM diagrams below illustrate the different situations.

With expectations of deflation π = −3%, before any policy is implemented,

76

the equilibrium would be :

Y = 120, i = 1%,C = 80, I = 0.

Here the equilibrium is correctly given as the intersection of the IS and LM curves,

but only just.

Monetary policy with expectations of deflation π =−3% :

Y = 130, i = 0%,C = 85, I = 5.

Here the economy is in a liquidity trap and equilibrium is “off the LM curve”.

Fiscal policy with expectations of deflation π =−3% :

Y = 180, i = 4%,C = 110, I = 0.

Here the equilibrium is on the vertical portion of the IS curve.

Note that if monetary policy could turn expectations around, it would become

more effective, say π =+3% :

Y = 180, i = 1%,C = 110, I = 30.

However, the effect of rising prices would, eventually, reduce the effectiveness of

the policy. For expectations of greater inflation to be an effective part of monetary

policy, it is best if they precede actual inflation for a time. You may, on rare

occasions, have expectations of inflation without some inflation down the road,

but it cannot be the norm for otherwise expectations would be irrational.

???

77

IS

LM

Y

r

Figure 13: Equilibrium in the IS-LM Model with Baseline Parameters

78

IS

LM

Y

r

Figure 14: Equilibrium in the IS-LM Model with Expected Deflation

79

IS

LM

Y

r

Figure 15: Fiscal Policy in the IS-LM Model with Expected Deflation

80

IS

LM

Y

r

Figure 16: Monetary Policy in the IS-LM Model with Expected Deflation : The Liquidity Trap

81

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