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# 1 An economy can produce either consumption or investment goods

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1 An economy can produce either consumption or investment goods

Econ 441 Alan Deardorff

Problem Set 4 - Answers Specific Factors Models

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Problem Set 4 - Answers

Specific Factors Models

1. In the Extreme Specific Factors Model,

a. What does a country’s excess demand curve look like?

The PPF in the Extreme Specific Factors Model is just a point in goods X,Y

space (X,Y space). Excess demand and supply are just the difference between

this point and the tangencies of indifference curves with various price lines:

b. What determines the relative price at which the excess demand curve crosses the

vertical axis?

This is the autarky price, of course, and it is given by the slope of the

indifference curve passing through . X,Y

c. Suppose a world of two countries that are trading freely, with the home country

importing good X, and exporting good Y to Foreign. Suppose now that Home

(only) experiences an improvement in its technology so that the factors employed

in its X industry become more productive by, say, 10%. What will this do to

i) its excess demand or supply curve,

ii) the world equilibrium relative price of X,

iii) the real wage of labor in Home’s X industry

iv) the real rental price of capital in Foreign’s Y industry?

XYThis will increase by 10% without changing . For any given price, this

will increase income, causing consumption of both goods to increase (assuming

they are normal goods). In order for consumption of Y to increase, however,

Xconsumption of X cannot rise by as much as . Therefore, for any price, the

country’s excess supply of X increases and its excess demand decreases, shifting

Econ 441 Alan Deardorff

Problem Set 4 - Answers Specific Factors Models

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its ED curve to the left throughout its length, as shown below. This, in turn,

causes the world equilibrium relative price of X to fall, as also shown.

p=p/pXY

ES*

* p

*’ ED p

ED’

X

In the Home X industry, labor is paid the value of its marginal product. That

marginal product has increased by 10% due to the improved technology, so

labor’s wage in units of X has gone up by that amount. However, the price of X

has fallen, and we don’t know by how much. If the price falls by less than 10%,

then this is necessarily an improvement in its real wage. But if it falls by more

than 10%, which it may (depending on the elasticities of the various curves),

then workers in the Home X industry can buy less of Y, and may be made worse

off if their demand for Y is a large enough part of their budgets. Thus the effect

on the wage in the Home X industry is ambiguous.

As for capital in the Foreign Y industry, its marginal product has not changed,

so it is paid the same in terms of Y. But that payment is worth more in terms of

X, due to the price change, so its real payment has increased.

2. In the (Standard) Specific Factors Model of a small open economy that initially exports good X, analyze the effects on

i) outputs of the goods, X and Y,

ii) the real wage of labor,

iii) the real rental price of capital in the X industry, and

iv) the quantity of X exported

due to the following changes (one at a time):

a. A fall in the price of good Y, holding the price of good X constant.

The fall in price of good Y causes the Y industry’s value of marginal product of

labor curve, VMP=pMP, to shift down, as shown below. Labor market LYYLY

equilibrium is found at a lower nominal wage, w’, with an increase in the

amount of labor employed in the X industry and a decrease in L. Therefore, Y

output of X rises and output of Y falls, moving the country down and to the right

on its PPF.

Econ 441 Alan Deardorff

Problem Set 4 - Answers Specific Factors Models

Page 3 of 5

The effect on the real wage is ambiguous. It has fallen in nominal terms, and

therefore with respect to the price of X, which has not changed. But the price of

Y has fallen, and the wage would have to have fallen to w shown below to equal 0

that fall in price (since that is the amount that the pMP curve has fallen), and yLY

it hasn’t done that. So w/p is down, but w/p is up. Xy

w w YX Y

C’

C

Q w w’ Q’ w0

pMP pMPYLY XLX p’MPYLY

O X O LX Y L X X

Nominal payments to capital in the X industry have increased, since the increase

in L increases capital’s marginal product (or note the increased area below X

VMP above w’). Since p is unchanged and p has fallen, this is a real LXXY

increase.

Looking at output and consumption in the PPF diagram, we see that output of X

has increased, while consumption of X may have risen or fallen (it is shown

having risen) depending on income and substitution effects. Without any

assumption about preferences, we can’t be sure that exports of X increase.

b. An increase in the size of the labor force.

This expands the diagram horizontally, since its horizontal dimension is the

labor endowment. It is drawn below keeping the O origin fixed, and therefore X

shifting to the right both the right-side vertical axis and the VMP curve, since LY

that is drawn with respect to that axis. The result is that the intersection with

the unchanged VMP curve also moves right, but not as much, and occurs at a LX

lower w.

This fall in the wage, holding prices and capital stocks fixed, requires a fall in

the marginal product of labor and therefore that employment expands in both

industries. Thus output rises in both industries. The fall in the wage is a real

decline, since both prices are fixed. And the increased employment increases

the returns to capital in both sectors, also in real terms since prices are fixed.

Econ 441 Alan Deardorff

Problem Set 4 - Answers Specific Factors Models

Page 4 of 5

Here again we cannot be sure what happens to exports, since both production

and consumption of X expand.

w w w YX Y Y

C’

Q’ C Q w w’

VMPLY

(VMP)’ VMPLYLX

O O X OX Y Y

c. Destruction of a part of the capital stock employed in the Y industry.

This reduces the marginal product of labor in the Y industry, shifting the VMP LY

curve down. The labor-market diagram looks the same as in part (a), except

that the curve has shifted due to changing the MP function instead of changing LY

p. This time, however, the PPF is shrunken inward, since less of Y (but not X) Y

can be produced. In fact, from the labor-market diagram we see again that

since L increases, output of X rises. Output of Y falls, of course, both because X

of the loss of capital and then also because of the fall in L as labor also leaves Y

the industry.

w w YX Y

C

C’ Q

w w’ Q’

pMP pMPYLY XLX pMP YLY

O X O LX Y L X X

The real wage falls, because the nominal wage falls and prices have not

changed.

Econ 441 Alan Deardorff

Problem Set 4 - Answers Specific Factors Models

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Capital that survives in the Y sector actually gains, though this is a bit hard to

see. Since w has fallen with no change in p, MP must have fallen. But YLY

marginal products depend on the ratio of capital to labor, so K/L must have YY

fallen. This in turn means that MP must have risen, and therefore that r has KYY

gone up. Since p is also unchanged, this is a real increase. X

This time we can say what happens to exports. The fall income reduces

consumption of X (if X is normal), while the output of X has increased.

Therefore exports of X rise.

3. The figure below shows an initial production equilibrium in the (standard) Specific Factors Model. It differs from what we usually assume, however, in that the marginal product of labor in the Y-industry is assumed to drop to zero at a certain industry size, and the initial equilibrium has it at that size.

a) Show what the production possibility frontier looks like for this economy, where

on the PPF it is operating in this initial equilibrium, and what the budget line of

consumers in the aggregate must look like.

The PPF is now kinked as shown at the

right by the curve AQB. That is, it is

horizontal at output A out to point Q,

then curved like a normal PPF down to

some point B on the horizontal axis. It is

initially operating at the kink, so the

budget line of consumers is the straight

line DQE reflecting the initial prices.

b) Suppose now that the relative price of good Y, , rises. What will happen to pY

outputs of X and Y and to the real wage of labor?

The rise in price shifts up the ;；pMPL curve as shown, but YY

this does not move its vertical

portion. Thus the intersection

with pMPL is unchanged XX

and the nominal wage

remains at its initial level.

However, since the price of

one of the goods has risen

Econ 441 Alan Deardorff

Problem Set 4 - Answers Specific Factors Models

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(and the other has not changed), the real wage is reduced.

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