Download Econ 4413 International Trade Keith Maskus

Econ 4413
International Trade
Brief answers to Problem Set 1
Keith Maskus
Textbook questions, chapter 2:
1. With identical production functions, when X and Y face the same factor prices, they will choose
the same K/L (kx = ky) ratios (which must both equal the economy's endowment ratio,k ). Thus, the
efficiency locus is the diagonal line of the Edgeworth Box. With CRS, the PPF will be a straight
line, showing constant marginal (opportunity) costs. With IRS, the PPF will be convex ("bowed in"
) to the origin. While the efficiency locus is the diagonal in both cases, the numerical values of the
isoquants would be different, with constant increases in output as X (or Y) rises for CRS, but rising
increases in output as X (or Y) rises for IRS.
2. Endowment changes would shift the size of the Edgeworth box, as below. This means the PPF
would shift in a biased way toward labor-intensive good X and away from Y. We can't tell from the
problem whether the endpoints of the PPF would be higher or lower, since one factor rises and one
falls.
Y PPF0
K0
K1
PPF1
L0
L1
X
3. In figure 2.8, draw a straight ray from Ox through point B. You will see that it cuts the X3
isoquant below the efficiency locus. Because these isoquants are homothetic, the slope of X3
isoquant at this intersection equals the slope of X1 isoquant at B. This means the slope of X3
isoquant at the tangency to Y0 isoquant on the efficiency locus is steeper, implying a higher w/r
(=) at this point. Note also that the ray from Ox through the tangency between X3 and Y0
isoquants is steeper than the original ray you drew, so kx rises as output of X rises and output of Y
falls. (You should show also that ky is higher at the new point than at B.) Thus,  rises as output of
labor-intensive good rises and output of capital-intensive good falls.
Problems on Problem Set.
2. a. Total Product of Labor is the set of figures provided. Marginal Product of Labor, for L rising
from 1 to 11, is: 200, 170, 130, 100, 75, 65, 60, 55, 45, 40, 30.
Spools
TPL
MPL
L
b. Both TPL and MPL rise (they do not double except for special cases).
3. a. on your own; same idea as above. b. When labor force is cut in half, the TPK and MPK both
fall (they do not get cut in half except in special cases).
c. As the capital-labor ratio rises, each laborer has more capital to work with, making each more
productive. Since average product of workers rises, so must total product (more output for given
labor force). But also the marginal worker is more productive. But if the capital-labor ratio rises, it
means each unit of capital has less labor to work with, so capital is less productive in both total and
marginal terms. A more technical answer is that any rise in the K/L ratio can be decomposed into
an equal percentage rise in K and L (which holds MPL and MPK constant) and then a further rise in
K for the new, fixed labor force (which then raises the MPL and lowers the MPK). Overall, we
have MPL rises as k rises, but MPK falls as k rises.
4.
a. Consider X. If you double L and K you get 2L + 4K = 2(L + 2K) = 2X. Same idea for Y.
Isoquant for X: K = X/2 - L/2, which is a straight line with slope (dK/dL) of -1/2 (this
holds X constant, which is the meaning of an isoquant). Isoquants are drawn below.
K
Y0
X0
L
b. Consider Y. If you double L and K you get (2K)1/2(2L)1/2 = 2(K)1/2(L)1/2 = 2Y, so there is
CRS. Same idea for X. Isoquant for Y: K = (Y/L1/2)2 = Y2/L. You need a bit of calculus
here to get dK/dL = -(Y2/L2), which is negatively sloped for a given Y (for example, if Y =
5 the isoquant slope is -25/L2). It is also convex, meaning the slope diminishes as L rises
and K falls. Isoquants are drawn in problem 5.
5.
a. the exponent on capital in the production function for Y is higher (in a Cobb-Douglas
production function this means the capital-labor ratio will be higher in Y than X).
b. for X: dK/dL = -(2X3/L3) for Y: dK/dL = -(Y2/L2).
c. Let C0 be the level of cost. Then the isocost line in X is C0 = wLX + rKX .
d. Drawn below.
K
C0/r
Y0
X0
C0/w
e. Slope of isocost: since K = C0/r - (w/r)L, the slope is -(w/r) = -. But we know from the
text and lecture that the slope of the X isoquant is -(MPLx/MPKx). If you can take partial
derivatives, it is easy to show that MPLx/MPKx = 2(Kx/Lx) = 2kx and that MPLy/MPKy =
(Ky/Ly) = ky. Note this means that in equilibrium we have  = 2kx = ky. Interestingly, then,
for these two production functions Y is twice as capital-intensive as X, meaning that X is
twice as labor-intensive as Y.
NOTE CAREFULLY: I DO NOT EXPECT YOU TO BE ABLE TO PERFORM CALCULUS TO
SUCCEED IN THIS CLASS. THIS EXERCISE WAS DESIGNED TO ILLUSTRATE SOME
POINTS ABOUT ISOQUANTS, ETC. YOU WILL BE EXPECTED TO BE ABLE TO USE
GRAPHICAL PROPERTIES OF PRODUCTION FUNCTIONS, PPFS, AND SO ON, BUT NOT
TO DIFFERENTIATE THEM MATHEMATICALLY.
6. Diagrams are below. For the Leontief function, ky does not change as  changes. For CobbDouglas, kx does change; in particular it gets less capital intensive as  falls.
K
K
Y0
kx2
ky
kx1
2
L
1
X0
L
7.
a. Box below. b. kx = ky = 40/100 = 0.4.
k = 0.4
40
(K)
OY
Efficiency locus
OX
100 (L)
Recall that the overall endowment ratio must be a weighted average of the capital-labor
ratios in X and Y. If these ratios are both the same, so must be the endowment ratio, so all 3 are
equal to 0.4
8.
a. for Y = 1000, Ly = 100, Ky = 40. for Y = 750, Ly = 75, Ky = 30. and so on.
b. kx = ky = 0.4 (except at zero outputs). c. PPF is a straight line (below). d. constant
marginal costs means you give up an equal amount of Y for every gain in X; implies a constant
MRT, or slope of PPF.
Y
X
Note you might want to figure out the endpoints of this PPF and also determine its slope.
9. On your own. Here the PPF is convex but kx and ky continue to be 0.4, so the efficiency locus in
the Edgeworth box is the diagonal. Decreasing costs means that for equal reductions in Y the
economy gains increasing amounts of X at the margin, so the MRT is declining.
10. On your own. This is the standard case discussed in the textbook and in some of the problems
above. Increasing costs (concave PPF) means that for equal reductions in Y the economy gains
decreasing amounts of X at the margin, so the MRT is rising.
NOTE: FOR QUESTIONS BELOW ASSUME A CLOSED ECONOMY WITH NO
INTERNATIONAL TRADE.
11. Note in the hint that the capital-labor ratio of the resources released by Y is smaller than the
original K-L ratio there, but it is also bigger than the original K-L ratio in X where the resources are
absorbed. Thus, both sectors become more capital-intensive. Because capital-labor ratios get
higher as  increases (recall the isoquant diagrams), it follows that  must be higher. Output of X
rose and that of Y fell. Intuition here is that as Y contracts and X expands, there is an excess
demand for labor and an excess demand for capital, causing w to rise and r to fall in order to restore
equilibrium at the new output levels.
12. PPF is below. Since it's concave it has increasing opportunity costs (rising MRT).
Y
100
70
40
0
50
75 100
X
13. Note the chord between points (X,Y) = (75,40) and (100,0) has slope of -(40/25), or -1.6. This
is flatter than the slope right at the point (90,25), which I would estimate to be -2.0 (a nice round
number). This means that p = (px/py) = 2, or X is twice as expensive as Y (one X is worth two Y).
Consumption in Y terms: pX + Y = 2(90) + 25 = 205 units of Y.
National Income in X terms: X + (1/p)Y = 90 + (1/2)25 = 102.5 units of X.
If px = $3 billion (and so py = $1.5 billion) we calculate
nominal national income = 3(90) + 1.5(25) = $407.5 billion
14. Note the chord between points (100,0) and (75,40) has slope of -1.6 and that between (75,40)
and (50,70) = -(30/25) = -1.2. Thus, if the slope of the PPF around (90,25) is perhaps -2.0, while
the slope around (75,40) to be around -1.5. So the change would be from -2.0 to -1.5, or a fall in p
from 2.0 to 1.5. Clearly, this is a very crude estimate.
15. The price ratio is -300/500 = -0.6. This means that one X is worth 0.6Y. So if real income rose
to 750 X it would be equivalent to a rise in real income to 750*0.6 = 450Y.
16. Questions 1, 2, 4, and 5 in Chapter 3:
1. This simple diagram will do. Equilibrium points exist at A and B.
Y
A
B
X
2. Consumers could be classified as laborers and capital owners, for example. In such a case, if a
change in commodity prices lowers the wage rate and raises the price of capital, laborers would
suffer a lower budget constraint while capital owners would enjoy a higher budget constraint.
4. The budget constraints would become steeper due to lower price of Y and higher price of X.
Individual 1 would enjoy this change since she prefers good Y, but individual 2 would be worse off.
5. Moving to a higher community indifference curve would imply a higher level of national income.
Thus, we could try to devise a policy in which we tax some of the gains away from the winners and
use the revenues to compensate the losers, so that no one is worse off and some people are better
off. Note for this policy to work we would like the tax-redistribution policy to have no effect on
incentives to work and save (that is, to be an efficient "lump-sum" redistribution, which is
extremely difficult to achieve).
Individual Utility Theory
17. a. Draw these yourself. The slope of the indifference curve is the Marginal Rate of Substitution
between goods Y and X; it indicates the increase in X consumption that is required to maintain
constant utility for each marginal loss in Y consumption.
b. Budget constraint: I0 = pxCx + pyCy, where the C's indicate consumption levels. In slopeintercept form the budget constraint is: Cy = I0/py -(px/py)Cx and it is graphed below. The slope is
thus the relative price ratio. If income rises, the budget constraint shifts out in a parallel movement.
If the price of X rises it pivots toward the origin along the X axis. If the price of Y falls it pivots
away from the origin along the Y axis.
Y
I0/py
A
X
c. Shown above, at A. At the equilibrium we see MRS = px/py, or the marginal rate at which the
consumer wishes to trade off Y for X exactly equals the price ratio, which is the rate at which she
can trade off Y for X in the market.
18. The substitution effect is the movement from A to B below and is associated with the change in
prices, holding utility fixed; note that because we move along the given indifference curve there
must be a rise in consumption of X and a fall in consumption of Y due to this effect. The income
effect is the movement from B to C due to the higher real income from the price change. A normal
good is one for which consumption rises as real income rises, but an inferior good is one for which
consumption falls as real income rises.
Y
A
C
B
X
Aggregation of Preferences
19. Aggregate demand for X would depend only on relative prices and total income, not the
distribution of income between people 1 and 2: Dx = D(p, NI), where NI = I1 + I2.
Y
(Y/X)*
Y
2
1
p*
X
X
For a given p*, homothetic preferences means that individuals consume along the ray (Y/X)* for
any income level. In the diagram, let point 1 indicate consumption for individual 1 (along her
budget constraint) and point 2 be consumption for individual 2. Person 2 has higher income than
person 1. Now note that if you reversed the points, so individual 1 had the higher income, the total
amounts of X and Y demanded would not be changed. But if preferences were identical but not
homothetic, we could have the kind of situation shown in the right diagram. Convince yourself that
shifting income from person 2 to person 1 would not necessarily result in the same demand for X
and Y.
*20. a. To solve this problem you would do the following:
Maximize U1 subject to the budget constraint: I1 = pxX1 + pyY1 and do the same for individual
2.
Solving this problem involves setting up the Lagrangean function:
L = (X1-X*)a(Y1-Y*)(1-a) + (I1-pxX1-pyY1)
Take partial derivatives with respect to X1,Y1, and  and set them equal to zero. Solve the
resulting three equations for the demand curves for X1 and Y1, which will be (after some algebra):
X1 = (1/px)[aI1 + pxX* - pyY*] - aX*
Y1 = (1/py)[(1-a)I1 - pxX* + pyY*] - (1-a)Y*
(Same expressions with "2" subscripts will hold for individual 2.)
b. Suppose we take income away from 2 and give it to 1. Then we can derive:
X1/I1 = -(X2/I2) = apx. Use the diagram in Figure 3.7.
21. On your own.
General Equilibrium and Excess Demand Curves
22. Suppose H imports X, then we have:
EXxH = IMxF.
But each country has balanced trade in value terms, so that:
pEXxH = IMyH and pIMxF = EXyF. Combine these facts to show:
IMyH = EXyF.
23. In the graph below, autarky production and consumption are at Ah, with relative price ph. In
free trade production point is Bh, consumption point is Ch, with relative price p*; note that H
imports good X and exports good Y. National income in terms of good X is point Vh in autarky
and point V* in free trade. Extend these same lines up to the Y axis to find national income in
terms of good Y.
Y
CICh
Bh
Z
Ch
CIC*
Ah
ph
p*
Vh
Quantity of exports is ZBh and quantity of imports is ZCh.
V*
24. Just using Figure 4.5, we see pa is autarky price in H and pa* is autarky price in foreign.
Equilibrium quantities of X are Ex (F's imports) and Ex* (H's exports). The rectangles formed
by these X quantities and the international price ratio p* would give trade quantities for Y (F's
exports and H's imports). Welfare rises because each country moves further and further away
from autarky as the terms of trade improve.
25. The "terms of trade" are defined as a country's export price divided by its import price (more
generally, an index of export prices divided by an index of import prices). Thus, in Figure 4.5, p is
H's terms of trade and 1/p is F's terms of trade. As H's excess demand curve shifts to the left, it is
supplying more exports at any relative price. This drives down the relative price p (show this),
implying a worsening in H's terms of trade. A shift to the right of F's excess demand curve means a
rise in its import demand at any price, therefore raising p, which is a worsening in F's terms of
trade.
Gains from Trade
26. is clear from the construction of excess demand curves. That is, the further away from autarky
on an excess demand curve, the higher is the associated community indifference curve.
27. See diagram below. Gains from exchange are the movement from A to Cs (holds output fixed
but allows consumers to trade at free-trade prices). Gains from specialization are the movement
from Cs to T (allows output to shift at free-trade prices). Total gains from trade are movement from
A to T.
p*
p*
Y
T
Cs
A
B
X
Note in this case I've drawn the country exporting X and importing Y.
28. I think I'll leave this one to you. Your answer should focus on what free trade would do to the
real incomes of workers in the textile sector (see Chapter 8 for a full discussion), while your
counter-argument should focus on the gains from trade theorem.