ECONOMICS 8413
ASSIGNMENT 1
FALL, 1999
Production Frontiers
1. Consider the two-good, two-factor neoclassical production model with constant returns and
different factor intensities:
(a) x1 = F(L1,K1) (b) x2 = G(L2,K2)
(c) L1 + L2 = L (d) K1 + K2 = K.
Allow each industry to minimize costs subject to an output constraint, say:
min wL1 + rK1 subject to F( ) = x10 .
Show that this minimization leads to Pareto efficiency in resource allocation:
 = w/r = MPL1/MPK1 = MPL2/MPK2 .
Use this result and totally differentiate system (a) - (d) to show that the slope of the PPF equals the
ratios of marginal products:
-dx2/dx1 = MPK2/MPK1 = MPL2/MPL1 .
Allow each industry to maximize profits, say, 1 = P1F( ) - wL1 - rK1 and derive the conditions:
PiMPLi = w PiMPKi = r i = 1,2
Show also that profit maximization in the competitive case implies:
Pi = MCi i = 1,2, where MCi indicates marginal costs. In turn, these results imply that
w/MPLi = r/MPKi = MCi .
Use the above results to show that the slope of the PPF also equals the ratio of marginal costs and,
therefore, the relative price:
-dx2/dx1 = MC1/MC2 = P1/P2 = p.
Finally, show that the MRT, or the slope of the PPF, exhibits rising marginal costs:
-(d2x2/dx12) > 0 if both goods are produced (hint: consider the sign of dp/dx1).
2. In the basic two-good, two-factor model, let the production functions for x1 and x2 be identical.
Derive the general shape of the production frontier under the following assumptions:
a. Both factors are mobile between sectors and there are constant returns to scale;
b. Both factors are mobile between sectors and there are increasing returns to scale;
c. Capital is sector-specific (the Ricardo-Viner model) and there are constant returns to
scale.
Factor Intensities
3. The (2x2) model is characterized by two equations systems:
1.
aL1( )x1 + aL2( )x2 = L aK1( )x1 + aK2( )x2 = K
2.
aL1( )w + aK1( )r = P1 aL2( )w + aK2( )r = P2
Let x2 be capital-intensive. Indicate what this ranking means in terms of the factor-output
coefficients. For a given set of a's, state and prove the Rybczynski theorem. Also demonstrate that
/p > 0.
4. Consider the following Cobb-Douglas production functions:
x1 = AL1aK11-a x2 = BL2bK21-b 0 < a,b < 1 A,B > 0.
Derive the demand functions in each sector for L and K. Show that the ratio of capital-labor ratios
(K1/L1)/(K2/L2) = k1/k2
equals some constant and that x2 is capital-intensive if a > b.
5. Consider the following CES production functions:
x1 = [A1L1-a + A2K1-a]-1/a
A1,A2 > 0, 1+a > 0.
-b
-b -1/b
x2 = [B1L2 + B2K2 ]
B1,B2 > 0, 1+b > 0.
Demonstrate that the ratio of capital-labor ratios becomes:
k1/k2 = C1-2 , where C is a constant and the  parameters are elasticities of substitution.
Let 1 > 2. Draw the associated isoquants and determine the critical * at which a factor-intensity
reversal occurs.
National Utility, SICs, and Demand
6. This is an exercise on Scitovsky SICs. Let A,B denote individuals, U denote utility, and x1,x2
denote goods. Then the Scitovsky problem is to
minimize x2 = x2a + x2b
(1)
subject to x1a + x1b = x1
(2)
Ua(x1a,x2a) = Ua
(3)
Ub(x1b,x2b) = Ub
(4)
a. Set up the Lagrangean, derive the first-order conditions and show that they imply:
dx1a/dx2a = dx1b/dx2b (or MRSa = MRSb) (5)
b. Use (1), (2), and (5) to show that:
dx2/dx1 = dx2a/dx1a = dx2b/dx1b
What is the significance of this equation for welfare analysis?
c. What must be true about the second-order conditions in b. to establish the solution as a
minimum?
d. Draw an Edgeworth Box and develop the Scitovsky SIC for a particular distribution.
Now suppose the dimensions of the box rise (there are more aggregate goods available). Provide a
rigorous argument that: (i) if both individuals are better off due to this increase the resulting SIC
does not intersect the original one and (ii) if one individual is better off and one is worse off due to
this increase the resulting SIC intersects the original one at some point.
7. Consider a Cobb-Douglas utility function with minimum consumption requirements for each
individual but the same exponents:
Ua = (c1a - c1a)(c2a - c2a)1-
Ub = (c1b - c1b)(c2b - c2b)1-
a. Using budget constraints, derive the demand functions for each individual for goods x1
and x2.
b. For either individual, show that her demand is homothetic if c1 = c2 = 0. Show that if c2
= 0 but c1 >0 then (c1/c2)/I < 0, so that her demand is not homothetic with respect to the origin.
Derive the income elasticities of her demand for both goods and show that the elasticity for x1 <
that for x2.
Offer Curves
8. Consider an endowment economy with fixed endowments x1* and x2*. Its utility function is
well-defined and Cobb-Douglas across the two goods:
U = x1x21- , which it maximizes subject to the budget constraint:
p(x1* - x1) + (x2* - x2) = 0, where p = P1/P2 .
a. Determine the demand functions for both goods.
b. Derive the offer curve (hint: determine expressions for m2 = (x2 - x2*) and e1 = (x1* - x1))
and verify that it satisfies balanced trade at all relative prices. Draw the offer curve (note: in this
case the offer curve will not be backward-bending).
c. Derive the import-demand elasticity () and the export-supply elasticity ():
 = (m2/p)(p/m2)
 = (e1/p)(p/e2)
and verify that  = -1. What are the values of these elasticities at autarky and as p   ?
Gains from Trade
9. Give a complete set of assumptions (that is, what must be ruled out?) sufficient to prove that free
trade is Pareto-superior to autarky. Develop a sufficient condition for restricted trade to be Paretosuperior to autarky.
Duality
10. Define the revenue (GNP) function carefully. Show and explain the following properties of the
revenue function:
a. rp(p,v) = x(p,v); (p'' - p')(x'' - x')  0
b. rv(p,v) = w(p,v); (w'' - w')(v'' - v')  0
c. rpv(p,v) = xv(p,v); rvp(p,v) = wp(p,v)
d. rvv(p,v) = 0 for a small open economy with fixed international factor prices
Ricardian Model
11. Consider a Ricardian model in which labor requirements per unit of output are
Good X1 Good X2
US
5
4
Japan
5
2
a. What is the structure of absolute advantage? What is the structure of comparative
advantage and what are the limits to the terms of trade, p* = p1/p2?
b. Assuming the US has 300 units of labor available and Japan has 400 and that preferences
are such as to split the labor force in half in autarky, develop the PPFs, autarky equilibrium, the
offer curves, and free trade equilibrium. What would the world PPF look like in this case?
c. Show that real wages rise for one country in the move to free trade. What are the limits
to the wage differential wJ/wUS in free trade and why? Which country has the higher real wages, if
either?
12. Draw the Dornbusch-Fischer-Samuelson continuum-of-goods diagram where A(z) = a*(z)/a(z)
gives foreign labor requirements per unit relative to home labor requirements per unit and  =
w/w*.
Suppose there is a general technology improvement in the foreign country. Indicate in the diagram
what happens to A, z, and . Let i index a representative good that shifts its production location
from H to F. Show that the real wage in terms of i is higher in both F and H after this shift.
Heckscher-Ohlin Model and Related Notions
13. We have shown that when there are smoothly convex isoquants with no factor intensity
reversals the resulting intensity diagrams in the right-hand diagram below have the indicated
relationship.
Develop the corresponding intensity diagrams for the following isoquant maps:
a. Straight lines:
b. Faceted:
c. Coincident over some range:
14. Consider the following assumptions of the (2x2x2) model: (i) identical technologies across
countries; (ii) identical homothetic demands across countries; (iii) no factor-intensity reversals.
Explain or show which of these restrictions are needed to prove each of the following three
theorems: a. the Heckscher-Ohlin theorem; b. the Factor Price Equalization theorem; c. the StolperSamuelson theorem.
Specific Factors Model
15. Consider a model with 2 goods X1 (labor-intensive) and X2 (capital-intensive) where intensity
rankings refer to a long-run equilibrium. In the short run there are 3 factors (mobile labor and
sector-specific capital types). Suppose a small open economy facing fixed international goods
prices is relatively capital abundant. Analyze the short-run and long-run impacts of the following
separate changes on domestic outputs, factor prices, and income distribution: a. an export tariff on
X2; b. an increase in the labor endowment; c. a decline in the X2-capital stock.
Verbally, what modifications (if any) to the analysis in part b. would be needed if this were a large
economy?
Integrated World Equilibrium
16. Use the FPE diagram to show the indeterminacy of production and trade patterns in the FPE set
with two factors and three goods. Does the addition of a third good make FPE more or less likely?
In the two-factor, n-good model with different factor prices, is there an indeterminacy in production
or trade? Why or why not?