The problems in this chapter focus primarily on the simple two-good general equilibrium model
in which “supply” is represented by the production possibility frontier and “demand” by a set of
indifference curves. One shortcoming of this approach is that students do not see the interaction
between output and input markets. Problems 13.7 and 13.8 seek to remedy this by using the
computer general equilibrium model presented in the chapter. The Analytical Problems in the
chapter illustrate a few general equilibrium “theorems,” but no very formal proofs are intended.
Comments on Problems
13.1 This problem repeats an example from Chapter 1 in which the production possibility
frontier is concave (a quarter ellipse). It is a good starting problem because it involves
very simple computations.
13.2 This problem is a simple example of general equilibrium with linear production functions
and differing preferences among the two people in the economy.
13.3 This problem is a fixed-proportions example that yields a concave production possibility
frontier. This is a good initial problem although students should be warned that calculus-
type efficiency conditions do not hold precisely for this type of problem.
13.4 This problem uses a quarter-circle production possibility frontier and a Cobb–Douglas
utility function to derive an efficient allocation. The problem then proceeds to illustrate
the gains from trade. It provides a good illustration of the sources of those gains.
13.5 This problem provides a numerical example of an Edgeworth Box in which efficient
allocations are easy to compute because one individual wishes to consume the goods in
fixed proportions.
13.6 This provides an example of efficiency in the regional allocation of resources. The
problem could provide a good starting introduction to mathematical representations of
comparative versus absolute advantage or for a discussion of migration. To make the
problem a bit easier, students might be explicitly shown that the production possibility
frontier has a particularly simple form for both the regions here (e.g., for region A it is
22
100xy+=
).
CHAPTER 13:
General Equilibrium and Welfare
WelfareExternalities and Public Goods
13.7 This problem uses the computer model to examine the consequences of various changes
in preferences or technology. Having students try to explain why things turn out the way
they do is a good way to build intuition.
Analytical Problems
13.8 Tax equivalence theorem. This problem uses the computer simulation model to shows
the formal equivalence between input and output taxes.
13.9 Returns to scale and the production possibility frontier. Here students are asked to use
Excel or some other software to illustrate the shape of production possibility frontiers
with varying degrees of returns to scale. One result is that frontiers can still be convex
with modest increasing returns providing input proportions are sufficiently different.
13.10 The trade theorems. This problem provides simple two-good graphical proofs of three
major trade theorems: (1) factor-price equalization; (2) the Stolper–Samuelson theorem;
and (3) the Rybczynski theorem. Although it requires only facility with the production
box diagram (and its underlying Edgeworth Box), it is a fairly difficult problem. Extra
credit might be given for the correct spelling of the discoverer of the third theorem.
13.11 An example of Walras’ law. This problem is a algebraic example of how Walras’ law
can be used to find the excess demand function for good 1.
13.12 Productive efficiency with calculus. This problem illustrates how the simple two-good
general equilibrium model of production can be solved for efficient allocations using
calculus. Especially important is to show how the tradeoffs implied by the calculus
results can be interpreted as providing equilibrium relative prices.
13.13 Initial endowments, equilibrium prices, and the first theorem of welfare economics.
This problem shows how initial endowments can constrain the possible prices that can
emerge from competitive bargaining. This would be a good opening to discussing the
concept of the “core” of a competitive economy, though that concept is not explicitly
covered in Chapter 13.
13.14 Social welfare functions and income taxation. This problem explores the complex
relationship between social welfare and the appropriate tax function.
Solutions
Chapter 13: General Equilibrium and Welfare
148
c. The slope of the frontier is given by
b. If the wage is 1, each person’s income is 10. Smith’s demand for
x
is
3x
xp=
13.3 Let f denotes food and c cloth.
d. The frontier is concave because it must satisfy both constraints. Since the RPT = 1
13.4 The PPF has the form
200.fc+=
Chapter 13: General Equilibrium and Welfare
150
b. Demand:
2.
pp=
For utility maximization,
2,MRS c f==
implying
2.cf=
13.5 a. Contract curve is straight line with slope of 0.5. The only price
Chapter 13: General Equilibrium and Welfare
151
b. An initial endowment for Smith of
80, 40
SS
ch==
is on the contract curve. At
c. An initial endowment of
80, 60
SS
ch==
for Smith is not on the contract curve.
Chapter 13: General Equilibrium and Welfare
152
13.7 a. By changing the utility of household 1 to
l
The utility-maximizing choices for household 1 are
118.10,x=
19.10,y=
and
b. After reversing the production functions, we obtain the following equilibrium
prices:
l
Chapter 13: General Equilibrium and Welfare
153
c. If the utility functions are changed to
The utility-maximizing choices for household 1 are
112.27,x=
16.48,y=
and
Analytical Problems
13.8 Tax equivalence theorem
Adding an ad valorem tax of 0.2 on goods x and y raises the same revenue (3.10) as an ad
valorem tax of 2.5 on capital and labor. Running the simulation with either tax yields the
same results:
Chapter 13: General Equilibrium and Welfare
154
13.9 Returns to scale and the production possibility frontier
We have never succeeded in deriving an analytical expression for all these cases. We
13.10 The trade theorems
For all of these proofs, draw the PPF and its underlying Edgeworth Box Diagram. The
world price ratio determines where production will occur on the PPF and where it will
occur in the Edgeworth Box. Given the assumption about factor intensities, the contract
curve in the Edgeworth Box will be concave (i.e., bowed upward).
Chapter 13: General Equilibrium and Welfare
155
13.11 An example of Walras’ law
b. Walras’s law states
0.
ii
Hence, if
23
0,ED ED==
then
11
0,p ED =
13.12 Productive efficiency with calculus
a. The problem for this society is to maximize utility subject to the technological
b. First-order conditions for a maximum are
c. A competitive equilibrium price ratio of
**
xy
pp
would cause utility-maximizing
consumers to choose
Chapter 13: General Equilibrium and Welfare
156
d. As discussed in the chapter, any factor that leads equilibrium prices to incorrectly
13.13 Initial endowments, equilibrium prices, and the first theorem of welfare economics
a. The value of A’s initial endowment is
.
AA
px y+
Hence, his or her demand for
b. Setting total demand equal to total supply for good x yields
c. With these initial endowments,
1p=
. Person A demands
d. Part (b) shows that increase in the endowment of either good for person A will
Chapter 13: General Equilibrium and Welfare
157
13.14 Social welfare functions and income taxation
a. The Lagrange problem for a welfare optimum is
b. A similar condition to that found in part (a) holds if the tax function is given by
c. If taxation is based on observed income we would need to model how
c
responds
d. If all individuals have the same relative weight in the social welfare function