© Michael E. Wetzstein, 2012
Chapter 6: Pure Exchange
Outline and Conceptual Inquiries
There are Gains from Trade
Consider a Two-Commodity and Two-Household Economy How do you feed the multitudes?
The World Viewed in an Edgeworth Box Can you put a whole economy in one box?
Application: Why the Edgeworth Box is Not a Pareto Box Understanding a Pareto-Efficient Allocation
Contract Curve (Pareto-Efficient Allocation) Why are Price-Signals Efficient?
Why should you love free markets?
Application: POW Camp
Illustration of the First Fundamental Theorem of Welfare Economics How much are you willing to exchange apples for oranges?
Walras’ Law and the Beauty of Price Signals Is there excess demand in an economy?
Improving Social Welfare
Why have equal opportunity legislation?
Application: Equitable Allocations and Equal Access
Equitable Endowments Pure Exchange
Pareto-Efficient Allocation
Competitive Price System
Fair Social Welfare
Allocation
© Michael E. Wetzstein, 2012
Summary
1. In partial-equilibrium analysis, only one segment of an economy is analyzed, without consideration of possible interactions with other segments. This is in contrast to general- equilibrium analysis, which investigates interactions among agents across market segments.
2. An efficient allocation of commodities is possible by households trading commodities, and an allocation of commodities is said to be Pareto-efficient if no one household can be made better off without making some other household worse off. The necessary condition for such an efficient allocation is that the marginal rates of substitution among all households are equal.
3. The set of Pareto-efficient allocations is represented by the contract curve in an Edgeworth box diagram. At every point on the contract curve, all gains from trade are exhausted.
4. A problem with the Pareto-efficient criterion is it does not result in a complete social ranking of alternative allocations. There are many policies that result in a gain to one agent but a loss to another. For such policies, the Pareto criterion will not aid in determining the social desirability of the policy.
5. As the number of households increases, the process of bartering for commodities becomes increasingly difficult. Since transaction costs escalate with increases in the number of
households, the competitive price system, where all commodities are valued in the market by their money equivalence, offers a Pareto-efficient mechanism for allocating commodities.
6. The Pareto efficiency of the competitive price system is established by the First Fundamental Theorem of Welfare Economics. Every perfectly competitive allocation is Pareto-optimal.
7. The Second Fundamental Theorem of Welfare Economics states that every Pareto-efficient allocation has an associated perfectly competitive set of prices.
8. The Walrasian equilibrium in a general equilibrium model is established through a
tâtonnement (trial-and-error) process. At equilibrium in every market, all markets clear, so aggregate demand is equal to aggregate supply.
9. A Pareto-efficient allocation of commodities does result from a Walrasian equilibrium;
however, it may not correspond to the optimal social-welfare allocation. A Walrasian equilibrium is based on a given initial allocation of endowments, so for a social-welfare optimal this distribution of endowments must be optimal.
10. An equitable distribution of initial endowments is where no household prefers any other household’s initial endowment. Given an equitable distribution of endowments, the
Walrasian equilibrium will result in a fair allocation of commodities. In such cases, a state of
© Michael E. Wetzstein, 2012
optimal social welfare may be obtained.
11. Providing equal opportunities for enriching a household’s endowments is an equitable distribution of endowments. As indicated by US history, a mating of equal opportunity with free markets can lead toward an optimal social-welfare allocation of commodities.
Key Concepts
contract curve core solution Edgeworth box equitable fair allocation feasible allocation
First Fundamental Theorem of Welfare Economics
general-equilibrium analysis Pareto-efficient allocation
Pareto improvement
partial-equilibrium analysis perfectly competitive price system price system
pure-exchange economy
Second Fundamental Theorem of Welfare Economics
tâtonnement Walras’ Law
Key Equations
MRS1 = MRS2 = … = MRSn
A necessary condition for a Pareto-efficient allocation of commodities.
An allocation is feasible if aggregate
consumption of each commodity is equal to aggregate supply.
MRS1 = MRS2 = … = MRSn = p1/p2
Walrasian equilibrium, where every
household’s rate of tradeoff equals society’s rate of tradeoff.
p1z1 + p2z2 = 0
Walras’ Law, where the value of the aggregate excess demand is zero.
© Michael E. Wetzstein, 2012
TEST YOURSELF
Multiple Choice
1. In a pure-exchange economy
a. Households produce and exchange commodities
b. Households exchange their endowments, but no production takes place
c. Households can trade their initial endowments of commodities, but no production takes place.
d. Prices are used for trading.
2. Suppose there are only three households in an economy. The necessary condition for an efficient allocation of commodities is
a. MRS1 = MRS2 = MRS3 = λ b. MRS1 + MRS2 + MRS3 = λ c. MRS1= MRS2 = MRS3
d. MRS1= MRS2 = MRS3 = MRS.
3. Suppose Robinson, R, and Friday, F, are the only two households in an economy. At the current allocation of x1 and x2, MRSR(x2 for x1) = ¾ while MRSF(x2 for x1) = 1. Which of the following is correct?
a. A Pareto improvement could be made if Robinson traded some x1 for x2
b. A Pareto improvement could be made if Friday traded some x1 for x2
c. A Pareto improvement could be made if Robinson received some x1 and x2 from Friday
d. A Pareto improvement could be made if Friday received some x1 and x2 from Robinson.
4. An Edgeworth box demonstrates
a. The supply and demand for two commodities
b. The feasible allocations of two commodities for two households c. Equitable allocations of commodities
d. All of the above.
5. What is the criterion for judging if an allocation is feasible?
a. The total quantity of each commodity consumed must be equal to the total available from endowments
b. MRS for each household must be equal
c. The price ratio must be equal to the ratio of the initial endowments d. Each household must not be envious of other households’ endowments.
6. A Pareto-efficient allocation occurs
a. When all gains from trade have been exhausted
© Michael E. Wetzstein, 2012
b. When there is no way to make someone better off without making someone else worse off
c. on the contract curve d. All of the above.
7. Suppose an economy consists of two households consuming two commodities (x1 and x2).
The contract curve for these households illustrates the a. Allocations for which a Pareto improvement is possible b. Initial endowments of commodities and contracts for trading c. Allocations that are Pareto-efficient
d. Prices for making a contract.
Refer to the following graph for Questions 8 and 9:
0R Fish x1
Bread x2
0
e2 C
e1 R R
e1 + e1
R F
e2 + e2
R F
A
B
U2R
U1 F U1
U2 R F
8. Which allocations are Pareto-efficient?
a. A b. C c. A and B d. A, B, and C.
9. The Pareto improvement is a. Between and b. below
c. Between and d. between and
© Michael E. Wetzstein, 2012
10. In a perfectly competitive price system,
a. A household has no control over the prices it faces b. Barter is used instead of exchanging money c. Endowments are efficient
d. A Pareto-efficient outcome is unattainable.
11. The First Fundamental Theorem of Welfare Economics requires that a. Households consume their initial endowments of commodities
b. Every Pareto-efficient allocation has an associated perfectly competitive set of prices c. A Pareto-efficient allocation maximizes social welfare
d. Every perfectly competitive allocation is Pareto-efficient.
12. A Walsrasian equilibrium occurs when n households’ MRS(x2 for x1) and prices yield a. MRS1 = MRS2 = …= MRSn = p2/p1
b. MRS1 + MRS2 + …+ MRSn = p2/p1
c. MRS1 + MRS2 + …+ MRSn = p1/p2 d. MRS1 = MRS2 = …= MRSn = p1/p2. 13. An offer curve is analogous to the
a. Price consumption curve b. Engel curve
c. Income consumption curve d. Contract curve.
14. At the Pareto-efficient allocation for two commodities two households (1 and 2), a. MRS1 = MRS2
b. The households’ offer curves intersect
c. The households’ indifference curves are tangent d. All of the above.
15. Suppose the demand functions for two households (S and J) are and The aggregate excess demand function for x1 is then
a.
b.
c.
d.
16. Walras’ Law implies that
a. Excess supply can occur but excess demand cannot b. Excess demand can occur but excess supply cannot c. The value of aggregate excess demand will equal zero
d. Excess demand for any one commodity must be equal to zero.
© Michael E. Wetzstein, 2012
Short Answer
1. Explain the difference between a partial equilibrium and a general equilibrium model.
Provide an example of each.
2. Suppose an economy consists of two households (Marjean and Kelly) that purchase two commodities (x1 and x2). Marjean’s MRSM(x2 for x1) is 3 while Kelly’s MRSK(x2 for x1) is 5.
Is this allocation of x1 and x2 efficient? Explain. If not, how could x1 and x2 be reallocated to improve efficiency?
3. Graph an Edgeworth box diagram explaining what is meant by a Pareto-efficient allocation.
Also, describe the core solution. What factors will determine the exact allocation of commodities chosen?
4. Comment on the following: “Every point on the contract curve results in economic efficiency, but social welfare is not maximized at every point.”
5. Describe the First and Second Fundamental Theorems of Welfare Economics.
6. Demonstrate how a household’s offer curve can be derived.
7. Explain Walras’ Law.
8. Alberta believes that society is better off when endowments of commodities are equitable.
Explain to her why this may be undesirable.
© Michael E. Wetzstein, 2012
Problems
1. Suppose Tom and Jerry have the utility functions and with and Find the contract curve (Pareto-efficient set).
2. Suppose Homer and Marge have the utility functions and
with initial endowments of and Find
the Walrasian equilibrium allocations of x1 and x2.
3. Refer to Problem 2. Find Homer’s and Marge’s offer curves.
4. Suppose Sara and Andrea have the utility functions and
with initial endowments and Find the
Walrasian equilibrium allocations for x1 and x2.
5. An economy consists of two households (Carly, C, and Katherine, K) and two commodities (x1 for bottled water and x2 for pizza). Carly’s initial endowments are 4 waters and 1 pizza, while Katherine’s initial endowments are 3 waters and 3 pizzas. Assume that Carly and Katherine have the same identical utility function U = 3x1x2.
a. Graph the associated Edgeworth box.
b. Determine each household’s marginal rate of substitution (pizza for water).
c. On the graph, illustrate indifference curves for Carly and Katherine at their initial endowments.
d. Determine the Pareto-efficient condition. On the graph, indicate the locus of Pareto- efficient allocations.
e. Determine Carly’s and Katherine’s consumption bundles in a competitive equilibrium.
What are the relative prices of bottled water and pizza at this equilibrium?
