1. Chapter 32 Exchange
Key Concept: Pareto optimum and Market Equilibrium
The First and Second Theorems

2. Chapter 32 Exchange
Before we focus on partial equilibrium analysis.
We now turn to general equilibrium analysis:
how demand and supply conditions
interact in several markets to determine
the prices of many goods.

3. Tax on oil imports, S shifts to the left, price of oil ↑, demand for natural gas ↑, price of natural gas ↑, oil demand ↑, price of oil ↑,…

4. We first look at the case of pure exchange where we ignore production side for the moment.
Consumers have endowments and they trade among themselves.
First, we introduce a graphical tool known as the Edgeworth box.

5. The Edgeworth box
2 goods (1, 2)
2 consumers (A, B)
xA=(xA1, xA2), xB=(xB1, xB2) (allocation)
wA=(wA1, wA2), wB=(wB1, wB2) (endowment)

6. Feasible allocation
xA1+ xB1= wA1+ wB1 and
xA2+ xB2= wA2+ wB2 (the allocation is in the box).

7. A feasible allocation x=(xA, xB) is Pareto optimal if there is no other feasible allocation x’ =(xA’, xB’) that Pareto dominates it.
That is, there is no feasible allocation x’ =(xA’, xB’) such that (xA’ wA xA and xB’ sB xB) or (xA’ sA xA and xB’ wB xB).

8. Set of all Pareto optimal allocations: Pareto set
The part of Pareto set where both are at least as well off as endowment: the contract curve

9. Fig. 31.1

10. Fig. 31.2

11. Notice that in general on the Pareto set, MRSA1,2=MRSB1,2 for the usual reason.

12. We now turn to the concept of equilibrium.
Suppose prices are (p1, p2).
Then A maximizes his utility and chooses xA=(xA1, xA2).
Similarly, B maximizes his utility and chooses xB=(xB1, xB2).
We say the market is an equilibrium if consumers maximizes utilities, producers maximizes profits and markets clear (demand equals supply).

13. We say the market is an equilibrium if
consumers maximizes utilities,
producers maximizes profits and
markets clear (demand equals supply).

14. In a pure exchange economy
suppose (p1, p2) are equilibrium prices
we have to check
1) consumers maximizes utilities
2) xA1(p1, p2,wA1,wA2)+xB1(p1, p2,wB1,wB2)
=wA1+wB1
xA2(p1, p2,wA1,wA2)+xB2(p1, p2,wB1,wB2)
=wA2+wB2

15. Note that only relative prices matter
Note that only relative prices matter. If (p1, p2) is good, so is (2p1, 2p2).
Related to Walras’ law.

16. Fig. 31.3

17. Fig. 31.4

18. p1(xA1-wA1)+p2(xA2-wA2)=0 and
p1(xB1-wB1)+p2(xB2-wB2)=0.
Thus p1(xA1+xB1-wA1-wB1)+p2(xA2+xB2-wA2-wB2)=0.
The value of aggregate excess demand is identically zero. This holds for any prices, not just the equilibrium prices.
If one market clears, the other clears too.

19. The First Theorem of Welfare Economics: every competitive equilibrium is Pareto optimal.

20. Fig. 31.7

21. Suppose we have an equilibrium (p1, p2), (xA, xB) which is not Pareto optimal.
Then there exists a feasible allocation (xA’, xB’) that Pareto dominates.
Without loss of generality suppose it is the case that (xA’ wA xA and xB’ sB xB).

22. Without loss of generality suppose it is the case that (xA’ wA xA and xB’ sB xB). Then we must have p1xB1’+p2xB2’>p1wB1 +p2wB2.
Under some mild condition, we will have p1xA1’+p2xA2’ ≧p1wA1 +p2wA2. So p1(xA1 ’ +xB1 ’ -wA1 -wB1 )+p2(xA2 ’ +xB2 ’ -wA2 -wB2 )>0.
But this violates feasibility of (xA’, xB’).

23. Draw a case where the first theorem is violated.

24. Fig. 31.8

25. Roughly, Pareto optimum requires MRSA1,2=MRSB1,2.
The market achieves this because by consumer utility maximization MRSA1,2=p1/p2=MRSB1,2.

26. The Second Theorem of Welfare Economics: every Pareto optimum is a competitive equilibrium for some initial allocation of goods.
Illustrate this. Draw a case where the second theorem is violated.

27. The two welfare theorems can be used to justify market mechanism.
First, equilibrium is Pareto optimal.
Second, even if we think a particular Pareto optimum is not equitable, and the society should aim for a more equitable Pareto optimum, then the second theorem tells us that the issue of distribution and efficiency can be separated. A lump sum tax is used to achieve equity and then market is used to achieve efficiency.

28. Minor point on demonstrating an ordinary monopolist and a perfectly discriminating monopolist in the Edgeworth box.

29. In the former, the monopolist chooses a point on another’s offer curve to max his utility.

30. Fig. 31.5

31. In the latter, the monopolist chooses a point on another’s indifference curve through the endowment to max his utility.
Hence it is generally inefficient in the former while efficient in the latter.

32. Fig. 31.6

33. Chapter 32 Exchange
Key Concept: Pareto optimum and Market Equilibrium
The First and Second Theorems