• 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.

• 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 ↑,…

• 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, a graphical tool known as 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).

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

• 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).

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

• Notice that in general on the Pareto set, MRSA

1,2=MRSB1,2 for the usual reason.

• 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).

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

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

• 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.

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

• 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). 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’).

• Draw a case where the first theorem is violated.

• Roughly, Pareto optimum requires MRSA

1,2=MRSB1,2. The market achieves

this because by consumer utility maximization MRSA

1,2=p1/p2=MRSB1,2.

• 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.

• 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.

• Minor point on demonstrating an ordinary monopolist and a perfectly discriminating monopolist in the Edgeworth box. In the former, the monopolist chooses a point on another’s offer curve to max his utility. 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.