• Chapter 32 Exchange
• Key Concept: Pareto optimum and
Market Equilibrium
• The First and Second Theorems
• 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, we introduce a graphical tool
known as the Edgeworth box.
• 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 (the allocation is 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: the contract curve
• Notice that in general on the Pareto set,
MRSA1,2=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).
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We say the market is an equilibrium if
consumers maximizes utilities,
producers maximizes profits and
markets clear (demand equals supply).
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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
• 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-wA2wB2)=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.
• The 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).
• 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
MRSA1,2=MRSB1,2.
• The market achieves this because by
consumer utility maximization
MRSA1,2=p1/p2=MRSB1,2.
• 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.
• 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.
• Chapter 32 Exchange
• Key Concept: Pareto optimum and
Market Equilibrium
• The First and Second Theorems