Lecture II: Exchange continued

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Lecture II: Exchange continued. Chapter Thirty-One Varian. An Economy. An economy Consists of l commodities and n consumers. Each consumer i is characterized by a endowment vector of length l where the value of element j is the initial endowment quantity of good j ;

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### Lecture II: Exchange continued

Chapter Thirty-One Varian

An Economy
• An economy

Consists of l commodities and n consumers.

Each consumer i is characterized by

• a endowment vector of length l where the value of element j is the initial endowment quantity of good j;
• and a utility function that maps commodity bundles to a real number, and represents consumer i’s preference ordering on the commodity space.
Pareto Optimality

An feasible allocation is an assignment of a commodity bundle to each consumer, such that

A Pareto optimal allocation a feasible

such that there exist no other feasible

Characterizing Pareto Optimal Allocations

Let’s assume each consumer’s utility function is differentiable and quasi-concave.

Now consider the following maximization of aggregate utility function

Let be the Lagrange multipliers

Characterizing Pareto Optimal Allocations

Our first order Khun –Tucker conditions

• For each , or

2. Feasibility constraints:

Economic Implications

1. If xji > 0 for j = 1, 2 then

Thus we get

2. . If xji > 0 for j = 1, 2 and i = 1, 2 then we get MRS121=MRS122

Corner Solutions and Pareto Optimality
• Interior Case: MRSA= MRSB
• In a corner solution x11 = 0 and xji > 0 for i and j not 1. and MRS121≤MRS122

With a simple two person Edgeworth box economy we have four possible PO corner solutions.

Corner Solution I: if xA=0, then MRSA< MRSB

Corner Solution II: if xB=0, then MRSA> MRSB

Corner Solution III: if yA=0, then MRSA> MRSB

Corner Solution II: if yB=0, then MRSA<MRSB

• Let’s look at the four corner solution on the board
Competitive Equilibrium

Consider an economy

A competitive equilibrium of E is a price system p*=(p1, …, pl) and an allocation x*=(x1*,…, xn*) that satisfies

1.

2.

Walras’ Law (Implications on Prices)
• Walras’ Law is an identity; i.e. a statement that is true for any positive prices (p1,p2).
• The value of aggregate excess demand is equal to zero.
• In an economy with K goods, if there is equilibrium in K – 1, then remaining goods market must also be in equilibrium
• A competitive equilibrium is unique in relative prices, but not in absolute prices
Terms of demand
• w: initial endowment allocation
• x: goods in the final allocation for x
• x: gross demand for the good x
• (x – w): net demand for the good x, excess demand
• (xA + xB): aggregate demand for two person economy for good x
Walras’ Law
• Every consumer’s preferences are well-behaved so, for any positive prices (p1,p2), each consumer spends all of his budget.
• For consumer A (budget constraint):
• For consumer B:
Walras’ Law

Summing gives

Walras’ Law

Rearranged,

Walras’ Law

This says that the summed marketvalue of excess demands is zero for

any positive prices p1 and p2-- this is Walras’ Law.

Walras’ Law
• We often name z the aggregate excess demand function and, because x is a demand function with p, we get
• It stands for any p(p1, p2) > 0. It is the Walras’ law.
Implications of Walras’ Law

Suppose the market for commodity 1

is in equilibrium; that is,

Then

implies

Implications of Walras’ Law
• So one implication of Walras’ Law for a two-commodity exchange economy is that if one market is in equilibrium then the other market must also be in equilibrium.
• In general, if there are k markets, then only k-1 prices are determined at equilibrium.
• I.e., the general equilibrium determines only relative prices.
Implications of Walras’ Law

What if, for some positive prices p1 and

p2, there is an excess quantity supplied

of commodity 1? That is,

Then

implies

Implications of Walras’ Law

So a second implication of Walras’ Law

for a two-commodity exchange economy

is that an excess supply in one market

implies an excess demand in the other

market.

It also means that not any p(p1, p2) > 0

can ensure z1 = 0 and z2 = 0 at the

same time, even if any p(p1, p2) > 0 can

ensure p1z1+p2z2 =0

Implications of Walras’ Law
• Because the equilibrium prices are prices which lead demand for and supply of every good to be equal at the same time, the Walras’ law does not mean there must be at least a set of prices pE(pE1, pE2) > 0 which could result in z1 = 0 and z2 = 0 simultaneously.
• Therefore: We must find the equilibrium prices pE(pE1, pE2) > 0 if it/they would exist.
Relative Prices
• Walras’ law means there are only k-1 prices that can be determined independently in an economy with k goods. That means also, we can only get pi for good i that
• pi = pi(pj) i=1,2,…,k, i≠j
• Only after pj is determined, can we get all pi.
• pj is called a numeraire price. Good j is the numeraire.
Relative prices
• All prices except the numeraire price are prices relative to the numeraire price. Making it as 1, all prices are relative to the numeraire.
• That is: All prices are relative prices. There are no absolute prices.
• Changing pj from being 1 to being 2, all other prices will be double. But the exchange relations between each pair of all kinds of goods remain the same.
• In the economy with paper money, governments can multiply the numeraire price. We refer it to change in price level or in absolute price level.
Solving for the General Equilibrium
• Step 1: Specify the individual’s problem of maximizing utility function subject to the budget constraint. Note, we will set p2 = 1, and thus p1 is really the ratio of relative prices or the slope of the budget constraint.
• Step 2: Solve for each consumer’s system of demand equations as a function of p1.
• Step3: Write down the aggregate excess demand function for good 1.
• Step 4: Set the aggregate demand excess demand function equal to zero and solve for p1. Now you have the equilibrium price vector (p1*, 1)
• Step 5: Solve for each consumers’ equilibrium commodity allocation by substituting (p1*, 1) into each consumer’s system of demand equations.
Solving General Equilibrium -Example

Let wA = (4,8) and wB = (8,0) and

Demand Functions:

Aggregate excess demand for x

Solving General Equilibrium -Example

Setting zx = 0 and solving for p yields, p* = 6/5

Substituting into Demand Functions yields competitive equilibrium allocations

Equilibrium doesn’t have to be unique

Example: Let wA = (1,3) and wB = (3,1)

A and B have same utility function u(x, y)= Min{x,y}

OB

OA

• Trading in competitive markets achieves a particular Pareto-optimal allocation of the endowments.
• This is an example of the First Fundamental Theorem of Welfare Economics.
First Fundamental Theorem of Welfare Economics
• Given that consumers’ preferences are well-behaved, trading in perfectly competitive markets implements a Pareto-optimal allocation of the economy’s endowment.
• It means that competitive equilibrium will necessarily be Pareto efficient.
The Equilibrium with Monopoly
• A price setting monopoly leaves unrealized gains from exchange
• A perfect price setting monopoly maximizes gains from exchange and is therefore Pareto optimal
Second Fundamental Theorem of Welfare Economics
• The First Theorem is followed by a second that states that any Pareto-optimal allocation (i.e. any point on the contract curve) can be achieved by trading in competitive markets provided that endowments are first appropriately rearranged amongst the consumers.
Second Fundamental Theorem

OB

OA

The contract curve

Second Fundamental Theorem

Implemented by competitive

OB

OA

Second Fundamental Theorem

Can this allocation be implementedby competitive trading from w?

OB

OA

Second Fundamental Theorem

Can this allocation be implementedby competitive trading from w? No.

OB

OA

Second Fundamental Theorem

But this allocation is implementedby competitive trading from q.

OB

Slope of the budget line is the price that “decentalizes” the Pareto allocation

OA