12. General equilibrium: An exchange economy Varian, Chapter 30

The simplest market Two people, or agents Two goods Agent A and Agent B Two goods Good x and Good y Agents interact by exchanging or trading goods There is no production of either good

Questions we’re interested in Will unregulated exchange lead to “good” outcomes? Under what conditions? How does trade occur? Bargaining? Using prices? Can an outsider (e.g., government) intervene to improve things?

Endowments Endowments A has (wAx, wAy) B has (wBx, wBy) y (wAy+wBy) Any possible bundle of goods for A is in this space Endowments A has (wAx, wAy) B has (wBx, wBy) Endowment wAy x Person A (wAx+wBx) wAx

Preferences y (wAy+wBy) Endowment A’s indifference wAy curves x Person A (wAx+wBx) wAx

The simplest market Allocation: A pair of consumption bundles (xA, yA), (xB, yB) Feasible allocation: pair of consumption bundles that add up to total endowment (xA, yA)+(xB, yB)=(wAx, wAy)+(wBx, wBy)

The Edgeworth Box: endowments Person B wBx y Endowments A has (wAx, wAy) B has (wBx, wBy) wBy Endowment Any feasible allocation of goods among the agents is a point in this box wAy Person A x wAx

The Edgeworth Box: preferences Person B wBx y B’s indifference curves wBy Endowment A’s indifference curves wAy Person A x wAx

Trade in the Edgeworth Box Person B wBx y B’s indifference curves wBy Points to which A would be willing to trade Endowment A’s indifference curves wAy Person A x wAx

Trade in the Edgeworth Box Person B wBx y B’s indifference curves wBy Points to which B would be willing to trade Endowment A’s indifference curves wAy Person A x wAx

Mutually beneficial trade Person B wBx y B’s indifference curves If trade is voluntary they should end up somewhere in here wBy Endowment A’s indifference curves wAy Person A x wAx

What does the result look like? Person B wBx y At there are no more gains from trade wBy Endowment wAy Person A x wAx

Describing potential outcomes Pareto set, or contract curve: The set of all points that could be the outcome of a bargain Depends on w Can only make one individual better off by making the other worse off

The Pareto Set Person B wBx y Potential bargaining outcomes from endowment Allocation if A has all the bargaining power wBy Pareto set Allocation if B has all the bargaining power Endowment wAy Person A x wAx

Example: Pareto set Consumer A: Consumer B: Find the contract curve (wAx, wAy)=(5,10) u(xA, yA)=1/3 ln(xA) + 2/3 ln(yA) Consumer B: (wBx, wBy)=(10,5) u(xB, yB)=1/2 ln(xB) + 1/2 ln(yB) Find the contract curve

The Pareto Set wBx Person B 10 y 15 Endowment wAy 10 5 wBy Pareto set Person A wAx x

Can we narrow down outcomes? So far we’ve said nothing about the mechanism or process by which people trade We’ve found that agents should get to the contract curve….. …..but we’re not sure what point they’ll reach on that curve If trading is via prices, this indeterminacy can be resolved

What about prices? Excess supply of x Excess demand of x A price-based process Set py=1 Try a value of px=p Gives slope of budget line for both consumers Find gross demands of both consumers Vary p until demand is a feasible allocation Excess supply of x Lower p Excess demand of x Raise p

Gross demand A Person B wBx y wBy Endowment wAy Person A x wAx

Gross demand B Person B wBx y wBy Endowment wAy Person A x wAx

Market equilibrium At these prices, excess demand and excess Person B wBx y At these prices, excess demand and excess supply are both zero wBy Endowment wAy Person A x wAx

Prices and excess supply Amount of x that A wants to sell wBx Person B y Amount of x that B wants to buy Excess supply of x wBy Endowment wAy Budget constraint Person A x wAx

Prices and excess demand wBx Person B y Excess demand for y Amount of y that A wants to buy wBy Endowment Amount of y that B wants to sell wAy Budget constraint Person A x wAx

A’s price offer curve Person B wBx y Agent A’s price offer curve wBy Endowment wAy x Person A wAx

B’s price offer curve Person B wBx y wBy Agent B’s price offer curve Endowment wAy Person A x wAx

Using price offer curves to find the market equilibrium Person B wBx y Agent A’s price offer curve wBy Agent B’s price offer curve Endowment wAy Person A x wAx

Example: Price offer curves Consumer A: (wAx, wAy)=(5,10) u(xA, yA)=1/3 ln(xA) + 2/3 ln(yA) Consumer B: (wBx, wBy)=(10,5) u(xB, yB)=1/2 ln(xB) + 1/2 ln(yB) Find the IOCs and the equilibrium allocation and price.

The Pareto Set wBx Person B 10 8.57 y 15 wAy 10 5 wBy 9 6 Endowment POC A POC B Pareto set 5 6.43 15 Person A wAx x

The First Fundamental Theorem of Welfare Economics All equilibria resulting from a competitive market are Pareto efficient There are no gains from trade available from the result of a price-based exchange There may be other undesirable properties of a market outcome

The First Fundamental Theorem of Welfare Economics Person B wBx y The market equilibrium allocation is on the contract curve, so is Pareto efficient wBy Contract curve Endowment wAy Person A x wAx

The Second Fundamental Theorem of Welfare Economics If preferences are convex, any Pareto efficient allocation can be reached by a competitive market, with the correct redistribution of the endowments Using redistribution then allowing price-based trade, a planner can choose any PE allocation

The Second Fundamental Theorem of Welfare Economics Person B wBx y Pareto efficient allocations Can the market get us to any Pareto efficient allocation we want? wBy Answer: Yes, with convex preferences and redistribution Trade Endowment wAy Redistribute good x from B to A Person A x wAx

Policy implications How to help the poor Give them money! Price controls, quantity controls, etc lead to inefficient outcomes