General General Equilibrium and Equilibrium and

Market EfficiencyMarket Efficiency

Production EconomyProduction Economy

PlanPlan

• Pure exchange economiesPure exchange economies– Definition of a Pareto Efficient allocationDefinition of a Pareto Efficient allocation– Competitive equilibrium allocationCompetitive equilibrium allocation– First Welfare TheoremFirst Welfare Theorem– Second Welfare TheoremSecond Welfare Theorem

• Production EconomiesProduction Economies– Production Possibilities frontierProduction Possibilities frontier– Equilibrium AllocationEquilibrium Allocation– First Welfare TheoremFirst Welfare Theorem

Exchange Economy Exchange Economy

• Imagine a world with two Imagine a world with two individuals, Ann and Bill, and two individuals, Ann and Bill, and two goods, clothing and food.goods, clothing and food.

• Each of the individuals has an Each of the individuals has an initial endowment represented as initial endowment represented as a pair a pair (c,f)(c,f) the quantities of the quantities of clothing and food correspondingly.clothing and food correspondingly.

• See Edgeworth boxSee Edgeworth box

Pareto EfficiencyPareto Efficiency

• An allocation of goods in an economy An allocation of goods in an economy is is Pareto efficientPareto efficient if there is no other if there is no other allocation that will make at least one allocation that will make at least one individual in the economy better off individual in the economy better off without worsening the well-being of without worsening the well-being of the others.the others.

• There may be several Pareto There may be several Pareto efficient allocations of goods.efficient allocations of goods.

The Big QuestionsThe Big Questions

• Will free markets allocate goods Will free markets allocate goods efficiently?efficiently?– It depends on production technology and It depends on production technology and

preferencespreferences– 11stst Welfare Theorem Welfare Theorem

• Can distributional equity and economic Can distributional equity and economic efficiency issues be separated?efficiency issues be separated?– The answer again, depends on technology The answer again, depends on technology

and preferencesand preferences– 22ndnd Welfare Theorem Welfare Theorem

A simple production A simple production economyeconomy

• Assume that two goods can be Assume that two goods can be produced in the economy, clothing and produced in the economy, clothing and foodfood

• Production of each one of the goods Production of each one of the goods requires capital and labor.requires capital and labor.

• Total quantities of the inputs are fixedTotal quantities of the inputs are fixed• What is the optimal allocation of What is the optimal allocation of

capital and labor to the two activities?capital and labor to the two activities?• How to allocate the goods to Geoffrey How to allocate the goods to Geoffrey

and Elizabeth?and Elizabeth?

The Edgeworth BoxThe Edgeworth Box

Condition 1 determining Pareto Condition 1 determining Pareto Efficient Allocation (Efficiency in Efficient Allocation (Efficiency in

Consumption)Consumption)• Assume that Elizabeth’s Assume that Elizabeth’s

and Geoffrey’s and Geoffrey’s preferences are preferences are (strictly) monotonic and (strictly) monotonic and convexconvex

• Then in a Pareto Then in a Pareto optimal allocation the optimal allocation the marginal rates of marginal rates of substitution between substitution between the two goods (apples the two goods (apples and raspberries) of and raspberries) of Elizabeth and Geoffrey Elizabeth and Geoffrey should be equal.should be equal.MRSR , A

1 R1 , A1 =MRSR ,A2 R 2 , A 2

Assume E. is ready toexchange at most 5 units of raspberries for 1 unit of apples, but G’s MRS between raspberries and apples is 3

Then a benevolent planner can offer to take 4 units of raspberries from E. and give it to G. in exchange for 1 unit of apples. Both will agree, as the will be happier under the new allocation. Thus, the initial allocation was not Pareto optimal

Contact CurveContact Curve

• Contract Curve is Contract Curve is a set of all Pareto a set of all Pareto efficient efficient allocationsallocations

• It also describes It also describes all allocations that all allocations that may result from a may result from a voluntary voluntary contracts between contracts between rational informed rational informed economic agents.economic agents.

Condition 2 determining Pareto Condition 2 determining Pareto Efficient Allocation (efficiency in Efficient Allocation (efficiency in

production)production)• Assume that Assume that

technologies are technologies are convexconvex

• Then in a Pareto Then in a Pareto optimal allocation optimal allocation the marginal rates the marginal rates of technical of technical substitution in substitution in production of the production of the two goods should two goods should be equal.be equal.

MPLC

MPKC=MPL

F

MPK F;

MRTSL ,KC LC , KC =MRTSL ,KF LF , K F

Assume MRTS for clothing is 5, while MRTS for food is 3

Then a benevolent planner can [fill the blank].Therefore more food and clothing can be produced with the same amount of resources.Thus, the initial allocation was not Pareto optimal

Generating Production Generating Production Possibilities FrontierPossibilities Frontier

• Contract Curve in the Contract Curve in the Edgeworth Edgeworth production box is a production box is a set of all Pareto set of all Pareto efficient allocations of efficient allocations of inputs to production inputs to production of clothing and foodof clothing and food

• The curve contains all The curve contains all the points the points (allocations) for (allocations) for which which

•Production possibilities frontier is the set of all possible output combinations that can be produced with a given endowments of factor inputs.•The slope of PPF is

MPLC

MPKC=MPL

F

MPK F;

MRTSL ,KC LC , KC =MRTSL ,KF LF , K F

MRT C ,F=MPK F

MPKC=MPL

F

MPLC

Efficiency in ProductionEfficiency in Production

Condition 3 determining Pareto Condition 3 determining Pareto Efficient Allocation (Efficient Efficient Allocation (Efficient

Production mix)Production mix)• Define the Marginal rate Define the Marginal rate

of transformation between of transformation between clothing and food, clothing and food,

•

• In a Pareto optimal In a Pareto optimal allocation the marginal allocation the marginal rates of substitution rates of substitution between the two goods between the two goods should equal to the should equal to the marginal rate of marginal rate of transformation:transformation:

MRSC ,F=MRT C , F

Assume MRT between clothing and food is 5 and MRS of both consumers is 3

Then a benevolent planner can [fill the blank]. Therefore, both consumers will be happier.Thus, the initial allocation of inputs across production of food and clothing was not Pareto optimal

MRT C ,F=MPK F

MPKC=MPL

F

MPLC

Equilibrium AllocationEquilibrium Allocation• Assume Elizabeth and Geoffrey own capital and a (fixed Assume Elizabeth and Geoffrey own capital and a (fixed

amount) of labor. They want to consume clothing and food.amount) of labor. They want to consume clothing and food.• Consider a Walrasian Auctioneer who announces prices for Consider a Walrasian Auctioneer who announces prices for

clothing, food, capital and labor in the economy clothing, food, capital and labor in the economy

• Once the prices are announced, the producers determine Once the prices are announced, the producers determine the amounts of labor and capital they want to employ, the amounts of labor and capital they want to employ, Elizabeth and Geoffrey announce their demand for Elizabeth and Geoffrey announce their demand for clothing and food.clothing and food.

• The procedure continues till the markets for capital, labor, The procedure continues till the markets for capital, labor, clothing and food clear (supply for each good equals to the clothing and food clear (supply for each good equals to the demand)demand)

PC , P F ,r , w

Condition 1 for Pareto Condition 1 for Pareto Efficient Allocation is Efficient Allocation is

satisfiedsatisfied• Consumers choose how much to Consumers choose how much to

consume (clothing, food for Elizabeth consume (clothing, food for Elizabeth and Geoffrey):and Geoffrey):– Marginal rate of substitution of Elizabeth is Marginal rate of substitution of Elizabeth is

equal to the ratio of output prices equal to the ratio of output prices – The same is true for the marginal rate of The same is true for the marginal rate of

substitution of Geoffreysubstitution of Geoffrey

• Thus, the marginal rates of substitution Thus, the marginal rates of substitution of Elizabeth and Geoffrey are equal to of Elizabeth and Geoffrey are equal to each other!each other!

MRSC ,F1 C 1 , F 1 =

PCP F

=MRSC ,F

2 C2 , F 2

Condition 2 for Pareto Condition 2 for Pareto Efficient Allocation is Efficient Allocation is

satisfiedsatisfied• Firms choose a combination of inputs Firms choose a combination of inputs

(labor, capital in each type of production):(labor, capital in each type of production):– Marginal rate of technical substitution Marginal rate of technical substitution

between labor and capital in production of between labor and capital in production of food equals to the ratio of input pricesfood equals to the ratio of input prices

– The same is true for MRTS in production of The same is true for MRTS in production of clothingclothing

• Thus, the marginal rates of technical Thus, the marginal rates of technical substitution in both activities are equal to substitution in both activities are equal to each other!each other!MRTSL ,K

C LC , KC =wr=MRTS L, K

F LF , K F

Condition 3 for Pareto Condition 3 for Pareto Efficient Allocation is Efficient Allocation is

satisfiedsatisfied• Firms choose the level of output (total clothing, total Firms choose the level of output (total clothing, total

food):food):– The market value produced by the last unit of input The market value produced by the last unit of input

(capital, labor) should equal to its rental price in (capital, labor) should equal to its rental price in production of food production of food

– The same is true for the market value of inputs in The same is true for the market value of inputs in production of clothingproduction of clothing

• Thus, the market value produced by the last unit of Thus, the market value produced by the last unit of capital is the same in both activities. The same is capital is the same in both activities. The same is true for labor. The market value of the last unit of true for labor. The market value of the last unit of capital is its marginal product times the price of capital is its marginal product times the price of output. Therefore,output. Therefore,PC MPK

C=P F MPKF=r ; so MRT C ,F=

MPK F

MPK C=PCP F

=MRSC ,F

First Welfare TheoremFirst Welfare Theorem

• If – consumers and producers act as price

takers;– there is a market for every commodity;– all the commodities are rival and

excludable (there are no externalities neither in consumption nor in production);

– consumers’ preferences and the production technologies are “well-behaved”

• Then a market allocation is Pareto Efficient

Second Welfare TheoremSecond Welfare Theorem

• Suppose that the assumptions for Suppose that the assumptions for the first welfare theorem holdthe first welfare theorem hold

• Then for every Pareto efficient Then for every Pareto efficient allocation there are prices that allocation there are prices that support a competitive equilibrium support a competitive equilibrium with transfers, which generates this with transfers, which generates this allocation.allocation.

From the Production Possibilities From the Production Possibilities Frontier to the Utility Possibilities Frontier to the Utility Possibilities

FrontierFrontier

Example 1Example 1

• Ann’s considers food and clothing as Ann’s considers food and clothing as perfect complements, for Geoffrey they perfect complements, for Geoffrey they are one-to-one substitutes. Elizabeth are one-to-one substitutes. Elizabeth has 20 units of food and 40 units of has 20 units of food and 40 units of clothing. Geoffrey has 20 units of each.clothing. Geoffrey has 20 units of each.

• Is the initial allocation PO? What is the Is the initial allocation PO? What is the set of Pareto Optimal Allocations?set of Pareto Optimal Allocations?

• What prices can support (efficient) What prices can support (efficient) allocation allocation C A=30 , F A=30

C B=30 , F B=10

PPF, an examplePPF, an example• There are 100 units of capital and labor There are 100 units of capital and labor

in country A.in country A.• A unit of capital produces a unit of food. A unit of capital produces a unit of food.

Capital and labor are perfect (1-to-1) Capital and labor are perfect (1-to-1) substitutes in the production of food. substitutes in the production of food. The same is true for clothing. The same is true for clothing.

• What is the PPF of country A? What is the PPF of country A?

Q = K L

PPF and gains from tradePPF and gains from trade• There are 100 units of capital and labor in country B.There are 100 units of capital and labor in country B.• Two units of capital produces a unit of clothing. Capital Two units of capital produces a unit of clothing. Capital

and labor are perfect (1-to-1) substitutes in the and labor are perfect (1-to-1) substitutes in the production of clothing. production of clothing.

• The food production technology is the same as in AThe food production technology is the same as in A• • What is the PPF of country B?What is the PPF of country B?• Will countries A and B gain from trade?Will countries A and B gain from trade?• What sector will support the trade agreement?What sector will support the trade agreement?• What sector will lose? Is it possible to compensate the What sector will lose? Is it possible to compensate the

losers?losers?

F=K F L F

C=2K C2Lc

Example 3Example 3

• There are 40 units of capital and labor.There are 40 units of capital and labor.• A unit of capital produces a unit of food. A unit of capital produces a unit of food.

Capital and labor are perfect (1-to-1) Capital and labor are perfect (1-to-1) substitutes in the production of food. substitutes in the production of food.

• What is the PPF? What is the PPF? • What are the Pareto optimal quantities of What are the Pareto optimal quantities of

clothing and food to be produced if the clothing and food to be produced if the preferences of Elizabeth and Geoffrey are the preferences of Elizabeth and Geoffrey are the same as in the previous example?same as in the previous example?

C=FC K c , LC =K C LC