General Equilibrium and Market Efficiency

1. General Equilibrium and Market Efficiency Production Economy

2. Plan • Pure exchange economies • Definition of a Pareto Efficient allocation • Competitive equilibrium allocation • First Welfare Theorem • Second Welfare Theorem • Production Economies • Production Possibilities frontier • Equilibrium Allocation • First Welfare Theorem

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

4. The Big Questions • Will free markets allocate goods efficiently? • It depends on production technology and preferences • 1st Welfare Theorem • Can distributional equity and economic efficiency issues be separated? • The answer again, depends on technology and preferences • 2nd Welfare Theorem

5. A simple exchange economy • Imagine an economy with two consumers, Elizabeth and Geoffrey • They consume two goods, apples and raspberries • Suppose that total quantities of raspberries and apples are fixed in the economy • Each one of the consumers has an endowment of apples an raspberries. • Let Elizabeth and Geoffrey trade. Will the resulting allocation be Pareto Efficient?

6. The Edgeworth Box

7. Conditions determining Pareto Efficient Allocation • Assume that Elizabeth’s and Geoffrey’s preferences are (strictly) monotonic and convex • Then in a Pareto optimal allocation the marginal rates of substitution between the two goods (apples and raspberries) of Elizabeth and Geoffrey should be equal. Assume E. is ready to exchange 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

8. Utility-Improving Trades

9. Contact Curve • Contract Curve is a set of all Pareto efficient allocations • It also describes all allocations that may result from a voluntary contracts between rational informed economic agents.

10. Contract Curve • Contract Curve is a set of All Pareto efficient allocations. • Note it is independent of prices and of distribution of initial endowments. • Describe the Contract Curve: use the condition determining Pareto efficient allocations (equality of the marginal rates of substitution) as well as the two resource constraint (describing total amounts of goods in the economy). Depict the contract Curve in the Edgeworth box.

11. Competitive Equilibrium Allocation • Consider a Walrasian Auctioneer who announces prices for apples and raspberries in the economy. • Both agents behave as price-takers. • Once the prices are announced, Elizabeth and Geoffrey announce their supply (amount they will sell) and demand (amount they will buy) of apples and raspberries. • The procedure continues till the markets for apples and raspberries clear (supply for each good equals to the demand)

12. Competitive Behavior: Geoffrey

13. Competitive Behavior: Elizabeth

14. Elizabeth and Geoffrey are Price Takers • Assume that Elizabeth’s and Geoffrey’s MRS between apples and raspberries are as follows • Assume that Elizabeth initially holds 50 units of raspberries and 80 units of apples • Geoffrey’s endowment is 50 units of raspberries and 20 units of apples • What are the market clearing prices that will be announced by the auctioneer? Set price for apples equal to $4 and price for raspberries equal to $2. Calculate optimal bundles for both individuals. Will the markets clear? Now set both prices equal to each other. Repeat the exercise.

15. A simple production economy • Assume that two goods can be produced in the economy, clothing and food • Production of each one of the goods requires capital and labor. • Total quantities of the inputs are fixed • What is the optimal allocation of capital and labor to the two activities? • How to allocate the goods to Geoffrey and Elizabeth?

16. Condition 1 determining Pareto Efficient Allocation (Efficiency in Consumption) • Assume that Elizabeth’s and Geoffrey’s preferences are (strictly) monotonic and convex • Then in a Pareto optimal allocation the marginal rates of substitution between the two goods (apples and raspberries) of Elizabeth and Geoffrey should be equal. Assume E. is ready to exchange 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

17. Condition 2 determining Pareto Efficient Allocation (efficiency in production) • Assume that technologies are convex • Then in a Pareto optimal allocation the marginal rates of technical substitution in production of the two goods should be equal. 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

18. Efficiency in Production

19. Condition 3 determining Pareto Efficient Allocation (Efficient Production mix) • Define the Marginal rate of transformation between clothing and food, • In a Pareto optimal allocation the marginal rates of substitution between the two goods should equal to the marginal rate of transformation: 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

20. Generating Production Possibilities Frontier • Contract Curve in the Edgeworth production box is a set of all Pareto efficient allocations of inputs to production of clothing and food • The curve contains all the points (allocations) for 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

21. Y Example: General Equilibrium PPF X

22. Y Example: General Equilibrium • XeB Ys YeB PPF X Xs

23. Y Example: General Equilibrium • XeB Ys Slope = -p1e/p2e • YeB PPF X XeA Xs

24. Y Example: General Equilibrium • XeB Ys Slope = -p1e/p2e • YeA YeB PPF X XeA Xs

25. The three conditions for Pareto Efficiency • Efficiency in exchange (allocation of final goods is on the contract curve: marginal willingness to pay for goods (MRS) is equalized across consumers) • Efficiency in use of inputs (combination of final outputs is on the PPF: marginal productivity of inputs (MRTS) is equalized across production processes) • Efficiency in product mix (slope of the PPF is equal to the MRS: marginal rate of transformation (marginal cost of producing an additional unit of output (MRT) is equal to the marginal willingness to pay for this unit (MRS) )

26. Equilibrium Allocation • Assume Elizabeth and Geoffrey own capital and a (fixed amount) of labor. They want to consume clothing and food. • Consider a Walrasian Auctioneer who announces prices for clothing, food, capital and labor in the economy • Once the prices are announced, the producers determine the amounts of labor and capital they want to employ, Elizabeth and Geoffrey announce their demand for clothing and food. • The procedure continues till the markets for capital, labor, clothing and food clear (supply for each good equals to the demand)

27. Condition 1 for Pareto Efficient Allocation is satisfied • Consumers choose how much to consume (clothing, food for Elizabeth and Geoffrey): • Marginal rate of substitution of Elizabeth is equal to the ratio of output prices • The same is true for the marginal rate of substitution of Geoffrey • Thus, the marginal rates of substitution of Elizabeth and Geoffrey are equal to each other!

28. Condition 2 for Pareto Efficient Allocation is satisfied • Firms choose a combination of inputs (labor, capital in each type of production): • Marginal rate of technical substitution between labor and capital in production of food equals to the ratio of input prices • The same is true for MRTS in production of clothing • Thus, the marginal rates of technical substitution in both activities are equal to each other!

29. Condition 3 for Pareto Efficient Allocation is satisfied • Firms choose the level of output (total clothing, total food): • The market value produced by the last unit of input (capital, labor) should equal to its rental price in production of food • The same is true for the market value of inputs in production of clothing • Thus, the market value produced by the last unit of capital is the same in both activities. The same is true for labor. The market value of the last unit of capital is its marginal product times the price of output. Therefore,

30. First Welfare Theorem The Invisible Hand • 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

31. Second Welfare Theorem • Suppose that the assumptions for the first welfare theorem hold, and assume that both preferences and technologies are convex. • Then for every Pareto efficient allocation there are prices that support a competitive equilibrium with transfers, which results in this allocation.

32. PPF, an example • There are 100 units of capital and labor in country A. • A unit of capital produces a unit of food. Capital and labor are perfect (1-to-1) substitutes in the production of food. The same is true for clothing. • What is the PPF of country A?

33. PPF and gains from trade • There are 100 units of capital and labor in country B. • Two units of capital produces a unit of clothing. Capital and labor are perfect (1-to-1) substitutes in the production of clothing. • The food production technology is the same as in A • What is the PPF of country B? • Will countries A and B gain from trade? • What sector will support the trade agreement? • What sector will lose? Is it possible to compensate the losers?

34. Example 2 • There are 40 units of capital and labor. • A unit of capital produces a unit of food. Capital and labor are perfect (1-to-1) substitutes in the production of food. • What is the PPF? • What should be the value of MRS between clothing and food for Elizabeth and Geoffrey in an optimal allocation?

35. Summary • If consumers and producers • act in their self-interest, • take prices as given, • know exactly what they are buying and selling; • all goods are rivalry and excludable, • there is a market for every commodity; • and all the markets clear, • then the resulting allocation is Pareto (economically) efficient, so that no benevolent planner can improve the efficiency of the market allocation.

36. Summary, cont. 2. In addition, if preferences and technologies are convex, redistributing initial endowments and then letting consumers and producers act in their self-interest can achieve an efficient allocation with the desired degree of “fairness.”