1. Chapter 10 General Equilibrium and Economic Welfare
Capitalism is the astounding belief that the most wickedest of men will do the most wickedest of things for the greatest good of everyone.
John Maynard Keynes

2. Chapter 10 Outline
Challenge: Anti-Price Gouging Laws 10.1 General Equilibrium 10.2 Trading Between Two People 10.3 Competitive Exchange 10.4 Production and Trading 10.5 Efficiency and Equity Challenge Solution

3. Challenge: Anti-Price Gouging Laws
Background:
After a disaster strikes, equilibrium prices tend to rise.
Many state governments enforce anti-price gouging laws to prevent prices from rising, while prices may be free to adjust in neighboring states.
Questions:
Does a binding price control that affects one state, but not a neighboring state, cause shortages?
How does it affect prices and quantities sold in the two states?
Which consumers benefit from these laws?

4. Chapter 10 Introduction
For a market equilibrium to be efficient, two conditions must be met:
Consumption must be efficient
Happens if goods cannot be reallocated among people so that at least someone is better off and no one is worse off
Production must be efficient
Happens if it is impossible to produce more output at current cost given current knowledge
A Pareto improvement is a change, such as a reallocation of goods between people, that helps at least one person without harming anyone else. An allocation is Pareto efficient if no Pareto improvement is possible.
For a market equilibrium to be equitable, we need to be willing to make a value judgment about whether everyone has their fair share.

5. 10.1 General Equilibrium
Partial-equilibrium analysis is an examination of equilibrium and changes in equilibrium in one market in isolation.
By contrast, general-equilibrium analysis addresses how equilibrium is determined in all markets simultaneously.
This is especially important for markets that are closely related
Example:
discovery of oil deposit in a small country
citizens’ income is raised
increased income affects all markets in that country simultaneously (spillover effects)

6. 10.1 Competitive Equilibrium in Two Interrelated Markets
Consider linear demand functions for two goods, Q1 and Q2, as functions of their prices, p1 and p2:
The supply functions (with positive coefficients) are:
What do we do with these equations?
Equate Qd and Qs in each market

7. 10.1 Competitive Equilibrium in Two Interrelated Markets
After equating Qd and Qs, two equations and two unknowns can be solved for the prices of both goods:
These expressions for p1 and p2 can be substituted back into either demand or supply equations to yield a solution for Q1 and Q2.
Note that both prices and quantities are functions of all of the demand and supply coefficients.

8. 10.1 Competitive Equilibrium in Two Interrelated Markets
Comparative statics under general equilibrium model:
Comparative statics under partial equilibrium model:

9. 10.1 Minimum Wage with Incomplete Coverage
Partial-equilibrium analysis of minimum wage laws predicted unemployment

10. 10.1 Minimum Wages with Incomplete Coverage
General-equilibrium analysis of minimum wage laws indicates that unemployment need not be created.
How many hours will be hired in the covered sector under the minimum wage laws setting 𝑤 ?
1. 𝐿 𝐶 𝐿 𝐶 not above
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11. 10.1 Minimum Wages with Incomplete Coverage
After the minimum wage 𝑤 set in the covered sector, what is the supply function in the uncovered sector?
1. 𝑆 𝑆− 𝐿 𝐶 𝑆− 𝐿 𝐶 not above
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12. 10.1 Minimum Wages with Incomplete Coverage
General-equilibrium analysis of 𝑤 predicts:
A decrease in hired labors in the covered sector.
A lower wage in the uncovered sector.
A decrease in overall hired labors

13. 10.1 Minimum Wages with Incomplete Coverage - example
EX: labor demands in two sectors:
Covered: L1d= w
Uncovered: L2d=400-w
Supply: Ls=w
Equilibrium before the minimum wage policy.
Equilibrium wage: 260
Covered: L1d=120
Uncovered: L2d=140
Equilibrium after the minimum wage policy of 𝑤 =300.
Covered: L1=100; w1=300
Residual Supply: Ls=w-100
Uncovered: L2d=150; w2=250
Total employment drops (from 260 to 250) along with a lower wage rate in the uncovered sector.

14. Solved Problem 10.1
The government starts taxing the cost of labor by 𝝉 per hour in a covered sector only, so that the wage that workers in both sectors receive is w, but the wage paid by firms in the covered sector is w + τ.
Total employment, employment in the covered sector, and equilibrium wage rate drops.
Employment in uncovered sectors rises.

15. Solved Problem 10.1

16. 10.1 Minimum Wages with Incomplete Coverage - example
EX: labor demands before labor tax of $50 per hour:
Covered: L1d= w
Uncovered: L2d=400-w
Supply: Ls=w
Equilibrium before the taxation.
Equilibrium wage: 260
Covered: L1d=120
Uncovered: L2d=140
Equilibrium after the taxation in covered sector.
L1d in covered sector after taxation: L1d= (w+50)
After taxation eq.: w=250; L1=100; L2=150
Total employment drops along with a lower wage rate in both sectors.

17. 10.2 Trading Between Two People
General-equilibrium model can be used to show that free trade is Pareto efficient.
After all voluntary trades have occurred, we cannot reallocate goods so as to make one person better off without harming another.
Consider example of neighbors, Jane and Denise, who each have an initial endowment of firewood and candy
Jane: 30 cords of firewood and 20 candy bars
Denise: 20 cords of firewood and 60 candy bars
These endowments can be shown graphically using indifference curves.

18. 10.2 Trading Between Two People
Jane and Denise before they engage in trade

19. 10.2 Trading Between Two People
If Jane and Denise do not trade, they can each only consume their initial endowments.
In order to see whether Jane and Denise would benefit from trading firewood and candy bars, we use an Edgeworth box.
An Edgeworth box illustrates trade between two people with fixed endowments of two goods.
An Edgeworth box is useful in general equilibrium models because both the firewood and candy bar markets are being affected simultaneously.

20. 10.2 Trading Between Two People
Initial endowments place Jane and Denise at point e, but area B holds more preferred bundles for both.
Area B provides room for Pareto Improvement.

21. 10.2 Trading Between Two People
When is endowment e not Pareto Efficient?
Resources are not equally divided
MRS are not the same
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22. 10.2 Trading Between Two People
Suppose |MRSJ|=3>|MRSD|=1, how can the two trade to benefit both?
Jane feels 1C=3W, while Denise feels 1C=1W.
1C=4W
1C=2W
1C=0.5W
NOT ABOVE
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23. 10.2 Trading Between Two People
Suppose |MRSJ|=3>|MRSD|=1, how can the two trade to benefit both?
Jane feels 1C=3W, while Denise feels 1C=1W.
If 1C=2W, Jane is willing to give W for acquiring more C, while Denise is willing to give C for getting more W.
Both are strictly happier after trade.

24. 10.2 Trading Between Two People
Should Jane and Denise trade? Yes.
We make four assumptions about their tastes and behavior in order to answer this question:
Utility maximization: Each person maximizes her utility.
Usual-shaped indifference curves: Each person’s indifference curves have the usual convex shape.
Nonsatiation: Each person has strictly positive marginal utility for each good (e.g. each wants as much of each good as possible).
No interdependence: Neither person’s utility depends on the other’s consumption and neither person’s consumption harms the other person

25. 10.2 Trading Between Two People
No further trade is possible at a bundle like f, which is:
A mutually beneficial trade (compared to e)
On the contract curve, and therefore Pareto optimal. Note that Jane’s MRS is equal to Denise’s MRS at point f.

26. 10.2 Trading Between Two People
The contract curve is the set of all Pareto-efficient bundles.
Name refers to the fact that Jane and Denise are unwilling to engage in further trades, or contracts, only at points along the contract curve.
These allocations are the final contracts.
The contract curve is derived by maximizing Jane’s utility subject to leaving Denise’s utility unchanged (or vice versa).
Calculus can be used to show that this maximization problem boils down to points where their indifference curves have the same slopes: MRSj = MRSd.

27. 10.2 Trading Between Two People
Example 1:
Suppose UA(x1, x2)=UB(x1, x2)=x11/2x21/2. Let the endowments be w1A+w1B=100= w2A+w2B
The allocation (x1A, x2A) (and (x1B, x2B) ) on the contract curve satisfied MRSA=MRSB, or
→ x2A/ x1A= x2B/ x1B=(100- x2A)/(100 –x1A)
→ (100 –x1A)x2A=(100- x2A)x1A
→ x2A=x1A, the diagonal line, is the contract curve

28. 10.3 Competitive Exchange
Without knowledge of the trading process, we only know that Jane and Denise trade to some allocation on the contract curve.
With knowledge of the exact trading process, we can determine their final allocation.
General-equilibrium models can show that a competitive market has two desirable properties:
Competitive equilibrium is efficient
First Theorem of Welfare Economics
Any efficient allocations can be achieved by competition
Second Theorem of Welfare Economics

29. 10.3 Competitive Exchange
Given prices of the two goods, a price line can be added to the Edgeworth box.
The price line is all the combinations of goods that Jane could get by trading, given her endowment.
If the price of firewood is $2 and the price of a candy bar is $1, then the price line indicates that Jane would choose to trade wood for candy and move from point e to f.
Similarly, given those prices, Denise would prefer to trade candy for wood and move from point e to f.

30. 10.3 Competitive Exchange
The prices of candy and wood establish the price line.
At these prices, Jane sells wood to Denise, and Denise sells candy to Jane. They trade to the Pareto optimal allocation, f.

31. 10.3 Competitive Equilibrium
The prices that set this price line are not consistent with a competitive equilibrium.

32. 10.3 Competitive Exchange
Let e1A (P1, P2)=x1A(P1, P2)- w1A be the excess (net) demand of good 1 of individual A under (P1, P2),
Similarly e1B(P1, P2)= x1B(P1, P2)- w1B for individual B under (P1, P2),
Excess demand of good 1 must sum up to zero in equilibrium (so that market clears)
e1A(P1, P2)+e1B(P1, P2)=0 at equilibrium

33. 10.3 Competitive Exchange
Recall that if 𝑈= 𝑥 1 𝛼 𝑥 2 𝛽 , given (P1, P2), income M.
𝑥 1 𝐷 = 𝛼𝑀 (𝛼+𝛽) 𝑃 1 ;
𝑥 2 𝐷 = 𝛽𝑀 (𝛼+𝛽) 𝑃 2 ;

34. 10.3 Competitive Exchange
Example 1 with (w1A, w2A) =(60, 20) and (w1B, w2B) =(40, 80) when (P1, P2)=(2, 1):
x1A(2, 1)=(1/2)(2x60+1x20)/2=35
x2A(2, 1)=(1/2)(2x60+1x20)/1=70
x1B(2, 1)=(1/2)(2x40+1x80)/2=40
x2B(2, 1)=(1/2)(2x40+1x80)/1=80
e1A (2, 1)+e1B (2, 1)=(35-60)+(40-40)=-25≠0
e2A (2, 1)+e2B (2, 1)=(70-20)+(80-80)=50≠0
So (P1, P2)=(2, 1) are not equilibrium prices.

35. 10.3 Competitive Exchange If (P1*, P2*) clears market 2
(P1*, P2*) must clear up both markets. In market 1
e1A (P1*, P2*)+e1B (P1*, P2*)=0 in market 1
[(1/2)(60P1*+20P2*)/P1*-60] +[(1/2)(40P1*+80P2*)/P1*-40]=0
P1*=P2*, or P1*/P2*=1
If (P1*, P2*) clears market 2
e2A (P1*, P2*)+e2B (P1*, P2*)=0 in market 2
[(1/2)(60P1*+20P2*)/P2*-20] +[(1/2)(40P1*+80P2*)/P2*-80]=0
One can only solve the equilibrium relative prices (P1*/P2*) because one market clearing condition implies the other one → Walras’ law.

36. 10.3 Competitive Exchange Why would Walras’ law be true?
For individual A who will spend up all his income,
P1x1A(P1, P2)+P2x2A(P1, P2)≡P1w1A(P1, P2)+P2w2A(P1, P2); All the spending are financed.
P1e1A(P1, P2)+P2e2A(P1, P2)≡0; [P1e1A(P1, P2) ≡-P2e2A(P1, P2)]; All the income will be spent.
For individual B,
P1e1B(P1, P2)+P2e2B(P1, P2)≡0
By summing up above,
P1[e1A(P1, P2)+e1B(P1, P2)]+P2[e2A(P1, P2)+e2B(P1, P2)]≡0 for all nonnegative (P1, P2), not just equilibrium prices.
When (P1*, P2*) induces the equilibrium in market 1, we have e1A(P1*, P2*)+e1B(P1*, P2*)=0. It thus implies e2A(P1*, P2*)+e2B(P1*, P2*)=0. In words, if market 1 clears, market 2 will also clear. As its consequence, the equilibrium condition in market 2 cannot provide any additional information.

37. 10.3 The Efficiency of Competition
In a competitive equilibrium, the indifference curves of both types of consumers are tangent at the same bundle on the price line, thus:
If the competitive equilibrium must lie on the contract curve, we have demonstrated the First Theorem of Welfare Economics
Any competitive equilibrium is Pareto efficient
By adjusting initial endowments so they lie along the price line, we demonstrate the Second Theorem of Welfare Economics
Any Pareto-efficient equilibrium can be obtained by competition given an appropriate endowment

38. 10.3 The Efficiency of Competition
Example 1c:
Suppose UA(x1, x2)=UB(x1, x2)=x11/2x21/2. Total endowments are 100 and 100. The endowments (x1A, x2A) (and (x1B, x2B) ) that will eventually reach the competitive equilibrium (50, 50) and (50, 50) are combinations of endowments (x1A, x2A) on the straight line (budget line) such that
its price ratio equal to 1
It goes through (50, 50)
x1A+x2A=100

39. 10.4 Production and Trading
So far our discussion of trade has been entirely about consumption, but what about production?
Production capabilities can be summarized with a production possibility frontier (PPF).
PPF shows the maximum combination of two outputs that can be produced from a given amount of input.
In our example, assume:
Jane can use her labor to produce up to 3 candy bars or 6 cords of firewood in a day
Denise can use her labor to produce up to 6 candy bars or 3 cords of firewood in a day

40. 10.4 Production and Trading
PPF curves can be combined to show joint productive capacity.

41. 10.4 Production and Trading
The slope of the production possibility frontier is the marginal rate of transformation (MRT).
MRT tells us how much more wood can be produced if the production of candy is reduced by one bar.
More generally, MRT shows how much it costs to produce one good in terms of the forgone production of the other good.
The comparative advantage in producing a good goes to the person who can produce the good at a lower opportunity cost.
Jane has comparative advantage in producing wood
Denise has comparative advantage in produce candy

42. 10.4 Benefits of Trade
Differences in MRTs imply that Jane and Denise can benefit from trade.
Assume both like to consume wood and candy in equal proportions.
Without trade, each produces 2 candy bars and 2 cords of wood each day
With trade:
Denise specializes in candy production and makes 6 candy bars
Jane specializes in firewood production and makes 6 cords of wood
If production is split equally, each gets 3 candy bars and 3 cords of wood each day!
Trade works when comparative advantage is followed.

43. 10.4 The Number of Producers
With just two producers – Jane and Denise – the PPF has one kink.
As other methods of production with different MRTs are added, the PPF gets more kinks.
As the number of different producers gets very large, the PPF becomes a smooth curve that is concave to the origin.
The MRT along this smooth PPF tells us about the marginal cost of producing one good relative to the other.

44. 10.4 Optimal Product Mix
Individual’s utility is maximized at point a, the point where the PPF touches the indifference curve (MRS = MRT).

45. 10.4 Competition
Each price-taking consumer picks a bundle of goods such that:
If all relative prices are the same for all individuals in competitive equilibrium, all will have equal MRSs and no further trades can occur.
The competitive equilibrium achieves consumption efficiency
Impossible to redistribute goods to make one person better off without making someone worse off.

46. 10.4 Competition
Each competitive firm sells a quantity of candy (c) and wood (w) such that price equals marginal cost:
Taking the ratio of these and combining with the fact that MRT is the ratio of marginal costs yields:
Thus, competition insures an efficient product mix:
The rate at which firms can transform one good into another equals the rate at which consumers are willing to substitute between goods.

47. 10.4 Competitive Equilibrium
At the competitive equilibrium, the relative prices that firms and consumers face are the same.

48. 10.5 Efficiency and Equity
How well various people in a society live depends on:
Efficiency (size of the pie)
Equity (how the pie is divided)
Role of the government
Wealth is redistributed with every government action
Agricultural price support programs transfer wealth to farmers
Income taxes transfer income from better-off to poor
Proceeds from the lottery (played by mostly lower-income people) funds merit-based college scholarships in many states

49. 10.5 Efficiency and Equity
A social welfare function combines various consumers’ utilities to provide a collective ranking for allocations.
Graphically summarized by a isowelfare curve, along which social welfare is constant.
A utility possibility frontier (UPF) is the set of utility levels corresponding to Pareto-efficient allocations along the contract curve.
Society maximizes welfare by choosing the allocation for which the highest possible isowelfare curve touches the UPF.

50. 10.5 Efficiency and Equity
Society maximizes welfare by choosing the allocation for which the highest possible isowelfare curve touches the UPF.

51. 10.5 Efficiency and Equity
Many rules by which society might decide among various allocations have been suggested.
These different social welfare functions yield different distributions of goods:
Utilitarian: equal weight to all people in society ( )
Generalized utilitarian: different weights assigned, perhaps to adults, hard workers, etc. ( )
Rawlsian: maximizes well-being of worst off individual ( )

52. 10.5 Efficiency versus Equity
Given a particular welfare function, society might prefer an inefficient allocation to an efficient one.
Example: one person has everything, which means any reallocation would make that one person worse off, but would likely be preferred by everyone else.
Sometimes, in an attempt to achieve greater equity, efficiency is reduced.
Example: advocates for the poor prefer providing them with public housing (equity), but this is inefficient because the poor would be better off with a cash transfer of equal value.

53. 10.5 Efficiency versus Equity
If competition maximizes efficiency and our usual welfare measure, shouldn’t we strive to eliminate any distortion (tariff, quota, tax, etc.)?
An economy with no distortions is a first-best equilibrium
Any distortion will reduce efficiency
Eliminating some distortions does not guarantee the same outcome as eliminating all of them.
The Theory of the Second Best says that if an economy has at least two market distortions, eliminating just one may either increase or decrease welfare!

54. 10.5 Efficiency and Equity
Permitting trade may raise welfare (as in panel (a)) or may lower it (as in panel (b)) depending on existing distortions.

55. Challenge Solution