Market Design. Econ University of Notre Dame

Transcription

1 Market Design Econ University of Notre Dame

2 What is market design? Increasingly, economists are asked not just to study or explain or interpret markets, but to design them. This requires different tools and ideas than classical economics, which started in the late 19th century and sought to explain observations or trends in the absence of data Market design problems are often subject to difficult constraints (money or prices cannot be used, certain quotas must be met, participants can misrepresent their objectives, there are legal or social customs to be respected)

3 What is market design? Increasingly, economists are asked not just to study or explain or interpret markets, but to design them. This requires different tools and ideas than classical economics, which started in the late 19th century and sought to explain observations or trends in the absence of data Market design problems are often subject to difficult constraints (money or prices cannot be used, certain quotas must be met, participants can misrepresent their objectives, there are legal or social customs to be respected)

4 What is market design? Increasingly, economists are asked not just to study or explain or interpret markets, but to design them. This requires different tools and ideas than classical economics, which started in the late 19th century and sought to explain observations or trends in the absence of data Market design problems are often subject to difficult constraints (money or prices cannot be used, certain quotas must be met, participants can misrepresent their objectives, there are legal or social customs to be respected)

5 What is this class about? Why do markets fail? (Diagnosis: Market power, externalities, incomplete information, missing markets) How can these failures be fixed or abated?

6 Selling a good There is a single unit of an item (say, a diamond) There are i = 1,..., N buyers, who each have a gross value of v i for the good, and get a payoff of v i t if they buy at a price of t and a payoff of zero otherwise How can the good be sold to maximize efficiency (it goes to the buyer who values it the most)? Revenue (the payment from the bidders is as large as we can make it)?

7 Selling a good The English auction: start with a price of zero, and raise it slowly. Buyers indicate at each moment whether they wish to continue or withdraw. The good is sold to the buyer who stays in the longest, at the price at which the second-to-last buyer withdrew. The Dutch auction: start with a very high price, and lower it slowly. At any moment, a buyer can purchase the good at the current price. Which auction raises more revenue/is more efficient? Are either of these auctions the most profitable/efficient among all possible ways of selling the good?

8 Selling a good The English auction: start with a price of zero, and raise it slowly. Buyers indicate at each moment whether they wish to continue or withdraw. The good is sold to the buyer who stays in the longest, at the price at which the second-to-last buyer withdrew. The Dutch auction: start with a very high price, and lower it slowly. At any moment, a buyer can purchase the good at the current price. Which auction raises more revenue/is more efficient? Are either of these auctions the most profitable/efficient among all possible ways of selling the good?

9 Selling a good The English auction: start with a price of zero, and raise it slowly. Buyers indicate at each moment whether they wish to continue or withdraw. The good is sold to the buyer who stays in the longest, at the price at which the second-to-last buyer withdrew. The Dutch auction: start with a very high price, and lower it slowly. At any moment, a buyer can purchase the good at the current price. Which auction raises more revenue/is more efficient? Are either of these auctions the most profitable/efficient among all possible ways of selling the good?

10 Outline of the class 1 Perfectly competitive markets, welfare, and efficiency 2 Market failures: market power, externalities, incomplete information, missing markets 3 Classical design principles: first- and second-price auctions, generalized second-price auctions, Vickrey auctions, the Gale-Shapley algorithm, top trading cycles 4 Mechanism design: the revelation principle, the Vickrey-Clarke-Groves mechanism, revenue equivalence

11 Perfectly Competitive Markets, Welfare, and Efficiency

12 The frictionless ideal: perfectly competitive markets Consumption and production don t impose costs or benefits on other parties (no externalities) Households and firms take prices as given and decide quantities (no market power) The market is centralized, trade occurs all at once (complete markets) All relevant information is known to all parties (complete information)

13 The frictionless ideal: perfectly competitive markets Consumption and production don t impose costs or benefits on other parties (no externalities) Households and firms take prices as given and decide quantities (no market power) The market is centralized, trade occurs all at once (complete markets) All relevant information is known to all parties (complete information)

14 The frictionless ideal: perfectly competitive markets Consumption and production don t impose costs or benefits on other parties (no externalities) Households and firms take prices as given and decide quantities (no market power) The market is centralized, trade occurs all at once (complete markets) All relevant information is known to all parties (complete information)

15 The frictionless ideal: perfectly competitive markets Consumption and production don t impose costs or benefits on other parties (no externalities) Households and firms take prices as given and decide quantities (no market power) The market is centralized, trade occurs all at once (complete markets) All relevant information is known to all parties (complete information)

16 The frictionless ideal: perfectly competitive markets We ll start with the firm/production side, then consider the consumer/consumption side. Once we understand both sides of the market, we ll put them together at look at equilibrium. We ll look at two representative agents (what we re really thinking of is a large number of identical firms and households who all behave similarly). The presentation might be more mathematical than you re used to, even though we ll get the same results and pictures (that s part of building up to more abstract models later).

17 The frictionless ideal: perfectly competitive markets We ll start with the firm/production side, then consider the consumer/consumption side. Once we understand both sides of the market, we ll put them together at look at equilibrium. We ll look at two representative agents (what we re really thinking of is a large number of identical firms and households who all behave similarly). The presentation might be more mathematical than you re used to, even though we ll get the same results and pictures (that s part of building up to more abstract models later).

18 The frictionless ideal: perfectly competitive markets We ll start with the firm/production side, then consider the consumer/consumption side. Once we understand both sides of the market, we ll put them together at look at equilibrium. We ll look at two representative agents (what we re really thinking of is a large number of identical firms and households who all behave similarly). The presentation might be more mathematical than you re used to, even though we ll get the same results and pictures (that s part of building up to more abstract models later).

19 The perfectly competitive market: Firm behavior A representative firm receives a price p for each unit of a good it sells, and it costs C(q) = c 2 q2 to produce q units. Then Total revenue of q units is pq Total costs are C(q) = c 2 q2 Profits are the difference between total revenue and total cost, π(q) = pq C(q) = pq c 2 q2 How do we find the profit-maximizing quantity, q, for which π(q ) π(q ) for any other q the firm could pick?

20 The perfectly competitive market: Firm behavior A representative firm receives a price p for each unit of a good it sells, and it costs C(q) = c 2 q2 to produce q units. Then Total revenue of q units is pq Total costs are C(q) = c 2 q2 Profits are the difference between total revenue and total cost, π(q) = pq C(q) = pq c 2 q2 How do we find the profit-maximizing quantity, q, for which π(q ) π(q ) for any other q the firm could pick?

21 The perfectly competitive market: Firm behavior A representative firm receives a price p for each unit of a good it sells, and it costs C(q) = c 2 q2 to produce q units. Then Total revenue of q units is pq Total costs are C(q) = c 2 q2 Profits are the difference between total revenue and total cost, π(q) = pq C(q) = pq c 2 q2 How do we find the profit-maximizing quantity, q, for which π(q ) π(q ) for any other q the firm could pick?

22 The perfectly competitive market: Firm behavior Let s start by thinking about searching for a maximum, starting from some arbitrary quantity q. Consider the change in profit from using some other quantity, q : ( π(q) π(q ) = pq c 2 q2) (pq c 2 q 2) Now, divide by q q to get = p(q q ) c 2 (q + q )(q q ) π(q) π(q ) q q = p c 2 (q + q ) This represents the normalized gain/loss to switching from q to q.

23 The perfectly competitive market: Firm behavior We now take the limit as q goes to q: or π(q) π(q ) lim q q q q = lim p c q q 2 (q + q ) π (q) = p cq So the change in profit π(q) when evaluated at the optimal quantity q equals p cq. If this is positive, it means that profits are increasing at q, so we should pick a larger value. If this is negative, it means that profits are decreasing at q, and we should pick a smaller value. Only when π (q ) = 0 do we know that there are no better options than q nearby.

24 The perfectly competitive market: Firm behavior We now take the limit as q goes to q: or π(q) π(q ) lim q q q q = lim p c q q 2 (q + q ) π (q) = p cq So the change in profit π(q) when evaluated at the optimal quantity q equals p cq. If this is positive, it means that profits are increasing at q, so we should pick a larger value. If this is negative, it means that profits are decreasing at q, and we should pick a smaller value. Only when π (q ) = 0 do we know that there are no better options than q nearby.

25 Maxima ( ) ( ) >0

26 Maximization in general So for x to solve max f (x), x it is necessary that f (x ) = 0; we call the equation f (x ) = 0 the first-order necessary conditions. For some problems, this approach doesn t work. Why not?

27 Maximization in general So for x to solve max f (x), x it is necessary that f (x ) = 0; we call the equation f (x ) = 0 the first-order necessary conditions. For some problems, this approach doesn t work. Why not?

28 The perfectly competitive market: Firm behavior Now we know that a necessary condition for q to be a maximum of π(q) is that π (q ) = p cq = 0 or q = p c In other words, the firm s supply curve is q S = p/c.

29 The perfectly competitive market: Firm behavior Now we know that a necessary condition for q to be a maximum of π(q) is that π (q ) = p cq = 0 or q = p c In other words, the firm s supply curve is q S = p/c.

30 The perfectly competitive market: Firm behavior : =

31 The perfectly competitive market: Consumer behavior There is a representative consumer who buy the good q along with another good, we ll call m, the numeraire or money. The consumer has a utility function over bundles (q, m) of u(q, m) = b log(q) + m The consumer also faces a budget constraint, w = pq + m, so that total expenditures on the two goods cannot be more than w. What bundle should the consumer pick?

32 The perfectly competitive market: Consumer behavior There is a representative consumer who buy the good q along with another good, we ll call m, the numeraire or money. The consumer has a utility function over bundles (q, m) of u(q, m) = b log(q) + m The consumer also faces a budget constraint, w = pq + m, so that total expenditures on the two goods cannot be more than w. What bundle should the consumer pick?

33 The perfectly competitive market: Consumer behavior There is a representative consumer who buy the good q along with another good, we ll call m, the numeraire or money. The consumer has a utility function over bundles (q, m) of u(q, m) = b log(q) + m The consumer also faces a budget constraint, w = pq + m, so that total expenditures on the two goods cannot be more than w. What bundle should the consumer pick?

34 (Quick aside: Quasilinear utility ) In the way we specified utility, we have u(q, m) = b log(q) + m, where m is money spent on goods besides q. We call this kind of utility function quasilinear since m enters additively or linearly. It implies that there are no substitution effects between consumption of this good and the others that the consumer purchases, or that this good is a small fraction of the consumer s total budget. We ll use this idea over and over later in class.

35 (Quick aside: Quasilinear utility ) In the way we specified utility, we have u(q, m) = b log(q) + m, where m is money spent on goods besides q. We call this kind of utility function quasilinear since m enters additively or linearly. It implies that there are no substitution effects between consumption of this good and the others that the consumer purchases, or that this good is a small fraction of the consumer s total budget. We ll use this idea over and over later in class.

36 The perfectly competitive market: Consumer behavior Rather than use Lagrange multipliers, we can rewrite the budget constraint as m = w pq and substitute it into the utility function to get max b log(q) + w pq q which is unconstrained in q, making it a much easier problem to solve. Then our necessary condition is giving us an demand curve, q D = b p. b q p = 0 q = b p

37 The perfectly competitive market: Consumer behavior : = /

38 Market Equilibrium Definition A price-quantity pair (p, q ) is a price-taking equilibrium or perfectly competitive equilibrium if 1 Firms maximize profits, taking the price as given. 2 Consumers maximize utility, taking the price as given. 3 Markets clear, so that supply equals demand, or q S = q D. We ve already solved for supply and demand above, so all that s left to do is equate them: so that p = bc and q = q S = p c = b p = qd b c.

39 Market Equilibrium Definition A price-quantity pair (p, q ) is a price-taking equilibrium or perfectly competitive equilibrium if 1 Firms maximize profits, taking the price as given. 2 Consumers maximize utility, taking the price as given. 3 Markets clear, so that supply equals demand, or q S = q D. We ve already solved for supply and demand above, so all that s left to do is equate them: so that p = bc and q = q S = p c = b p = qd b c.

40 Market Equilibrium : = : = /

41 Market Equilibrium This model gives all the basic predictions of principles micro: Demand is downward sloping (LOD), supply is upward sloping (LOS) If there s a positive shock to demand (i.e., b increases), quantity demanded shifts up at all prices, and the market-clearing price and quantity increase. If there s a positive shock to supply (i.e., c increases), quantity supplied shifts down at all prices, and the market-clearing price goes up while the market-clearing quantity goes down. If b and c change at the same time, it s difficult to say much about what happens.

42 A Fundamental Economic Question Suppose we want to maximize the performance of an economy: is a free-market in which firms and individuals interact through a price system better or worse than a command and control economy in which a central authority governs economic activity? (Link)

43 What are reasonable assumptions to make about what v i (q i ) and C j (q j ) look like? Perfectly competitive markets in general There are i = 1, 2,..., I consumers who each choose a bundle of goods (q i, m i ) to maximize its utility, v i (q) + m i, subject to the constraint pq i + m i = w i, taking prices as given. There are j = 1, 2,..., J firm who each choose how much quantity q j to supply to maximize its profits, π j (q j ) = pq j C j (q j ), taking prices as given. A perfectly competitive equilibrium is then a price p, a set of quantities demanded (qi=1,..., q i=i ), and a set of quantities supplied (qj=1,..., q j=j ) such that households are maximizing their utilities taking the price as given, firms are maximizing their profits taking the price as given, and markets clear: I qi = i=1 J j=1 q j

44 What are reasonable assumptions to make about what v i (q i ) and C j (q j ) look like? Perfectly competitive markets in general There are i = 1, 2,..., I consumers who each choose a bundle of goods (q i, m i ) to maximize its utility, v i (q) + m i, subject to the constraint pq i + m i = w i, taking prices as given. There are j = 1, 2,..., J firm who each choose how much quantity q j to supply to maximize its profits, π j (q j ) = pq j C j (q j ), taking prices as given. A perfectly competitive equilibrium is then a price p, a set of quantities demanded (qi=1,..., q i=i ), and a set of quantities supplied (qj=1,..., q j=j ) such that households are maximizing their utilities taking the price as given, firms are maximizing their profits taking the price as given, and markets clear: I qi = i=1 J j=1 q j

45 The perfectly competitive market: Firm behavior in general Each firm j then solves max q j pq j C j (q j ), which has the FONC p = C j (q j ), or price equals marginal cost. Each consumer then solves max q i v i (q i ) pq i + w i, which has the FONC v i (q i ) = p, or marginal benefit equals price. At the perfectly competitive equilibrium, for every household i and firm j, v i (q i ) = p = C j (q j ), so the price, marginal cost, and marginal benefit of the last units produced or consumed are equated across all firms and consumers.

46 The perfectly competitive market: Firm behavior in general Each firm j then solves max q j pq j C j (q j ), which has the FONC p = C j (q j ), or price equals marginal cost. Each consumer then solves max q i v i (q i ) pq i + w i, which has the FONC v i (q i ) = p, or marginal benefit equals price. At the perfectly competitive equilibrium, for every household i and firm j, v i (q i ) = p = C j (q j ), so the price, marginal cost, and marginal benefit of the last units produced or consumed are equated across all firms and consumers.

47 The perfectly competitive market: Firm behavior in general Each firm j then solves max q j pq j C j (q j ), which has the FONC p = C j (q j ), or price equals marginal cost. Each consumer then solves max q i v i (q i ) pq i + w i, which has the FONC v i (q i ) = p, or marginal benefit equals price. At the perfectly competitive equilibrium, for every household i and firm j, v i (q i ) = p = C j (q j ), so the price, marginal cost, and marginal benefit of the last units produced or consumed are equated across all firms and consumers.

48 The Welfare Theorem Rather than letting the market determine the outcome, what if a benevolent social planner decided the quantities (q i=1,..., q i=i ) and (q j=1,..., q j=j )? How would the outcome be different? Definition The social planner s problem is found by summing the payoffs of all the agents, and maximizing the resulting social welfare function. The outcome that maximizes the social welfare function is efficient. Here, the social welfare function is the sum of profits and utility, giving us W (q) = = I J (v i (q i ) pq i + w i ) + (pq j C j (q j )) i=1 j=1 I J (v i (q i ) + w i ) C j (q j ) i=1 j=1

49 The Welfare Theorem So we want to maximize subject to I (v i (q i ) + w i ) i=1 I q i = i=1 J C j (q j ) j=1 J q j. This is a constrained maximization problem, and we need a Lagrangian to solve it. j=1

50 The Welfare Theorem So we want to maximize subject to I (v i (q i ) + w i ) i=1 I q i = i=1 J C j (q j ) j=1 J q j. This is a constrained maximization problem, and we need a Lagrangian to solve it. j=1

51 Lagrangians Suppose you want to maximize f (x 1,..., x N ) subject to a constraint like p 1 x p N x N = w. The Lagrangian function works like this: 1 Move all the terms in your constraint to the same side of the equality: p 1 x p N x N w = 0. 2 Multiply by λ, where λ is the Lagrange multiplier, and add to your objective function: L(x, λ) = f (x 1,..., x N ) λ (p 1 x p N x N w) 3 Take derivatives with respect to each variable you are maximizing and the Lagrange multiplier: for each i, f (x) x i λp i = 0, and (p 1 x p N x N w) = 0.

52 Lagrangians Suppose you want to maximize f (x 1,..., x N ) subject to a constraint like p 1 x p N x N = w. The Lagrangian function works like this: 1 Move all the terms in your constraint to the same side of the equality: p 1 x p N x N w = 0. 2 Multiply by λ, where λ is the Lagrange multiplier, and add to your objective function: L(x, λ) = f (x 1,..., x N ) λ (p 1 x p N x N w) 3 Take derivatives with respect to each variable you are maximizing and the Lagrange multiplier: for each i, f (x) x i λp i = 0, and (p 1 x p N x N w) = 0.

53 Lagrangians Suppose you want to maximize f (x 1,..., x N ) subject to a constraint like p 1 x p N x N = w. The Lagrangian function works like this: 1 Move all the terms in your constraint to the same side of the equality: p 1 x p N x N w = 0. 2 Multiply by λ, where λ is the Lagrange multiplier, and add to your objective function: L(x, λ) = f (x 1,..., x N ) λ (p 1 x p N x N w) 3 Take derivatives with respect to each variable you are maximizing and the Lagrange multiplier: for each i, f (x) x i λp i = 0, and (p 1 x p N x N w) = 0.

54 The Lagrange Multiplier If we consider the maximized Lagrangian, we have: L(x, λ ) = f (x 1,..., x N ) λ (p 1x p N x N w) How does the agent s payoff vary with respect to w? dl N dw = {f xi (x ) λ p i } dx i dw i=1 λ dw (p 1x p N x N w) λ = λ The Lagrange multiplier is the shadow value of relaxing the constraint; how much more utility does the household get from another dollar of wealth?

55 The Lagrange Multiplier If we consider the maximized Lagrangian, we have: L(x, λ ) = f (x 1,..., x N ) λ (p 1x p N x N w) How does the agent s payoff vary with respect to w? dl N dw = {f xi (x ) λ p i } dx i dw i=1 λ dw (p 1x p N x N w) λ = λ The Lagrange multiplier is the shadow value of relaxing the constraint; how much more utility does the household get from another dollar of wealth?

56 The Lagrange Multiplier If we consider the maximized Lagrangian, we have: L(x, λ ) = f (x 1,..., x N ) λ (p 1x p N x N w) How does the agent s payoff vary with respect to w? dl N dw = {f xi (x ) λ p i } dx i dw i=1 λ dw (p 1x p N x N w) λ = λ The Lagrange multiplier is the shadow value of relaxing the constraint; how much more utility does the household get from another dollar of wealth?

57 Lagrangians The Lagrangian function works like this: 1 The constraint becomes: I q i i=1 J q j = 0. 2 The Lagrangian is: I L(q, λ) = i (q i ) + w i ) i=1(v I C j (q j ) λ q i j=1 i=1 3 The FONCs are v i (q i ) λ = 0 for each household i, and C j (q j ) + λ = 0 for each firm j, and for the Lagrange multiplier, ( I i=1 q i ) J j=1 q j = 0 j=1 J j=1 q j

58 Lagrangians The Lagrangian function works like this: 1 The constraint becomes: I q i i=1 J q j = 0. 2 The Lagrangian is: I L(q, λ) = i (q i ) + w i ) i=1(v I C j (q j ) λ q i j=1 i=1 3 The FONCs are v i (q i ) λ = 0 for each household i, and C j (q j ) + λ = 0 for each firm j, and for the Lagrange multiplier, ( I i=1 q i ) J j=1 q j = 0 j=1 J j=1 q j

59 Lagrangians The Lagrangian function works like this: 1 The constraint becomes: I q i i=1 J q j = 0. 2 The Lagrangian is: I L(q, λ) = i (q i ) + w i ) i=1(v I C j (q j ) λ q i j=1 i=1 3 The FONCs are v i (q i ) λ = 0 for each household i, and C j (q j ) + λ = 0 for each firm j, and for the Lagrange multiplier, ( I i=1 q i ) J j=1 q j = 0 j=1 J j=1 q j

60 The Welfare Theorem But the Lagrangian FONCS imply and v i (q i ) = λ = C j (q j ) I qi = i=1 J qj, so the Lagrange multiplier at the optimum must be the same as the price in the perfectly competitive market, and the outcomes must be the same. Theorem Perfectly competitive markets maximize (utilitarian) social welfare. j=1

61 The Welfare Theorem But the Lagrangian FONCS imply and v i (q i ) = λ = C j (q j ) I qi = i=1 J qj, so the Lagrange multiplier at the optimum must be the same as the price in the perfectly competitive market, and the outcomes must be the same. Theorem Perfectly competitive markets maximize (utilitarian) social welfare. j=1

62 The Welfare Theorem What does the welfare theorem say? If 1 no agents consumption/production decisions have a direct impact on the payoffs of other agents except through prices... 2 no agents have market power... 3 all agents meet and trade at the same time... 4 all agents are completely informed... then markets implement the same outcome a benevolent government would pick.

63 The Welfare Theorem What does the welfare theorem say? If 1 no agents consumption/production decisions have a direct impact on the payoffs of other agents except through prices... 2 no agents have market power... 3 all agents meet and trade at the same time... 4 all agents are completely informed... then markets implement the same outcome a benevolent government would pick.

64 Interpretations of the Welfare Theorem Laissez-faire capitalism: Governments actually don t and couldn t aggregate all the information available to firms and consumers and pick outcomes, so it s best to leave the job to markets despite their other flaws. Sure, these flaws create inefficiencies, but entrepreneurs will step in to resolve them and pick up the rents themselves. Traditional socialism: Markets are only really optimal when all of the problems have been assumed away. Market power, incomplete information, and search frictions are huge problems. The obvious fix for these flaws is for the government to intervene directly through regulation.

65 Interpretations of the Welfare Theorem Laissez-faire capitalism: Governments actually don t and couldn t aggregate all the information available to firms and consumers and pick outcomes, so it s best to leave the job to markets despite their other flaws. Sure, these flaws create inefficiencies, but entrepreneurs will step in to resolve them and pick up the rents themselves. Traditional socialism: Markets are only really optimal when all of the problems have been assumed away. Market power, incomplete information, and search frictions are huge problems. The obvious fix for these flaws is for the government to intervene directly through regulation.

66 Institutions We see both entrepreneurs and regulation throughout the economy: Signaling: credentials, accreditation, resumes, guilds; university degrees Screening: discounts, coupons, price schedules/discrimination; business vs coach Intermediaries: universities, headhunters, market makers, e-commerce; Google, Uber, Walmart Financial instruments: derivatives, futures, insurance; MacDonald s and the McRib Policy: licensing, auditing, inspections, taxes/subsidies, artificial markets, black markets, crime, tariffs; farm subsidies

67 Welfare Why did our social planner maximize the sums of household utilities and firm profits? The answer is: it seems like a reasonable thing to do, but there are many other reasonable things to do As a society, the way we decide how to evaluate welfare is important, and (despite its role in economics) utilitarianism is a questionable philosophy in general

68 Welfare Why did our social planner maximize the sums of household utilities and firm profits? The answer is: it seems like a reasonable thing to do, but there are many other reasonable things to do As a society, the way we decide how to evaluate welfare is important, and (despite its role in economics) utilitarianism is a questionable philosophy in general

69 Welfare Why did our social planner maximize the sums of household utilities and firm profits? The answer is: it seems like a reasonable thing to do, but there are many other reasonable things to do As a society, the way we decide how to evaluate welfare is important, and (despite its role in economics) utilitarianism is a questionable philosophy in general

70 Classic Notions of Welfare Suppose there are agents i = 1,..., N with utility functions u i (x) over some outcome x. We wish to discuss the relative social desirability of different outcomes x. Utilitarian: The best x solves Rawlsian: The best x solves W u = max x W R = max x N u i (x) i=1 min u i (x) i Pareto dominance: An outcome x Pareto dominates x if for all i, u i (x) u i (x ) and for at least one i, u i (x) > u i (x ). An outcome x is Pareto optimal if no outcome x Pareto dominates it.

71 Classic Notions of Welfare Suppose there are agents i = 1,..., N with utility functions u i (x) over some outcome x. We wish to discuss the relative social desirability of different outcomes x. Utilitarian: The best x solves Rawlsian: The best x solves W u = max x W R = max x N u i (x) i=1 min u i (x) i Pareto dominance: An outcome x Pareto dominates x if for all i, u i (x) u i (x ) and for at least one i, u i (x) > u i (x ). An outcome x is Pareto optimal if no outcome x Pareto dominates it.

72 Classic Notions of Welfare Suppose there are agents i = 1,..., N with utility functions u i (x) over some outcome x. We wish to discuss the relative social desirability of different outcomes x. Utilitarian: The best x solves Rawlsian: The best x solves W u = max x W R = max x N u i (x) i=1 min u i (x) i Pareto dominance: An outcome x Pareto dominates x if for all i, u i (x) u i (x ) and for at least one i, u i (x) > u i (x ). An outcome x is Pareto optimal if no outcome x Pareto dominates it.

73 Classic Notions of Welfare Suppose there are agents i = 1,..., N with utility functions u i (x) over some outcome x. We wish to discuss the relative social desirability of different outcomes x. Utilitarian: The best x solves Rawlsian: The best x solves W u = max x W R = max x N u i (x) i=1 min u i (x) i Pareto dominance: An outcome x Pareto dominates x if for all i, u i (x) u i (x ) and for at least one i, u i (x) > u i (x ). An outcome x is Pareto optimal if no outcome x Pareto dominates it.

74 Where does the firehouse go? Households are uniformly distributed between zero and one Suppose we are deciding where to place a firehouse, x [0, 1] For each i in [0, 1], the payoff to that household is u i (x) = x i, the distance from i to x. Utilitarian welfare is W u = x 0 (i x)di + 1 What x maximizes utilitarian welfare? Rawlsian welfare is W R = What x maximizes Rawlsian welfare? x (x i)di = x x { x, x 1/2 (1 x), x < 1/2. What selections of x are Pareto optimal?

75 Where does the firehouse go? Households are uniformly distributed between zero and one Suppose we are deciding where to place a firehouse, x [0, 1] For each i in [0, 1], the payoff to that household is u i (x) = x i, the distance from i to x. Utilitarian welfare is W u = x 0 (i x)di + 1 What x maximizes utilitarian welfare? Rawlsian welfare is W R = What x maximizes Rawlsian welfare? x (x i)di = x x { x, x 1/2 (1 x), x < 1/2. What selections of x are Pareto optimal?

76 Where does the firehouse go? Households are uniformly distributed between zero and one Suppose we are deciding where to place a firehouse, x [0, 1] For each i in [0, 1], the payoff to that household is u i (x) = x i, the distance from i to x. Utilitarian welfare is W u = x 0 (i x)di + 1 What x maximizes utilitarian welfare? Rawlsian welfare is W R = What x maximizes Rawlsian welfare? x (x i)di = x x { x, x 1/2 (1 x), x < 1/2. What selections of x are Pareto optimal?

77 Where does the firehouse go? Households are uniformly distributed between zero and one Suppose we are deciding where to place a firehouse, x [0, 1] For each i in [0, 1], the payoff to that household is u i (x) = x i, the distance from i to x. Utilitarian welfare is W u = x 0 (i x)di + 1 What x maximizes utilitarian welfare? Rawlsian welfare is W R = What x maximizes Rawlsian welfare? x (x i)di = x x { x, x 1/2 (1 x), x < 1/2. What selections of x are Pareto optimal?

78 Where does the firehouse go? Households are uniformly distributed between zero and one Suppose we are deciding where to place a firehouse, x [0, 1] For each i in [0, 1], the payoff to that household is u i (x) = x i, the distance from i to x. Utilitarian welfare is W u = x 0 (i x)di + 1 What x maximizes utilitarian welfare? Rawlsian welfare is W R = What x maximizes Rawlsian welfare? x (x i)di = x x { x, x 1/2 (1 x), x < 1/2. What selections of x are Pareto optimal?

79 Where does the firehouse go? Households are uniformly distributed between zero and one Suppose we are deciding where to place a firehouse, x [0, 1] For each i in [0, 1], the payoff to that household is u i (x) = x i, the distance from i to x. Utilitarian welfare is W u = x 0 (i x)di + 1 What x maximizes utilitarian welfare? Rawlsian welfare is W R = What x maximizes Rawlsian welfare? x (x i)di = x x { x, x 1/2 (1 x), x < 1/2. What selections of x are Pareto optimal?

80 How does economics work? Economics has two modes: Positive: how the world is Normative: how the world should be Especially in a class like this one, it is easy to confuse a few things: The consequences of equilibrium analyses are intended to be positive claims, not normative ones: the concept of a price-taking equilibrium is not prescriptive, it is an attempt to describe how the world works When we do welfare and efficiency analyses, we are saying, if you are a profit-maximizer/utilitarian/rawlsian/etc, the best thing to do is X, not X is what everyone should do In (actual) social science, you need to conceptualize the world to understand it, but also to understand how that model or theory leads to conclusions. Do not assume the conclusions you prefer and attempt to justify them.

81 How does economics work? Economics has two modes: Positive: how the world is Normative: how the world should be Especially in a class like this one, it is easy to confuse a few things: The consequences of equilibrium analyses are intended to be positive claims, not normative ones: the concept of a price-taking equilibrium is not prescriptive, it is an attempt to describe how the world works When we do welfare and efficiency analyses, we are saying, if you are a profit-maximizer/utilitarian/rawlsian/etc, the best thing to do is X, not X is what everyone should do In (actual) social science, you need to conceptualize the world to understand it, but also to understand how that model or theory leads to conclusions. Do not assume the conclusions you prefer and attempt to justify them.