5. Externalities and Public Goods

Transcription

1 5. Externalities and Public Goods Welfare properties of Walrasian Equilibria rely on the hidden assumption of private goods: the consumption of the good by one person has no effect on other people s utility, but prevents them from consuming that good Externalities (or external effects) and Public Goods are the most relevant cases of market failures because of the existence of non-private goods Externalities appear when the decisions of one agent (consumer or firm) has a direct effect on the welfare of other agents (consumers or firms) Public Goods are goods such that the consumption by one agent does not prevent others from consuming them as well

2 5. Externalities and Public Goods Externalities Loud music at 3:00 am (negative consumption externality) Well kept garden (positive consumption externality) Paper factory upstream on a river (negative production externality) Bee farming next to a flower production field (positive production externality)

3 5. Externalities and Public Goods Public Goods Highways National Defense Pollution TV signal Rural community commons

4 5. Externalities and Public Goods Public Goods types Non-Rivalrous Rivalrous Non-Excludable (Pure) Public Goods Common-Pool Resources Excludable Club Goods Private Goods

5 5. Externalities and Public Goods Plan Externalities The problem: Market inefficiency The solutions: Quotes, Taxes, Bargaining and Property Rights On missing markets Public Goods The problem: The inefficiency of private provision The solutions: Lindahl Equilibria Common-Pool Resources

6 5. Externalities and Public Goods Externalities Definition 5.1 An externality is present whenever the well-being of a consumer or the production possibilities of a firm are directly affected by the actions of another agent in the economy Direct externalities (the effect of a paper factory on a fishery downstream) are not to be mistaken with pecuniary externalities (the effect of the price of paper on the costs of the fishery). While the later are negotiated in the market, the former are not.

7 5. Externalities and Public Goods 5.1 Bilateral Externalities 5.1 Bilateral Externalities I = {1, 2} n traded commodities (x 1 is a numeraire) p (prices) are given. Consumers are price-takers and cannot affect prices At prices p, individual s initial endowments generate a wealth of w i The externality h is an action taken by individual 1 at 0 monetary cost Each individual has preferences over the commodities he consumes and over h u i (x i 1,...,x i n, h)

8 5. Externalities and Public Goods 5.1 Bilateral Externalities From the point of view of individual 2, h represents an external effect of consumers 1 s action. We therefore assume that u 2 (x 2 1,...,x 2 n,h) 0 h The indirect utility function is defined by v i (p, w i, h) = max u i (x i,h) s.t. p x i w i

9 5. Externalities and Public Goods 5.1 Bilateral Externalities A usual simplifying assumption is that preferences are quasilinear with respect to the numeraire commodity x 1. By doing so, it can be proven that the demand functions of the n 1 nonnumeraire commodities do not depend on w. Then, the indirect utility function takes the form v i (p,w i,h) = φ i (p, h) + w i Since we assume that prices p are constant, we may write v i (w i,h) = φ i (h) + w i We further assume that φ i is twice differentiable and that φ i (h) < 0

10 5. Externalities and Public Goods 5.1 Bilateral Externalities Competitive Equilibrium Outcome Choice of h by individual 1: Assuming that we are at a competitive equilibrium (in the n commodities) at prices p, individual 1 will choose h to maximize v i. Hence, he will choose the level h such that it satisfies the First Order Condition (assuming interior solution) φ 1(h ) = 0 Even though consumer 2 s utility depends on h, it cannot affect the choice of h. Herein lies the problem

11 5. Externalities and Public Goods 5.1 Bilateral Externalities Efficient Outcome In contrast, any Pareto efficient level of hmust maximize the joint surplus, that is, the sum of the utility functions (because of the quasilinearity assumption) max φ i (h) + φ 2 (h) The First Order Condition for an interior maximum is φ 1(h ) + φ 2(h ) = 0 (1) That is, if h is a (interior) Pareto efficient solution, then φ 1(h ) = φ 2(h )

12 5. Externalities and Public Goods 5.1 Bilateral Externalities Analysis Clearly, the efficient level h given by φ 1(h ) = φ 2(h ) and the competitive level h given by will never coincide unless which would mean NO externality φ 1(h ) = 0 φ 2(h ) = 0

13 5. Externalities and Public Goods 5.1 Bilateral Externalities If φ 2( ) < 0 (negative externality) we have φ 1(h ) = φ 2(h ) > 0 φ 1( )decreasing φ 1(h ) = 0 h > h If φ 2( ) > 0 (positive externality) we have φ 1(h ) = φ 2(h ) < 0 φ 1( )decreasing φ 1(h ) = 0 h < h

14 5. Externalities and Public Goods 5.1 Bilateral Externalities Negative Externality φ 2(h) h h φ 1(h) + φ 2(h) h φ 1(h)

15 5. Externalities and Public Goods 5.1 Bilateral Externalities Positive Externality φ 1(h) + φ 2(h) φ 2(h) h φ 1(h) h h

16 5. Externalities and Public Goods 5.1 Bilateral Externalities The efficient outcome does not mean that the externality is eliminated (h 0) In the case of a negative externality, the marginal benefit to individual 1 equals the marginal cost to individual 2 In the case of a positive externality, it must be the case that the sum of the marginal benefit of both individuals equals zero The efficient level of a negative externality is greater that zero even in cases such as pollution, noise, etc. It is difficult to value the benefits-costs of such externalities as, generally, those that produce the negative externality (polluters) think that the optimal level of the externality should be larger than those who suffer from the externality

17 5. Externalities and Public Goods 5.2. Classical Solutions 5.2 Classical Solutions There are 3 classical approaches to solve the inefficiencies produced by externalities Quotas: The government imposes a constraint on the level of the activity producing the externality Taxes: The government charges with a tax the activity producing the externality Bargaining: Provide conditions so that the individuals reach an optimal agreement by themselves on the level of the activity producing the externality

18 5. Externalities and Public Goods 5.2. Classical Solutions Quotas In the presence of a negative externality (h > h ) the government can simply pass a law prohibiting levels beyond h In the presence of a positive externality (h < h ) the government can simply pass a law requiring individual 1 to produce at least h Although this is a simple solution, it is difficult to implement. It requires the government to enforce the quota and to monitor the producer, which can be difficult and costly It would be better if the government could make some adjustments to the market so that it worked properly

19 5. Externalities and Public Goods 5.2. Classical Solutions Taxes The government can impose a tax on the level of production, h, of the (negative) externality Let t h be a tax that individual 1 must pay per unit of h produced. His maximization problem, then, becomes max φ 1 (h) t h h whose First Order Condition is φ 1(h t ) = t h (2) We know (efficient solution 1) that φ 1(h ) + φ 2(h ) = 0 Thus, by setting t h = φ 2(h ) > 0

20 5. Externalities and Public Goods 5.2. Classical Solutions we make sure that h t = h satisfies the First Order Condition 2 Thus, the tax t h = φ 2(h ) known as Pigouvian tax, implements the efficient outcome

21 5. Externalities and Public Goods 5.2. Classical Solutions φ 2(h) φ 2(h ) = t h h h h φ 1(h)

22 5. Externalities and Public Goods 5.2. Classical Solutions The optimal tax equals the marginal externality at the efficient level h It is equal to the amount that individual 2 would be willing to pay to slightly reduce h from its efficient level h Due to tax cost beard by individual 1, he is required to internalize the externality that he imposes on individual 2 Tax must be imposed on the externality producing activity directly To implement quotas/taxes the government must have detailed knowledge of the benefits and costs of the externality to the individuals

23 5. Externalities and Public Goods 5.2. Classical Solutions Bargaining and Enforceable Property Rights A decentralized possibility is to let the two individuals to negotiate a solution to the problem by themselves The success of such procedure depends on the whether property rights are clearly assigned Does individual 1 has the right to produce h? If so, how much? Can individual 2 prevent individual 1 from producing h? If so, how much? Coase Theorem: As long as property rights are clearly established, the two parties will negotiate in such a way that the optimal level of h is implemented

24 5. Externalities and Public Goods 5.2. Classical Solutions Example Suppose that individual 2 has the right (by law) to a smoke-free environment Individual 2 can prohibit individual 1 from smoking But individual 1 can buy from individual 2 the right to smoke a few cigarettes a day in exchange of some money transfer The two individual can bargain both over the size of the transfer and over the number of cigarettes that are allowed to smoke

25 5. Externalities and Public Goods 5.2. Classical Solutions Bargaining Mechanism (Individual 2 rights) Suppose that individual 2 has the right to an externality-free environment. The bargaining procedure could work as follows: 1. Individual 2 offers individual 1 a take-it-or-leave-it contract specifying a payment T 2 and an activity level h 2 2. Individual 1 decides on whether to accept such offer (a) If the offer is accepted then the contract is implemented (h = h 2 ) (b) If the offer is rejected then individual 1 cannot produce any externality (h = 0)

26 5. Externalities and Public Goods 5.2. Classical Solutions Bargaining Solution For an offer (h,t) to be accepted by individual 1, it must be the case that it produces higher utility than the disagreement level h = 0. That is, individual 1 will accept (h, T) if and only if φ 1 (h) T φ 1 (0) Given this (incentive compatible) constraint, individual 2 will choose (h 2, T 2 ) in order to solve max h.t φ 2 (h) + T s.t. φ 1 (h) T φ 1 (0)

27 5. Externalities and Public Goods 5.2. Classical Solutions Since individual 2 will prefer higher T, the constraint will be binding at the optimum. Hence, the problem becomes max h φ 1(h) + φ 2 (h) φ 1 (0) The First Order Condition is given by, φ 1(h 2 ) + φ 2(h 2 ) = 0 which is, precisely, the condition 1 that defines the Pareto efficient level of h.

28 5. Externalities and Public Goods 5.2. Classical Solutions Hence, individual 2 will offer h 2 = h in exchange of a transfer of T 2 = φ 1 (h ) φ 1 (0) Individual 1 will accept such offer (h 2, T 2 ) Thus, the bargaining mechanism implements the socially efficient outcome

29 5. Externalities and Public Goods 5.2. Classical Solutions Bargaining Mechanism (Individual 1 rights) Suppose now that individual 1 has the right to produce a level h of the externality as high as he wants The bargaining procedure could work as before: 1. Individual 2 offers individual 1 a take-it-or-leave-it contract specifying a payment T 1 and an activity level h 1 2. Individual 1 decides on whether to accept such offer (a) If the offer is accepted then the contract is implemented (h = h 1 ) (b) If the offer is rejected then individual 1 can produce as much as he wants (h = h )

30 5. Externalities and Public Goods 5.2. Classical Solutions Bargaining Solution As before, for an offer (h,t) to be accepted by individual 1, it must be the case that it produces higher utility than the disagreement level h = h. That is, individual 1 will accept (h, T) if and only if φ 1 (h) T φ 1 (h ) Given this (incentive compatible) constraint, individual 2 will choose (h 2, T 2 ) in order to solve max h.t φ 2 (h) + T s.t. φ 1 (h) T φ 1 (h )

31 5. Externalities and Public Goods 5.2. Classical Solutions Again, since individual 2 will prefer higher T, the constraint will be binding at the optimum. Hence, the problem becomes The First Order Condition is given by, max h φ 1(h) + φ 2 (h) φ 1 (h ) φ 1(h 2 ) + φ 2(h 2 ) = 0 which, again, is the condition 1 that defines the Pareto efficient level of h.

32 5. Externalities and Public Goods 5.2. Classical Solutions Hence, individual 2 will offer h 1 = h in exchange of a transfer of T 1 = φ 1 (h ) φ 1 (h ) Individual 1 will accept such offer (h 1, T 1 ) Thus, the bargaining mechanism implements, again, the socially efficient outcome

33 5. Externalities and Public Goods 5.2. Classical Solutions The fact that, regardless of how property rights are allocated, bargaining leads to a Pareto efficient (h = h ) outcome is known as the Coase Theorem The way property rights are established has distributional consequences: The transfer is larger in the case individual 2 has the property right than when individual 1 has. Notice, though, that under the quasilinearity assumption wealth transfers have no effect on social welfare Well-defined property rights, and quasilinearity, are essential to the result The government does not need to have any information on the benefits-costs of the externality to any of the individuals. It only has to ensure that property rights are clear and enforceable

34 5. Externalities and Public Goods 5.2. Classical Solutions Externalities and missing markets The problem generated by the presence of externalities is often seen as a problem of the absence of a market for the externality generating activity Suppose that There is a market for activity h Individual 2 has the right to prevent all activity h, but can sell the right to produce 1 unit of h at the unit price p h In this case, individual 1 faces the problem max h φ 2(h) + p h h

35 5. Externalities and Public Goods 5.2. Classical Solutions The First Order Condition is φ 2(h) = p h which implicitly defines a supply function h 2 (p h ) In turn, when deciding how many rights to purchase, individual 1 solves the problem max h φ 1(h) p h h

36 5. Externalities and Public Goods 5.2. Classical Solutions The First Order Condition in this case is, φ 1(h) = p h which implicitly defines a demand function h 1 (p h ) The market-clearing condition establishes that h 1 (p h ) = h 2 (p h ), that is, φ 1(h) = φ 2(h) This condition, once more, is the equation 1 determining the efficient outcome The existence of this market, therefore, induces an equilibrium level of the externality generating activity equal to its social optimum h

37 5. Externalities and Public Goods 5.3 Public Goods Definition 5.2 A Public Good is a commodity such that its consumption by one individual does not preclude consumption by other individuals as well It should be noted that the definition refers to the same unit of the commodity The name public refers to the particular characteristic of the commodity that it is available to the everybody at the same time, not that it is provided by a public organism (public school) or community (public television) or that it is free (public domain or public licenses)

38 5. Externalities and Public Goods 5.3 Public Goods Public Goods types Non-Rivalrous Rivalrous Non-Excludable Pure Public Goods (Police) Common-Pool Resources Excludable Club Goods (Satellite TV) Private Goods (Apples) A Public Bad can usually be redefined as a Public Good Pollution Clean Air Noise Silence

39 5. Externalities and Public Goods 5.3 Public Goods Pure Public Goods I = {1, 2,...,I} n traded commodities (x 1 is a numeraire) p (prices) are given. Consumers are price-takers and cannot affect prices. Decisions on the Public Good cannot affect prices of the traded commodities. At prices p, individual s initial endowments generate a wealth of w i The quantity of the Public Good is denoted by x. Each individual has preferences over the commodities he consumes and over x u i (x i 1,...,x i n, x)

40 5. Externalities and Public Goods 5.3 Public Goods As before, we assume that preferences are quasilinear with respect to the numeraire commodity x 1. Then, the indirect utility function takes the form v i (p,w i,x) = φ i (p, x) + w i Since we assume that prices p are constant, we may write v i (w i,x) = φ i (x) + w i We further assume that φ i is twice differentiable with φ i (x) < 0, and that φ i (x) > 0 in the case of a public good (negative if it is a public bad) On the supply side, the cost of supplying q units of the public good is c(q), where c (q) > 0 and c (q) > 0

41 5. Externalities and Public Goods 5.3 Public Goods Efficient Outcome A Pareto efficient level of provision of the public good must maximize the aggregate surplus, max q I φ i (q) c(q) i=1 whose corresponding First Order Condition is given by, I φ i(q ) c (q ) = 0 i=1 Thus, at the efficient level, the total marginal utility is equal to the marginal cost

42 5. Externalities and Public Goods 5.3 Public Goods c (q) I φ i (q) i=1 q q

43 5. Externalities and Public Goods 5.3 Public Goods Private Provision of a Public Good Suppose now that the public good is to be provided by a profit-maximizer private company that sells 1 unit of the public good at the unit price p. At this price p, each consumer decides how much of the public good to buy, x i Consumer are utility-maximizers and thus the problem max x i φ i (x i + j i x j ) px i has, for each consumer, the First Order Condition φ i(x i + x i) = p

44 5. Externalities and Public Goods 5.3 Public Goods The firm maximizes profits max pq c(q) so that p = c (q ) The equilibrium condition requires that supply equals demand, that is, x = I x i = q i=1 Thus, for any individual that purchases a positive amount of the public good it must be the case that φ i(x ) = c (x )

45 5. Externalities and Public Goods 5.3 Public Goods Analysis Notice that for all individuals, even for those that do not contribute to (do not purchase) the public good, we have that φ i (x ) > 0 Suppose that individuals 1 through K do not contribute while individuals K + 1 through I do. Then, I φ i(x ) = i=1 = K I φ i(x ) + φ i(x ) i=1 i=k+1 K φ i(x ) + (I (K + 1))c (x ) > c (x ) i=1

46 5. Externalities and Public Goods 5.3 Public Goods so, I φ i(x ) > c (x ) i=1 whenever a positive amount of the public good is supplied, which does not coincide with the Pareto efficient level I φ i(x ) = c (x ) i=1 Thus, when people make voluntary contributions the market will provide too little of the public good

47 5. Externalities and Public Goods 5.3 Public Goods c (q) I φ i (q) i=1 q q q

48 5. Externalities and Public Goods 5.3 Public Goods Explanation: The Free-Rider Problem Given x, the purchase of one more unit of the public good by consumer i involves a private cost of p a private benefit of φ i (x ) but also a public benefit of j i φ j (x ) Nevertheless, when deciding on x i individual i considers only the private benefit against the private cost For the efficient solution, the society values the social benefit against the social cost (which is equal to the private cost as p = c (q))

49 5. Externalities and Public Goods 5.3 Public Goods Example Suppose that individuals can be ordered according to their marginal utilities φ 1(x) < φ 2(x) < < φ I(x) for all x In this case, the individual First Order Condition φ i(x ) = φ i(x i + x i) = p can hold for at most one individual, call it m. All i m will choose x i = 0. This implies that, i m, φ i (0 + x i ) < p = φ m(x m + 0). Thus, m must be individual I since φ I (x) > φ i (x) for all i I

50 5. Externalities and Public Goods 5.4 Lindahl Equilibrium There are several approaches to the solution of the insufficient provision of a public good in the market The government could impose (by law) to each individual his contribution The government could take over the provision of the public good The government could tax/subsidize the provision of public good in such a way that provides incentives so that private benefits/cost are in line with public benefits/costs

51 5. Externalities and Public Goods 5.4 Lindahl Equilibrium For instance, the government can set a per unit subsidy to each individual equal to s i = j i φ i(x ) (3) Then, the individual i s problem becomes (assuming all other individual choose x j ) max φ i (x i + x i) + s i x i px i The corresponding First Order Condition is, in this case, φ i(x i + x i) + s i = p

52 5. Externalities and Public Goods 5.4 Lindahl Equilibrium Substituting the values of the subsidy in 3 and adding the market-clearing equilibrium condition we have, φ i(x i + x i) + φ j(x ) = p = c (x) j i which is satisfies when x = x, so that I φ i(x ) = c (x ) i=1 Thus, the efficient level is implemented (but might be quite expensive!!)

53 5. Externalities and Public Goods 5.4 Lindahl Equilibrium Lindahl Equilibrium Suppose that the public good can be unbundled into I private goods x i (i = 1...,I), where each good can be interpreted as the individual i s enjoyment of the public good. Each of these goods will have its own price p i, known as the Lindahl price. Given an equilibrium price, the individual s problem is max φ i (x i ) p i x i which has a First Order Condition φ i(x i ) = p i for each i

54 5. Externalities and Public Goods 5.4 Lindahl Equilibrium On the supply side, when the firm produces one single unit of the public good x, it is producing one unit of each of the personalized good x 1,...,x I. Hence, for each unit of x the firm produces it earns p 1 + p p I. Thus, the firm s problem becomes max I p i x c(x) i=1 and the corresponding First Order Condition I p i = c (x) i=1

55 5. Externalities and Public Goods 5.4 Lindahl Equilibrium Combining the optimality conditions of the consumers and the firm we get I φ i (x) = c (x) i=1 which, again, is satisfied at the efficient level x = x. Thus, the Lindahl Equilibrium implements the socially efficient level of public good, yet Individuals act as price takers of its own personalized good For the equilibrium to work, individuals must believe that if they do not purchase any of the public good, then they will not be able to consume any of it It could be a good model for Club goods

56 5. Externalities and Public Goods 5.5 Common-Pool Resources A common-pool resource is a kind of good whose consumption is rivalrous and non-excludable. Examples are: local fishing grounds common grazing land irrigation systems common wood forests The typical problem in these cases is the overuse of the resource

57 5. Externalities and Public Goods 5.5 Common-pool resources Example: A Local Fishery Ground There are I fishers, each owning k i fishing boats k = I i=1 k i denotes the total number of operating fishing boats All boats are equal, they can catch the same number of fishes f(k) represents the total number of fishes caught by k boats (f > 0, f < 0, f(0) = 0) Fishing boats are produced at a cost c(k) (c > 0, c > 0) The price of fish is normalized to 1 The quantity used of the public good is tied (by f) to the number of boats k

58 5. Externalities and Public Goods 5.5 Common-pool resources Pareto efficient number of boats The Pareto efficient solution is found from the problem max f(k) c(k) whose corresponding First Order Condition is f (k ) = c (k ) (4)

59 5. Externalities and Public Goods 5.5 Common-pool resources The Market equilibrium number of boats Supply If p is the market-clearing price of a fishing boat, the boat producers must solve max pk c(k) whose solution is given by p = c (k) (5) Demand Each fisher solves the problem the First Order Condition is s max k i k i k i + k i f(k) pk i f (k) k i k + f(k) k k i k = p (6)

60 5. Externalities and Public Goods 5.5 Common-pool resources Market-clearing conditions imply that f (k ) k i k + f(k ) k i k k = c (k ) (7) By symmetry, at equilibrium we must have ki = kj for all i, j. Thus, condition 7 may be rewritten as f (k ) 1 I + f(k ) I 1 k = c (k ) (8) I

61 5. Externalities and Public Goods 5.5 Common-pool resources Notice that the left-hand side of condition 8 is a convex combination of the marginal product, f (k), and the average product, f(k) k Since f is concave, we know that f (k) < f(k) k Thus, c (k ) = f (k ) 1 I + f(k ) I 1 k I > f (k )

62 5. Externalities and Public Goods 5.5 Common-pool resources Altogether we have, At the Pareto efficient level k we have f (k ) = c (k ) At the market equilibrium level k we have f (k ) < c (k ) Since f is decreasing and c is increasing, we have k > k That is, the Market overuses the fishery

63 5. Externalities and Public Goods 5.5 Common-pool resources c (k) f (k) k k k

64 5. Externalities and Public Goods 5.5 Common-pool resources Solutions Quotes Taxes Government control Community Norms, promoting cooperation, punishment... (Game theoretical machinery!)