Welfare Analysis in Partial Equilibrium.

Transcription

1 Welfare Analysis in Partial Equilibrium. Social welfare function: assigns social welfare value (real number) to each profile of utility levels (u 1,u 2,...u I ): W (u 1,u 2,...u I ) (Utilitarian welfare). Assume: W is strongly monotonic in its arguments.

2 For any given consumption and production levels of good l, (x 1,...x I,q 1,...,q J ), where i=1 x i = vectors that are attainable are given by: {(u 1,u 2,...,u I ): i=1 u i i=1 JX j=1 φ i (x i )+ω m q j,the utility JX j=1 c j (q j )}. As the boundary of this set expands, the maximum social welfare W attainable on this set (through redistribution of the numeraire good) increases (strictly).

3 Thus, *For any strongly monotonic social welfare function W, a change in the consumption and production of good l leads to an increase in (the maximum attainable) social welfare if and only if it increases the Marshallian surplus: S(x 1,...x I,q 1,...q J )=[ i=1 φ i (x i ) JX j=1 c j (q j )]. Thus, social welfare analysis of changes in the consumption and production of good l can be carried out exclusively in terms of the Marshallian surplus.

4 Indeed, as we have seen, Pareto efficiency also requires that the consumption and production of good l must satisfy max S(x 1,...x I,q 1,...q J ) (x 1,...,x I ) 0 (q 1,...q J ) 0 s.t. i=1 x i = JX j=1 q j.

5 Consider a consumption and production vector of good l, (bx 1,...bx I, bq 1,...bq J ) such that for by = (i) (bx 1,...bx I ) solves: max [ x i,i 1,..I s.t. i=1 i=1 φ i (x i )] i=1 bx i x i = by, x i 0,i=1,..I. (ii) (bq 1,...bq J ) solves min q j,j=1,...j s.t. JX j=1 JX j=1 c j (q j ) q j = by, q j 0,j =1,..J.

6 We have seen that: φ 0 i ( bx i )=P (by) =B 0 (by), i such that bx i > 0 c 0 j ( bq j )=C 0 (by), j such that bq j > 0, where P is the inverse aggregate demand function, B 0 (.) is the industry marginal benefit andc 0 (.) is the industry marginal cost (or the aggregate inverse supply function).

7 S(bx 1,...bx I, bq 1,...bq J ) = [ i=1 φ i (bx i ) = B(by) C(by) = = = [ byz 0 byz 0 byz 0 JX j=1 c j (bq j )] B 0 (y)dy C(0) P (y)dy byz 0 byz 0 C 0 (y)dy C 0 (y)dy C(0) [P (y) C 0 (y)]dy] S(0)

8 Note: [ byz 0 [P (y) C 0 (y)]dy] is the area between the aggregate demand and supply surves and can be written as : [ = [ byz 0 byz 0 [P (y) C 0 (y)]dy] [P (y) byp(by)] + [byp(by) (C(by) C(0))] = CS(P (by)) + PS(P (by)) where CS(p) and PS(p) denote the aggregate consumer and producer surplus generated in a (hypothetical) market with price taking consumers and producers at price market price p.

9 Therefore, in partial equilibrium analysis, social welfare maximization, Marshallian surplus maximization and Pareto efficiency are roughly equivalent in their implication for the production and consumption of "the good" and eventually reduce to maximization of CS + PS. It is easy to see that [ at the output where: byz 0 [P (y) C 0 (y)]dy] is maximized P (y )=C 0 (y ) i.e., social marginal benefit equates industry s marginal cost. As C 0 (y) is inverse aggregate supply curve, this is also the aggregate output consumed and produced in a competitive equilibrium (supply=demand). Thus, competitive equilibrium outcome is equivalent to Marshallian surplus maximization. Allofthisassumesnoexternalitiesorotherdistortions (taxes, subsidies etc).

10 Welfarelossduetodistortionsismeasuredbythechange in CS +PS i.e., the area between aggregate demand and the supply (or industry MC curve). Sometmes, called deadweight loss.

11 Example. Welfare loss due to a distortionary tax (in a competitive market). Sales tax on good l: t per unit paid by consumers. Tax revenue returned to consumers through lump sum transfer (non distortionary spending). Let (x 1 (t),..., x I (t),q 1 (t),...,q J (t)) and p (t) be the competitive equilibrium allocation and price given tax rate t.

12 FOC: φ 0 i (x i (t)) = p (t)+t, foralli such that x i (t) > 0. c 0 j (q j (t)) = p (t), for all j such that x j (t) > 0.

13 Let x (t) =x(p (t)+t) = i=1 x i (t). Market clearing: x(p (t)+t) =q(p (t)) * Easy to check that p (t) is strictly decreasing in t and that (p (t)+t) is non-decreasing in t Then, x (t) is non-increasing in t so that x (t) x (0). Strict inequality if c j strictly convex.

14 Let S (t) =S(x 1 (t),..., x I (t),q 1 (t),...,q J (t)). We have that S (t) =[ x (t) Z 0 [P (y) C 0 (y)]dy] S(0) W elfare change = S (t) S (0) = x (t) Z x (0) [P (y) C 0 (y)]dy] which is negative since x (t) x (0) and P (y) > C 0 (y) for all y [0,x (0)).

15 Market failure: Situations in which the some assumptions of the fundamental theorems of welfare economics do not hold - usually the first theorem (i.e., market does not lead to Pareto efficiency). An important factor behind market failure: EXTERNAL- ITIES.

16 An externality is present whenever a consumer s utility or the technological possibilities of a firm are directly affected by the actions of some other agents (consumers or firms) in the economy. Exclude effects mediated through prices.

17 A Simple Model of Bilateral Externality. Two price taking agents, i =1, 2. Each agent i s utility depends on her consumption of L traded goods (x 1i,...x Li ) as well as some other action h R + taken by agent 1: where u i (x 1i,...x Li,h) = x 1i + g i (x 2i,...x Li,h) = x 1i + g i (x 1i,h) x 1i =(x 2i,...x Li ) and g i is a differentiable function with g i h 6=0.

18 The possible range over which h may be chosen is not affected by the budget constraint. Let the price of the numeraire good 1 be equal to 1. Given prices p R L 1 and wealth w i, the indirect utility of agent i for each level of h is given by v i (p, w i,h) = maxx 1i + g i (x 1i,h) s.t. x 1i + p x 1i w i,x 1i 0. equivalent to v i (p, w i,h)=max x 1i [w i p x 1i + g i (x 1i,h)]

19 The Walrasian demand for goods 2,...Lare independent of w i so that v i (p, w i,h)=w i p x 1i (p, h)+g i (x 1i (p, h),h)] which can be written as being of the form: v i (p, w i,h)=w i + φ i (p, h) As prices are assumed to be fixed and unaffected by the choice of h by the agents, we can write φ i (p, h) =φ i (h). The preferences of each agent regarding choice of h is summarized by φ i (h).

20 [The analysis remains unchanged if the two agents are firms and φ i (h) is the derived profit function.]

21 Assume: φ i (h) is twice differentiable with φ i (h) < 0 (strictly concave).

22 In a competitive equilibrium in which the price vector is p, both agents maximize utility subject to constraints imposed by p and wealth w i. Therefore, in equilibrium consumer 1 chooses h so as to maximize φ i (h): φ 0 i (h ) 0 = 0,if h > 0..

23 In any Pareto optimal allocation, it must be the case that h = h o must solve: max [φ 1(h)+φ 2 (h)]. h 0 To see this note that if this is not the case, then - one can choose a different value of h to increase the joint welfare - and then compensate the person who loses as a result of this change by a transfer of wealth (numeraire good) so that the welfare of one person can be increased without reducing the welfare of the other person, and this would violate Pareto optimality.

24 FOC: φ 0 1 (ho ) φ 0 2 (ho ) = φ 0 2 (ho ), if h o > 0. Thus, if externality is present so that φ 0 2 (h) 6= 0at all h 0, the equilibrium value h is not Pareto optimal unless h o = h =0. Suppose h o > 0,h > 0. Then, φ 0 1 (h ) = 0 φ 0 1 (ho ) = φ 0 2 (ho ) If externality is negative - in particular, φ 0 2 < 0, then φ 0 1 (ho ) > 0 so that strict concavity of φ 1 implies h > h o. If externality is positive - in particular, φ 0 2 > 0, then φ 0 1 (ho ) < 0 so that strict concavity of φ 1 implies h < h o.

25 Traditional Solutions: 1. Government Intervention: Quotas and Taxes. Suppose externality is negative so that h >h o. - Regulate quantity of h directly and set a maximum permissible level (ceiling) or quota at h o. [Command and control] - Impose a tax on externality generating activity: Pigouvian taxation.

26 Set tax of t h per unit of h on agent 1.Consumer 1 then maximizes: max φ 1(h) t h h h 0 which has the FOC: φ 0 1 (h) t h = t h,ifh>0. So if tax is set at a level so that t h = φ 0 2 (ho ) then h = h o solves consumer 1 s maximization problem (uniquely). Note Pigouvian tax = marginal externality at the optimal solution = amount consumer 2 is willing to pay to reduce h marginally from its optimal level h o. Taxation internalizes an externality.

27 [However, if h o =0, then any tax > φ 0 2 (ho ) also ensures Pareto optimality.] All of the above holds when externality is positive so that h <h o. In that case the quota is replaced by a minimum quantity restriction on h and the tax is a subsidy per unit of h.

28 Even in the case of negative externality, one can subsidize reduction in h (instead of taxing h) to achieve optimality. For example, a subsidy of s h to agent 1 for each unit of reduction below h. Agent 1 maximizes φ 1 (h)+s h (h h) = [φ 1 (h) hs h ]+s h h which is equivalent to imposing a tax per unit of h equal to s h and combine this with a lump sum subsidy of s h h. So setting s h = φ 0 2 (ho ), restores optimality. Similarly, for a positive externality where h <h o,atax on reduction of h below h o can work to reduce optimality.

29 Note: important to tax externality generating activity directly. Also note: though direct quantity control (quotas) and Pigouvian taxes/subsidies are equivalent above, they are not necessarily so if government has less than full information about benefits and costs of the externality to individual agents.

30 2. Decentralized Solution: Bargaining with Enforceable Property Rights. Establish enforceable property rights over externality generating activity. Take the case of negative externality. Suppose we give right to externality free environment to agent 2. Agent 1 can try to get permission from agent 2 to generate any level of h>0 in return for a payment (in terms of the numeraire good).

31 Bargaining process. Take a very simple process: agent 2 makes agent 1 an offer which agent 1 can either accept or reject. The offer specifies payment T in return for permission to generatesomelevelofh. If agent 1 rejects the offer, no further negotiation takes place and h =0.

32 Agent 1 will accept the offer as long as φ 1 (h) T φ 1 (0). So, in a SPNE, of the bargaining game, agent 2 will set (h, T ) so as to max 2(h)+T ] h,t 0 s.t. φ 1 (h) T φ 1 (0) and this is equivalent to max [φ 2(h)+φ 1 (h) φ 1 (0)] h 0 which implies h = h o, the socially optimal level.

33 What if agent 1 is given property right over externality generating activity? Agent 2 can make an offer to pay agent 1 an amount y in return for reducing the level of externality to some h<h. Agent1willagreetodosoonlyif φ 1 (h)+y φ 1 (h ). So, in SPNE of the bargaining game, agent 2 will set (h, y) so as to max 2(h) y h,y 0 s.t. φ 1 (h)+y φ 1 (0) and this is equivalent to max [φ 2(h)+φ 1 (h) φ 1 (0)] h 0 which again implies h = h o, the socially optimal level.

34 Coase theorem (Coase, 1960): If trade of the externality can occur, bargaining will lead to an efficient outcome no matter how property rights are allocated. Better than government intervention as it does not require government to have information about individuals. But does not work well if individuals have incomplete information about each other. Also, transaction costs in decentralized bargaining where many agents are involved.

35 3. Market for right to generate externality. Missing market - at the heart of the externality problem. Suppose: - property rights are well defined and enforceable - competitive market for right to engage in externality generating activity exists.

36 Again, take the case of negative externality. Suppose agent 2 has right to externality free environment. Agent 2 can sell units of the right to generate externality in a competitive market. Agent 1 can buy. Let p h be the price per unit. Both agents price taking.

37 Agent 1 s demand: max h which has the FOC: φ 1(h) hp h φ 0 1 (hd ) p h = p h, if h d > 0. Agent 2 s supply: max h which has the FOC: φ 2(h)+hp h φ 0 2 (hs ) p h = p h, if h s > 0.

38 In competitive equilibrium: h d = h s = h b and φ 0 1 (b h) p h φ 0 2 (b h) φ 0 1 (b h) = p h = φ 0 2 (b h),if h>0. b This implies h b maximizes [φ 1 (h)+φ 2 (h)] i.e., bh = h o and the rights are traded at equilibrium price p h = φ0 1 (ho )= φ 0 2 (ho ).

39 Public Goods. A public good is a commodity for which use of a unit of the good by one agent does not preclude its use by other agents. No rivalry in consumption. If commodity not desirable: Public bad: consumption of a unit of the good by one agent does not decrease its consumption by other agents * Non-depletable: consumption by one individual does not affect supply available for other agents. Knowledge, air quality... Intermediate cases: consumption by one affects availability to others to some degree.

40 Ex. Congestion effects. Partially depletable. If entirely non-depletable: pure public good. If entirely depletable: pure private good. Public goods: excludable or non-excludable. Excludable public goods are those for which it is possible to exclude some individuals from accessing or consuming the public good. Non-excludable: technologically impossible or very costly to do so. We focus on non-excludable pure public goods.

41 Model. I price taking consumers, L traded private goods and one public good. Partial eqm: quantity of public good has no effect on the demand or prices of the L traded goods. Each consumer i s utility: u i (x 1i,...x Li,x)=x 1i + g i (x 1i,x) where x is the provision of (consumption of) the public good and x 1i =(x 2i,...x Li ). Let the price of the numeraire good 1 be set equal to 1, and let p be the price vector of goods 2,...L.

42 Then, consumer i solves max[x 1i + g i (x 1i,x)] s.t.p x 1i w i,x 1i 0. where w i is the wealth of consumer i (in terms of the numeraire good) available for expenditure on private goods 2,...L. Note w i may include profits from shares of firms and may therefore depend on p. The demand for goods 2,...L, is independent of w i. Given p, the indirect utility for each given level of x: v i (p,w i,x) = w i p x 1i (p, x)+g i (x 1i (p,x),x) = w i + φ i (p,x) and fixing the prices of goods 2,...L,wecanwriteφ i (p,x)= φ i (x) as the derived utility to individual i from the public good (willingness to pay for the public good).

43 We assume φ i (x) is twice differentiable with φ 0 i (x) > 0, φ i (x) < 0 on R +. Cost of producing or supplying q units of the public good is c(q) units of the numeraire good. We assume c(q) is twice differentiable with c 0 (q) > 0 and c (q) > 0 at all q 0. Note: for public bad whose reduction is costly: φ 0 i (x) < 0,c 0 (q) < 0.

44 In any Pareto optimal allocation, the provision of the public good must solve: max [{ φ i (q)} c(q)] q 0 i=1

45 To see this, suppose to the contrary that there is a PO allocation where bq units of the public good are produced and for some q 6= bq i.e., i=1 φ i (q) c(q) > i=1 φ i (bq) c(bq) i=1 φ i (q) (c(q) c(bq)) > i=1 φ i (bq) which implies that if we modify the PO allocation by changing the quantity of the public good (produced and consumed) from bq to q and pass on the difference in production cost to the consumers as change in their consumption of the numeraire good (leaving everything else unchanged), then the total utility of all individuals increase. By a suitable transfer of the endowment of numeraire good between, we can now ensure that there is another allocation which makes everyone better off relative to the initial PO allocation. A contradiction.

46 Therefore, the FOC for socially optimal quantity of public good q o : i=1 φ 0 i (qo ) c 0 (q o ), if q o =0, = c 0 (q o ), if q o > 0. At an interior solution, the sum of consumers marginal benefits from the public good (marginal willingness to pay for the public good) must equal the marginal cost of production.

47 Private Provision of Public Goods. Assume market exists for public good and each consumer i decides how much of the public good x i she wants to buy at its market price p. The total amount of the public good purchased is x = i=1 x i. Assume a single price taking competitive firm produces the public good at cost c(.). [Alternatively, J price taking firms with industry cost function c(.).]

48 At a competitive equilibrium with price p,each consumer i s purchase x i solves: max[φ x i (x i + X x i k ) p x i ] k6=i taking the quantity of the public good purchased by other agents as given.

49 To see this recall that the indirect utility for any level of total consumption of the public good x (given w i,the wealth available for expenditure on private goods, and the price vector p of private goods, which is independent of the consumption of the public good) is given by v i (p,w i,x) = w i p x 1i (p, x)+g i (x 1i (p,x),x) = w i + φ i (p,x)=w i + φ i (x) and given the equilibrium consumption of the public good by other agents, x = x i + X k6=i x k and the wealth available for expenditure on private goods is w i = bw i px i where bw i is the total income of consumer i (endowment + share income) so that agent i must choose x i to maximize bw i px i + φ i (x i + X k6=i x k ) which yields the above.

50 First order necessary and sufficient condition: φ 0 i (x i + X x k ) k6=i = φ0 i (x ) p, if x i =0, = p, if x i =0, where x = i=1 x i.

51 Firm s supply q satisfies max q 0 [p q c(q)] so that first order necessary and sufficient condition: Market clearing: p c 0 (q ), if q =0 = c 0 (q ), if q > 0. q = x.

52 Thus, if q = x > 0,then x k > 0 for some k so that φ 0 k (x )=p = c 0 (q ) and since φ 0 i (x ) > 0 for all i, i 1 φ 0 i (q ) >c 0 (q ). In contrast, the socially efficient (Pareto optimal) level of provision of the public good x o satisfies so that i 1 φ 0 i (qo ) c 0 (q o ), if q o =0, = c 0 (q o ), if q o > 0. q o >q.

53 Market provides too little of the public good. Reason: purchase of the public good by one agent creates positive externality for others. Individual s private purchase decision does not take into account this externality. Free-rider problem.

54 Suppose that consumers are ordered in their marginal benefit from the public good i.e., φ 0 1 (x) <φ0 2 (x) <... < φ0 I (x), x 0. Then, φ 0 i (x )=p can hold for at most one consumer i and since φ 0 k (x ) p for all k 6= i, i = I. Therefore, only the consumer who derives the largest marginal benefit from the public good will provide at and the total provision of the public good would be exactly equal to what it would be if agent I was the only consumer in the economy. The other agents free-ride perfectly.

55 Government Intervention. Quantity based intervention: directly provide x (funded throughlumpsumtaxes). Price-based intervention: taxes and subsidies: Subsidytoeachconsumeri: s i = X k6=i φ 0 k (qo ) per unit of the public god purchased. In that case, at any price p of the public good, each consumer i maximizes: so that FOC: φ i (x i + X k6=i x k ) (p s i )x i φ 0 i (x i + X k6=i x k ) p s i, if x i =0 = p s i, if x i > 0.

56 so that (assuming q o > 0) we have a competitive equilibrium at price p o = c 0 (q o ), x o = choosing i for which x o i = so that x o i i=1 = q o,and > 0 we have from the FOC: φ 0 i (xo i + X x o k )+X φ 0 k (qo ) k6=i k6=i k=1 k=1 φ 0 k (qo )=p o φ 0 k (qo )=c 0 (q o ) which ensures Pareto optimality. All of this requires that the government know the benefits derived by consumers from the public good.

57 Lindahl Equilibria. For each consumer i, design a market for the public good as experienced by consumer i. Assume that you can price a consumer for all the public good she enjoys. Each consumer i s consumption of the public good is a distinct good with its own price p i.

58 Given price p i each consumer decides on her total consumption x i of the public good so as to maximize: FOC: max x i 0 [φ i(x i ) p i x i ] φ 0 i (x i) p i = p i, if x i > 0.

59 Think of the firm as producing a bundle of I goods, one for each consumer, (q 1,...,q I ) subject to a technological constraint q 1 =... = q I which reflects the fact that by its very nature, you can t produce different levels of public good consumption for different consumers. Thus, in effect, firm solves: with FOC: max [( p i q) c(q)] q 0 i=1 i=1 p i c 0 (q) = c 0 (q), if q>0.

60 In market equilibrium with price p i, the optimal consumption of the public good by each consumer must equal the total production of the public good: and the FOCs imply: x i = q and i=1 φ 0 i (q ) i=1 p i c 0 (q ) i=1 φ 0 i (q )= which means that i=1 p i = c 0 (q ), if q > 0, q = q o and the market equilibrium is Pareto optimal.

61 This type of equilibrium with personalized markets for the public good is called a Lindahl equilibrium. In essence, each consumer is priced for her entire consumption of the public good, so there are no externalities or free riding. However, it requires excludability. Should be able to exclude consumers from consumption of the public good if they don t pay according to their personalized price. Also, single buyer in each personalized market makes the price taking assumption unrealistic.

62 Multilateral Externalities. Numerous parties generate and are affected by externalities. Depletable externalities (private, rivalrous) : experience of the externality by one agent reduces the amount felt by other agents. Somewhat like a private good. Non depletable externalities are like public goods. Air pollution. Ex.

63 Lindahl Equilibria. For each consumer i, design a market for the public good as experienced by consumer i. Assume that you can price a consumer for all the public good she enjoys. Each consumer i s consumption of the public good is a distinct good with its own price p i.

64 Given price p i each consumer decides on her total consumption x i of the public good so as to maximize: FOC: max x i 0 [φ i(x i ) p i x i ] φ 0 i (x i) p i = p i, if x i > 0.

65 Think of the firm as producing a bundle of I goods, one for each consumer, (q 1,...,q I ) subject to a technological constraint q 1 =... = q I which reflects the fact that by its very nature, you can t produce different levels of public good consumption for different consumers. Thus, in effect, firm solves: with FOC: max [( p i q) c(q)] q 0 i 1 i=1 p i c 0 (q) = c 0 (q), if q>0.

66 In market equilibrium with price p i, the optimal consumption of the public good by each consumer must equal the total production of the public good: and the FOCs imply: x i = q and i=1 φ 0 i (q ) i=1 p i c 0 (q ) i=1 φ 0 i (q )= which means that i=1 p i = c 0 (q ), if q > 0, q = q o and the market equilibrium is Pareto optimal.

67 This type of equilibrium with personalized markets for the public good is called a Lindahl equilibrium. In essence, each consumer is priced for her entire consumption of the public good, so there are no externalities or free riding. However, it requires excludability. Should be able to exclude consumers from consumption of the public good if they don t pay according to their personalized price. Also, single buyer in each personalized market makes the price taking assumption unrealistic.

68 Multilateral Externalities. Numerous parties generate and are affected by externalities. Depletable externalities : experience of the externality by one agent reduces the amount felt by other agents. Like a private good. Non depletable externalities are like pure public goods. Ex. Air pollution.

69 Assume agents who generate externalities are different from those who experience them. (Simplifying) In particular, suppose externality is generated by firms and experienced by consumers. Also, assume externality generated by firms is homogenous (consumers do not care about source of externality). Price taking agents.

70 L traded goods with price vector p. J firms that generate externality in the process of production. Given p, theprofit function of firm j can be defined as a function of the externality h j 0 it generates: π j (h j ). I>1consumers with quasi-linear utility. Given price vector p, the preferences of consumer i with respect to the externality e h i experienced by her is given by her derived utility φ i ( e h i ). Her indirect utility is actually φ i ( e h i )+w i. Assume: j =1,...J,i =1,...I,π j,φ i are twice differentiable with π j (.) < 0,φ i (.) < 0. To fix ideas, assume that the externality is negative i.e., φ 0 i (.) < 0,π0 j > 0.

71 Depletable Externality. At any unrestrained competitive equilibrium where there is no market for the externality, each firm j generates externality h j so that π 0 j (h j ) 0, if h j =0, = 0, if h j > 0.

72 Any PO allocation involves the levels (h o j,j =1,..J,e h o i,i= 1,...I) that solve: s.t. i=1 eh i = max[ h j,e hi i=1 JX j=1 h j. φ i ( e h i )+ JX j=1 π j (h j )] The constraint reflects the depletable nature of the externality - one more unit of the externality experienced by one person implies there is one unit less of the externality to be experienced by others.

73 If a PO allocation does not solve the above problem, then onecanmodifytheprofile of externality generated and consumed so as to increase φ i ( e h i )+ JX i=1 j=1 π j (h j ). Since JX j=1 π j (h j ) is returned to consumers as share income the total indirect utility i=1 (φ i ( e h i )+w i ) of all individuals can be increased i.e., a higher utility possibility frontier can be reached which violates the initial hypothesis of Pareto optimality.

74 If μ is the multiplier for this constraint, FOC (necessary &sufficient) : φ 0 i (e h o i ) μ = μ, if h eo i > 0. μ π 0 j (ho j ), = π 0 j (ho j ), if ho j > 0. If well defined and enforceable property rights can be specified over externality and competitive market for the externalityexists(withi,j being large enough so that all agents are price takers) with price =μ, then Pareto optimality can be attained by decentralized decision making.

75 Note: here it does not matter whether - consumers are awarded the property rights so that each firm has to buy units of the externality generated by it from the consumers OR - firms are awarded the property rights so that each consumer has to buy units of reduction in the externality generated by firms.

76 Non-depletable Externality. Each consumer experiences an externality level equal to JX j=1 h j. Essentially, the externality generated by firms is pure public bad.

77 In any Pareto optimal allocation, the level of externality generated (h o j,j =1,...J) satisfies: max h j i=1 φ i ( JX k=1 h k )+ FOC (necessary and sufficient): JX j=1 π j (h j )] φ 0 JX i ( h o k ) i=1 k=1 π0 j (ho j ) = π 0 j (ho j ), if ho j > 0. Note: This is exactly the condition for optimal provision of the public good if we interpret π 0 j as a firm s marginal cost of production of public good. [In our specific discussion of public good, we took J =1].

78 As in the case of public good, if we have a competitive market for the externality we will not get the Pareto optimal level of externality. Free rider problem. This is a fundamental difference of the multilateral externality problem from the bilateral case.

79 Recall that in an unrestrained competitive eqm, each firm j generates externality h j so that π 0 j (h j ) 0, if h j =0, = 0, if h j > 0. Suppose we give property rights to firms and let consumers buy from any firm each unit of externality reduction below its unrestrained level. If consumers buy zero from firm j, it would generate h j amount of externality. Given price p for unit reduction in externality, firm j chooses h j so as to solve: max π j(h h j [0,h j )+(h j ] j h j)p and so FOC: π 0 j (h j ) p = p, if h j > 0.

80 Each consumer i buys reduction of amount x i solve: so as to so that FOC: JX max φ x i ( h i j x i X j=1 k6=i x k ) p x i φ 0 i ( J X j=1 In equilibrium, h j x i X k6=i x k ) p = p,ifx i > 0. JX j=1 (h j h j )= Suppose x > 0.Then, x i i.e., i=1 x i = x X φ 0 J i ( h j x )=p j=1 φ 0 i ( J X j=1 > 0 for some i. Inthatcase h j )= p

81 and since φ 0 i < 0, i, i=1 φ 0 i ( JX j=1 h j ) < p This violates Pareto optimality. π 0 j (h j ),j =1...J

82 Market based solution can work only if we have personalized markets for externality as in the Lindahl equilibrium concept. But that, as we have seen, is problematic. Similar free rider problems would also arise in any bargaining solution.

83 Government intervention can secure efficiency (as for other pure public goods). Quotas: Government can regulate a quota equal to h o j on externality generation. (ex. emission standards).

84 Taxes: Impose a tax per unit of externality generation: t = φ 0 JX i ( h o k ) i=1 k=1 Each firm j will then set h j so as to solve: which yields FOC: max h j 0 π j(h j ) th j π 0 j (h j) t = t, if h j > 0 and comparing to necessary and sufficient condition for Pareto efficiency in externality generation, we have h j = h o j.

85 Both taxes and quota require government to know utility and profit functions of consumers & firms. Mixed solution: Tradeable Externality permit. Government gives h o = JX h o k k=1 permits to firms. Each permit allows a firm to generate a unit of externality. Total amount of externality generated is at the optimal level. Question: will each firm generate the optimal level of externality?

86 Permits distributed arbitrarily among firms. Firm j given h j permits. Firms then trade these permits in a competitive market. Let p h be the equilibrium price. Each firm j s demand for permits (and its externality generation) at this price is given by solving: so that from the FOC: max h j π j (h j )+p h (h j h j ) π 0 j (h j) p h = p h, if h j > 0. Market clearing: JX j=1 h j = h o.

87 Comparing to the FOC for Pareto optimality, it can be shown that the above equations can be satisfied only if and p h = i=1 φ 0 i (ho ) h j = h o j. This scheme is useful when government has limited information on individual profit functions of firms & cannot say which firm should bear how much of the externality reduction. Of course, it requires government to be able to compute the aggregate externality level h o.