Market Failure: Externalities

Transcription

1 Market Failure: Externalities Ram Singh Lecture 21 November 10, 2015 Ram Singh: (DSE) Externality November 10, / 18

2 Questions What is externality? What is implication of externality for efficiency of competitive equilibrium? What are the corrective measures available? What are the relative merits of the corrective measures? Can the market itself take care of externality? Ram Singh: (DSE) Externality November 10, / 18

3 WE with FOPs I Assume There are no intermediate goods; There are pure inputs (factors of production) and pure consumptions goods; Pure inputs/fops are l = 1,..., L. Set of FOPs is L = {1,.., L} total endowments of factors is z = ( z 1,..., z L ) >> 0 and is initially owned by consumers. Consumers do not directly consume these FOP endowments. One firm produces only one good; good j is produced by firm j. So, k = j, and j = 1,..., J. Set of firms and also the consumption goods is J = {1,.., J} Ram Singh: (DSE) Externality November 10, / 18

4 WE with FOPs II Let, p = ( p 1,..., p J ) be the given the output price vector, consider any arbitrary allocation of FOPs across firms, say w = (w1,...w L ) be the given input price vector (y 1,..., y J ) be the profit maximizing output production level of FOPs for firms j = 1,..., J. (z 1,..., z J ) be the profit maximizing demand of FOPs for firms j = 1,..., J. Ram Singh: (DSE) Externality November 10, / 18

5 WE with FOPs III Recall, Proposition The Competitive equilibrium (WE) is Pareto optimum. Proposition The equilibrium factor allocation, (z 1,..., z J ), is Pareto optimum. Proposition The equilibrium factor demand, (z 1,..., z J ), maximizes the aggregate/total profit for the economy. Ram Singh: (DSE) Externality November 10, / 18

6 WE with FOPs IV Proposition The production plan (y 1,..., y J ) maximizes the aggregate/total profit for the economy if and only if it is a Pareto optimal plan. However, in presence of externality, all these results breakdown in fact, the existence of WE cannot be guaranteed anymore Government intervention is needed - generally but not always Ram Singh: (DSE) Externality November 10, / 18

7 A simple illustration I Assume There are two competitive firms Firm 2 uses only one FOP, say l, to produce one marketable output Firm 1 also uses the FOP l but it produces a marketable output along with another non-marketable output/input e There is no market in e Firm 1 decides on the level of e; firm 2 has no direct control over choice of e The profit functions are π 2 (y 1, l 1, e, p, w) and π 2 (y 2, l 2, e, p, w), respectively Ram Singh: (DSE) Externality November 10, / 18

8 A simple illustration II Note for given p and w, we have Note: You can think of Assume φ i (e, p, w) = max π i (y i, l i, e, p, w) φ i (e) φ 1 (e) as the maximum profit for 1 given the level of e opted by firm 1. φ 2 (e) as the maximum profit for 2 given the level of e opted by firm 1. φ 1(e) > 0, φ 1 (e) < 0, φ 2(e) < 0, φ 2 (e) > 0, i.e., e is good for firm 1 but bad for firm 2. Ram Singh: (DSE) Externality November 10, / 18

9 A simple illustration III Moreover, there exists ē, such that φ 1 (ē) < 0. Question Which firm is the cause behind the externality? Firm 1 will solve max e {φ 1 (e)}. It will choose e p that solves the following FOCs: φ 1(e) = 0 (1) That is, φ 1 (ep ) = 0. However, the total profit maximization problem is max e {φ 1(e) + φ 2 (e)} (2) For this OP, the FOCs is: Let e solve (3). φ 1(e) + φ 2(e) = 0 (3) Ram Singh: (DSE) Externality November 10, / 18

10 A simple illustration IV That is, φ 1 (e ) + φ 2 (e ) = 0. Clearly, e p > e. Question What is wealth maximizing level of externality - e p or e? What is Kaldor efficient level of externality - e p or e? Ram Singh: (DSE) Externality November 10, / 18

11 WE with Externality I Assume There are J firms; j = 1, 2,..., J FOPs are l = 1,..., L, e. Set of FOPs is L = {1,.., L, e} There is no market in e Firm 1 decides how much of e to use/produce total endowments of factors is z = ( z 1,..., z L ) >> 0 and is initially owned by consumers. Consumers do not directly consume these endowments. One firm produces only one good; good j is produced by firm j. So, k = j, and j = 1,..., J. Set of firms and also the consumption goods is J = {1,.., J} Ram Singh: (DSE) Externality November 10, / 18

12 Individual Production Levels I For simplicity, assume there are only two firms. Take any price vector p = ( p 1,..., p J ) for outputs and w = ( w 1,..., w L ) for inputs. Firm 1 will choose e and ȳ 1 that solves = max y 1,e { p 1 f 1 (z 1, e) } L w k.zk 1 where f 1 (z 1, e) is the output level, when input vector used is z 1 = (z 1 1,..., z1 L ) along with e. When f 1 (.) is strictly increasing and strictly concave, the demanded (z 1, e) will solve the following FOCs: k=1 Ram Singh: (DSE) Externality November 10, / 18

13 Individual Production Levels II The 2nd firm will solve: zl 1 f solves : p 1 (z 1, e) 1 zl 1 f e solves : p 1 (z 1, e) 1 e max z 2 { p 2 f 2 (z 2, e) = w l for all l = 1,..., L, (4) = 0. (5) } L wk.zk 2, k=1 The demanded (z 2 ) will solve the following FOCs: z 2 l solves : p 2 f 2 (z 2, e) z 2 l = w l for all l = 1,..., L, (6) So, the WE factor allocation is characterized by (4), (5) and (6). Ram Singh: (DSE) Externality November 10, / 18

14 Pareto Optimal Levels The total profit maximization problem is 2 max ( p j f j (z j, e) w.z j ) e,z 1,z 2 j=1, i.e., max { p1 f 1 (z 1, e) w.z 1 + p 2 f 2 (z 2, e) w.z 2} (7) e,z 1,z 2 For this OP, the FOCs are: zl 1 f solves : p 1 (z 1, e) 1 zl 1 f e solves : p 1 (z 1, e) 1 e z 2 l solves : p 2 f 2 (z 2, e) z 2 l = w l for all l = 1,..., L, (8) + p 2 f 2 (z 2, e) e = 0. (9) = w l for all l = 1,..., L, (10) So, the PO allocation of factors is characterized by (8), (9) and (10). Ram Singh: (DSE) Externality November 10, / 18

15 Corrective Measure: Quantity Regulation Quantity Regulation: Firm is allowed to produce up to e and not beyond Sever penalty for production beyond e In equi, Firm 1 will choose e = e The outcome is Pareto efficient. Ram Singh: (DSE) Externality November 10, / 18

16 Corrective Measure: Pigouvian Tax (Price Regulation) Let us go back to the simple case. Suppose, There are two firms Firm 1 causes negative externality for Firm 2 Govt imposes tax on externality creator Firm 1 pays a per unit tax t = φ 2 (e ). Now, 1 will choose e p that solves: max {φ 1(e) te}, i.e., max {φ 1(e) φ 2(e ).e} e e φ 1(e) t = 0, i.e., φ 1(e) φ 2(e ) = 0, i.e., e p = e. Ram Singh: (DSE) Externality November 10, / 18

17 Corrective Measures: Subsidy Suppose, Govt offers subsidy to the externality creator for a reduction in externality level below e Gross subsidy is: s(e) = (e e)φ 2 (e ). Now, 1 will choose e p that solves: max {φ 1(e) + s(e)}, i.e., max {φ 1(e) + (e e)φ 2(e )} e e Again, e p = e. φ 1(e) + s (e) = 0, i.e., φ 1(e) φ 2(e ) = 0, i.e., Ram Singh: (DSE) Externality November 10, / 18

18 Corrective Measures: Liability Let φ 2 be the profit in the absence of externality, i.e., φ 2 = φ 2 (e = 0) Suppose, The externality creator is required to compensate the victim of externality Firm 1 pays a compensation equal to loss; l(e) = φ 2 φ 2 (e). Now, 1 will choose e p that solves: max {φ 1(e) l(e)}, i.e., max {φ 1(e) [ φ 2 φ 2 (e)]} e e φ 1(e) l (e) = 0, i.e., φ 1(e) φ 2(e) = 0, i.e., e p = e. Ram Singh: (DSE) Externality November 10, / 18