Externalities

1. Prerequisites Almost essential Welfare and Efficiency Externalities MICROECONOMICS Principles and Analysis Frank Cowell June 2006

2. Externalities Overview... The nature of externality A special type of transaction Production externalities Consumption externalities Connections

3. The nature of externality • An externality is a kind “involuntary” transaction • A case where market allocation methods don’t work • Agents cannot be excluded from the transaction using conventional price mechanism • An example of “market failure”? • Externalities can be detrimental or beneficial • We will deal with two broad types: • Production externalities • Consumption externalities

4. Production externality • One firm influences another’s production conditions • Affects other firms’ cost curves. • Not effect of wage or input price changes… • …externality is outside the market mechanism. • Model this as a parameter shift • If firm f’s output produces an externality… • …production function of firm k has f’s output as a parameter… • … or MC curve of firm k has f’s output as a parameter. • Example: networking • One firm’s activity creates pool of skilled workers from which neighbouring firms may benefit. • Example: pollution • One firm’s activity (glue production) causes emissions that are to the detriment of its neighbours (restaurants who must filter the air).

5. Consumption externality • One agent’s consumption of a good directly affects another • Alf’s consumption of good 1 is an argument of Bill’s utility function • Related to the analysis of public goods • Public goods are non-excludable and non-rival • These properties are mutually independent • Consumption externalities are non-excludable but rival • Example: Scent from fresh flowers • Nonexcludable: you can’t charge for the scent • Rival: more scent requires more flowers

6. Externality questions • How can we model different types of externality? • How can we quantify an externality? • How can we value an externality? • How will the externality modify the efficiency conditions? • How can we implement an efficient outcome if there are externalities?

7. Externalities Overview... The nature of externality How production externalities work; how they are evaluated • Basics • Efficiency • Simple implementation • Private initiative Production externalities Consumption externalities Connections

8. Production: the framework • There is a known collection of firms • Indexed by f = 1,2, ..., nf. • Identities of firms exogenously determined • Describe each firm’s activities using net-output vector. • Net output by firm f of good i is qif, i = 1,2,...,n • Usual sign convention • Net output vector is qf = (q1f,q2f,q3f,...,qnf) • Firm f’s production possibilities are known • Implicit production function Ff(•) • Argument is net output vector is qf , and possibly other things. • Set of feasible net outputs given byFf(qf) ≤ 0 • Transformation curve given by net outputs such thatFf(qf) = 0 • Now introduce externality...

9. Quantifying an externality • Consider a polluting firm f. • Case of a positive externality follows easily • Just reverse signs appropriately... • …and rename “victim” as “beneficiary.” • When f produces good 1 it causes the pollution • could affect other firms k = 1, 2, ..., f – 1, f + 1, ..., nf. • the more f produces good 1, the greater the damage to k. • How much damage? • Consider the impact of pollution on firm k • Will enter the production function Fk(•) • Use the firm’s transformation curve... Standard diagram.... Jump to “Multi-output firm”

10. q2 k k q1` Externality: Production possibilities • Production possibilities, firm k Fk(•) > 0 • Production possibilities, if firm f’s emissions increase Fk(•) = 0 Fk(•) < 0 low emissions by firm f • If Fk(•) = 0 then an increase in the negative externality will result in Fk(•) > 0. high emissions by firm f

11. Valuing an externality • What is value to victim firm k of pollution by f ? • Need quantification of pollution: • Identify source of externality – production of good 1 • Then use units of output of good 1. • Use same approach as for “value of an input” • Focus on impact of marginal amount: • How much impact on activity of firm k? • Need the derivative of production function Fk. • Measure effect in terms of a numéraire: • Here we take this to be good 2. • But could be any other good.

12. vanishes if there is no externality Production externality • Characteristics of production generates inefficiency • Firm k may be affected by others' output of good 1: Fk(qk; q11,q12,...,q1k-1, q1k+1...) net output of firm k this is positive for a negative externality: it is shifting “inwards” firm k’s feasible set. • Now evaluate the marginal impact of some firm f on others: • Direct impact of f on production possibilities of firm k nf å k=1 1 —— F2k ¶Fk(•) ——— ¶q1f e21f := – • …evaluated in terms of good 2… • …and summed over all k • Value of the marginal externality imposed through production by f of good 1. Marginal product of good 2 for firm k

13. Externalities Overview... The nature of externality Deriving the conditions for a PE allocation • Basics • Efficiency • Simple implementation • Private initiative Production externalities Consumption externalities Connections

14. Externality and efficiency • Take the problem of efficient allocation with externality • Two main subproblems are treated separately... • Characterisation • Implementation • Characterisation uses standard efficiency model • introduce production/consumption externality features • examine impact on the FOCs • Implementation may follow on from this... Jump to “Welfare: efficiency”

15. The approach • Use a maximisation procedure to characterise efficiency: • Specify technical and resource constraints • Fix all persons but one at an arbitrary utility level • Then max utility of remaining person • So problem is to maximise U1(x1) subject to: • Uh(xh) ≥ uh, h = 2, …, nh • Ff(qf; q11,q12,...,q1f1,q1f+1...) ≤ 0,f = 1, …, nf • xi ≤qi + Ri ,i= 1, …, n • where • xh = (x1h,x2h,x3h,...,xnh) • xi = åh xih , i = 1,...,n • qi = åf qi f technical feasibility materials' balance

16. Lagrangean method: • Introduce Lagrange multipliers: • lh for each utility constraint • mffor each firm’s technology constraint • ki for materials’ balance on good i. • Then maximise U1(x1) + åhlh[Uh(xh)  uh]  åf mfFf (qf; q11,q12,...,q1f1,q1f+1...) + åi ki[qi + Ri  xi] • First-order conditions for an interior maximum: • lhUih (xh) = ki, i = 1,...,n nhf ¶Fk(•) • mf Fif(qf) + mfå ——— = k1 • k=1¶q1f • mfFif(qf) = ki , i = 2,3,...,n only good 1 generates an externality

17. From the FOC... • Consider tradeoff between goods 1 and 2 • From the first of the FOCs: U1h(xh) k1 ——— = — U2h(xh) k2 • Use the definition of e21f. Then other FOCs give F1f(qf) k1 ——— –e21f= — F2f(qf) k2 • This is the efficiency criterion: • Instead of the condition “MRT=shadow price ratio”… • …we have a modified marginal rule.

18. q2 f ~ qf ^ qf Efficiency with production externality • Production possibilities • If externality is ignored... • Taking account of externality... F1 k1 — = — + externality F2 k2 • Produce less of good 1 for efficiency... F1 k1 — = — F2 k2 f q1`

19. Externalities Overview... The nature of externality Corrective taxes and other devices... • Basics • Efficiency • Simple implementation • Private initiative Production externalities Consumption externalities Connections

20. Implementation • Use the efficiency criterion for guidance on policy design • The simple marginal rule suggests a method of implementation • We can use it to modify the market mechanism: • MRT – producer prices • MRS – consumer prices • …how to connect the two of these?

21. Towards a policy rule (2) value ofexternality shadow prices F1f(qf) k1 ——— –e21f= — F2f(qf) k2 • Take the modified FOC (private) marginal cost of producing 1 F1f(qf) k1 ——— = — + e21f F2f(qf) k2 • Rearrange: consumer prices F1f(qf) p1 ——— = — + e21f F2f(qf) p2 • Introduce the market: t = – e21f • Corrective tax (negative externality) or subsidy (positive externality): F1f(qf) p1 ——— = — – t F2f(qf) p2

22. Production externality: policy • From the FOC a simple corrective tax can be designed • Called “Pigovian” (from A.C. Pigou’s Economics of Welfare) • Needs information about production functions. • Both for victim and perpetrator. • Alternative 1: merger. • Merging the firms “internalises” the externality. • Combined firm takes into account interdependence of production. • Alternative 2: public issue of “pollution rights” • Again the externality is internalised. • Polluter takes account of true its activity because of new market • Equilibrium price determined as for the Pigovian tax. • However could there be a purely private solution?

23. Externalities Overview... The nature of externality Development of a “pseudo market” • Basics • Efficiency • Simple implementation • Private initiative Production externalities Consumption externalities Connections

24. Private solution: A model • Efficient outcome through individual initiative? • Assume that there are just two firms and two goods. • The second of these assumptions is unimportant. • First assumption may be important. • Firm 1’s output of good 1 imposes costs on firm 2. • Full information: • Each firm knows the other’s production function. • Externality is common knowledge. • Activity can be monitored. • Communication is costless. • Firm 2 (victim) has an interest in communicating • Does this by setting up a financial incentive for firm 1. • How should this be structured?

25. The victim’s problem • Firm 2 offers firm1 a side-payment (Bribe) b. • This payment needs to be accounted for in the computation of profits. • It can be treated as a control variable for firm 2. • Optimisation problem of firm 2 (the victim) is: n max S piqi2− b − m2F2(q2, q11) {q2, b} i=1 • Solve this in the usual way…

26. The victim’s problem: interpretation • Firm 2 designs incentive for firm 1 • a “side-payment schedule” • or “conditional bribe function.” • Incentive scheme captures costs to firm 2 • Slope equals marginal cost of pollution. • The higher is the level of the polluting output... • ...the lower is the level of the conditional bribe. • Should influence actions of perpetrator (firm 1) • Analyse firm 1’s behaviour in same framework

27. Solving the victim’s problem • FOC for net outputs of firm 2 is pi − m2Fi2 (q2, q11) = 0 • FOC for the side payment b is: dF2(q2, q11) dq11 − 1 +m2 ─────── ── = 0 dq11 db • Using the definition of the externality: dq11 − 1 +m2F22(q2, q11) e211 ── = 0 db • Rearranging the FOC then gives: db ── = m2F22(q2, q11) e211 = p2e211 dq11

28. The perpetrator’s problem • For firm 2’s “schedule” to work, firm 1 has to know about it • It rationally incorporates this into its profit calculation • It will note that the bribe is conditional on a variable under its own control • The optimisation problem for firm 1 is: • n • max S piqi1+ b(q11)− m1F1(q1) • q1i=1 • Again solve this in the usual way…

29. Solving the problem Feedback effect from 1’s net output on 2’s bribe offer • FOC for net outputs of firm 1 is: • d b(q11) • p1qi1 + ───── − m1F11(q1) = 0 • dq11 • p2− m1F21(q1) = 0 • Substituting in for the slope of the bribe function: • F11(q1) p1 • ──── = ── +e211 • F21(q1)p2 • This condition same as FOC for efficiency!

30. Private solution: result • Bribe function has internalised the externality • Firm 2 conditions side-payment on observable output of good 1 • Firm 1’s responds rationally to the side-payment. • FOC conditions same as before • Private solution induces an efficient allocation • Implements the same allocation as the Pigovian tax • But no external guidance is required. • It should be independent of where the law places the responsibility for the pollution (Coase’s result)

31. Private solution: difficulties • Solution makes important informational requirements • Imposed on both firms. • There may be an incentive for firms to misrepresent costs, leading to loss of efficiency. • It requires a special notion of participation. • What determines the set of participants? • What if there is free entry? • It focuses only on marginal impacts • If the polluter is allowed to sell pollution rights there could be problems with this private sector “solution” • This is similar to the nonconvexity problem

32. ~ q ^ q A fundamental nonconvexity q2 • Production possibilities… • If firm 1’s pollution could drive the other out of business • The optimal point? • If polluter can sell pollution rights indefinitely... 0 q1

33. Externalities Overview... The nature of externality Interactions between consumers Production externalities Consumption externalities Connections

34. vanishes if there is no externality Consumption externality • Characteristics of goods generates inefficiency • Household ℓ affected by others’ consumption of good 1: Uℓ(xℓ; x11,x12,...,x1ℓ1, x1ℓ+1,...) consumption of household ℓ • Now evaluate the marginal impact of some household h on others: • Direct impact of h on utility of ℓ… nh å ℓ=1 1 —— U2ℓ ¶Uℓ(•) ——— ¶x1h • …evaluated in terms of good 2… e21h:= • …and summed over all ℓ MU of good 2 for household ℓ • Gives the value of the marginal externality imposed through consumption by h of good 1.

35. Lagrangean method: • Use same method as for production externalities • Introduce Lagrange multipliers: • lh for each utility constraint • mffor each firm’s technology constraint • ki for materials’ balance on good i. • Then maximise U1(x1;,x12,x13, ...) + åhlh[Uh(xh; x11,x12,...,x1h-1, x1h+1,...)  uh]  åf mfFf (qf) + åi ki[qi + Ri  xi] • First-order conditions for an interior maximum: nh ¶U1ℓ(•) • lhU1h (x1;,x12,x13, ...) + lh å ——— = k1 • ℓ=1¶x1h • lhUih (x1;,x12,x13, ...) = ki , i = 2,3,...,n • mfFif(qf) = ki, i = 1,2,...,n only good 1 generates the externality

36. FOC has a similar interpretation... • From the FOC for production: F1f(qf) k1 ——— = — F2f(qf) k2 • Substituting in the value of the externality we also have U1h(xh) k1 ——— +e21h =— U2h(xh) k2 • Again we have a modified marginal rule • Again it can give us useful guidance on policy

37. Negative consumption externality • Production possibilities • Competitive equilibrium (with consumption externality) x2 U1h F1 — = — – externality U2hF2 • Efficiency with consumption externality • Produce less of good 1 for efficiency... U1hF1 — = — U2hF2 x1`

38. Towards a policy rule value ofexternality shadow prices U1h(xh) k1 ——— +e21h= — U2h(xh) k2 • Take the modified FOC h willingness to pay for 1 in terms of 2 U1h(xh) k1 ——— = — – e21h U2h(xh) k2 • Rearrange: Producer prices U1h(xh) p1 ——— = — – e21h U2h(xh) p2 • Introduce the market: t = −e21h U1h(xh) p1 ——— = — + t U2h(xh) p2 • A Pigovian tax/subsidy (for negative/positive externalities) .

39. Externalities Overview... The nature of externality Lessons and applications Production externalities Consumption externalities Connections

40. Externalities: lessons • The analysis of externality is not a peripheral issue in microeconomics • Connects to other key topics... • Industrial organisation: • Production externalities and industry supply • Merger as a solution to inefficiency with externality • Public goods: • An extreme form of consumption externality

41. Externalities: summary modify the MRS = MRT rule by the marginal cost of externality • Characterisation problem: • Implementation problem: For production externalities – encourage private resolution through extended markets? Otherwise introduce a tax/subsidy corresponding to the marginal cost of externality