Prerequisites Almost essential Welfare and Efficiency Frank Cowell

Prerequisites Almost essential Welfare and Efficiency Frank Cowell: Microeconomics June 2006 Externalities MICROECONOMICS Principles and Analysis Frank Cowell

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality A special type of transaction Production externalities Consumption externalities Connections

The nature of externality Frank Cowell: Microeconomics n n An externality is a kind “involuntary” transaction A case where market allocation methods don’t work u u n n 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: u u Production externalities Consumption externalities

Production externality Frank Cowell: Microeconomics n One firm influences another’s production conditions u u u n Model this as a parameter shift u u u n 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 u n Affects other firms’ cost curves. Not effect of wage or input price changes… …externality is outside the market mechanism. One firm’s activity creates pool of skilled workers from which neighbouring firms may benefit. Example: pollution u One firm’s activity (glue production) causes emissions that are to the detriment of its neighbours (restaurants who must filter the air).

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

Externality questions Frank Cowell: Microeconomics n How can we model different types of externality? n How can we quantify an externality? n How can we value an externality? n How will the externality modify the efficiency conditions? n How can we implement an efficient outcome if there are externalities?

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality How production externalities work; how they are evaluated Production externalities Consumption externalities Connections • Basics • Efficiency • Simplementation • Private initiative

Production: the framework Frank Cowell: Microeconomics n There is a known collection of firms u u n Describe each firm’s activities using net-output vector. u u u n Net output by firm f of good i is qif, i = 1, 2, . . . , n Usual sign convention Net output vector is qf = (q 1 f, q 2 f, q 3 f, . . . , qnf) Firm f’s production possibilities are known u u n Indexed by f = 1, 2, . . . , nf. Identities of firms exogenously determined Implicit production function Ff( • ) n output vector is qf , and possibly other things. Argument is net Set of feasible net n outputs given by Ff(qf) ≤ 0 n outputs such that Ff(qf) = 0 Transformation curve given by net Now introduce externality. . .

Quantifying an externality Frank Cowell: Microeconomics n Consider a polluting firm f. u u u n When f produces good 1 it causes the pollution u u n 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? u Jump to “Multi-output firm” Case of a positive externality follows easily Just reverse signs appropriately. . . …and rename “victim” as “beneficiary. ” Consider the impact of pollution on firm k Will enter the production function Fk( • ) Use the firm’s transformation curve. . . Standard diagram. .

Externality: Production possibilities Frank Cowell: Microeconomics q 2 k Fk( • ) §Production possibilities, firm k >0 §Production possibilities, if firm f’s emissions increase =0 Fk( • ) < 0 low emissions by firm f §If Fk( • ) = 0 then an high emissions by firm f increase in the negative externality will result in Fk( • ) > 0. k q 1`

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

Production externality Frank Cowell: Microeconomics n Firm k may be affected by others' output of good 1: Fk(qk; q 11, q 12 , . . . , q 1 k-1, §Characteristics of production generates inefficiency vanishes if there k+1 q 1. . . ) is no externality net output of firm k this is positive for a negative n Now evaluate the marginal impact of externality: it is shifting some firm f on others: “inwards” firm k’s feasible set. nf e 21 f : = – å k=1 Marginal product of good 2 for firm k 1 ¶Fk( • ) —— ——— F 2 k ¶q 1 f §Direct impact of f on production possibilities of firm k §…evaluated in terms of good 2… §…and summed over all k §Value of the marginal externality imposed through production by f of good 1.

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality Deriving the conditions for a PE allocation Production externalities Consumption externalities Connections • Basics • Efficiency • Simplementation • Private initiative

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

The approach Frank Cowell: Microeconomics n Use a maximisation procedure to characterise efficiency: u u u n So problem is to maximise U 1(x 1) subject to: u u u n Specify technical and resource constraints Fix all persons but one at an arbitrary utility level Then max utility of remaining person Uh (xh ) ≥ uh , h = 2, …, nh Ff(qf; q 11, q 12 , . . . , q 1 f 1, q 1 f+1. . . ) ≤ 0, f = 1, …, nf xi ≤ qi + Ri , i= 1, …, n where u u u xh = (x 1 h, x 2 h, x 3 h, . . . , xnh) xi = åh xih , i = 1, . . . , n qi = åf qi f materials' balance technical feasibility

Lagrangean method: Frank Cowell: Microeconomics n Introduce Lagrange multipliers: u u u n n lh for each utility constraint mf for each firm’s technology constraint ki for materials’ balance on good i. Then maximise U 1(x 1) + åhlh [Uh(xh) uh] åf mf F f (qf; q 11, q 12 , . . . , q 1 f 1, q 1 f+1. . . ) + åi ki[qi + Ri xi] First-order conditions for an interior maximum: only good 1 u lh. Uih (xh) = ki, i = 1, . . . , n generates an k externality nhf ¶F ( • ) u mf Fif(qf) + mf ——— = k 1 k=1 ¶q 1 f u mf. Fif(qf) = ki , i = 2, 3, . . . , n å

From the FOC. . . Frank Cowell: Microeconomics n n Consider tradeoff between goods 1 and 2 From the first of the FOCs: U 1 h(xh) k 1 ——— = — U 2 h(xh) k 2 Use the definition of e 21 f. Then other FOCs give F 1 f(qf) k 1 ——— – e 21 f = — F 2 f(qf) k 2 This is the efficiency criterion: u u Instead of the condition “MRT=shadow price ratio”… …we have a modified marginal rule.

Frank Cowell: Microeconomics Efficiency with production externality §Production possibilities §If externality is ignored. . . q 2 f ~ qf F 1 k 1 — = — + externality F 2 k 2 ^qf §Taking account of externality. . . §Produce less of good 1 F 1 k 1 — =— F 2 k 2 for efficiency. . . f q 1`

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality Corrective taxes and other devices. . . Production externalities Consumption externalities Connections • Basics • Efficiency • Simplementation • Private initiative

Implementation Frank Cowell: Microeconomics n n n 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: u u u MRT – producer prices MRS – consumer prices …how to connect the two of these?

Towards a policy rule (2) value of shadow prices externality Frank Cowell: Microeconomics n n F 1 f(qf) k 1 Take the modified FOC ——— – e 21 f = — F 2 f(qf) k 2 (private) marginal cost of producing 1 Rearrange: F 1 f(qf) k 1 ——— = — + e 21 f F 2 f(qf) k 2 consumer prices F 1 f(qf) p 1 ——— = — + e 21 f F 2 f(qf) p 2 n Introduce the market: n t = – e 21 f Corrective tax (negative f(qf) F p 1 1 externality) or subsidy ——— = — – t (positive externality): f f F 2 (q ) p 2

Production externality: policy Frank Cowell: Microeconomics n From the FOC a simple corrective tax can be designed u u u n Alternative 1: merger. u u n Merging the firms “internalises” the externality. Combined firm takes into account interdependence of production. Alternative 2: public issue of “pollution rights” u u u n Called “Pigovian” (from A. C. Pigou’s Economics of Welfare) Needs information about production functions. Both for victim and perpetrator. 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?

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality Development of a “pseudo market” Production externalities Consumption externalities Connections • Basics • Efficiency • Simplementation • Private initiative

Private solution: A model Frank Cowell: Microeconomics n n Efficient outcome through individual initiative? Assume that there are just two firms and two goods. u u n n Firm 1’s output of good 1 imposes costs on firm 2. Full information: u u n The second of these assumptions is unimportant. First assumption may be important. 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 u u Does this by setting up a financial incentive for firm 1. How should this be structured?

The victim’s problem Frank Cowell: Microeconomics n n Firm 2 offers firm 1 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 piqi 2 − b − m 2 F 2(q 2, q 11) {q 2, b} i=1 n Solve this in the usual way…

The victim’s problem: interpretation Frank Cowell: Microeconomics n Firm 2 designs incentive for firm 1 u u n Incentive scheme captures costs to firm 2 u u u n n a “side-payment schedule” or “conditional bribe function. ” 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

Solving the victim’s problem Frank Cowell: Microeconomics n n FOC for net outputs of firm 2 is pi − m 2 Fi 2 (q 2, q 11) = 0 FOC for the side payment b is: d. F 2 (q 2 , q 11 ) dq 11 − 1 + m 2 ─────── ── = 0 dq 11 db Using the definition of the externality: dq 11 − 1 + m 2 F 22(q 2, q 11) e 211 ── = 0 db Rearranging the FOC then gives: db ── = m 2 F 22(q 2, q 11) e 211 = p 2 e 211 dq 11

The perpetrator’s problem Frank Cowell: Microeconomics n n 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 q 1 n S piqi 1 + b(q 11) − m 1 F 1(q 1) i=1 Again solve this in the usual way…

Solving the problem Frank Cowell: Microeconomics n n n Feedback effect from 1’s net FOC for net outputs of firm 1 is: output on 2’s bribe offer d b(q 11) p 1 qi 1 + ───── − m 1 F 11(q 1) = 0 dq 11 p 2 − m 1 F 21(q 1) = 0 Substituting in for the slope of the bribe function: F 11(q 1) p 1 ──── = ── + e 211 F 21(q 1) p 2 This condition same as FOC for efficiency!

Private solution: result Frank Cowell: Microeconomics n Bribe function has internalised the externality u u n FOC conditions same as before u u u n Firm 2 conditions side-payment on observable output of good 1 Firm 1’s responds rationally to the side-payment. 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)

Private solution: difficulties Frank Cowell: Microeconomics n Solution makes important informational requirements u u n It requires a special notion of participation. u u n Imposed on both firms. There may be an incentive for firms to misrepresent costs, leading to loss of efficiency. What determines the set of participants? What if there is free entry? It focuses only on marginal impacts u u 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

A fundamental nonconvexity Frank Cowell: Microeconomics q 2 §Production possibilities… §If firm 1’s pollution could drive the other out of business § The optimal point? l 0 §If polluter can sell pollution rights indefinitely. . . ~ q l ^q q 1

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality Interactions between consumers Production externalities Consumption externalities Connections

Consumption externality Frank Cowell: Microeconomics n Household ℓ affected by others’ §Characteristics of goods generates inefficiency consumption of good 1: consumption of household ℓ Uℓ(xℓ; n vanishes if there x 11, x 12 , . . . , x 1ℓ 1, x 1ℓ+1, . . . ) is no externality Now evaluate the marginal impact of some household h on others: nh e 21 h: = å ℓ=1 ¶Uℓ( • ) 1 —— ——— U 2ℓ ¶x 1 h MU of good 2 for household ℓ §Direct impact of h on utility of ℓ… §…evaluated in terms of good 2… §…and summed over all ℓ § Gives the value of the marginal externality imposed through consumption by h of good 1.

Lagrangean method: Frank Cowell: Microeconomics n n Use same method as for production externalities Introduce Lagrange multipliers: u u u n n lh for each utility constraint mf for each firm’s technology constraint ki for materials’ balance on good i. Then maximise U 1(x 1; , x 12 , x 13, . . . ) + åhlh [Uh(xh; x 11, x 12 , . . . , x 1 h-1, x 1 h+1, . . . ) uh] åf mf F f (qf) + åi ki[qi + Ri xi] only good 1 First-order conditions for an interior maximum: generates the externality nh ¶U 1ℓ( • ) u lh. U 1 h (x 1; , x 12 , x 13, . . . ) + lh ——— = k 1 å u u ¶x 1 h lh. Uih (x 1; , x 12 , x 13, . . . ) = ki , i = 2, 3, . . . , n mf. Fif(qf) = ki , i = 1, 2, . . . , n ℓ=1

FOC has a similar interpretation. . . Frank Cowell: Microeconomics n n From the FOC for production: F 1 f(qf) k 1 ——— = — F 2 f(qf) k 2 Substituting in the value of the externality we also have U 1 h(xh) k 1 ——— + e 21 h = — U 2 h (xh) k 2 n Again we have a modified marginal rule n Again it can give us useful guidance on policy

Negative consumption externality Frank Cowell: Microeconomics §Production possibilities §Competitive equilibrium (with consumption externality) x 2 U 1 h F 1 — = — – externality U 2 h F 2 §Efficiency with consumption externality §Produce less of good 1 U 1 h F 1 — = — U 2 h F 2 for efficiency. . . x 1`

Towards a policy rule value of shadow prices externality Frank Cowell: Microeconomics n n Take the modified FOC h willingness to pay for 1 in terms of 2 Rearrange: Introduce the market: A Pigovian tax/subsidy (for negative/positive externalities) U 1 h(xh) k 1 ——— + e 21 h = — U 2 h (xh) k 2 U 1 h(xh) k 1 ——— = — – e 21 h U 2 h (xh) k 2 Producer prices U 1 h(xh) p 1 ——— = — – e 21 h U 2 h (xh) p 2 U 1 h(xh) p 1 ——— = — + t U 2 h (xh) p 2 t = −e 21 h .

Externalities Overview. . . Frank Cowell: Microeconomics The nature of externality Lessons and applications Production externalities Consumption externalities Connections

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

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