Prerequisites Almost essential Welfare and Efficiency EXTERNALITIES MICROECONOMICS

Prerequisites Almost essential Welfare and Efficiency EXTERNALITIES MICROECONOMICS Principles and Analysis Frank Cowell November 2018 Frank Cowell: Externalities 1

Overview Externalities The nature of externality A special type of transaction Production externalities Consumption externalities Connections November 2018 Frank Cowell: Externalities 2

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 November 2018 Frank Cowell: Externalities 3

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) November 2018 Frank Cowell: Externalities 4

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 • 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 November 2018 Frank Cowell: Externalities 5

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? November 2018 Frank Cowell: Externalities 6

Overview Externalities The nature of externality How production externalities work; how they are evaluated Production externalities • Basics • Efficiency • Simplementation • Private initiative Consumption externalities Connections November 2018 Frank Cowell: Externalities 7

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 = (q 1 f, q 2 f, q 3 f, …, qnf) § Firm f’s production possibilities are known • implicit production function f( ) • argument is net output vector is qf , and possibly other things • set of feasible net outputs given by f(qf) ≤ 0 • transformation curve given by net outputs such that f(qf) = 0 § Now introduce externality November 2018 Frank Cowell: Externalities 8

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 k( ) § Use the firm’s transformation curve November 2018 Frank Cowell: Externalities Standard diagram 9

Externality: Production possibilities q 2 k §Production possibilities, firm k k( ) > 0 k( ) = 0 k( ) < 0 §Production possibilities, if firm f’s emissions increase low emissions by firm f §If k( ) = 0 an increase in negative externality results in k( ) > 0 high emissions by firm f k q 1 November 2018 Frank Cowell: Externalities 10

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 k § Measure effect in terms of a numéraire: • here we take this to be good 2 • but could be any other good November 2018 Frank Cowell: Externalities 11

Production externality § Firm k may be affected by others' output of good 1: k(qk; q 11, q 12 , …, q 1 k‒ 1, §Characteristics of production generates inefficiency q 1 k+1…) net output of firm k vanishes if there is no externality § Now evaluate the marginal impact of some firm f on others: nf e 21 f : = – å k=1 Marginal product of good 2 for firm k November 2018 1 ¶ k( ) —— ——— 2 k ¶q 1 f this is positive for a negative externality: it is shifting “inwards” firm k’s feasible set §Direct impact of f on production possibilities of firm k §Evaluated in terms of good 2 §Summed over all k §Value of marginal externality imposed through production by f of good 1 Frank Cowell: Externalities 12

Overview Externalities The nature of externality Deriving the conditions for a PE allocation Production externalities • Basics • Efficiency • Simplementation • Private initiative Consumption externalities Connections November 2018 Frank Cowell: Externalities 13

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 November 2018 Frank Cowell: Externalities 14

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 U 1(x 1) subject to: technical • Uh(xh) ≥ uh, h = 2, …, nh feasibility f f 1 2 f 1 f+1 • (q ; q 1 , …, q 1 …) ≤ 0, f = 1, …, nf • xi ≤ qi + Ri , i= 1, …, n § where • xh = (x 1 h, x 2 h, x 3 h, …, xnh) • xi = åh xih , i = 1, …, n • qi = åf qi f November 2018 materials' balance Frank Cowell: Externalities 15

Lagrangian method: § Introduce Lagrange multipliers: • 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 (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 • lh. Uih (xh) = ki, i = 1, …, n generates an externality nf ¶ k( ) • mf if(qf) + å mk ——— = k 1 k=1 ¶q 1 f • mf if(qf) = ki , i = 2, 3, …, n November 2018 Frank Cowell: Externalities 16

From the FOC § Consider tradeoff between goods 1 and 2 § From 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 1 f(qf) k 1 ——— – e 21 f = — 2 f(qf) k 2 § This is the efficiency criterion: • instead of the condition “MRT=shadow price ratio” • we have a modified marginal rule November 2018 Frank Cowell: Externalities 17

Efficiency with production externality §Production possibilities §If externality is ignored §Taking account of externality q 2 f ~ qf 1 k 1 — = — + externality 2 k 2 ^qf §Produce less of good 1 1 k 1 — =— 2 k 2 for efficiency f q 1` November 2018 Frank Cowell: Externalities 18

Overview Externalities The nature of externality Corrective taxes and other devices Production externalities • Basics • Efficiency • Simplementation • Private initiative Consumption externalities Connections November 2018 Frank Cowell: Externalities 19

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? November 2018 Frank Cowell: Externalities 20

Towards a policy rule (2)value of shadow prices externality § Take the modified FOC § Rearrange: (private) marginal cost of producing 1 1 f(qf) k 1 ——— – e 21 f = — 2 f(qf) k 2 1 f(qf) k 1 ——— = — + e 21 f 2 f(qf) k 2 consumer prices § Introduce the market: § Corrective tax (negative externality) or subsidy (positive externality): November 2018 1 f(qf) p 1 ——— = — + e 21 f 2 f(qf) p 2 1 f(qf) p 1 ——— = — – t 2 f(qf) p 2 Frank Cowell: Externalities t = – e 21 f 21

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 § Could there be a purely private solution? November 2018 Frank Cowell: Externalities 22

Overview Externalities The nature of externality Development of a “pseudo market” Production externalities • Basics • Efficiency • Simplementation • Private initiative Consumption externalities Connections November 2018 Frank Cowell: Externalities 23

Private solution: A model § Efficient outcome through individual initiative? § Assume (1) just two firms (2) just two goods • assumption (1) may be important • assumptions (2) is unimportant § 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? November 2018 Frank Cowell: Externalities 24

The victim’s problem § 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 {q 2, b} S piqi 2 − b − m 2 2(q 2, q 11) i=1 § Solve this in the usual way November 2018 Frank Cowell: Externalities 25

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 November 2018 Frank Cowell: Externalities 26

Solving the victim’s problem § FOC for net outputs of firm 2 is pi − m 2 i 2 (q 2, q 11) = 0 § FOC for the side payment b is: d 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 22(q 2, q 11) e 211 ── = 0 db § Rearranging the FOC then gives: db ── = m 2 22(q 2, q 11) e 211 = p 2 e 211 dq 11 November 2018 Frank Cowell: Externalities 27

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 Spiqi 1 + b (q 11) − m 1 1(q 1) q 1 i=1 § Again solve this in the usual way November 2018 Frank Cowell: Externalities 28

Solving the problem § Feedback effect from 1’s net FOC for net outputs of on firm 1 is: output 2’s bribe offer d b(q 11) p 1 qi 1 + ───── − m 1 11(q 1) = dq 11 0 p 2 − m 1 21(q 1) = 0 § Substituting in for the slope of the bribe function: 11(q 1) p 1 ──── = ── + e 211 21(q 1) p 2 § This condition same as FOC for efficiency! November 2018 Frank Cowell: Externalities 29

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) November 2018 Frank Cowell: Externalities 30

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 November 2018 Frank Cowell: Externalities 31

A fundamental nonconvexity q 2 §Production possibilities §If firm 1’s pollution could drive the other out of business § The optimal point? l 0 November 2018 §If polluter can sell pollution rights indefinitely ~ q l ^q q 1 Frank Cowell: Externalities 32

Overview Externalities The nature of externality Interactions between consumers Production externalities Consumption externalities Connections November 2018 Frank Cowell: Externalities 33

Consumption externality § Household ℓ affected by others’ consumption of good 1: Uℓ(xℓ; x 11, x 12 , …, x 1ℓ 1, x 1ℓ+1, …) §Characteristics of goods generates inefficiency vanishes if there is no externality consumption of household ℓ § Now evaluate the marginal impact of some household h on others: nh e 21 h: = å ℓ=1 1 ¶Uℓ( ) —— ——— U 2ℓ ¶x 1 h MU of good 2 for household ℓ November 2018 §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 Frank Cowell: Externalities 34

Lagrangian method: § Use same method as for production externalities § Introduce Lagrange multipliers: • 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 (qf) only good 1 + åi ki[qi + Ri xi] generates the § First-order conditions for an interior maximum: externality nh ¶U 1ℓ( ) • lh. U 1 h (x 1; , x 12 , x 13, …) + lh å ——— = k 1 ℓ=1 ¶x 1 h • lh. Uih (x 1; , x 12 , x 13, …) = ki , i = 2, 3, …, n • mf if(qf) = ki , i = 1, 2, …, n November 2018 Frank Cowell: Externalities 35

FOC has a similar interpretation § From the FOC for production: 1 f(qf) k 1 ——— = — 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 § Again we have a modified marginal rule § Again it can give us useful guidance on policy November 2018 Frank Cowell: Externalities 36

Negative consumption externality §Production possibilities §Competitive equilibrium (with consumption externality) x 2 U 1 h 1 — = — – externality U 2 h 2 §Efficiency with consumption externality §Produce less of good 1 for U 1 1 — = — U 2 h efficiency x 1` November 2018 Frank Cowell: Externalities 37

Towards a policy rule § Take the modified FOC h willingness to pay for 1 in terms of 2 § Rearrange: shadow prices value of externality 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 § Introduce the market: § A Pigovian tax/subsidy (for negative/positive externalities) November 2018 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 Frank Cowell: Externalities t = −e 21 h . 38

Overview Externalities The nature of externality Lessons and applications Production externalities Consumption externalities Connections November 2018 Frank Cowell: Externalities 39

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 November 2018 Frank Cowell: Externalities 40

Externalities: summary § Characterisation problem: modify the MRS = MRT rule by the marginal cost of externality § Implementation problem: For production externalities – encourage private resolution through extended markets? Otherwise introduce a tax/subsidy corresponding to the marginal cost of externality November 2018 Frank Cowell: Externalities 41