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 Production transaction 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

•Basics How production Production •Efficiency externalities work; how externalities •Simple implementation •Private initiative they are evaluated

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 f • net output by firm f of good i is qi , i = 1,2,…,n • usual sign convention f f f f f • net output vector is q = (q1 , q2 , q3 , …, qn ) . Firm f’s production possibilities are known • implicit production function Φf(⋅) • argument is net output vector is qf , and possibly other things • set of feasible net outputs given by Φf(qf) ≤ 0 • transformation curve given by net outputs such that Φf(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 k Standard • will enter the production function Φ (⋅) diagram . Use the firm’s transformation curve

November 2018 Frank Cowell: Externalities 9 Externality: Production possibilities

k q2 .Production possibilities, firm k Φk ⋅ ( ) > 0 .Production possibilities, if firm k Φ (⋅) = 0 f’s emissions increase Φk(⋅) < 0 low emissions .If Φk(⋅) = 0 an increase in negative by firm f externality results in Φk(⋅) > 0 high emissions by firm f

k q1

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 Φk . 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 .Characteristics of production others' output of good 1: generates inefficiency

vanishes if there k k 1 2 k‒1 k+1 Φ (q ; q1 , q1 , …, q1 , q1 …) is no externality net output of firm k this is positive for a negative . Now evaluate the marginal impact of externality: it is shifting some firm f on others: “inwards” firm k’s feasible set .Direct impact of f on production n f 1 ∂Φk(⋅) possibilities of firm k e f := – —— ——— .Evaluated in terms of good 2 21 ∑ Φ k f k=1 2 ∂q1 .Summed over all k . Marginal product of Value of marginal externality imposed good 2 for firm k through production by f of good 1

November 2018 Frank Cowell: Externalities 12 Overview Externalities

The nature of externality

•Basics Deriving the conditions Production •Efficiency for a PE allocation •Simple implementation externalities •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 U1(x1) subject to: • Uh(xh) ≥ υh, h = 2, …, n technical h feasibility f f 1 2 f−1 f+1 • Φ (q ; q1 ,q1 ,…,q1 ,q1 …) ≤ 0, f = 1, …, nf

• xi ≤ qi + Ri , i= 1, …, n . materials' where balance h h h h h • x = (x1 , x2 , x3 , …, xn ) h • xi = ∑h xi , i = 1,…,n f • qi = ∑f qi

November 2018 Frank Cowell: Externalities 15 Lagrangian method:

. Introduce Lagrange multipliers:

• λh for each utility constraint • µf for each firm’s technology constraint • κi for materials’ balance on good i . Then maximise 1 1 h h h U (x ) + ∑hλh [U (x ) − υ ] f f 1 2 f−1 f+1 − ∑f µf Φ (q ; q1 ,q1 ,…,q1 ,q1 …) + ∑i κi[qi + Ri − xi] . First-order conditions for an interior maximum: λ h h κ only good 1 • hUi (x ) = i, i = 1,…,n generates an k externality nf ∂Φ (⋅) • µ Φ f(qf) + ∑ µ ——— = κ f i k f 1 k=1 ∂q1 f f • µfΦi (q ) = κi , 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: h h U1 (x ) κ1 ——— = — h h U2 (x ) κ2 f . Use the definition of e21 . Then other FOCs give f f Φ1 (q ) κ1 f ——— – e21 = — f f Φ2 (q ) κ2 . 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 f .Taking account of externality q2

Φ1 κ1 — = — + externality ~ Φ2 κ2 qf

. Φ κ Produce less of good 1 1 1 for efficiency ^qf — = — Φ2 κ2

f q1`

November 2018 Frank Cowell: Externalities 18 Overview Externalities

The nature of externality

•Basics Corrective taxes and Production •Efficiency other devices •Simple implementation externalities •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 f f Φ1 (q ) κ1 f . Take the modified FOC ——— – e21 = — f f Φ2 (q ) κ2 (private) marginal cost f f of producing 1 Φ1 (q ) κ1 . Rearrange: f ——— = — + e21 f f Φ2 (q ) κ2 consumer prices f f . Introduce the market: Φ1 (q ) p1 f ——— = — + e21 f f Φ2 (q ) p2 . t = – e f Corrective tax (negative f f 21 externality) or subsidy Φ1 (q ) p1 ——— = — – t (positive externality): f f Φ2 (q ) p2 November 2018 Frank Cowell: Externalities 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

•Basics Development of Production •Efficiency a “pseudo •Simple implementation externalities •Private initiative market”

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 firm1 a side-payment (Bribe) β . 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 2 2 2 1 max Σ piqi − β − µ2 Φ (q , q1 ) {q2, β} 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 2 2 1 pi − µ2 Φi (q , q1 ) = 0 . FOC for the side payment β is: 2 2 1 1 dΦ (q , q1 ) dq1 − 1 + µ2 ─────── ── = 0 1 dq1 dβ . Using the definition of the externality: 1 dq1 2 2 1 1 − 1 + µ2Φ2 (q , q1 ) e21 ── = 0 dβ . Rearranging the FOC then gives: dβ 2 2 1 1 1 ── = µ2Φ2 (q , q1 ) e21 = p2 e21 1 dq1

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 1 1 1 1 max Σpiqi + β (q1 ) − µ1Φ (q ) q1 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 outputsoutput of onfirm 2’s bribe 1 is: offer 1 d β(q1 ) 1 1 1 p1qi + ───── − µ1 Φ1 (q ) = 0 1 dq1 1 1 p2 − µ1 Φ2 (q ) = 0 . Substituting in for the slope of the bribe function: 1 1 Φ1 (q ) p1 1 ──── = ── + e21 1 1 Φ2 (q ) p2 . 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

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  q

^q  0 q1

November 2018 Frank Cowell: Externalities 32 Overview Externalities

The nature of externality

Interactions between Production consumers externalities

Consumption externalities

Connections

November 2018 Frank Cowell: Externalities 33 Consumption externality

. Household ℓ affected by others’ .Characteristics of goods consumption of good 1: generates inefficiency vanishes if there ℓ ℓ 1 2 ℓ−1 ℓ+1 is no externality U (x ; x1 ,x1 , …, x1 , x1 ,…)

consumption of household ℓ . Now evaluate the marginal impact of some household h on others: .Direct impact of h on utility of ℓ nh 1 ℓ ∂U (⋅) .evaluated in terms of good 2 e h:= —— ——— 21 ∑ ℓ h .and summed over all ℓ ℓ=1 U2 ∂x1 MU of good 2 . Gives the value of the marginal for household ℓ externality imposed through consumption by h of good 1

November 2018 Frank Cowell: Externalities 34 Lagrangian method:

. Use same method as for production externalities . Introduce Lagrange multipliers: • λh for each utility constraint • µf for each firm’s technology constraint • κi for materials’ balance on good i . Then maximise 1 1 2 3 U (x ;,x1 , x1 , …) h h 1 2 h-1 h+1 h + ∑hλh [U (x ; x1 ,x1 , …, x1 , x1 ,…) − υ ] f f − ∑f µf Φ (q ) + ∑ κ [q + R − x ] only good 1 i i i i i generates the . First-order conditions for an interior maximum: externality ℓ nh ∂U1 (⋅) h 1 2 3 • λhU1 (x ;,x1 , x1 , …) + λh ∑ ——— = κ1 h ℓ=1 ∂x1 h 1 2 3 • λhUi (x ;,x1 , x1 , …) = κi , i = 2,3,…,n f f • µfΦi (q ) = κi , i = 1,2,…,n

November 2018 Frank Cowell: Externalities 35 FOC has a similar interpretation

. From the FOC for production: f f Φ1 (q ) κ1 ——— = — f f Φ2 (q ) κ2 . Substituting in the value of the externality we also have h h U1 (x ) κ1 h ——— + e21 = — h h U2 (x ) κ2 . 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

x2 consumption externality) h U1 Φ1 — = — – externality .Efficiency with consumption h externality U2 Φ2

. h Produce less of good 1 for U1 Φ1 — = — efficiency h U2 Φ2

x1`

November 2018 Frank Cowell: Externalities 37 Towards a policy rule value of shadow prices externality h h U1 (x ) κ1 . Take the modified FOC h ——— + e21 = — h h U2 (x ) κ2 h willingness to pay h h for 1 in terms of 2 U1 (x ) κ1 . Rearrange: h ——— = — – e21 h h U2 (x ) κ2 Producer prices h h . Introduce the market: U1 (x ) p1 h ——— = — – e21 h h U2 (x ) p2 h t = −e21 . h h A Pigovian tax/subsidy (for U1 (x ) p1 negative/positive ——— = — + t externalities) h h U2 (x ) p2 . November 2018 Frank Cowell: Externalities 38 Overview Externalities

The nature of externality

Lessons and Production applications 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 For production externalities – encourage private resolution problem: through extended markets?

Otherwise introduce a tax/subsidy corresponding to the marginal cost of externality

November 2018 Frank Cowell: Externalities 41