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Externalities 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 .
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