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

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

4. Frank Cowell: Microeconomics The nature of externality An externality is a kind “involuntary” transaction An externality is a kind “involuntary” transaction A case where market allocation methods don’t work 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 Externalities can be detrimental or beneficial We will deal with two broad types: We will deal with two broad types:  Production externalities  Consumption externalities

5. Frank Cowell: Microeconomics Production externality One firm influences another’s production conditions 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 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 Example: networking  One firm’s activity creates pool of skilled workers from which neighbouring firms may benefit. Example: pollution Example: pollution  One firm’s activity (glue production) causes emissions that are to the detriment of its neighbours (restaurants who must filter the air).

6. Frank Cowell: Microeconomics Consumption externality One agent’s consumption of a good directly affects another 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 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 Consumption externalities are non-excludable but rival Example: Scent from fresh flowers Example: Scent from fresh flowers  Nonexcludable: you can’t charge for the scent  Rival: more scent requires more flowers

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

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

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

10. Frank Cowell: Microeconomics Quantifying an externality Consider a polluting firm f. 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 When f produces good 1 it causes the pollution  could affect other firms k = 1, 2,..., f – 1, f + 1,..., n f.  the more f produces good 1, the greater the damage to k. How much damage? How much damage?  Consider the impact of pollution on firm k  Will enter the production function  Will enter the production function  k () Use the firm’s transformation curve... Use the firm’s transformation curve... Jump to “Multi-output firm” Standard diagram....

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

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

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

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

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

16. Frank Cowell: Microeconomics The approach Use a maximisation procedure to characterise efficiency: 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: So problem is to maximise U 1 (x 1 ) subject to:  U(x h ) ≥ , h = 2, …, n h  U h (x h ) ≥  h, h = 2, …, n h  f = 1, …, n f   f (q f ; q 1 1,q 1 2,...,q 1 f  1,q 1 f+1...) ≤ 0, f = 1, …, n f  i= 1, …, n  x i ≤ q i + R i, i= 1, …, n where  x 1 h, x 2 h, x 3 h,..., x n h  x h = (x 1 h, x 2 h, x 3 h,..., x n h )   x i =  h x i h, i = 1,...,n   q i =  f q i f technical feasibility materials' balance

17. Frank Cowell: Microeconomics Lagrangean method: Introduce Lagrange multipliers: Introduce Lagrange multipliers:  for each utility constraint  h for each utility constraint  for each firm’s technology constraint   f for each firm’s technology constraint  for materials’ balance on good i.   i for materials’ balance on good i. Then maximise U 1 (x 1 ) +  h h [U h (x h )   h ]   f  f  f (q f ; q 1 1,q 1 2,...,q 1 f  1,q 1 f+1...) +  i  i [q i + R i  x i ] First-order conditions for an interior maximum: First-order conditions for an interior maximum: , i = 1,...,n  h U i h (x h ) =  i, i = 1,...,n n hf  k ()    f  i f (q f ) +  f   ———  =  1   k=    q 1 f , i = 2,3,...,n   f  i f (q f ) =  i, i = 2,3,...,n only good 1 generates an externality

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

19. Frank Cowell: Microeconomics Efficiency with production externality  1  1 — = —  2  2  1  1 — = —  2  2  1  1 — = — + externality  2  2  1  1 — = — + externality  2  2 q 1` q 2 f f qfqf ^ qfqf ~   Production possibilities   Taking account of externality...   If externality is ignored...   Produce less of good 1 for efficiency...

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

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

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

23. Frank Cowell: Microeconomics Production externality: policy From the FOC a simple corrective tax can be designed 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. Alternative 1: merger.  Merging the firms “internalises” the externality.  Combined firm takes into account interdependence of production. Alternative 2: public issue of “pollution rights” 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? However could there be a purely private solution?

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

25. Frank Cowell: Microeconomics Private solution: A model Efficient outcome through individual initiative? Efficient outcome through individual initiative? Assume that there are just two firms and two goods. 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. Firm 1’s output of good 1 imposes costs on firm 2. Full information: 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 Firm 2 (victim) has an interest in communicating  Does this by setting up a financial incentive for firm 1.  How should this be structured?

26. Frank Cowell: Microeconomics The victim’s problem Firm 2 offers firm1 a side-payment (Bribe) . Firm 2 offers firm1 a side-payment (Bribe) . This payment needs to be accounted for in the computation of profits. This payment needs to be accounted for in the computation of profits. It can be treated as a control variable for firm 2. It can be treated as a control variable for firm 2. Optimisation problem of firm 2 (the victim) is: Optimisation problem of firm 2 (the victim) is: n max  p i q i 2 −  −  2  2 (q 2, q 1 1 ) max  p i q i 2 −  −  2  2 (q 2, q 1 1 ) {q 2,  } i=1 Solve this in the usual way… Solve this in the usual way…

27. Frank Cowell: Microeconomics The victim’s problem: interpretation Firm 2 designs incentive for firm 1 Firm 2 designs incentive for firm 1  a “side-payment schedule”  or “conditional bribe function.” Incentive scheme captures costs to firm 2 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) Should influence actions of perpetrator (firm 1) Analyse firm 1’s behaviour in same framework Analyse firm 1’s behaviour in same framework

28. Frank Cowell: Microeconomics Solving the victim’s problem FOC for net outputs of firm 2 is FOC for net outputs of firm 2 is p i −  2  i 2 (q 2, q 1 1 ) = 0 FOC for the side payment  is: FOC for the side payment  is: d  2 (q 2, q 1 1 ) dq 1 1 d  2 (q 2, q 1 1 ) dq 1 1 − 1 +  2 ─────── ── = 0 dq 1 1 d  dq 1 1 d  Using the definition of the externality: Using the definition of the externality: dq 1 1 dq 1 1 − 1 +  2  2 2 (q 2, q 1 1 ) e 21 1 ── = 0 d  d  Rearranging the FOC then gives: Rearranging the FOC then gives: dddd ── =  2  2 2 (q 2, q 1 1 ) e 21 1 = p 2 e 21 1 dq11dq11dq11dq11

29. Frank Cowell: Microeconomics The perpetrator’s problem For firm 2’s “schedule” to work, firm 1 has to know about it For firm 2’s “schedule” to work, firm 1 has to know about it It rationally incorporates this into its profit calculation It rationally incorporates this into its profit calculation It will note that the bribe is conditional on a variable under its own control It will note that the bribe is conditional on a variable under its own control The optimisation problem for firm 1 is: The optimisation problem for firm 1 is: n max  p i q i 1 +  (q 1 1 )  −  1  1 (q 1 ) max  p i q i 1 +  (q 1 1 )  −  1  1 (q 1 ) q 1  i=1 q 1  i=1 Again solve this in the usual way… Again solve this in the usual way…

30. Frank Cowell: Microeconomics Solving the problem FOC for net outputs of firm 1 is: FOC for net outputs of firm 1 is: d  (q 1 1 ) d  (q 1 1 ) p 1 q i 1 + ───── −  1  1 1 (q 1 ) = 0 dq 1 1 dq 1 1 p 2 −  1  2 1 (q 1 ) = 0 Substituting in for the slope of the bribe function: Substituting in for the slope of the bribe function:  1 1 (q 1 ) p 1 ──── = ── + e 21 1  2 1 (q 1 ) p 2 This condition same as FOC for efficiency! This condition same as FOC for efficiency! Feedback effect from 1’s net output on 2’s bribe offer

31. Frank Cowell: Microeconomics Private solution: result Bribe function has internalised the externality 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 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) It should be independent of where the law places the responsibility for the pollution (Coase’s result)

32. Frank Cowell: Microeconomics Private solution: difficulties Solution makes important informational requirements 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. 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 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

33. Frank Cowell: Microeconomics A fundamental nonconvexity q2q2 0 q1q1 q ~ q ^   Production possibilities…   The optimal point?   If polluter can sell pollution rights indefinitely...   If firm 1’s pollution could drive the other out of business

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

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

36. Frank Cowell: Microeconomics Lagrangean method: Use same method as for production externalities Use same method as for production externalities Introduce Lagrange multipliers: Introduce Lagrange multipliers:  for each utility constraint  h for each utility constraint  for each firm’s technology constraint   f for each firm’s technology constraint  for materials’ balance on good i.   i for materials’ balance on good i. Then maximise U 1 (x 1 ;,x 1 2, x 1 3,...) +  h h [U h (x h ; x 1 1,x 1 2,..., x 1 h-1, x 1 h+1,...)   h ]   f  f  f (q f ) +  i  i [q i + R i  x i ] First-order conditions for an interior maximum: First-order conditions for an interior maximum: n h  U 1 ℓ ()   h U 1 h (x 1 ;,x 1 2, x 1 3,...) + h  ———  =  1   ℓ=    x 1 h , i = 2,3,...,n  h U i h (x 1 ;,x 1 2, x 1 3,...) =  i, i = 2,3,...,n , i = 1,2,...,n   f  i f (q f ) =  i, i = 1,2,...,n only good 1 generates the externality

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

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

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

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

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

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