Download presentation
Presentation is loading. Please wait.
1
MICROECONOMICS Principles and Analysis Frank Cowell
Prerequisites Almost essential Welfare and Efficiency Externalities MICROECONOMICS Principles and Analysis Frank Cowell November 2018
2
Overview A special type of transaction Externalities
The nature of externality A special type of transaction Production externalities Consumption externalities Connections November 2018
3
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
4
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
5
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
6
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
7
Overview How production externalities work; how they are evaluated
The nature of externality How production externalities work; how they are evaluated Production externalities Basics Efficiency Simple implementation Private initiative Consumption externalities Connections November 2018
8
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 = (q1f, q2f, q3f, …, qnf) Firm f’s production possibilities are known implicit production function Ff() argument is net output vector is qf , and possibly other things set of feasible net outputs given by Ff(qf) ≤ 0 transformation curve given by net outputs such that Ff(qf) = 0 Now introduce externality November 2018
9
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 Fk() Use the firm’s transformation curve Standard diagram November 2018
10
Externality: Production possibilities
q1 k q2 Production possibilities, firm k Fk() > 0 Production possibilities, if firm f’s emissions increase Fk() = 0 Fk() < 0 low emissions by firm f If Fk() = 0 an increase in negative externality results in Fk() > 0 high emissions by firm f November 2018
11
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 Fk Measure effect in terms of a numéraire: here we take this to be good 2 but could be any other good November 2018
12
Production externality
Firm k may be affected by others' output of good 1: Characteristics of production generates inefficiency vanishes if there is no externality Fk(qk; q11, q12 , …, q1k‒1, q1k+1…) net output of firm k this is positive for a negative externality: it is shifting “inwards” firm k’s feasible set Now evaluate the marginal impact of some firm f on others: Direct impact of f on production possibilities of firm k nf å k=1 1 —— F2k ¶Fk() ——— ¶q1f e21f := – Evaluated in terms of good 2 Summed over all k Value of marginal externality imposed through production by f of good 1 Marginal product of good 2 for firm k November 2018
13
Overview Deriving the conditions for a PE allocation Externalities
The nature of externality Deriving the conditions for a PE allocation Production externalities Basics Efficiency Simple implementation Private initiative Consumption externalities Connections November 2018
14
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
15
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) ≥ uh, h = 2, …, nh Ff(qf; q11,q12 ,…,q1f1,q1f+1…) ≤ 0, f = 1, …, nf xi ≤ qi + Ri , i= 1, …, n where xh = (x1h, x2h, x3h, …, xnh) xi = åh xih , i = 1,…,n qi = åf qi f technical feasibility materials' balance November 2018
16
Lagrangian method: Introduce Lagrange multipliers: Then maximise
lh for each utility constraint mf for each firm’s technology constraint ki for materials’ balance on good i Then maximise U1(x1) + åhlh [Uh(xh) uh] åf mf F f (qf; q11,q12 ,…,q1f1,q1f+1…) + åi ki[qi + Ri xi] First-order conditions for an interior maximum: lhUih (xh) = ki, i = 1,…,n nf ¶Fk() mf Fif(qf) + å mk ——— = k1 k= ¶q1f mfFif(qf) = ki , i = 2,3,…,n only good 1 generates an externality November 2018
17
From the FOC Consider tradeoff between goods 1 and 2
From first of the FOCs: U1h(xh) k1 ——— = — U2h(xh) k2 Use the definition of e21f . Then other FOCs give F1f(qf) k1 ——— – e21f = — F2f(qf) k2 This is the efficiency criterion: instead of the condition “MRT=shadow price ratio” we have a modified marginal rule November 2018
18
Efficiency with production externality
Production possibilities If externality is ignored Taking account of externality q2 f k1 — = — + externality k2 qf ~ Produce less of good 1 for efficiency F k1 — = — F k2 qf ^ q1` f November 2018
19
Overview Corrective taxes and other devices Externalities
The nature of externality Corrective taxes and other devices Production externalities Basics Efficiency Simple implementation Private initiative Consumption externalities Connections November 2018
20
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
21
Towards a policy rule (2)
value of externality shadow prices F1f(qf) k1 ——— – e21f = — F2f(qf) k2 Take the modified FOC (private) marginal cost of producing 1 F1f(qf) k1 ——— = — + e21f F2f(qf) k2 Rearrange: consumer prices F1f(qf) p1 ——— = — + e21f F2f(qf) p2 Introduce the market: t = – e21f Corrective tax (negative externality) or subsidy (positive externality): F1f(qf) p1 ——— = — – t F2f(qf) p2 November 2018
22
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
23
Overview Development of a “pseudo market” Externalities
The nature of externality Development of a “pseudo market” Production externalities Basics Efficiency Simple implementation Private initiative Consumption externalities Connections November 2018
24
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
25
The victim’s problem Firm 2 offers firm1 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 piqi2 − b − m2 F2(q2, q11) {q2, b} i=1 Solve this in the usual way November 2018
26
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
27
Solving the victim’s problem
FOC for net outputs of firm 2 is pi − m2 Fi2 (q2, q11) = 0 FOC for the side payment b is: dF2(q2, q11) dq11 − 1 + m2 ─────── ── = 0 dq db Using the definition of the externality: dq11 − 1 + m2F22(q2, q11) e211 ── = 0 db Rearranging the FOC then gives: ── = m2F22(q2, q11) e211 = p2 e211 November 2018
28
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 Spiqi1 + b (q11) − m1F1(q1) q i=1 Again solve this in the usual way November 2018
29
Solving the problem FOC for net outputs of firm 1 is: d b(q11)
Feedback effect from 1’s net output on 2’s bribe offer FOC for net outputs of firm 1 is: d b(q11) p1qi1 + ───── − m1 F11(q1) = 0 dq11 p2 − m1 F21(q1) = 0 Substituting in for the slope of the bribe function: F11(q1) p1 ──── = ── + e211 F21(q1) p2 This condition same as FOC for efficiency! November 2018
30
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
31
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
32
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 ^ q1 November 2018
33
Overview Interactions between consumers Externalities
The nature of externality Interactions between consumers Production externalities Consumption externalities Connections November 2018
34
Consumption externality
Household ℓ affected by others’ consumption of good 1: Uℓ(xℓ; x11,x12 , …, x1ℓ1, x1ℓ+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: Direct impact of h on utility of ℓ nh å ℓ=1 1 —— U2ℓ ¶Uℓ() ——— ¶x1h evaluated in terms of good 2 e21h:= and summed over all ℓ MU of good 2 for household ℓ Gives the value of the marginal externality imposed through consumption by h of good 1 November 2018
35
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 U1(x1;,x12 , x13, …) + åhlh [Uh(xh; x11,x12 , …, x1h-1, x1h+1,…) uh] åf mf F f (qf) + åi ki[qi + Ri xi] First-order conditions for an interior maximum: nh ¶U1ℓ() lhU1h (x1;,x12 , x13, …) + lh å ——— = k1 ℓ= ¶x1h lhUih (x1;,x12 , x13, …) = ki , i = 2,3,…,n mfFif(qf) = ki , i = 1,2,…,n only good 1 generates the externality November 2018
36
FOC has a similar interpretation
From the FOC for production: F1f(qf) k1 ——— = — F2f(qf) k2 Substituting in the value of the externality we also have U1h(xh) k1 ——— + e21h = — U2h (xh) k2 Again we have a modified marginal rule Again it can give us useful guidance on policy November 2018
37
Negative consumption externality
Production possibilities Competitive equilibrium (with consumption externality) x2 U1h F1 — = — – externality U2h F2 Efficiency with consumption externality Produce less of good 1 for efficiency U1h F1 — = — U2h F2 x1` November 2018
38
Towards a policy rule U1h(xh) k1 ——— + e21h = — U2h (xh) k2 U1h(xh) k1
value of externality shadow prices U1h(xh) k1 ——— + e21h = — U2h (xh) k2 Take the modified FOC h willingness to pay for 1 in terms of 2 U1h(xh) k1 ——— = — – e21h U2h (xh) k2 Rearrange: Producer prices U1h(xh) p1 ——— = — – e21h U2h (xh) p2 Introduce the market: t = −e21h A Pigovian tax/subsidy (for negative/positive externalities) U1h(xh) p1 ——— = — + t U2h (xh) p2 . November 2018
39
Overview Lessons and applications Externalities
The nature of externality Lessons and applications Production externalities Consumption externalities Connections November 2018
40
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
41
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
Similar presentations
