Theory of Externalities Partial Equilibrium Analysis

Transcription

1 Theory of Externalities Partial Equilibrium Analysis Definition: An externality is resent whenever the well being of a consumer or the roduction ossibilities of a firm are directly affected by the actions of another agent in the economy. The choice of the word directly is imortant because it distinguishes between ecuniary externalities (external effects that indirectly affect actions of other agents any effects that are mediated by rices) and technological externalities. For examle, a roduction externality is resent if a downstream fishery s roductivity is affected by the emissions of a nearby chemical lant, but not simly because the fishery s rofitability is affected by the rice of chemicals (which, in turn, is affected by the chemical lant s outut of chemicals). Pecuniary or indirect externalities are resent in any cometitive market and do not create any inefficiency. Technological or direct externalities, on the other hand, do affect cometitive equilibria.

2 Examles:. Consumtion externality: music from neighbour s stereo x j m individual j s consumtion of music m U ( x, y ) j j j individual j s utility function m U ( x, x, y ) i i j i individual i s utility function If If i m <, neighbour s music generates a negative consumtion externality j i m >, neighbour s music generates a ositive consumtion externality j. Production externality: industrial emissions y i f( x ) i firm i s roduction function y i z i ollution roduction function y j f( x j, zi) firm j s roduction function ( x j, zi) < z i firm i s emissions reduce firm j s outut, generating a negative externality

3 Lecture Plan. Simle Bilateral Externality Model introduce concets of externality and external costs. Derive Pareto Otimum. 3. Derive Cometitive Equilibrium 4. Comarative Analysis discuss subotimality of cometitive equilibrium 5. Centralized (standards, taxes) and decentralized solutions (environmental negotiation and marketable ermits) Simle Bilateral Partial Equilibrium Model assume erfect cometition two agents constitute a small art of the economy so their actions do not affect rices assume assive ollution victim victim cannot reduce damages at margin by changing own inut (only by shutting down) firm : ul and aer lant dums effluents into river effluents cause damages to downstream river users (loss in aquatic life due to reduction in dissolved oxygen, chemical level sufficiently high to be hazardous for swimming and drinking, scum, detergent suds, discolouration which degrades aesthetic beauty of water) firm : hotel resort which rovides water recreational activities (swimming, kayaking, windsurfing) effluents dumed by lant makes water unsafe for swimming and unattractive for other water recreation activities and, as a result, resort s business declines as effluents level rise as river becomes more olluted, demand for water recreation falls and resort s revenues, hence rofits fall lant does not bear costs it imoses on resort (or any other river users) these costs are external to lant do not directly affect lant s rofits

4 Pul and aer lant π P π Ch ( ) lant s rofits where π are zero ollution rofits and h is ollution level P P Ch ( ) < h> π ( h) > π ( ) h> Ca () > a h h> abatement costs nonnegative P P C ( h) <, C ( h) > π >, π < h hh C () a >, C () a > abatement costs increasing at increasing rate Hotel Resort π H h π Dh ( ) hotel s rofits where π h are zero ollution rofits Dh ( ) monetary value of damages are nonnegative D ( h) > marginal damages are increasing D ( h) > marginal damages are increasing at increasing rate D ( h) ( h) φ Aside: Mas Colell notation C ( h) ( h) φ φ ( h) π C( h) φ ( h) C ( h) > φ ( h) C ( h) < φ h ( h) π D( h) φ ( h) D ( h) < φ ( h) D ( h) <

5 Pareto Otimum Social lanner will choose level of externality to maximize social welfare: max W π P + π H π C( h) + π h D( h) h Can rewrite roblem as social cost minimization roblem: min Cs C( h) + D( h) h C h s C ( h) + D ( h) h * : C ( h) D ( h)

6 Cometitive Equilibrium max P π π Ch ( ) h π h P C ( h) h: C ( h) Clearly, h > h * since environmental damage costs are not internalized at cometitive equilibrium. At h >, the lant imoses imoses excessive damages on the resort: * Dh ( ) Dh ( ) D ( hdh ) > h h * Social losses will arise: * W W( h ) W( h ) C ( h) + D ( h) dh < h h *

7 can measure cost of uncontrolled ollution and benefit of (or WTP for) controlling ollution using D ( h) can measure benefit of uncontrolled ollution and cost of controlling ollution using C ( h)

8 Key Results. h > in general, otimality does not entail eliminating externality zero ollution is not otimal otimality requires balancing costs and benefits of ollution at the margin. h > h * excessive level of ollution generated in a market economy source of market failure is failure of economic agents to internalize full social costs of actions when making rivately otimal decisions failure to internalize external costs results in a breakdown of invisible hand * 3. Dh ( ) > Dh ( ) excessive level of environmental degradation in market economy * 4. W W( h ) W( h ) C ( h) + D ( h) dh < social welfare loss arises h h *

9 Centralized Solutions. Command and Control: Standard Set h h *.. Incentive Based: Pigouvian tax imose tax on externality generating activity ul and aer lant must ay tax t er unit of h what is otimal tax? t D ( h ) > * * tax is set equal to marginal damages at Pareto otimum tax forces olluter to internalize cost of actions imosed on others t * will induce Pareto otimum: max P π * π Ch ( ) th h π h P * C ( h) t ht: C ( h) t * () Substituting t D ( h ) * * into () yields h: C ( h ) D ( h ) h h * * t t t hence t * will induce the Pareto otimum.

10 First Welfare Theorem: CE are PO iff comlete set of markets exist roerty rights are well defined no externalities no strategic behaviour erfect information Three conditions must be met for Pareto otimality: A Efficient consumtion MRS MRS B,, Efficient roduction MRTS MRTS,, Efficient roduct mix MRS MRT,,

11 Theory of Externalities General Equilibrium Model Consider an economy with x i two consumtion goods indexed by i, consumed by reresentative consumer y i k two roduction goods indexed by i,, roduced by two firms indexed by k,, where suerscrits denote identity of firm and subscrits denote tye of good There are two externalities affecting firm : one generated by consumer s consumtion of good, x one generated by roduction of good by firm, y Illustrative examle: ollution of a river by city inhabitants (municial sewage) and a firm (industrial effluents) that affect a downstream water using firm

12 Consumer U( x, x ) ( ω, ω ) U( ) is differentiable, increasing and strictly quasi concave initial endowment Producers y f ( y ) firm 's roduction technology, where y is firm 's outut, is differentiable and concave and is firm f y 's inut (negative outut) of good y f ( y, y, x ) firm 's roduction technology, where y is firm 's outut, is differentiable and concave, is firm 's f y inut (negative outut) of good, y is firm 's outut of good and x is consumer s consumtion of good y, y inuts negative oututs Since inuts have negative sign, Hence, firm 's roduction of good and consumer s consumtion of good both enter firm 's roduction function, generating two negative externalities.

13 Pareto Otimal Allocation max U( x, x ) { x, x, y, y, y, y } s.t. y + y + ω x scarcity contraints y + y + ω x y + f ( y ) technological constraints y + f ( y, y, x) We can ignore the inequality constraints because of the convexity and concavity assumtions these assumtions are sufficient conditions for interior solutions. The Lagrangian can be written as max { x, x, y, y, y, y } ( ) ( ) ( y f ( y )) ( y f ( y, y, x )) L U( x, x ) + λ y + y + ω x + λ y + y + ω x + µ + + µ + L L λ µ U f + λ () () L λ µ µ f + (3) L λ µ f + L λ µ (4) (5) λ µ L f + (6)

14 Solution λ Eliminate µ and µ. Exress efficiency conditions in terms of. λ Efficient consumtion Rewrite equations () and (). Substitute λ µ from equation (5): λ + λ f () λ () Substituting equation () into equation () and dividing yields: λ λ + + (7) where the RHS of equation (7) is the social MRS,. The social MRS, takes into account all effects of consumtion activities (marginal benefits to consumer as well as marginal external costs imosed on others as a result of consumtion activity). The social MRS, must take into account that substituting one unit of good for good affects the roduction of good by and therefore the individual s utility level is changed by.

15 Efficient roduction for firm Rewrite equations (3) and (4). Substitute λ µ from equation (5): λ µ λ f (3) λ µ f (4) Substitute equation (4) into equation (3). Dividing the equations yields: λ λ + (8) The RHS of equation (8) is the social MRT,. By using one additional unit of good as an inut, firm roduces roduction of good by. and consequently affects the

16 Efficient roduction for firm Rewrite equations (5) and (6): λ µ (5) λ µ f (6) Dividing equation (6) by equation (5) yields: λ λ f y (9) Note that firm 's roduction activity does not generate an externality, hence the RHS of equation (9), the social MRT,, equals the rivate MRT,.

17 Pareto Otimal Allocation λ λ + U relative scarcity rices social MRS social MRT social MRT,,, Key results otimal organization of economy does not necessarily require total elimination of externalities, even when they are negative consumtion and roduction good affects roduction of good negatively but otimality does not require eliminating roduction of good in general, zero ollution is not otimal external costs (or direct effects of one agent s actions uon others) must be internalized in consumtion and roduction decisions

18 Consumer s decision roblem Cometitive Equilibrium max x, x U( x, x ) s.t. B ω + ω + π(, ) x + x where is the value of the consumer s initial endowment and ω + ω π(, ) are industrial rofits max x, x L U( x, x ) + λ( B x x ) L L λ λ Eliminating 8 yields: relative rices rivate MRS,

19 Firm 's decision roblem max π y + y s.t. y f ( y ) y max π f ( y ) + y y π + f f

20 Firm 's decision roblem max π f ( y, y, x ) + y y π + f f

21 Cometitive Equilibrium A cometitive equilibrium is a vector of rices (, ) and an allocation ( x, x, y, y, y, y ) such that firms rofits and consumer s utility are maximized (first order necessary and sufficient conditions are satisfied) suly and demand is equalized in both markets At a cometitive equilibrium, self interest maximization leads each agent to equate his/her rivate marginal rate of substitution or transformation to the rice ratio, resulting in the equalization of rivate rates: relative rices rivate MRS rivate MRT rivate MRT,,,

22 Comaring Cometitive Equilibrium and Pareto Otimum In contrast, a Pareto otimal allocation requires equalization of social marginal rates of substitution and transformation to relative scarcity or shadow rices. In general, CE with externalities not PO! divergence of rivate and social valuations economic decisions too decentralized at cometitive equilibrium agents generating negative externality will roduce/consume too much rices of negative externality generating roduction or consumtion goods are too low (since full social costs of economic activity not reflected in market rices)

23 Decentralized Solutions Otimal Taxation Is there a tax structure that will sustain a cometitive equilibrium as a Pareto otimum? Can we find a set of taxes for olluters and victim damage or comensation ayments that will induce consumers and firms to choose Pareto otimal levels of economic activities? Define: t c garbage tax er unit of good consumed t f J emission tax er unit of good roduced damage comensation er unit of good roduced Assume taxes collected are redistributed to consumer as lum sum transfer T. Consumer s decision roblem max x, x ( ) L U( x, x ) + λ ω + ω + T + π(, ) ( + t ) x x c L λ( + t c ) L λ Eliminating 8 yields: tc

24 Firm 's decision roblem max π ( t ) f ( y ) + y y f π ( t f ) + + t f + t f

25 Firm 's decision roblem Damage comensation rates deend on victim s activity level. For examle, if exogenous shift in demand leads to an increase in outut, damage caused by firm and consumers will necessarily increase and hence comensation ayment must increase. max π ( + τ) f ( y, y, x ) + y y π τ f ( + ) + τ τ

26 Cometitive Equilibrium with Taxes and Comensation U x U x t f y t f y t f x t f y c f c f + + τ τ * * * set of taxes and comensation ayments which will sustain a { } t t c f * * *,,τ cometitive equilibrium which is Pareto otimal otimal taxes are standard Pigouvian taxes taxes are set equal to the value of the marginal damage of economic activities evaluated at the Pareto otimal level of these activities taxes are based on a olluters ay rincile Pigouvian taxes are asymmetric no subsidization or comensation of victims necessary to sustain Pareto otimum rovided t c and t f set otimally market rices are now equal to shadow rices in social lanner s roblem

27 Creation of markets by secifying roerty rights Social lanner wishes to establish a comlete set of cometitive markets which incororates externalities must assign roerty rights. Suose that social lanner assigns firm the right to an unolluted river. The initial assignment of roerty rights creates two ollution rights markets: Consumer must urchase the right to ollute from firm : rice of right for consumer to ollute one unit Firm must also urchase the right to ollute from firm : q rice of right for firm to ollute one unit

28 Consumer s decision roblem max x, x U( x, x ) s.t. ω + ω + π(, ) x + x + x x x where, the quantity of rights demanded by consumer, must equal x, x the number of units of ollution created. max x, x ( ) L U( x, x ) + λ B ( + ) x x L λ( + ) L λ Eliminating 8 yields:

29 Firm 's decision roblem max π y + y q y s.t. y f ( y ) y y y where y is the quantity of ollution rights demanded by firm and y is firm 's level of ollution. Note that, institutionally, firm is constrained to urchase as many rights as it creates units of ollution. max π ( q ) f ( y ) + y y π ( q ) + + q + q

30 Firm 's decision roblem Firm will suly quantities of ollution rights y$, x$ at rices q,, resectively. max y, y$ x$ π y + y + q y$ + x$ s.t. y f ( y, y$, x$ ) max y, y$ x$ π f ( y, y$, x$ ) + y + q y$ + x$ π + π q + q q $ $ $ $ π + $ $ $ $ Note that firm will sell the right to ollute so as to equate the market value of the roerty right (MB of selling the right) to the value of the damage created by selling the right (MC of selling the right).

31 Cometitive equilibrium with marketable ollution ermits q + Definition of cometitive equilibrium requires market clearing in both ollution ermit markets: x$ x x y$ y y From firm 's rofit maximization roblem and market clearing conditions we have that $ t * c q $ t * f ermit rices marginal damage at Pareto otimum Pigouvian taxes Substituting market clearing ermit rices into cometitive equilibrium conditions will yield conditions for Pareto otimal allocation. by defining roerty rights, social lanner can create missing markets can sustain a cometitive equilibrium as Pareto otimum with a comlete set of markets!

32 Marketable Permits versus Taxes two decentralized solutions to externality roblem rice set by central authority in tax regime rice determined by market forces in marketable ermit system economic agents find otimal rice of ollution in decentralized, atomistic economy invisible hand works rovided there is a comlete set of markets both solutions force economic agents to fully internalize costs of actions solutions have identical allocative consequences but different distributional consequences Caveats models assume externalities the only source of institutional failure other ossible sources of institutional failure market failure transaction costs administrative costs may differ strategic behaviour trading restrictions regulatory failure other distortionary olicies may be in lace such as subsidization of externality generating activity incomlete information regulator must know MB and MD curves exactly to set olicies efficiently incomlete enforcement models assumes full comliance highly unlikely regulator has the resources and knowledge to design erfect ex ost governance structure regulatory cature global failure