Week 6 Market Failure due to Externalities

Transcription

1 Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to a consumpton externalty, or a producer, gvng rse to a producton externalty. Suppose there are two frms. Frm 1 produces output y at a cost C(y) and mposes an externalty n the form of a cost E(y) on frm 2. If p s the prce of frm 1 s output, profts for the frms are: π = py C( y) and π = E( y) (1) 1 2 Frm 1 chooses output by maxmsng π 1 and the proft maxmsng level of output occurs when: p = C ( y). However, snce Frm 1 takes account only of prvate costs, the proft-maxmsng output, y s, from a socal perspectve, too hgh. If the two frms merged n order to nternalse the externalty, the merged frm would maxmse: π = py C( y) E( y) (2) and the proft maxmsng output would occur when: p = C ( y) + E ( y) that s when prce was equal to margnal socal cost. The soluton to (2) would yeld the socal optmum, trangle, BC. y. In Fgure 1, below, the socal cost of the externalty s the area of the Prce Margnal Socal Cost tax { B C Margnal Prvate Cost Demand Y Y Output Fgure 1: The Socal Cost of n Externalty

2 2. Modellng Externaltes There are two consumers (=1,2) and N commodtes (r=1 N). The utlty of consumer depends upon the quanttes consumed by hm/her of the commodtes ( X, r = 1,.., N) and upon the level, denoted r consumer : 1 2 N It s assumed that: () consumer can take the acton, for any level cost., of some acton taken by the other U = U ( X, X,..., X ; ) (3), wthout any monetary U () The acton taken by mposes an externalty upon, f 0. The U U externalty s postve f > 0 and negatve f < 0 where: If V ( p, W, ) represents 's ndrect utlty functon, then: N (,, ) = ( 1,..., 1; ) st r r = X r r= 1 V p W Max U X X p X W (4) W s the wealth of the consumer, p = { } s the prce vector. p r It s assumed that the utlty functon, pr s the prce of the commodty r and U s quas-lnear wth respect to some numerare commodty: U ( X1,.., X ; ) = X1 + G ( X2,.., X ; ). Then, holdng prces constant, the ndrect utlty functon takes the form: N V ( W, ) = φ ( ) (5) N Consumer wll choose so as to maxmse ( ) φ and the frst order condton for ths s: φ ( ) = 0 (6) The acton level (droppng the superscrpt ) represents 's optmal acton and the combned utltes at hs prvate optmum s: where: satsfes condton (6). φ ( ) φ ( ) 0 + = (7) The socally optmal acton, however, would be for to choose so as to maxmse: φ ( ) + φ ( ) (8) 2

3 and the frst order condton for ths s: φ ( ) + φ ( ) (9) Condton (9) yelds the socally optmal acton by, prvately optmal acton,. If ( ) 0 n contrast to hs/her φ <, so that the externalty s negatve, by condtons (6) and (9): φ ( ) = 0, but φ ( ) > 0 <. By contrast, f φ ( ) > 0, so that the externalty s postve, by condtons (6) and (9): φ ( ) = 0, but φ ( ) < 0 >. φ '( ) φ '( )+ φ '( ) -φ '( ) 0 Fgure 1 Negatve Externalty φ '( )+ φ '( ) φ '( ) 0 Fgure 3 1 Postve Externalty

4 3. Solutons to Externaltes: Taxes and Subsdes The externalty problem arses because consumer - who s takng the externalty-producng acton - faces the wrong prces and these lead t to take acton at level nstead of at. It s assumed that the externalty s negatve so that <. correctve tax (frst proposed by Pgou, 1920) can be mposed whch, by yeldng the rght prces, wll nduce t to act at the level. If t s the tax rate per unt of acton taken, then consumer 's optmsaton problem would be: Max φ ( ) t (10) and the frst order condton for ths s: φ ( ) = t. If the tax rate, t, s set at t = φ ( )(.e. to consumer 's margnal damage when consumer takes acton = ), then =. The problem wth ths soluton s that t requres the taxng authorty to know the externalty damage functon φ ( ) and, hence, the socally optmal acton level,. But, f ths was known, consumer could smply be regulated to produce. Under a subsdy, consumer s pad an amount s per unt of acton to reduce hs/her externalty-producng acton,. Now consumer 's obectve functon s: φ ( ) + ( ) s (11) and the frst order condton for solvng ths problem s: φ ( ) = s. If the subsdy rate, $s, s set at s= φ ( )(.e. to consumer 's margnal damage when consumer takes acton = ), then =. The dfference between a tax and a subsdy s that, under a tax, consumer makes payment whle, under a subsdy, he/she receves payment. Under a subsdy, consumer, by reducng hs/her acton by one unt, forgoes φ ( ) of utlty but gans $s by way of subsdy. 4. Mssng Markets The reason that the externalty problem arses s that there s no market for the externalty-producng acton. Consumer undertakes the externalty-producng acton, up to any level that he/she chooses, wthout any cost. The acton s, therefore, a free good because there does not exst a market n whch t s traded. The problem 4

5 s, therefore, one of mssng markets and pso facto may be solved by creatng a market for the externalty-producng acton. Suppose that consumer has the rght to be free of the externalty but can sell to consumer the rght to undertake the externalty-producng acton at a prce of $q per unt. Then the optmsng problem facng consumer s to decde how many unts of the acton to allow to undertake. Ths s acheved by solvng: Max φ ( ) + q (12) The frst order condton for solvng (12) s: whch mplctly defnes a supply functon: = S( q). φ ( ) = q (13) Consumer has to decde how many unts of the acton to buy and he/she does so by maxmsng: The frst order condton for solvng (14) s: whch mplctly defnes a demand functon: = D( q). The equlbrum prce, S φ ( ) q (14) φ ( ) = q (15) q, clears the market so that: Dq D ( ) Sq ( ) = (16) and at the equlbrum prce, by equatons (13) and (15): φ ( ) = φ ( ) (17) whch s the condton for a socal optmum. 5. Property Rghts Ths soluton s due to Coase (1960) and s often referred to as the Coase Theorem. Ths theorem says that when partes can bargan to ther mutual advantage wthout cost, then the resultng outcome wll be effcent, regardless of how property rghts are dstrbuted. Put dfferently, the Coasan soluton to the problem of externaltes s to establsh nsttutons whch wll defne and enforce property rghts and whch wll allow partes to bargan at zero transacton cost. Coase s frst pont was that externaltes are the ont product of the offender and the vctm and the most effcent system of avodng an externalty s to put the onus for avodance on the party whch can avod t at the least-cost. The Pgovan 5

6 soluton of penalsng the generator of externaltes would only be effcent f ths party was the lowest cost avoder; otherwse, t would be neffcent. Coase s second pont was that n order to remove the ll-effect of an externalty, nether regulaton nor taxes were necessary. If transacton costs were zero that s, any agreement that s n the mutual nterest of the partes can be arrved at costlessly then barganng between the partes would lead to an effcent outcome, regardless of how property rghts were defned. Coase s thrd pont was that the problem was not one of externaltes but, rather, one of transacton costs whch prevented externaltes beng barganed out of exstence. When we observed externaltes n the real world, Coase would have us enqure about the level of transacton costs whch prevented externaltes beng barganed away. Suppose consumer has the rght to be free of the externalty. Then consumer can sell to consumer the rght to undertake the externalty producng acton, at a level, n exchange for a payment, T. If consumer reects ths offer, then he/she cannot take the externalty producng acton and = 0. Then consumer wll accept ths offer f, and only f: φ ( ) T φ (0) (18) Gven constrant (18), consumer wll choose (, T ) so as to maxmse: φ ( ) + T st φ ( ) T φ (0) (19) Snce consumer prefers a hgher, to a lower, T, the constrant wll bnd at the optmum and the optmsng problem of (18) becomes: and the frst order condtons for solvng ths are: Max φ ( ) + φ ( ) φ (0) (20) φ ( ) = φ ( ) (21) whch s the condton for a socal optmum. Then consumer chooses ˆ φ ( ) φ (0) T =. Ths offer (, T ˆ ) s accepted by consumer. = and Suppose consumer has the rght to undertake the externalty producng acton up to any desred level. Consumer can offer consumer the level of acton n exchange for a payment, from to, of T. If consumer reects ths offer, 6

7 consumer has the rght to undertake the acton to any desred level,, where ths wll satsfy: φ ( ) = 0. So consumer wll accept ths offer f, and only f: φ ( ) T φ ( ) Gven constrant (22), consumer wll choose, T + (22) and T so as to: Mn φ ( ) + T st φ ( ) + T φ ( ) (23) Snce consumer wll prefer a smaller, to a larger, and (23) can be rewrtten as: and the frst order condtons for solvng ths are: T, constrant (22) wll be bndng φ ( ) + φ ( ) φ ( ) (24) φ ( ) = φ ( ) (25) whch s the condton for a socal optmum. Then consumer offers consumer a payment of = φ ( ) φ ( ) f he/she would restrct hs/her acton to T =. Ths offer (, T ) s accepted by consumer. The optmal level of acton,, s the same, rrespectve of whether consumer or have "property rghts". However, the dstrbutonal mplcatons are very dfferent: when consumer has the rght to be free of the externalty, consumer has to make a payment of ˆT to consumer ; however, f consumer has the rght to the acton, consumer has to make a payment of T to consumer. If property rghts are yet to be assgned, they should be assgned to the party who has to pay the hgher cost to avod the externalty (.e. avodance costs should be met by the "least-cost avoder"): to consumer, f ˆ T < T, and to consumer, f T ˆ > T. 7