Unt 5: Government polcy n compettve markets I E cency Prof. Antono Rangel January 2, 2016 1 Pareto optmal allocatons 1.1 Prelmnares Bg pcture Consumers: 1,...,C,eachw/U,W Frms: 1,...,F,eachw/C ( ) Consumers and frms nteract through an nsttuton (e.g., the market) to produce a feasble allocaton Bg queston: Is the resultng allocaton desrable? Three notons of desrablty: 1. E cency: measures lack of waste n producton/consumpton 2. Dstrbutve ustce: measures farness of dstrbuton of resources, e.g. maxmn 3. Procedural ustce: measures farness of process used to reach allocaton, e.g. equal treatment In ths course, we look at 1 and 2. Procedural ustce s studed n more advanced courses. 1
1.2 Pareto optmalty Pareto mprovement test: A feasble allocaton 1 Pareto mproves over feasble allocaton 0 f: 1. U ( 1 ) >U ( 0 )foratleastone 2. U ( 1 ) U ( 0 )forall I.e., need to make somebody better o wthout makng anybody worse o Afeasbleallocaton s Pareto optmal (PO) f there doesn t exst another feasble allocaton ˆ that Pareto mproves over. Example 1: 2consumerswthpreferencesU(q, m) =m W of good m No producton Feasble allocaton: m 1 + m 2 = W 1 + W 2. All feasble allocatons Pareto Optmal Ths llustrates orthogonalty between PO and equalty Example 2 : C = F =1 Look at cost functon wth SFC > 0, but FC =0 See vdeo for graphcal characterzaton of the set of feasble and PO allocatons. 1.3 Propertes of Pareto optmal allocatons RESULT: Let be a feasble allocaton. Then the followng statements are equvalent (under the manatned assumpton that preferences are quas-lnear and m s unbounded below): s Pareto optmal 2
solves max U ( ) feasble Proof outlne: If not PO then there exsts a feasble Pareto mprovng allocaton, whch mples that t also ncreases P U If not max P U,thenthereexstsafeasbleallocaton whch yelds hgher P U. Then possble to construct another feasble allocaton,byredstrbutngm from the allocaton,thatmakes everyone strctly better o. Ths mples that s a Pareto mprovement over the ntal allocaton. RESULT: Let be a feasble allocaton. Then the followng are equvalent (under the manatned assumpton that preferences are quas-lnear and m s unbounded below): Proof: s Pareto optmal solves max feasble SS( ) From above, P.O max P U ( ) From Unt 4, max P U ( ) max SS( )+constant REMARK: Propertes of a PO allocaton don t depend on dstrbuton of m REMARK: To fnd set of PO allocatons, solve max B (q c ) C (q f ) q1,...,q c C c q f 1,...,q f F s.t. Necessary condtons for PO: q c = q f 3
1. E cent allocaton of producton to frms: For any, h, k wth q f > 0,qf h > 0, and qf k =0, C 0 (q f )=C0 h(q f h ) <C0 k(q f k ) 2. E cent allocaton of consumpton: For any,, k wth q c > 0,q c > 0, and q c k =0, B 0 (q c )=B 0 (q c ) >B 0 k(q c k) 3. Overall producton e cency: For any, wth q c > 0andq f > 0, Intuton for Condton 1: B 0 (q c )=C 0 (q f ) Suppose C 0 (q f ) >C0 h (qf h ), for frms, h wth qf,qf h > 0, Then can decrease q f by dq and ncrease qf h by dq Ths leaves total producton unchanged, whle decreasng total costs of producton Intuton for Condton 2: Suppose B 0 (q c ) <B 0 (q c ), for consumers wth q c,q c > 0 Then can transfer dq from to and dm = dqb 0 (q c )from to nd erent between old and new allocatons strctly better o snce du = dq(b 0 (q c ) B 0 (q c )) > 0 Intuton for Condton 3: Suppose B 0 (q c ) >C 0 (q f )atannterorpont Frm can produce dq more and gve t to consumer n exchange for dm = dqc 0 (q f ) Ths s feasble, because the dm transfer covers exactly the extra costs of the frm 4
But consumer s utlty ncreases snce du = dqb 0 (q c ) dqc 0 (q f ) > 0 Su cent condtons for PO: D cult problem to derve general su cent condtons, especally n presence of FCs or SFCs If MB > 0,MB # and C( ) s DRS, then the three necessary condtons for PO gven above are also su cent 1.4 Example Example of nteror PO allocaton C = F =1 Cost functon: DRS, FC = SFC =0 Feasble set s graph of ponts (q, W c(q)) n qm-plane, w/ slope c 0 (q) Ind erence curves satsfy B 0 (q)dq +dm =0,sohaveslope B 0 (q) PO allocatons satsfy B 0 (q c )=c 0 (q f )(Condton3) Example of corner PO allocaton As before: C = F =1 Cost functon: DRS, FC = SFC =0 Feasble set s graph of ponts (q, W c(q)) n qm-plane, w/ slope c 0 (q) Ind erence curves satsfy B 0 (q)dq + dm =0,sohaveslope B 0 (q) However, nd erence curves nearly flat and cross m-axs wth c 0 (0) < B 0 (0) c 0 (0) >B 0 (0), so no mprovement possble from (0,W)andthss the unque PO allocaton 5
1.5 Example C =10,U (q, m) = ln(q)+m, > 0, W F =10,C (q) = q 2, > 0 Condton 1: MC =2 q f =) 1. Interor solutons 2. For all pars of frms k, : 2 q f =2 kq f k =) qf q f k = k.e. frms wth lower margnal costs produce more Condton 2: MB = q c =) 1. nteror solutons 2. For all pars of consumer, : q c = q c =) qc q c =.e. consumers wth hgher margnal beneft consume more Condton 3: MB = MC for all, =) for any consumer-frm par q c =2 q f Feasblty constrant: By Condton 1, and q c = q c = q f = 6 q f. (1) 1 q c 1 1 q f 1
So (1) s equvalent to 1 q c 1 = 1 q f 1. (2) Set A = P 1 and B = P 1 and wrte (2) as Aq c 1 = Bq f 1. (3) From Condton 3, we have 1 q c 1 =2 1 q f 1. (4) Solvng (3) and (4) for q f 1 and q c 1 yelds r 1 q f 1 = 2B 1 s q1 c B 1 = 2A 1 These are the quanttes frm 1 must produce and consumer 1 must consume n a PO allocaton. From here we can dentfy the full allocaton by usng the expressons above. NOTE: The allocaton of q, but not of m, s unquely determned n PO allocaton. Snce all consumers have same MB for m, wecanshft m around wthout a ectng the optmalty of the allocaton. 2 Frst Welfare Theorem 2.1 Result Frst Welfare Theorem (FWT): Any compettve market equlbrum allocaton s Pareto optmal Proof: 7
By contradcton Let,p be a cme p > 0, snce otherwse some consumer would consume an nfnte amount of q Suppose that s not PO Then there exsts another feasble whch s Pareto mprovng. Ths mples that: 1. for at least one, U ( ) >U ( ) =) p q c ( )+m c ( ) >p q c ( )+m c ( ),.e., what consumes at must cost more that what she consumed at,snceotherwsetheconsumerwouldnothave been maxmzng her utlty at. 2. For every consumer, U ( ) U ( ) =) p q( c )+m c ( ) p q( c )+m c ( ), by a parallel argument. Together, ths mples: p q c ( )+ m c ( ) > p q c ( )+ m c ( ) (5) By feasblty, p q c ( )+ m c ( )= p q c ( )+ W C (q f ( )) and p q c ( )+ m c ( )= p q c ( )+ W C (q f ( )) It follows that (5) s equvalent to p q c ( ) C (q f ( )) > p q c ( ) C (q f ( )), after cancellng the W terms from both sdes. 8
Snce the sum of consumers expendtures must equal the sum of frms revenues, ths s equvalent to ( ) > ( ), where = profts. But ths means at least one frm earns hgher profts at than they dd at, whch contradcts the assumpton that frms maxmze profts at the cme. Therefore s not a CME, whch s a contracton. Intuton 1: The nvsble hand of the market Market forces: market settles at p, wth: 1. s feasble (market clearng) 2. MB = p from utlty maxmzaton by consumers 3. p = MC from proft maxmzaton by frms But ths nduces the necessary condtons for PO, even though consumers only care about maxmzng ther own utlty, and frms only care about maxmzng ther own profts: 1. snce MC k = p = MC 2. snce MB = p = MB 3. snce MB = p = MC Bottom lne: If frms and consumers all try to do ther best, gnorng each other s needs, market forces wll lead them to settle on an allocaton that satsfes PO!!!!!!!!!!! Intuton 2: RESULT: SS maxmzed over feasble allocatons at a cme allocaton Proof: FWT =) s P.O. 9
PO max SS( ) (under the mantaned assumptons of feasble quas-lnear preferenes and m unbounded below) Crtcal graph: Please see vdeo lecture. The equlbrum market quantty s the optmal level of producton of good x The equlbrum market prce p equals the margnal socal beneft and the margnal socal cost of producng and consumng another unt 2.2 Dscusson Remark 1: FWT underles economsts wdespread belef n free-markets. Remark 2: FWT shows that compettve markets lead to PO allocatons at very low nformatonal demands. Requred nformaton: Consumers: know only ther own U( ) andp Producers: know only ther own C( ) andp Compare to requred nformaton for a dctator or central planner: Need to know all U( )s Need to know all C( )s Then must solve a computatonally d cult optmzaton problem Most economsts beleve that the d culty of accurately gatherng the requred nformaton and then solvng the necessary optmzaton problem s the man reason centrally planned economes lke the Sovet Unon have typcally faled to produce the same levels of economc growth as free market economes. Key assumptons behnd the FWT, and consequences when they fal: 10
Assumptons Optmal decson makng by consumers and frms Every actor s a prce taker No externaltes Consequences FWT fals due to DM mstakes (e.g. marketng, myopa) Imperfect competton, FWT fals (monopoly, olgopoly, brands) Publc goods & externaltes, FWT fals (envronment, R&D) Perfect nformaton Asymmetrc nformaton, FWT fals (nsurance, used cars, contracts) 3 Taxes and e cency 3.1 Deadweght loss Let SS opt, opt denote the soluton to the socal surplus maxmzaton problem over all feasble allocatons The Deadweght Loss (DWL) s gven by: DWL( ) =SS opt SS( ) =measureofne cencyatallocaton [n $s] = U ( opt ) U ( ) " # " = B (q c,opt ) C (q f,opt ) B (q c ( )) # C (q f ( )) Graphcal representaton: Assume that any quantty that s produced s allocated e cently among producers and consumers Ths leads to an mportant graphcal represenaton of the DWL. (See vdeo for detals) REMARK: DWL ncreases non-lnearly (and often as the square) wth devatons from opt 11
3.2 Lump-sum taxes Basc taxonomy of taxes Lump sum: Specfes fxed amount to be pad by each consumer/frm ndependent of ther actons T>0: taxes T<0: transfers Non-lump sum: Tax owed depends on consumer/frm actons Example: p/unt sales tax pad by consumer Note: to solate the e cency e ects of tax polces, we focus on revenue neutral tax polces n whch all revenue rased s returned to consumers usng lump-sum transfers RESULT: Lump-sum taxes do not ntroduce ne cences Suppose T = T1 c,...,tc c,tf 1,...,T f F wth P T c + P T f =0 Clam: DWL( T )=0 Why? Consumers: max q 0 B (q)+w T c pq =) D T = D not Frms: max q 0 pq C (q) T f =) S T = S not I.e., lump-sum taxes do not a ect optmzaton problem for consumer or frm, so cme unchanged Therefore, DWL( T )=0. Unfortunately, lump-sum taxes have serous lmtatons, as we ll see later n the course 3.3 Per-unt taxes Look at mpact of per-unt tax on demand and supply p/unt sales tax on consumers [$] revenue returned usng a lump-sum transfer: T = q C S = S no 12
Consumer: max q 0 B(q)+W + T q(p + ) =) D (p) = D no (p + ) Ths assumes that C s very large, so consumers take the sze of the lump-sum transfer as fxed Equlbrum e ects and DWL Aggregate supply curve stays the same Aggreagate demand curve shfts down by Therefore: q <q no and p <p no Comparatve statcs At equlbrum: no D (p ( )+ ) =no S (p ( )) =) dd no dp dp d +1 = ds no dp dp d =) dp d = d S no dp d D no dp d D no dp < 0 Ths formula shows how responses to taxes depend on the relatve senstvty of aggregate demand and aggregate supply to prce 4 Fnal remarks Key concepts: 1. Feasble allocaton s PO f there sn t another feasble wth U ( ) >U ( )forsome U ( ) U ( )forall 2. FWT: allocatons generated by compettve markets are PO 3. Lump-sum taxes redstrbute wthout ntroducng ne cences Non lump-sum taxes are dstortonary (.e. they have DWL > 0) 13