More Ramsey Pricing. (firm willing to produce) Notation: 1 of 6

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1 EP 7426, Problem Set 3 Len abrera More Ramey Prng regulated frm rodue two rodut, and, and ell them dretly to fnal utomer. onumer demand for thee erve known erfetly, a are the frm' roduton ot. Produt rodued wth ontant margnal ot, and rodut rodued wth ontant margnal ot. onumer demand for eah regulated rodut ndeendent of onumer demand for the other rodut. ngle regulator et the re for both of thee erve.. haratere n term of relevant re elatte of demand the Ramey re that mame the um of onumer' urlu and roft, whle enurng the frm nonnegatve roft.. What are the Ramey re n th ettng f the frm nur no fed ot of roduton? [Prove your anwer formally.]. Now uoe that the frm nur trtly otve fed ot of roduton. lo uoe that the frm the ole roduer of rodut, but that omettor alo rodue rodut. Thee omettor take the regulated re of rodut a gven, and elet ther uly of rodut n order to mame ther roft. The regulated frm ule the redual demand n the market lae.e., the dfferene between market demand and the uly of the omettve frnge at the regulated re. Relatve to the re of rodut that along wth the otmal re of rodut mame the um of onumer' urlu and roft n the abene of ometton, wll the orreondng re of rodut to be hgher or lower n the reene of ometton? Jutfy your anwer. [You do not need to rove your anwer rgorouly, but you mut elan fully the ba eonom log that underle your anwer.] D. Return to the ettng where the regulated frm a monooly uler of both rodut and. In th ettng, uoe that the demand for rodut le re elat than the demand for rodut. Suoe the regulator oberved to et the re of rodut below t margnal ot of roduton and the re of rodut above t margnal ot of roduton n th ettng. ould thee re obly be oally otmal re n the ene of mamng a oal welfare funton whle enurng nonnegatve roft for the frm? Jutfy your anwer. [gan, you do not need to rove your anwer rgorouly, but you mut elan fully the ba eonom log that underle your anwer.]. Sne the demand are ndeendent, onumer urlu from eah rodut an be omuted earately and then added together Regulator' otmaton roblem: ma S S π, ma total urlu,.t. π, 2 0 frm wllng to rodue Notaton: of 6

2 2 of 6 demand for rodut, d S onumer urlu for rodut at re fed ot of roduton, π Sub thee nto the roblem: d d ma,.t. 0 Lagrangan: [ ] d d Kuhn-Tuker ondton for re : 0, Note : 0 beaue re not retrted Note 2: eaue of roblem formulaton ontant M, otve fed ot, and downward long demand, the nonnegatve roft ontrant wll be bndng o 0 >. It bnd beaue urlu mamed wth margnal ot rng, but n th ae average ot eeed margnal ot o frm long money at P M. Manulate 0 to get term on ame de alo ure argument: [ ] Now move to RHS: Move to LHS & multly by / to get own re elatty of demand: Sub and : < 0 o f we ut t on RHS a, we dro the : Ramey Rule

3 . The re harateraton from art doe not hange.e., re marku nverely roortonal to re elatty of demand. y removng the fed ot, the only thng beng hanged n the formulaton are drong a ontant from the objetve no effet on the oluton and hangng the rght hand de of the nonnegatve roft ontrant whh wll affet the value of the lagrange multler. Lowerng the fed ot wll effetvely eand the feable regon veru havng a fed ot o the objetve value hould nreae.e., more total urlu. Takng fed ot to ero wll drve the lagrange multler to ero.e., the nonnegatve roft ontrant wll not be bndng beaue the frm make ero roft at the frt-bet oluton, P M. If we aume λ > 0.e., π 0, the Ramey Rule ay > for,. Th volate the ero roft ontrant. Therefore, by ontradton λ > 0. That how the re are the ame a the frt-bet re:,,.. rom the modfed Ramey Rule derved below: where In the frt term, / < 0 o we an take the abolute value and negate the gn; for the eond term, multly by / to get an elatty: re elatty of demand for rodut from hangng ; > 0 beaue the rodut are ubttute meaure reonvene of from hange n ; > 0 from aumton about trateg omlement <, whh mean the marku wll be larger than tandard Ramey re from art ; mlar reaonng a ubttute n the deendent demand ae; eentally have le elat demand for rodut beaue t' a trateg omlement to the omettor' rodut Dervng the modfed Ramey Rule: Notaton: ume and omettor' rodut are trateg omlement o > 0 th mean we an vew a a funton of, demand for omettor' rodut 3 of 6

4 To mlfy notaton:, vetor of all re [, ] [,, ] vetor of demand for regulated frm, omettor' ot funton S gro urlu for rodut S, gro urlu for rodut & Note: for ue n K-T ondton later S S, S and Heurt: More ormal: the value of rodut on the margn S S d 0 $ rea S * Margnal value of to onumer re D Regulator' Problem: ma S S, ma total urlu,.t. 0 frm wllng to rodue Lagrangan: S S, [ ] Kuhn-Tuker ondton for re S 0 No Dfferene - wth a lttle manulaton, th the ame ereon from art : 0 Note all the lae where enter, eeally, and,, where t atually enter twe 4 of 6

5 5 of 6 S S 0 To Smlfy, defne ly mlfaton, then try to get th to look lke the Ramey rule olate a term for the marku [ ] [ ] [ ] 0 M n ombne [ ] term and move term to RHS: [ ] [ ] M Dvde by : [ ] [ ] M Smlfy notaton further: and ly th and dvde by and olate marku for rodut : [ ] M eaue the omettor are mamng ther roft, they wll re at margnal ot; th elmnate the eond term: Let elatty of demand aountng for all effet; th gve the famlar Ramey Rule:

6 D. In th ae, the reult eem to volate the Ramey Rule, whh ay to rae re on the le elat good, but onder the regulator' roblem wth oal welfare weght: ma α S α S π, ma total urlu,.t. π, 0 frm wllng to rodue Th reult n a modfed Ramey Rule: M α Gven the roer oal welfare weght on the two rodut, any ombnaton of re that enure nonnegatve roft ould mame oal welfare. Therefore, the enaro reented lauble. 6 of 6