Hotelling s Rule. Therefore arbitrage forces P(t) = P o e rt.

Transcription

1 Htelling s Rule In what fllws I will use the tem pice t dente unit pfit. hat is, the nminal mney pice minus the aveage cst f pductin. We begin with cmpetitin. Suppse that a fim wns a small pa, a, f the ttal amunt f an exhaustible esuce. his small cmpetitive fim can sell sme all at the cuent pice, P, in a cmpetitive maket. Nw cnside a futue time, say t. he pice at that time will be P( and we can assume it is knwn with ceainty. Nw cmpetits have an abitage situatin. Suppse that the futue pice is knw with ceainty t be P( > P e. It fllws that anyne culd bw P at inteest ate and buy a unit f the esuce n the pen cmpetitive maket nw. his pesn wuld then hld the esuce until futue time t and then sell it f P( at time t. He wuld then etun the bwed mney P and the inteest (e -1)P and wuld make a iskless pfit f (P( - P e ). Since anyne can d this and thee ae n tansactins csts in u simple example, the pice P( must equal P e. If P( < P e, a iskless pfit can again be made by anyne wning the esuce nw. he esuce wne wuld meely sell the esuce nw and get P and then lend the P in cedit makets at inteest ate. By time t, the value lent wuld have isen t P e. he esuce culd then be bught back at P( and the speculat wuld have a pfit f (P e.- P(). heefe abitage fces P( = P e. his is the pf f the P( = P e cmpetitive picing elatin in Htelling s pape. his picing elatin des nt hld f a mnply since the mnplist must cnside that the amunt he sells may influence the pice f the pduct. A cmpetitive fim des nt have this pblem since each fim is small cmpaed t the ttal industy. Nte that Htelling des nt tell us (at this time) hw that P is detemined. F the fim, the ttal amunt f the esuce that can be extacted is equal t a. his means that we have the cnstaint

2 a qdt P( ) dt P e ) dt whee is when we have finished extacting the esuce. Nw, Htelling ightly ntes that at time, the quantity extacted is ze thee is n me that can be extact at time. If thee was any esuce left at time, it culd then be sld f a pfit and theefe wuld nt be the culminating time f extactin. All this means that P e ) =, which is an impant bunday cnditin Htelling genealizes the demand functin P() t allw the demand t change ve time. He wites it as P(,, and theefe the teminal cnditin becmes P(),) = his equatin abve hlds f any demand functin q and f any path f extactin. Htelling takes the example f P(, = 5 P( = 5 - P e which implies at time that P = 5e - Plugging the demand functin int the integal cnstaint abve gives { 5 P e P } dt 5 (1 e ) a which simplifies t P 5e 5 5e 5 a (1 e ) 5 a (1 e ) 5 a his last equatin can be witten as

3 e a 1 ( ) 5 Htelling then daws a gaph t shw hw that can be calculate fm this equatin. (see next page) Htelling s analysis thus fa shwed that cmpetitive picing wuld lead t P'( P whee this diffeential equatin has a teminal cnditin given implicitly by q ( P( ), ) his detemines P and the futue path f P(. he picing elatin in the bx abve is called Htelling s Rule.

4 Htelling next cnsides the scially ptimum ate f extactin f the esuce. d this, he makes libeal use f the cncept f utility being the aea unde a demand cuve. Clealy, this is nly an appximatin. Htelling s Rule -- the Optimal Extactin f a Nnenewable Resuce Htelling next cnsides the pblem f detemining the scially ptimal ate f extactin. He begins by nting that the aea unde a demand cuve is clsely elated t the ttal utility enjyed by cnsumes. his is because in equilibium Maginal Utility f X = (Pice f X)*(Maginal Utility f Incme) And, if the Maginal Utility f Incme is elatively cnstant, then and theefe in diffeentiable fm. U U = N i1 u( q) P i X i q P( s) ds Cnside the maximizatin f inteempal utility subject t the exhaustible esuce cnstaint. q ( Max u( ) e dt subject t dt. he Lagangian can nw be witten as

5 L u( e dt { a dt he fist de Eule Equatin is u e q u but p( ) q this as by the definitin f utility u(q) abve. heefe, we can wite p ( ) e and thus, p( e p( p e his shws that the scially ptimum ate f extactin and use f an exhaustible esuce is the same as that fund unde cmpetitin. Htelling cautins that this des nt mean that cmpetitin is clealy ptimal, since we have made many unealistic assumptins in deiving the esult. Hweve, this is smething which makes cmpetitin lk vey favable cmpaed t the types f wneship and usage. Htelling cntinues n t analyze mnply. It is nt difficult t undestand his analysis. he student shuld ty t calculate the ptimal pice and quantity that esults fm a mnply wning a fixed esuce. Assume demand is given by P = 5 q, as befe.