Seminar 4: Hotelling 2
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1 Seminar 4: Hoelling 2 November 3, Exercise Par 1 Iso-elasic demand A non renewable resource of a known sock S can be exraced a zero cos. Demand for he resource is of he form: D(p ) = p ε ε > A a known fuure dae = s a compeiively supplied perfec subsiue becomes available. Is producion causes consan uni coss c. 1.1 Compeiive marke resource marke (1) Derive he ime pahs for resource exracion and price when he resource marke is perfecly compeiive and: (a) s = (b) s. I undersand he case (a) as ha he backsop is always here, so ha he inverse demand funcion has a kink a p = c, and I undersand he case (b) as ha he backsop is never here, so ha here is always some demand. In oher words, even as p, demand will approach, bu never acually reach zero. I is useful o sar deriving he price pah for his case. Backsop is never here The problem for a supplier of he non-renewable resource under compeiive marke condiions is: m x R,T p R e r d s.. R = Ṡ S R S T T given free 1
2 Solve by maximum principle, using curren-value Hamilonian H = (p μ )R. Necessary condiions for opimaliy include: H R = p = μ (A) μ = rμ H S μ μ = r μ T (= if S T > ) (B) From (B) we know ha μ = μ e r and hence p = p e r So we know ha he price is rising a rae r. Moreover, we know ha i will rise wihou bounds. Wha we don know ye is he iniial level from which i sars, p. In order o pin down he iniial price level, we can use he condiion ha he enire resource will be used up. Technically, his follows from he (C). Inuiively, i canno be opimal o leave somehing of value in he ground. Since here are no exracion coss and he price can rise wihou bound, every uni of he resource is valuable. So, we have: S = Now, use he marke equilibrium condiion: R = D(p ) = p ϵ Backsop from beginning S = S = S = p ϵ ϵr R d (1) p ϵ ϵr e ϵr = [p e r ] ϵ : [p e r ] ϵ d (2) (3) p = (ϵrs ) 1 ϵ (4) Le us now urn o he case when here is a backsop-echnology from he beginning. The firs-order condiions will be he same. Only now, here is an upper bound o he price ha he producer can ask. Due o he presence of he backsop echnology, he resource owners canno sell anyhing a a price above p = c. Again, (A) and (B) imply ha he price rises a rae r, and hence i will evenually hi he level c. Wha we have o find, is he poin in ime when his happens, and he (new) iniial level ha he price sars from. Bu firs, we should argue ha indeed he price reaches he level c a ime T when he las uni of he resource is exraced. I canno be opimal o exhaus he resource before he price has reached c (a higher price pah resuling in more profis is feasible). Bu i can also no be opimal o have resource sock remaining afer he price has reached c (hese resource unis can only be 2
3 sold a he price c, bu due o discouning, heir presen value is declining he laer hey are sold). Hence, as before S = R d (5) Noe however ha now T is finie. To pin down T, we can use he fac ha we know p T = c. Once we know T, we know p since: p = p e r = p T e r(t ) p T = p e rt = c p = ce rt So, again using he marke equilibrium condiion R = D(p ) and insering i in (5), we ge an equaion wih T as he only unknown variable: S = ce r(t ) ϵ d So, T and consequenly p are uniquely defined. 1 Wha is lef is o argue ha he iniial price p will be lower han in he case wih no backsop, as he same amoun of resource is supplied o he marke, only during a shorer ime, so a each poin in ime demand has o be more, and hence price has o be lower. 1.2 Monopolis Now urning o he monopolis, i will again be pedagogical o firs consider he case when he backsop is never here, and hen o see how he exisence of a backsop changes hings. The essence of monopolisic behavior is o ry o exrac rens by suppressing supply. However, in he non-renewable resource case, oal supply is given by he resource consrain S R d, and he only hing ha he monopolis can do is o shif supply over ime, from relaively price-elasic periods o relaively less price-elasic periods. 1 In fac, closed form soluions can be found: T = 1 ϵr ln ϵr c ϵ S + 1 3
4 backsop never here Formally, he monopoliss problem is: m x p,t s.. R = Ṡ p (R )R e r d (6a) (6b) S given (6c) S T (6d) T free (6e) R = p ε For a change, we will use he presen value Hamilonian H = p (R )R e r λ R. The corresponding firs order condiions include: (6f) H R = (p + p R )e r λ = H S = = λ (A ) (B ) Again (and for he same reasons) i is rue ha all resources will be used and ha exracion will go on forever. From insering he demand funcion in (A ) we ge: R 1 ε (p + p R )e r = λ (p + p R ) = (p + p R )e r 1 1 ε R ε 1 R = R 1 ε 1 1 ε R ε 1 R R 1 ε 1 1 ε = R 1 ε 1 1 ε e r e r R 1 ε = R 1 ε er p = p e r Looks familiar? The reason why he monopoly follows he sandard Hoelling rule is ha i can do no beer. Demand elasiciy is consan hence shifing is of no meaning and he monopolis is lef wih no power o play around wih. 4
5 backsop here from he beginning When he backsop is here from he beginning, he problem (6) remains he same, only condiion (6f) is changed o: p ε for p < c R = (, c ) for p = c for p > c The necessary condiions from maximizing he presen-value Hamilonian are: (6f ) H R = (p + p R )e r λ (< if R = c ε ) (A ) H S = = λ (B ) λ T (= if S T > ) H(T ) = p T R T e rt λ T R T = (ransvers.) Due o he changed consrain (6f ), he firs-order-condiion (A ) now feaures an inequaliy sign. For an inerior soluion we have as before ha he marginal revenue should rise a he rae of ineres. Bu due o he backsop, his is no longer feasible when he price would have o be above c. We have wo cases: (1) The iniial sock of he resource is so low ha he monopolis would wan o se an iniial price above c. As his is no feasible, he bes ha he monopolis can do is o sell all resource a price c, unil i is empy, so ha R = c ϵ. (2) The iniial sock is large so ha here will be an iniial phase where (A ) holds wih equaliy, followed by a phase where he price says a c and he monopoliss supplied his resource unil i is empy. Denoe he end of he firs phase, T Ho, marking he end of Hoelling s days. Hence here are now hree unknowns: p, T and T Ho. Sar by using (ransvers.). We know ha p T = c and hence R T = c ε >. We ge: c = λe rt (7) From (A ) we also know ha a T Ho i is he case ha: (p THo + p T Ho R THo )e rt Ho = λ (8) Insering for he demand funcion and he limi price: c 1 1 e rt Ho = λ (9) ε 5
6 Puing (7) ino (9) les us solve for he lengh of he second phase (T T Ho ): c = c 1 1 e rt Ho e rt ϵ e rt T Ho = ϵ ϵ 1 T T Ho = 1 r ln ϵ ϵ 1 (1) The oal amoun exraced during his second phase is: S m = (T T Ho )c ϵ Deducing his amoun from he iniial sock we can find he lengh of he Hoelling phase in he ususal way by: S S m = = = Ho Ho Ho R d p ϵ d ce r(t Ho ) ϵ d (11) Hence, we have wo equaions for he wo unknowns T (equaion 1) and T Ho (equaion 11) and since p = ce rt Ho, he problem is solved. The iniial price wih a backsop will be higher han when he monopolis is unconsrained: As such a maximal price depresses he monopolis s fuure price, some of his price reducion may be compensaed for by raising he he price early in he period of exracion (Hoel, 1978, p.35). The inermediae case: he backsop appears Skech of soluion: Demand from he monopolis s viewpoin akes he form: R = p ε p ε for s for p c > s for p > c > s Similarly o 2(a), ha means here is an addiional consrain R c ε, bu now only for > s. How could we solve his? In wo seps. Firs we solve he problem from = s onwards. This is almos he same as in 2(a), excep ha we do no ye know how much of he resource here is iniially (S s ). So we solve he problem condiional on some unknown S s and ge a funcion V(S s ) describing wha any size of sock is worh o he monopolis a = s. 6
7 V(S s ) = m x p,t s p (R )R e r d (15) As argued above, i migh eiher be ha here is only a limi pricing phase (S s small ) or ha here is a Hoelling phase followed by a limi pricing phase. In he former case we have ha marginal revenue a = s is equal o c, in he laer o p e rs (1 1 ε ). Nex hen, we solve he fixed horizon problem unil = s aking V(S s ) ino accoun. V(S s ) is a so called scrap value. m x p s s.. R = Ṡ S S s p (R )R e r d + V(S s )e rs given R = p ε The presen value Hamilonian looks as usual. Bu we ge an addiional condiion for he scrap value, namely: λ s V (S s ) (= V if S s > ) (16) This is quie naurally inerpreed: on he LHS we have wha an addiional uni of he resource would be worh if used a s( ), on he RHS wha i is marginally worh if ransferred and hose should be equal if anyhing is ransferred. Oherwise, i mus be beer o use everyhing now. We know from he FOCs ha λ s = λ = (p + p R )e r. The envelope heorem ells us ha V (S ) = μ, where μ is he shadow price on he resource for he second problem s ( > s) which again from he FOC we know o be μ (p + p R )e r. Informally, wo hings can happen. (I) I is opimal o leave so much resource ha when = s( ), p s( ) < c and hence we have he same soluion as in 2(a) ha he consrain comes laer does no maer, here is nohing o gain from shifing producion o earlier periods. The second case (II) is p s( ) > c. In ha case, we mus have p s(+) = c Here we can ge wo sub-cases: (IIA) suppose p s (1 1 ) > c. Then marginal revenue ε righ before we reach = s is higher han righ afer here is no poin in exracing anyhing afer ha poin in ime. Or we ge (IIB) ha c(1 1 ε ) < p s(1 1 ) < c. In ha ε 7
8 case marginal revenue righ before = s is lower han righ afer and hence exracion coninues and price jumps down o c from somehing c < p s < c(1 1 ε ) 1 2 Exercise Par 2 Polluion This problem is very vague and specifies no model. Hence, do i yourself! The easies would be o ake he case of no exracion cos and a choke price b. The main poin wih quesion (1) is o recognize ha he producers will no ake he exernaliy ino accoun and hence produce as if here was no polluion. Since Ṗ = R and R d = S we will have ha P T = S. All of he resource will be exraced and end up as polluion. The main poin wih (2) is o recognize ha avoiding P o go beyond P essenially means prohibiing ha he enire resource is exraced, bu leaving S = S P in he ground. This can e.g. be achieved by levying a ax of a leas he size of he presen value of he resource ren a he ime when P has reached P. The effec on he price pah is he same as a reducion of he iniial reserve size: The new price pah sill grows a he rae of ineres, bu from a higher saring poin, and exracion ends earlier. Using he model wih sock depending coss makes your life a lo harder. An good discussion of his case can be found in Hoel (212, secion 3 and 4). References Hoel, M. (1978). Resource exracion, subsiue producion, and monopoly. Journal of Economic Theory, 19(1): Hoel, M. (212). Carbon axes and he green paradox. In Hahn, R. and Ulph, A., ediors, Climae Change and Common Sense: Essays in Honour of Tom Schelling. Oxford Universiy Press. 8
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