INTERTEMPORAL PRICE CAP REGULATION UNDER UNCERTAINTY By Ian M. Dobbs The Business School University of Newcastle upon Tyne, NE1 7RU, UK.

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1 INTERTEPORAL PRICE CAP REGULATION UNDER UNCERTAINTY By Ian. Dobbs The Bsiness Shool Universiy of Newasle on Tyne, NE 7RU, U. Eqaion Seion ADDITIONAL NOTES AND DERIVATIONS The blished aer omis deailed derivaions. These addiional noes give fll derivaions for all he reored resls. The following are very mh se by se derivaions, so eah se shold be easy and qik o follow. A. The Prie Proess The rie roess, on ime inervals on whih here is no enry/invesmen is given by he demand fnion: Q η A η (A. Wrie f ( A, Q Q η A η so f ( A, Q (A. Alying Iô s lemma, d f. da + f. dq + f da + f dq + f da dq (A.3 where A Q AA QQ AQ f ( A, Q η / A (A.4 A A (, / A η η f A Q + A (A.5 f ( A, Q η / Q (A.6 Q fq (, / Q A Q η η Q f A (, / Q A Q η Q A (A.7 (A.8 whils (sing he Iô rles, we ge da α Ad + σ A dϖ (A.9 da α Ad + σ A dϖ σ A d (A. dq θq d (A. dq θ Q d (A. and da dq. (A.3 Hene d f. da + f. dq + f da + f dq + f da dq A Q AA QQ AQ ( Q dq ( η Q A da dq ( A σ A d ( η / A da + ( η / Q dq + η( η + / A da + η( η / + / ( η / A α A d + σ A dϖ + ( η / Q θ Q d + η( η + / (A.4 Original Draf: /8/ Las daed: 4/7/3

2 Noes d η αd + σ dϖ θη d + η η + σ d (A.5 ( d η α + θ η + σ d ησ dϖ (A.6 Wriing µ ( η α + θ η + σ (A.7 hen his beomes d µ d ησ dϖ (A.8 whih is eqaion (5 in he aer. A Derivaion of he fndamenal differenial eqaion for vale: The arbirage eqaion was (reroded here for onveniene: θ + r vd d + E dv (A.9 Iô s lemma gives dv v d + v d + v d + v d + v dd (A. Now, v ψ ( x where x / (A. so v ψ ( / ψ ( / (A. ψ ( / v (A.3 v ψ ( / ψ ( / ψ ( / ψ ( / (A.4 v ψ ( / + ψ ( / + ψ ( / ψ 3 ( / 3 (A.5 v / ψ / (A.6 From (A.8 d µ d ησ dϖ (A.7 d µ d ησ dϖ η σ d (A.8 Sine d δ d i follows ha d. Hene he erm dv is given as dv v d + v d + v d + v d + v d + v dd ( / ψ ( / ( / ψ [ ] d + ψ ( / d + d + ψ ( / ψ ( / + ψ ( / 3 d d + dd and sbsiing frher (and abbreviaing he noaion a lile (A.9

3 ψ dv ψ d + d + ψ ψ d + ψ d + ψ dd ( / ψ µ ησ ϖ + ( η σ + ψ ψ ( δ ψ ψ d d d d dv ψ µ d ησ dϖ + η σ d + ψ ψ δ d Ths aking exeaions, η σ E( dv ψ + ( µ + δ ψ δ ψ d ψ η σ µ ψ δ ψ δ ψ d [ ησ ψ ] d ϖ θ + ψ + µ + δ ψ δ ψ (A.3 (A.3 Hene he arbirage eqaion beomes η σ ( r vd d ( d (A.3 η σ θ + r ψ ψ µ + δ ψ + δ ψ (A.33 ( ( σ x ψ + x µ + δ ψ θ + r + δ ψ + x (A.34 Noes 3 whih is eqaion (A. in he aendix o he aer. The general solion o (A.34 an be wrien as he sm of he general solion o he homogeneos eqaion ( σ x ψ + x µ + δ ψ θ + r + δ ψ (A.35 and a arilar solion o (A.34. A arilar solion, easily verified, is where ψ Bx (A.36 B ( r ( θ + r µ θ + + η α + θ η + σ, (A.37 and he general solion o he homogenos eqaion (A.35 an be wrien as ( x B x ψ + B x. (A.38 where he roos are defined as and where ( R R η σ + (A.39 / ( R R η σ, (A.4 / ( R µ + δ η σ. (A.4

4 Noes 4 R ( µ δ η σ η σ θ r δ (A.4 Hene he solion o (A.34 akes he form ( x B x B x ψ + + B x. (A.43 Noie ha η σ ( θ r δ + + > if σ >, so he roos are real and of oosie sign when nerainy is resen. The arbirary onsans B, B are deermined by bondary ondiions. As x, if vale is o be finie, i ms be ha B or he solion exlodes (see Dixi, 993. The oher onsan is deermined by an analysis of smooh asing ondiions a he bondary (a whih new invesmen is riggered. A3 Analysis of smooh asing ondiions Given B, from (A.43, he solion is ψ ( x Bx + B x (A.44 where B is given by (A.37. (a The Comeiive ase: In his ase i was esablished ha he smooh asing ondiions are ψ ( (A.45 and ψ (. (A.46 From hese eqaions we ge ψ ( B + B, (A.47 ( B B ψ + B B /, (A.48 and hene ( ψ ( B, (A.49 ( r. (A.5 θ + µ whih is Resl (i. (b The monooly ase In he monooly ase, invesmen ommenes a a ime a whih rie reahes he level, where is he relaive rie a whih new aaiy is added. Sine is a free hoie by he firm, smooh asing involves firs and seond derivaive ondiions (see Dmas, 99. The firs derivaive ondiion is ha, wih rese o he onrol variable, he rae of hange of vale shold js eqal he rae of hange of os; V (,, Q / ( Q / (A.5 where V (,, Q ψ ( x Q, Q A γ and / x. Sbsiing hese ino (A.5 gives

5 !, (A.5 γ { [ ψ ] } A ( / / γ γ [ ] + whih gives γ A ψ ( / A ψ ( / (/ γ A { γ [ ψ ( / ] + ψ ( / } γ ψ ( x + x ψ ( x (A.53 (A.54 [ ] γ [ ψ ] + ψ ( (A.55 The seond derivaive ondiion is V (,, Q / Q / Noes 5 ( (A.56 { } (A.57 { γ ψ ψ ψ } γ { γ ψ ( x + x ψ ( x } + x {( + γ ψ ( x + xψ ( x } (A.58 γ γ ψ ( x γ x ψ ( x (A.59 + ( + γ x ψ ( x + x ψ ( x whih gives γ γ A γ ψ ( / + ψ ( / [ ] γ [ ] ( [ ] + ( + A ( / (/ + ( / + ( / (/ { } [ ] { } γ γ ψ ( + γ ψ ( + ( + γ ψ ( + ψ ( (A.6 [ ] ψ + γ ψ + γ ( γ [ ψ ] γ γ ψ + γ ψ + + γ ψ + ψ ( ( ( ( (A.6 Now, from (A.44, ψ ( x Bx + B x (A.6 so ψ ( x B B x + (A.63 ( x ( B x (A.64 ψ so, sing (A.6-(A.64, eqaion (A.55 gives γ [ ψ ( ] ψ ( " # + $ γ B + B + ( B + B " ( + γ B + B ( + γ γ (A.65 % γ ( + γ B B (A.66 + γ &( ' &( ' and, from (A.6, again sing (A.6-(A.64, ( B + γ B + B + γ ( γ B + B * & ( ' & ( B γ B γ B γ ( γ ( ' (A B + B + B ( + γ + γ γ + γ + γ γ B γ γ (A.68 { } { }

6 @ ( + γ + Sbsiing for B sing (A.66, his gives γ ( + γ B + { γ + γ ( γ } B γ ( γ (A.69 ( γ + γ ( γ { } ( γ ( + γ B ( + γ + γ ( γ + { γ + γ ( γ }( + γ B γ ( γ ( + γ B ( ( ( + γ ( + γ + γ + γ γ γ + γ γ + γ + γ γ + γ + γ γ + γ { } γ ( + γ ( ( γ γ ( γ γ γ ( γ + γ + γ γ (A.7 (A.7 + γ B γ (A.7 γ + γ ( ( γ + γ B γ (!# "# %&$! " #' (# ' ( % (A.73 γ γ + γ + γ ( γ ( γ B γ ( γ + ( + + (A.74 ( + γ B γ (A.75,* -/.+* -. Sbsiing for B sing (A.37, his gives γ, + ( r (A.76 θ + µ + γ whih is Resl (ii. 43 Noes 6 A4 Relaionshi beween alernaive oion vale mliliers (omeiive ase Sar wih he original resl (A.5 ha ( r + θ µ (A.77 9 : > <?;> <? < >? σ σ σ 9 : >? 9 :;9 : R + R R R R R 4 and noe ha, from (A.39-(A.4 ha and ( θ + r + δ < σ θ r δ σ (A / (A.79

7 @ Noes 7 Wrie R + R R R R + + σ σ σ ( µ δ σ + σ (A.8 ( µ + δ σ ( ( +. (A.8 ( r + θ µ ( r + θ + δ ( δ + µ (A.8 so, from (A.79 and (A.8 r + θ + δ δ + µ ( ( σ σ ( ( + ( σ (A.83 (A.84 σ From (A.79 σ θ + + δ (A.85 so r ( r (A.86 θ + + δ Tha is,!! "# ( r " # ( r (A.87 θ + + δ + θ µ $&% '( $% ' ( whih is reored in he aer in eqaions (, (3 and resl (i. Hene also, from (A.76, γ &* * ( r. (A.88 θ + + δ + γ A5 The limiing ase: Under erainy, he erainy relaive rie a whih enry akes lae is (,. θ + r + ax η α + θ, δ or eqivalenly, ha (a if δ η( α θ < +, ( θ r η( α θ (b if δ η( α + θ, hen θ + r + δ. This seion shows ha as σ. Tha is, wriing Lim Lim σ, ha Lim Lim θ + r + η ( α + θ ( η + σ ( (A.89 [, ] θ + r + ax η α + θ δ

8 Noes 8 Define where σ σ φ σ ( σ r ( (A.9 φ σ θ + + η α + θ η + σ (A.9 Clearly, from (A.9, Lim φ σ θ + r + η α + θ (A.9 Case (a δ < η( α + θ Firs, examine Lim and Lim (. From (A.39-(A.4, sbsiing for µ sing (A.7: µ + δ µ + δ η σ ( θ r δ η σ η σ ( r η σ η ( θ + α + η ( η + σ η ( θ + α + η ( η + σ + + η σ θ + + δ + δ η σ + δ η σ η σ (A.93 ( ( δ η ( θ α ( ( r δ η θ + α + ησ + δ η θ + α + ησ + η σ θ + + δ η σ (( δ η ( θ α ησ η σ θ r δ η σ η σ (A.94 Now, if δ < η( α + θ hen ( δ η ( θ α + > (A.95 and, aking he limi in (A.94, as σ so + (whaever haens o he oher erms, hey are non-negaive, and he firs erm +. Ths, if δ η ( θ + α > hen Lim Lim (A.96 ( ( and hene Lim Lim Limφ θ + r + η( α + θ (A.97 Case (b δ > η( α + θ Then δ η θ + α <, (A.98 (

9 Noes 9 and behaves raher differenly in he limi. To see his, define ( ( r f ( σ δ η θ + α + ησ + δ η θ + α + ησ + η σ θ + + δ and so ha (A.99 g( σ η σ (A. ( σ f ( σ / g( σ (A. Now, when (A.98 holds, hen f ( and g ( so he limi for ( σ is no immediae from (A.. However, i an be obained by alying l Hoial s rle; hs g ( σ η and (( ( r ( δ η ( θ + α + ησ η + η ( θ + r + δ f ( σ η + δ η θ + α + ησ + η σ θ + + δ so ha and Lim g σ (A. η (A.3 Lim f ( σ η + ( η ( δ η ( θ + α + η ( θ + r + δ δ η ( θ + α ( ( ( r ( δ η ( θ + α η δ η θ + α + η δ η θ + α + η θ + + δ ( r ( δ η ( θ + α ( + r + ηδ + η θ + α + ηδ η θ + α + η θ + + δ η θ δ ( δ η θ + α Hene, by l Hoial s rle f Lim f Lim ( σ Lim g Lim g ( r ( σ ( σ ( σ ( σ η θ + + δ θ + r + δ + ( δ η θ + α η δ η ( θ α Ths given Lim ( σ exiss, i follows ha Lim Lim ( ( Lim θ + r + δ δ η ( θ + α θ + r + δ θ + r + δ θ + r + η ( θ + α δ η θ + α (A.4 (A.5 (A.6

10 Noes Hene σ φ Lim Lim Lim ( θ + r + δ θ + r + η θ + α ( θ r δ + + ( θ r η ( θ α This omlees he derivaion for (A.89 and hene resl in he aer. (A.7 A6. onooly firm sbje o ineremoral rie a. Here ha denoes he demand rie (he rie whih lears he marke; ha is, sh Q A (A.8 η η However, he firm is reqired o se a rie whih saisfies he rie a ; s s denoing he se rie as, hen his reqires ha. Ths s in[, ] in[, ]. (A.9 where is a onsan hosen by he reglaor. For low vales of he relaive rie x, he monoolis ilises exising aaiy, whils he rie is nonsrained. One x reahes a erain level, he rie a binds, b invesmen may sill be deferred (so here is qaniy raioning. Finally, if x inreases sffiienly, i will be oimal for he firm o add o aaiy. There is hs a ransiion bondary beween he noinvesmen regimes, and a frher bondary a whih invesmen ommenes; a eah of hese bondaries, smooh asing ondiions aly. Le ψ denoe he solion when here is no invesmen and no rie onsrain and le ψ denoe he solion when he rie onsrain alies, b here is no invesmen. The firs ask is o haraerise he roess in eah of he regimes. Following his, smooh asing ondiions are sdied. Regime : Unonsrained rie, no invesmen. The solion here is idenial o ha already esablished for he nonsrained monooly firm. Tha is, from (A.34, reeaed for onveniene, σ x ψ + ( µ + δ xψ ( δ + θ + r ψ + x (A. and he solion is ( x B x B x ψ + + B x (A. where B is given by (A.37. As before, noe ha, as x, if vale is o be finie, i ms be ha B. Dixi (993 dissses his sor of bondary ondiion in more deail. Hene he solion beomes ψ ( x B x + B x (A.

11 . Noes Regime : Unonsrained rie, no invesmen. The arbirage ondiion in his ase is, sine in his regime, ha θ + r vd d + E dv (A.3 where v ψ ( /. Alying Io s lemma, aking exeaions and simlifying, ( (A.4 σ x ψ µ δ xψ ( δ θ r ψ A arilar solion is learly ψ ( δ + θ + r (A.5 and so defining C /( θ + δ + r, (A.6 he general solion in his regime o akes he form ψ ( x C + C x + C x (A.7 where, are as before. The arbirary onsans C, C are deermined by a onsideraion of bondary ondiions (see below. Analysis of bondary ondiions: Le denoe a ime a whih here is a ransiion from a regime of nonsrained ries and no invesmen o a regime where ries are onsrained b here is sill no invesmen. Le denoe a ime where here is a ransiion from rie onsrained non-invesmen o a regime in whih here is rie-onsrained osiive invesmen. Regime / bondary: A he ime, smooh asing ondiions ms aly. Sine he roblem is no one in whih here is a free hoie for he level of he bondary (i is se by he reglaor, he smooh asing ondiions involve eqaliy only of vale and he firs derivaive of he vale fnions (Dmas, 99. Also, by definiion, he rie a binds, so x Ths v( v( and v ( v ( ms hold, and so ψ ( ψ (, (A.8 ψ ( ψ ( (A.9 where ψ ( B + B (A. B + B (A. ψ ψ ( C C C + + (A. C C ψ + (A.3 The analysis of hese ondiions is deferred nil he oher ondiions have been idenified. The osiive Invesmen bondary: A he hiing ime, smooh asing ondiions again aly. These are idenial wih hose for he nonsrained monooly roblem exe ha hey are now evalaed a he relaive rie. The firs derivaive ondiion is ha, [ ] γ ψ ( + ψ ( (A.4 The vale fnion an sill be wrien in his form in he rie onsrained region, simly bease he rie onsrain is also homogenos of degree in,.

12 Noes - omare wih (A.55 and he seond derivaive ondiion is ha. ψ ( + γψ ( + γ γ ψ ( (A.5 [ ] as in (A.6. Here C C ψ + + C (A.6 C C ψ + (A.7 ( C + ( C (A.8 ψ Analysis of smooh asing ondiions: From (A.8, sing (A. and (A., B + B C + C + C (A.9 Unknowns are B, C, C. From (A.9, sing (A. and (A.3, and mlilying by, B B C C + + (A.3 Unknowns are B, C, C. (A.3 From (A.4, sing (A.6 and (A.7, γ C + C + C + C + C Unknowns are C, C,. (A.3 From (A.5, sing (A.6, (A.7 and (A.8, ( C + ( C + γ C + C + γ γ C + C + C Unknowns are C, C,. Toal nknowns are B, C, C,. However he variable of ineres is. To find his, firs eliminae B. Eqaion (A.9 imlies B C B + C + C (A.33 whils from (A.3 C + C B B (A.34 C + C B B ( C B + C + C C + C B ( C B + C + C C + C B ( C B + C + C ( C B B C + (A.35

13 ' Noes 3 This gives C. I does no deermine B, C b links hem bease B C C B ( C B + B B C B ( B C ( B C ( + ( ( C B B B ( C B B + Tha is, one C is esablished, so oo is B. Now se he osiive invesmen bondary ondiions: from (A.4-(A.8, γ C + C + C + C + C and (A.36 (A.37 (A.38 ( C + ( C + γ C + C + γ γ C + C + C Firs find C from (A.37: γ ( C γ + C C γ + Then se his in (A.38: ( C + ( C + γ C + γ C C C C + γ γ + γ γ + γ γ γ γ ( + γ + γ γ C + ( + γ + γ γ C + γ γ C (A.39 (A.4 # $ # $ # $ ( + γ ( γ C * + Sbsiing for C sing (A.39, % & γ ( C γ + C ( + γ + γ γ + ( + + ( C γ γ γ * + (* + ( γ + { C } C ( + γ + γ γ γ ( γ + C + γ + + γ + γ γ C + γ γ γ +

14 ' ' ' & A ( C ( + + ( ( + C ( + + ( ( γ ( γ γ ( γ C γ ( γ ( γ C γ γ γ γ γ γ γ γ ( γ + ( + γ + γ ( γ ( γ + ( + γ + γ ( γ C γ γ γ + C γ ( C ( + γ + γ ( γ ( C {( ( + ( ( } γ γ γ γ γ γ C Now, from (A.35, ( C B + B C so ( γ + ( + γ + γ ( γ ( γ + ( + γ + γ ( γ ( C ( ( ( + ( ( " ( γ ( γ γ ( γ % # ( C B & # B # ( γ + ( + γ + γ ( γ * { } γ γ γ γ γ γ! $ % &$ $ $ Noes 4 (A.4 (A.4 (A.43 where from (A.6, C /( θ + δ + r (A.44 and, from (A.37, B r + θ µ (A.45 / Simlifying (A.43, 6 γ ( C ( ( + γ. ( γ + ( + γ + γ ( γ (. ( γ + ( + γ + γ ( γ - +,. / / C + B / 4 5/ : > ;? γ ( C ( ( + γ ( γ ( γ ( ( γ + γ ( + ( C + B + γ γ + + γ γ

15 Noes 5 ( C γ ( + γ ( γ γ + ( γ γ + C + B ( + ( γ + γ + γ ( γ ( + ( γ γ γ ( C γ ( + γ γ γ + γ + γ + C + B ( + γ + γ + γ γ γ + γ ( C γ γ γ γ ( + γ γ γ + C + B ( + γ γ γ ( C ( ( + γ #! " $ ( C + B ( + + ( + {} γ γ ( C γ ( + γ #! " $ (A.46 C + B γ + + γ + { } ( The aim now is o simlify his o obain he formla in resl 3. Denoe δ + θ + r (A.47 as he erainy omeiive enry rigger relaive rie, ( ( r ( ( r ( δ + θ + θ + µ (A.48 as he nerainy omeiive enry rigger relaive rie and γ ( + γ (A.49 as he nerainy nonsrained monooly enry rigger relaive rie, hen he aim is o show ha he rie a monooly relaive enry marke learing rie an be wrien as in resl 3; ha is, as % % & & / '' ' * ( * ( (. (A.5 So, rerning o (A.46, and dividing by C gives γ ( + γ (( / C ( B / C ( + -,.. (A.5 + γ + + γ + {} Dividing by ( and sbsiing for B, C hen gives

16 ! Noes 6 ( θ + δ + r ( ( ( r + θ µ ( γ ( + γ {( γ γ ( } ( r + ( + { } ( r + θ µ θ + δ + r γ + + γ + γ θ µ γ Using (A.47 and (A.48, his gives ( γ + + γ + ( γ ( + γ so ( γ ( + γ + ( + γ ( ( ( ( + η γ ( + γ (where η / γ. Wriing ( + η, hen ( ( + η ( (( + η ( + η " $! %# + η + η Hene (A.5 (A.53 (A.54 (A.55 ( ( " η η!& #("!# $ % " '! " $'% # # " $ %# + η + η + η + η " η + " # + η + η + η # whih is Resl 3. ( " ( η η # $ % + + η + * ( - + η + η ( ( /. ( / (A.56

17 Noes 7 A7 Proof for Resl 4. From he formla for ( in Resl 3 was {( ( } / (A.57 ( / Differeniaing wih rese o gives d (( (A.58 d ( d d / where, sing he definiion for, d d (( ( ( d d ( (A.59 so hene ( / $!!$%$!!$%"!#" (A.6 ( d d ( + ( + ( + d > d < as > <. (A.6 This omlees he roof for resl 4(i. As in eqaion (A.57, he erm in brakes { } ; sine / <, i follows ha +, whih is resl 4(ii. Leing in (A.57, learly (, whih is resl 4(iii. Seing, from (A.57, { } (A.6 Now, ( > < as ( < > (sine <. Using (A.6 his imlies ( > < as ( ( < > & & ( < ( > > < In fa > and hene ( >, whih is Resl 4 (iv. To esablish Resl 4 (v, firs sbsie in (A.6 sing resl, for ( / ( / ( η and he definiion for, noe ha Lim σ + o ge ( / η, and so ( Limσ +. From resl 3, Lim σ. Also Limσ ( η Lim ( η / Hene Lim ( Lim ( η Lim whih is Resl 4 (v. σ σ σ. Hene from / / (A.63