a) No constraints on import- export, no limit on reservoir, all water in the first period The monopoly optimisation problem is:

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1 Monopoly and rade Monopoly conrol impors, bu akes expor price as given. a No consrains on impor- expor, no limi on reservoir, all waer in he firs period he monopoly opimisaion problem is: Max p ( x x + p e s.. = = x = e e, e W, p = given > 0 ( ere p is he expor/impor price (prices are equal; no ransmission cos and e is expor if posiive and impor if negaive. Insering he energy balance ha holds as an equaliy yields he Lagrangian: L= p ( e e ( e e + p e = λ( e W = he necessary firs-order condiions are: = p ( e e ( e e + p ( e e λ 0 e 0 = p ( e e ( e e p ( e e + p = 0 λ 0( = 0 for e < W = We assume ha he amoun of elecriciy is posiive in all periods and ha he expor/impor prices are all differen. Since here is an expor opporuniy o a posiive price waer will no be wased and he shadow price on waer will be posiive. If hydro is used in an impor period hen he firs condiion in (3 holds wih an equaliy implying ha he flexibiliy-correced home marke price, p ( + η, is equal o he shadow price on waer. he second condiion ells us ha he flexibiliy-correced price is always equal o he impor price. Bu since he expor/impor prices are differen he shadow price on waer can only be deermined by one flexibiliy- (2 (3

2 2 correced price. We know ha in an expor period we mus also use hydro a home. herefore in an expor period he flexibiliy-correced price is also equal o he shadow price on waer. I is he highes expor price period ha will become he only expor period, and in all oher periods here will be impors and no use of hydro a home. his means ha in impor periods he flexibiliycorreced price is less han he shadow price on waer. An illusraion is provided in Figure. Since he impor price is lowes in period his period will be he impor period. he amoun of impor is deermined by he inersecion of he marginal revenue curve and he impor price line. he home marke price will be higher han he impor price in he sandard way of a monopoly. Impor may be regarded as an alernaive way o using hydro of producing elecriciy. In he expor period he use a home of hydro is deermined by he inersecion of he marginal revenue curve and he expor price line. Expor is residually deermined as he res of he available waer. he shadow price of waer is equal o he expor price. Period 2 Period λ p p ( + η p ( e p ( e p ( + η p Impor e Expor oal available waer e 2 Figure. Monopoly and rade wihou resricions

3 3 Comparing he monopoly soluion wih he socially opimal soluion he laer is indicaed by he verical broken lines. he impor and expor periods will be he same. he amoun of impor is deermined by he inersecion of he demand curve and he price line, and he amoun of hydro used a home in he expor period is deermined by he inersecion of he demand curve in period 2 and he price line for period 2. he shadow price on waer will be he same in he wo soluions, bu impor will be reduced in he monopoly case resuling in a higher home price han he impor price. In he expor period he monopoly will expor more waer and resric correspondingly he use of waer for elecriciy producion a home resuling in a home price higher han he expor price. he monopolis is playing price discriminaion beween wo markes. b Monopoly and rade wih an upper resricion on rade he monopoly opimisaion problem in he case of resricions on rade is: Max p ( x x + p e s.. = = x = e e, e W, e e e, p = given> 0 Insering he energy balance ha holds as an equaliy yields he Lagrangian: L= p ( e e ( e e + p e = = = λ( e W = α ( e e β ( e e he necessary firs-order condiions are: (4 (5

4 4 = p ( e e ( e e + p ( e e λ 0 e 0 = p ( e e ( e e p ( e e α + β + p = 0 λ 0( = 0 for e < W (6 = α 0( = 0 for e < e β 0( = 0 for e < e Since we eiher have impor or expor in a period he shadow prices on he upper and lower consrain canno boh be posiive a he same ime, bu hey may boh be zero if he consrains are no binding. We assume ha he amoun of elecriciy is posiive in all periods and ha he expor/impor prices are all differen. If hydro is used in an impor period hen he firs condiion in (3 holds wih an equaliy implying ha he flexibiliy-correced home marke price p ( + η is equal o he shadow price on waer. he second condiion ells us ha he flexibiliy-correced price is always equal o he impor price plus he shadow price on he upper consrain on impor. If he upper consrain is no binding hen he shadow value on waer is deermined by his impor price. We can herefore a mos have only one impor period wihou he consrain binding where hydropower is also used in he home marke. We know ha in an expor period we mus also use hydro a home. herefore in an expor period he flexibiliy-correced price is also equal o he shadow price on waer. he second condiion in (6 ells us ha he flexibiliy-correced price is equal o he expor price minus he shadow price on he expor consrain. I will be arbirary if expor in each period of expor is exacly equal o he consrain. In general here will herefore be a period when he expor possibiliy is no fully uilised. We will call his period he marginal expor period. Bu in his period he shadow price on waer is equal o he expor price. Expor prices mus be higher han impor prices. herefore we can conclude ha when we have a marginal expor period, hen i is no opimal o use waer in an impor period unless he impor consrain is binding wih a posiive shadow price on he impor consrain. his means ha in impor periods he flexibiliy-correced price is less han he shadow price on waer.

5 5 An illusraion is provided in Figure 2. Since he impor price is lowes in period his period will be he impor period. he original bahub walls are drawn wih solid lines. Boh impor- and expor- capaciies will be fully uilised. Since he impor/expor price is lowes in period his will be he impor period. he impor capaciy is added o he hydro wall o he lef and marked wih he broken verical line. he demand- and marginal revenue- curves are shifed o he lef and anchored on he impor wall. In he expor period 2 he hydro wall is shifed o he lef wih he lengh of he expor consrain marked wih he broken verical line o he lef of he hydro wall. his amoun will be expored. he demand- and marginal revenue- curves are shifed o he lef wih he lengh of he expor consrain and anchored on he broken verical line. he flexibiliy-correced prices are equal and equal o he shadow price in waer. he home price becomes higher han he expor price in he expor period, and he home price becomes higher han he impor price in he impor period. he connecion beween he shadow price on waer, he impor/expor prices and he shadow prices on he rade consrains are shown in he figure. Period Period 2 p ( e p ( e p p ( + η α 2 p Impor e p ( + η e e 2 β Expor e 2 λ p e e2 oal available waer Figure 2. Monopoly and rade wih consrains

6 6 c Monopoly wih rade- and reservoir-consrains he monopoly opimisaion problem in he case of resricions on rade and reservoir is: Max p ( x x + p e s.. = x = e e, e e e, R R + w e, R R, p = given> 0 (7 ere R is he waer (elecric energy in he reservoir, w he inflow and R is he capaciy limi. Insering he energy balance ha holds as an equaliy yields he Lagrangian: L= p ( e e ( e e + p e = = = = = λ ( R R w + e γ ( R R α ( e e β ( e e he necessary firs-order condiions are: = p ( e e ( e e + p ( e e λ 0 e 0 = p ( e e ( e e p ( e e α + β + p = 0 = λ + λ+ γ 0 R 0 R λ 0( = 0 for R < R + w e γ 0( = 0 for R < R α 0( = 0 for e < e β 0( = 0 for e < e he change form he previous case wihou reservoir resricion is ha he waer values are now period specific. wo consecuive waer values are conneced hrough he value of he shadow price on he reservoir consrain, as seen from he hird condiion in (9. he possibiliy of (8 (9

7 7 overflow may resric impor of elecriciy since waer is used unil he marginal revenue becomes zero if ha is necessary o avoid overflow. In expor periods home price may be driven furher up because here is a limi on he ransfer from he previous period. If he reservoir consrain does no become binding we are back o he soluion wihou a reservoir consrain. A bahub illusraion for wo periods is provided in Figure 3. Since he impor price is lowes in period his period will again be he impor period. he figure is based on Figure 2. Inflow o he reservoir in period is AC and in period 2 CD. he size of he reservoir is BC, indicaed by R, and he hinner solid verical lines from B and C represens he reservoir. he reservoir is inroduced from C o he lef o B because our problem for wo periods is how much waer o leave o period 2. he impor consrain is indicaed o he lef of he hydropower wall drawn wih a solid line. In our case he full impor capaciy will no be uilised. Bu he full expor capaciy will be used, and his capaciy is indicaed by he firs broken line o he lef of he righhand hydro wall drawn wih a solid line. he final layou of he figure is he resul of wo sages for he wo periods curves. In he firs sage he demand-and marginal revenue- curves are anchored o he hydropower walls. he opimaliy condiions for he impor period ell us ha he marginal revenue curve should pass hrough he inersecion beween he impor price line and he hydro wall hrough B. he demand- and marginal revenue curves are shifed horizonally o he lef o allow his, indicaed by he arrow, and he sopping poin is where he impor wall is ereced. If more impor is ried he marginal revenue will become smaller han he impor price. A leas waer AB has o be used in period, and he marke price maching his amoun is higher han he impor price. herefore impor is inroduced unil he marginal revenue is equal o he impor price. Remember he analogy beween impors and anoher echnology for producing elecriciy. he final marke price is found he usual way of moving up o he demand curve. Since he impor capaciy is no fully uilised he shadow price β on his capaciy is zero. he waer value becomes equal o he impor price for his period. he maximal amoun of waer is ransferred o period 2. Checking period 2 here in he firs sage is enough waer o uilise he expor capaciy fully. he verical broken line o he lef of he hydropower wall hen indicaes he reduced availabiliy for

8 8 hydropower a home, and he demand- and marginal revenue-curve are shifed horizonally o he lef and anchored o he new wall indicaed by he arrow. he inersecion of he hydropower wall hrough B and he marginal revenue curve for period 2 hen gives he waer value for period 2. he home price is found by he inersecion of he hydropower wall and he demand curve. he shadow price on he reservoir capaciy is he difference beween he wo periods waer values and is indicaed in he figure. Since he expor capaciy is fully uilised he shadow price is posiive and indicaed as he difference beween he expor price and he waer value for period 2. Comparing he monopoly soluion wih he social soluion we have ha period 2 price and quaniy on he expor marke and he home marke will be he same. he monopolis canno earn higher profi han in he social soluion. Bu he shadow prices α 2, λ 2 will be differen. he shadow price on waer is se equal o he expor price, and he shadow price on he expor consrain is equal o he difference beween he home price and he expor price. Bu in period he soluion will be differen for impors and he home price. he social planner will impor unil he impor price plus shadow price on he impor consrain becomes he home price. For he siuaion in Figure 3 he social planner will uilise he full impor capaciy, i.e. he firs sage horizonal shif of he demand curve o he lef will coninue o he impor resricion is reached. he home price will become lower han he monopoly home price, bu sill greaer han he impor price. he difference is he shadow price β on he impor consrain. he waer value in period will be he home price. Finally, he new shadow price on he reservoir consrain is he difference beween he waer values, i.e. he difference beween he marke prices.

9 9 Period Period 2 p ( e p ( e p ( + η α 2 λ 2 p p ( + η γ λ = p e e e 2 e 2 p Impor capaciy A e B R C e D 2 Inflow period Inflow period 2 Figure 3. Monopoly and rade-and reservoir consrains

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