1. seminar 41 ECON 4925 Autumn 2006 Resource Economics Seminar 6 Lecturer: Finn R. Førsund

2. seminar 4 2 Seminar 6 exercise 1. Consider a hydropower monopolist and specify only a total water constraint. Argue why such a simplification can be made. Demonstrate that the monopolist shifts the pattern of water use to using more water in periods with relatively elastic demand and less water in periods with relatively less elastic demand. You can use a bathtub diagram for two periods for illustration. Argue why the shadow price of water is always less in the monopoly case than in the social planning case. Discuss the relationship between the social planning case and the monopoly case when the demand functions are identical for all periods. 2. Consider a monopolist that has both hydropower and thermal power plants with a total limited capacity. Analyse the profit-maximising adjustment of the monopolist under the assumption of the constraint on the reservoir never becoming binding. Assume identical demand curves. What can you say about the water value in the monopoly case compared with the socially optimal case? Try to discuss the difference in the mix of technologies in each of the two periods when the hydro constraint is binding both in the social planning case and in the monopoly case.

3. seminar 4 3 The basic monopoly model Problem formulation The Lagrangian First-order condition

4. seminar 4 4 Interpretation of the first-order condition Marginal revenue can be expressed as demand-flexibility corrected prices Market prices will vary with relative elastic periods having lower prices than relative inelastic periods

5. seminar 4 5 Social optimum, and monopoly p 1 (e 1 H ) p2Mp2M Period 1 Period 2 p 2 S = λ S p 1 S = λ S p1Mp1M λMλM p 2 (e 2 H )

6. seminar 4 6 Social and monopoly water values Assume no spill, then at least one monopoly price must be less than the social price  The flexibility-corrected prices are equal and equal to the common monopoly water value  The monopoly water value must then be less than the social water value Assume spill  Monopoly water value is then zero

7. seminar 4 7 Spill Equal demand functions λ Period 2 A λ p1Sp1S p1Mp1M Period 1 p2Sp2S p2Mp2M M e1He1H D Total available energy e2He2H p 1 (e 1 H ) p 2 (e 2 H )

8. seminar 4 8 Hydro and thermal, the model Problem formulation The Lagrangian

9. seminar 4 9 Hydro and thermal model, cont First-order conditions

10. seminar 4 10 Interpretation of first-order conditions Flexibility-corrected prices equal and equal to the water value Flexibility-corrected price equal to the marginal cost of thermal plus shadow price on capacity If water value zero (spilling) then no thermal capacity will be used

11. seminar 4 11 The bathtub for mixed technologies without binding resrvoir constraint λλ p2p2 p1p1 a A M’ M D e c’ p 1 (x 1 ) p 2 (x 2 ) c’ Period 1 Period 2 Thermal extensions Hydro energy e 1 Th e 2 Th e1He1H e2He2H

12. seminar 4 12 p2p2 Mix of hydro and thermal with reservoir constraint binding, social planning case c’ γ1γ1 Period 1 Period 2 λ1λ1 λ2λ2 B C D A Hydro energy Thermal extensions a d c’ p1p1 p 1 (x 1 ) p 2 (x 2 )

13. seminar 4 13 p2p2 Mix of hydro and thermal with binding reservoir constraint, monopoly c’ γ1γ1 Period 1 Period 2 λ1λ1 λ2λ2 B C D A Hydro energy Thermal extensions a d c’ p1p1 p 1 (x 1 ) p 2 (x 2 )