1. Term paper 20071 ECON 4930 Term paper Finn R. Førsund

2. Term paper 2007 2 1a. Define the situation of overflow of the reservoir The water accumulation equation Strict inequality means that the amount of water at the end of period t is less than the sum of what was received from period t-1 plus inflow during period t subtracted the release during period t → overflow Overflow implies that, the reservoir capacity

3. Term paper 2007 3 1b. What is the unit of measurement of the fabrication coefficient a? Explain the calibration of the coefficient Unit for a: Calibration:  The height of the fall from the reservoir to the turbine, called head. Gravity gives the energy of water  The efficiency losses due to friction in pipes, turbine not perfect, totalling 10-15% loss

4. Term paper 2007 4 1c. Converting variables measured in water to variables measured in kWh Inserting the production function into the water accumulation equation:

5. Term paper 2007 5 2. The social planning problem

6. Term paper 2007 6 2a. Discuss the objective function for the planning problem The objective function is the area under the inverse demand curve (NB! Choke price finite)  Demand function can be linked to utility function  The model is partial because there are no links to other activities, goods, etc. in the economy  A typical general objective function is the consumer plus producer surplus. Because variable production costs are zero we are left with the area under the demand curve.  Discounting is neglected due to short total time period

7. Term paper 2007 7 2b. Why is the planning problem formulated as a dynamic problem? Having a reservoir means that water used today can alternatively be used tomorrow, water has an opportunity cost

8. Term paper 2007 8 2c. Discuss reasons for a constraint on production to be realistic Production measured in kWh can have an upper limit for a period due to technical reasons  The flow of water through the pipe hitting the turbines is constrained by the diameter  The conversion to electricity is constrained by installed turbine capacity  The production of electricity may be constrained by the size of the generator

9. Term paper 2007 9 2d. Kuhn – Tucker conditions The Lagrangian function

10. Term paper 2007 10 2d., cont. The Kuhn – Tucker conditions

11. Term paper 2007 11 2d., cont. Interpretation of shadow prices:  Change in the optimised objective function of a marginal change in the constraint, found by partial differentiation of the optimised Lagrangian  Shadow price on the water accumulation constraint Change in the objective function of a marginal change in the constraint (i.e., change in R t-1,w t )

12. Term paper 2007 12 2d., cont.  Shadow price on the reservoir capacity constraint  Shadow price on the production constraint

13. Term paper 2007 13 2e. Circumstances that may lead to a binding constraint for production. Concept of locking in of water and manoeuvrability of the reservoir Constraining production  Satisfying consumption in a high-demand period  Producing in order to prevent overflow Locking in of water  Impossible to prevent overflow physically Manoeuvrability  The rate of maximal production relative to reservoir size

14. Term paper 2007 14 2f. Kuhn – Tucker conditions for period T Realistic assumptions No satiation of demand: price positive Binding production constraint in period T  Not realistic unless T is a high-demand period, prevention of overflow is not realistic in the last period

15. Term paper 2007 15 2g. A bathtub diagram illustration p T-1 λ T-1 M D CB A Period TPeriod T-1 λTλT Total available water pTpT

16. Term paper 2007 16 2h. Events that may lead to social price and shadow price changes Threat of overflow t Emptying the reservoir t Binding production constraint t

17. Term paper 2007 17 2i. Shadow prices on stored water for period u+2, u+1 and u Production constraint binding for period u+1, but not for period u, and u+2 to T Reservoir in between full and empty from T-1 to u

18. Term paper 2007 18 2i. Illustration: two-period bathtub diagram for periods u and u+1 pupu p u =λ u =p T D C A Period u+1Period u ρ u+1 P u+1 p T =λ u+1 P u+1 =λ u+1 +ρ u+1 B B’’