Term paper ECON 4930 Term paper Finn R. Førsund

Term paper a. 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

Term paper b. 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

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

Term paper The social planning problem

Term paper a. 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

Term paper b. 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

Term paper c. 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

Term paper d. Kuhn – Tucker conditions The Lagrangian function

Term paper d., cont. The Kuhn – Tucker conditions

Term paper d., 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 )

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

Term paper e. 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

Term paper f. 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

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

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

Term paper i. 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

Term paper i. 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’’