Exercise 1. Jan Abrell Centre for Energy Policy and Economics (CEPE) D-MTEC, ETH Zurich. Exercise

Transcription

1 Exercise 1 Jan Abrell Centre for Energy Policy and Economics (CEPE) D-MTEC, ETH Zurich Exercise

2 Outline Reminder: Constraint Maximization Minimization Example: Electricity Dispatch Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise

3 Corner Solutions: Maximization df max f( x) = 0 x > 0 Inner Solution x dx s.t. x 0 df < 0 x = 0 Corner Solution dx

4 Corner Solutions: Minimization df min f( x) = 0 x > 0 Inner Solution x dx s.t. x 0 df > 0 x = 0 Corner Solution dx

5 Complementarity: KKT Conditions Note that FOCs can be described as: Maximization: df df x= 0; 0; dx dx x 0 Minimization: df df x= 0; 0; dx dx x 0 This is called a complementarity condition and often stated as (only max case): Similar for the case without non-negativity constraint: df 0 dx x 0 free means that x is unrestricted in sign df 0 dx = x free

6 Deriving MCP formulations via KKT conditions Steps to take: Bring problem in standard form (constraints in greater-equal form) Set up Lagrangian Form first-order conditions

7 Remarks What is the interpretation of the Lagrangian multiplier Multiplier or dual variables are the change in the value function when marginally relaxing the corresponding constraint Can be interpreted as (shadow) price of the constraining factor in units of the objective function What about equality restrictions Equality restriction are really two inequality restriction Multiplier becomes free in sign Figure out a supply-greater-demand formulation in order to get economically meaningful signs

8 Outline Reminder: Constraint Maximization Minimization Example: Electricity Dispatch Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise

9 Model Setup An electricity company possess different generation facilities (i) Each generation facility is characterized by its installed capacity (cap_i) Marginal cost are constant and given (c_i) The demand for electricity is constant The company wants to determine the least-cost power plant mix to meet demand

10 Derive KKT conditions Steps to take: Bring problem in standard form (constraints in greater-equal form) Set up Lagrangian Form first-order conditions

11 KKT Conditions Lagrangian Zero-profits generation Market clearing electricity Market clearing capacity

12 Outline Reminder: Constraint Maximization Minimization Example: Electricity Dispatch Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise

13 Exercise 1 Exercise

14 Derive Socially Efficient Solution Maximize net benefit Defined as area under the demand curve net of cost Area under demand curve: Cost: Net benefit: Solution: Exercise

15 Perfect Competition Producer maximize profit given price Price is given by inverse demand function Exercise

16 Monopoly Producers takes price reaction into account Marginal revenues equal to cost Producer abuses market power: Consumer (producer) surplus decrease (increases) Exercise

17 Outline Reminder: Constraint Maximization Minimization Example: Electricity Dispatch Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise

18 Exercise 2 Exercise

19 The Marginal Rate of Substitution (MRUS) Graphically MRUS is the slope of an indifference curve Intuitively MRUS tells how much of y is needed to compensate a change in x (holding utility constant) Use total differential of utility function Welfare Economics

20 Derive KKT conditions Steps to take: Bring problem in standard form (constraints in greater-equal form) Set up Lagrangian Form first-order conditions

21 KKT Conditions The Lagrangian is given as: Zero-profits donuts (x): Zero-profits beer (y): Market clearing for income:

22 Demand functions Price ratio equal to inverse MRUS Demand functions: Own price elasticity: Cross price elasticities are zero Exercise

23 Outline Reminder: Constraint Maximization Minimization Example: Electricity Dispatch Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise

24 Exercise 3 Exercise

25 Nash Equilibrium No player has an incentive to deviate from his action given the actions of the other player Find column (row) maxima for player A (B) Nash equilibria where joint maxima occur Here: Nash equilibrium in dominant strategy for A Exercise

26 Nash Equilibrium Find column (row) maxima for player A (B) Nash equilibria where joint maxima occur Exercise

27 Outline Reminder: Constraint Maximization Minimization Example: Electricity Dispatch Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise

28 Exercise 4 Exercise

29 Cartel Solution Producers jointly maximize profits First order condition Solution: Exercise

30 Cartel Equilibrium is not a Nash Equilibrium Cartel solution: Nash equilibrium: No player has an incentive to deviate from his action given the actions of the other player Check whether one producer has an incentive to deviate given action of the other Suppose deviates producing 40 (given B still produces 30) Thus, incentive to deviate Exercise

31 Cournot-Nash Equilibrium Each player maximizes profit taking reaction of other player into account First order condition: Reaction functions (assuming interior solution): Jointly solve reaction functions for: Cartel profit was 1800 (but unstable) Exercise

32 Questions? Exercise