1. 6. Optimization with 2 variables: Price discrimination Econ 494 Spring 2013 See Handout 5 for general case with two variables

2. Agenda 2

3. Calculus review: Optimization with 2 variables MaximumMinimum FONC (both must hold) SOSC (all 3 must hold) 3 These are the optimality conditions when there are only 2 variables. Later, you will see a more general version of this using matrices.

4. Calculus review: Young’s theorem The order of differentiation does not matter Result extends to any 2 nd partial derivative of a function of many variables This symmetry result will come in handy… 4

5. Application: price discrimination 5 See Silb p. 71, example 3

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8. 1. Set up objective function 8 Total revenue: Objective function:

9. 2. Find FONC 9 Interpret FONC: The profit-maximizing monopolist will choose a level of production such that the marginal revenue in each market is the same, and equals the marginal cost of producing the good. MC is exactly the same in both equations

10. 3. Find SOSC 10 As long as the SOSC hold, we know profits are maximized. By the IFT, as long as the SOSC hold, we know that we can, in principle, solve the FONC simultaneously for the explicit choice functions.

11. Extend this example 11

12. Price discriminating monopolist 12 Gross revenue in each market To keep notation simple, we will keep the general form of the revenue function. Net revenue in each market Step 1: Objective Function:

13. Optimality conditions 13

14. 4. Find explicit choice functions 14 5a. Substitute explicit choice function into FONC to get identities:

15. Understanding  i The 2 first derivatives of the objective fctn. are: 15 Note that we do not use the superscript *. This is because these are just first derivatives, and hold for any (y 1, y 2 ). The maximizing the objective function requires that we set these equal to zero, but this relationship is still an equality. We have not yet substituted the optimal solution back into the FONC. Note that  i are a function of both variables y 1 and y 2.

16. Understanding  i 16

17. Understanding  i 17

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20. Aside: More on the cost function 20 These are all identical ways of expressing the same relationship.

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22. 22 Now combine the 2 results into matrices…

23. Express in matrix form: 23 Notice that the terms above in black are all the 2 nd partial derivatives,  ij, evaluated at y i *(). Using this, rewrite the equations above: The 2 comparative static results are the simultaneous solution to this pair of equations.

24. In matrix form 24 This pair of simultaneous equations can be expressed in matrix form Ax = b: Use Young’s theorem  12 =  21 “Hessian” matrix (H): contains all 2 nd partial derivatives Determinant of the Hessian:

25. Review: Cramer’s rule (2 eqns) 25 The terms of the matrix equation Ax = b are: Let A(i) be the matrix obtained by replacing column i of the matrix A with the column vector b: Then the solution for x i is:

26. Apply Cramer’s rule 26

27. Interpret result 27

28. On your own… 28

29. Apply Cramer’s rule 29