Profit Maximization Single variable unconstrained optimization Econ 494 Spring 2013 Hands Ch

Agenda Single variable, unconstrained optimization First and second order conditions Application: Profit maximization Silb., Section 1.5 Problem set 2 due Mon, Feb 4 2

Calculus Review: Single variable optimization MaximumMinimum 1. Objective Function 2. FONC (First-order necessary conditions) 3. SOSC (Second-order sufficient conditions) 3

Application: Firm’s behavior 4

Examples Some optimization examples: Firms maximize profits Firms maximize total revenue Firms minimize costs Individuals maximize utility subject to a budget constraint Each of these theories can be tested by developing refutable hypotheses and testing whether results conform to the predictions. 5 Also see Silb. Examples 1-5, p

Profit Maximization 6

Generic Profit Maximization Problem Step 1: Set up the objective function Step 2: Find FONC Step 3: Find SOSC 7 KEEP THIS SLIDE HANDY !

Profit Maximization The preceding problem was in a general form. Most profit max problems we encounter will be variations on this theme. *** A key to most economic problems will be correctly specifying the objective function. *** What would the objective function look like if: the firm sells its output in a perfectly competitive market (i.e., price taker)? OR: there is only one firm supplying many buyers (monopolist)? OR: the firm faces some sort of tax per-unit of output? OR: the firm maximizes total revenue? Each of these theories about the firm will result in a different objective function, and possibly, a different set of refutable hypotheses. 8

Begin with some assertions 9

Two Alternative Theories: Monopoly vs. perfect competition MonopolistPerfect competition Total taxes paid Objective Function Endogenous Vbls (Choice) Exogenous Vbls. (parameters) 10

Example 1: Perfect Competition 11

Objective function Step 1: Set up objective function for a profit-maximizing, perfectly competitive firm facing a per-unit tax. 12 Parameters are: p>0; k>0; t>0 Choice variable is: y>0

First-order necessary conditions (FONC) Step 2: Find FONC 13 NOTE: The FONC are the implicit form of the firm’s choice function.

A note on the FONC 14

Second-order sufficient conditions (SOSC) Step 3: Find SOSC: 15 Remember, the FONC alone do not guarantee maximum profits!

16 IMPORTANT!! The explicit choice function will always be a function of all the parameters in the FONC, but not necessarily all parameters in the original objective function.

17

Implicit Function Theorem (IFT) The FONC are an implicit relationship between the variables and parameters If the SOSC are non-zero, by the IFT, we can, in principle, solve the FONC for the explicit choice function. This is a sufficient condition (not necessary) For a more thorough discussion, see Silb section 5.3. Key: The IFT allows us to assert that a solution to the FONC exists. 18

19 Look at the denominator. Why is the SOSC ≠ 0 important?

A comment on the SOSC Note that the SOSC serve some important purposes Confirm that a solution to the FONC exists Confirm that the solution is a maximum or a minimum Later you will see that the sign of the SOSC is useful in determining the sign of comparative statics 20

What have we accomplished? 21

Review Step 1. Set up objective function Step 2. Find FONC. Interpret. Step 3. Find SOSC. Interpret. Step 4. Use IFT to solve FONC for choice function Step 5. Comparative statics (next class…) 22

Example 2: Monopolist (from practice quiz) 23

Step 1: Set up objective function 24

Step 2: Find FONC 25

Step 3: Find SOSC 26 y* is a maximum

27

28

Solving problems with generic notation On your own, review handout “Solving problems with generic notation” 29