Economics 202: Intermediate Microeconomic Theory

1. Economics 202: Intermediate Microeconomic Theory 0. Welcome! 1. Go over class website (syllabus, requirements, etc.) • Take attendance and go over attendance policy • Powerpoint slides – if you’re the type of student who prefers to write everything down from the board, may want to take another section • I assume that you’ve done the reading before class, if you don’t, it’ll seem like I’m going really, really, really fast. Again, if you prefer not to have a high pre-class expectation, may want to take another section. No hard feelings :) . 2. Please print out and bring Student Information Sheet to class 3. HW#1 (due Tuesday, in class) 4. For next time, read Ch. 1 & 2 in Nicholson. 5. Begin math lecture

2. Review of Relevant Mathematics • Maximization of function of 1 variable  = f(q)   “change in” Change in profits for change in output: /q If /q > 0, owner should do what? • Derivative The limit of /q for small changes in q is the derivative of  = f(q). Notation: df/dq or f ´(q) or d/dq. Derivative of  = f(q) at q1 is

3. Review of Relevant Mathematics • Thus, d/dq = 0 indicates maximum profits. • Or does it? When would d/dq = 0 not maximize ? • When we have a U-shaped profit function • When we have a S-type curve • d/dq = 0 is only a necessary condition for -maximization. • It does not ensure maximum , so it is not sufficient. • To ensure -maximization, the derivative of d/dq must be negative at q*. • The derivative of a derivative is called a second derivative. • A second derivative is denoted d2/dq2 or f ´´(q). • So the conditions d/dq |q=q* = 0 & d2/dq2|q=q* < 0 together are sufficient conditions for q* to be a (local) -maximum.

4. Review of Relevant Mathematics • Rules for finding derivatives 1. If b is constant, then db/dx = 0. 2. If a and b are constants and b 0, then 3. d ln x/dx = 1/x where ln = loge 4. dax/dx = ax ln a, for any constant a.

5. Review of Relevant Mathematics • Rules for finding derivatives Let f(x) and g(x) be two functions. Then Let y = f(x) and x = g(z) be two functions. Then

6. Examples • Consider the following relationship:  = 1,000q - 5q2 What value of q maximizes profits? What is the maximum level of profits? Is this a global maximum? • Suppose the firm’s production function was given by q = 2L½. What is the -max level of Labor?

7. Partial Derivatives • Most economic relationships depend on several variables y = f (x1, x2, x3, …, xN) • How does y change if we change x1 , ceteris paribus? • This is a partial derivative and denoted y/x1 or f/x1 • Marshallian demand, Q/P < 0

8. Calculating Partial Derivatives • Consider the function: y = f (x1, x2) = ax12 +bx1x2 + c x22 • What are f/x1 and f/x2? • Consider • What are f/x1 and f/x2? • Consider y = f (x1, x2) = a ln x1+ b ln x2 • What are f/x1 and f/x2? • Second-order partial derivatives • Find f11, f12, f21, and f22 for the above functions.

9. Total Differential • Most economic relationships depend on several vars y = f (x1, x2, x3, …, xN) • Recall that the partial derivative, y/x1, is the change in y when we change x1 , ceteris paribus. • Now we’re interested in the total effect on y when all the x’s are changed by a small amount. • This is the Total Differential of f and is denoted by

10. FOC’s • First-order conditions for a maximum (or minimum) • This says that any economic activity should be continued until its marginal contribution to the objective is 0. Anything else does not maximize y! • These are necessary conditions for maximizing y.

11. SOC’s • Second-order conditions for a maximum (or minimum) • Intuitively, we want y to decrease when we move away (i.e., change the x’s) from our critical point where • Consider case of a single indep-var: y = f(x) • What condition must we place on 2nd derivative of f to ensure that x* is the x which (locally) maximizesy? • That is, what do we need to ask of 2nd derivative of f to make sure d2y is negative? • f(x) < 0 • f(x)=0 and f(x) < 0 together are sufficient conditions for maximizing y.

12. SOC’s • Consider case of two indep-var: y = f(x1, x2) • What condition must we place on 2nd partial derivatives of f to ensure that (x1*, x2*) is the point which (locally) maximizesy? • That is, what do we need to ask of 2nd partial derivatives of f to make sure d2y is negative? • f11 < 0 and f22 < 0 and f11f22 - (f12)2 > 0 • Recall the necessary conditions are f1 = 0 and f2 = 0 . • Sufficient conditions for unconstrained maximization of y are then • (1) f1 = 0 and f2 = 0 and • (2) f11 < 0 and f22 < 0 and f11f22 - (f12)2 > 0

13. Constrained Optimization • Find the xi’s that maximize y = f (x1, x2) subject to a constraint c – b1x1 – b2x2 = 0 • FOC are still f1 = 0 and f2 = 0 • SOC is now only f11f22 – 2f12f1f2 + f22f12 < 0 • Lagrangian multiplier method • Interpretation of the Lagrangian multiplier,  •  = MB of xi / MC of xi for all i • Example: max xy2 s.t. 1 = x + y