Lecture 2 : Economics and Optimization

1. Lecture 2: Economics and Optimization AGEC 352 Spring 2011 – January 19 R. Keeney

2. This week’s assignment • Did not materialize due to flu • It would have been busy work anyway just to get you used to going to the webpage to find material and begin working on your own

3. Next lab • I’ll give you instructions on Monday • Lab will be posted by 12 on Tuesday • Questions on discussion must be answered before the 1:20 timeframe to get credit • Some will be volunteer, some not… • The only wrong answers are 1) no response and 2) I don’t know. Take your best guess at the simplest short explanation and we’ll work from there…

4. Functions • A function f(.) takes numerical input and evaluates to a single value • This is just a different notation • Y = aX + bZ … is no different than • f(X,Z) = aX + bZ • For some higher mathematics, the distinction may be more important • An implicit function like G(X,Y,Z)=0

5. Basic Calculus • y=f(x)= x2 -2x + 4 • This can be evaluated for any value of x • f(1) = 3 • f(2) = 4 • We might be concerned with how y changes when x is changed • When ∆X = 1, ∆Y = 1, starting from the point (1,3)

6. Marginal economics • An instance in economics where we focus on changes in functions…

7. An Example

8. Graphical Analysis

9. Differentiation (Derivative) • Instead of the average change from x=1 to x=2 • Exact change from a tiny move away from the point x = 1 • We call this an instantaneous rate of change • Infinitesimal change in x leads to what change in y?

10. Power rule for derivatives • Basic rule • Lower the exponent by 1 • Multiply the term by the original exponent • If f(x) = axb • Then f’(x) = bax(b-1) • E.g. • f(x) = 6x3 • f’(x) = 18x2

11. Examples • f(x) = 5x3 + 3x2 + 9x – 18 • f(x) = 2x3 + 3y • f(x) = √x

12. Applied Calculus: Optimization • If we have an objective of maximizing profits • Knowing the instantaneous rate of change means we know for any choice • If profits are increasing • If profits are decreasing • If profits are neither increasing nor decreasing

13. Profit function Profits p

14. A Decision Maker’s Information • Objective is to maximize profits by sales of product represented by Q and sold at a price P that the producer sets • 1. Demand is linear • 2. P and Q are inversely related • 3. Consumers buy 10 units when P=0 • 4. Consumers buy 5 units when P=5

15. More information • **Demand must be Q = 10 – P • The producer has fixed costs of 5 • The constant marginal cost of producing Q is 3

16. More information • Cost of producing Q (labeled C) • **C = 5 + 3Q • So • 1) maximizing: profits • 2) choice: price level • 3) demand: Q = 10-P • 4) costs: C= 5+3Q • What next?

17. We need some economics and algebra • Definition of ‘Profit’? • How do we simplify this into something like the graph below?

18. Graphically the producer’s profit function looks like this

19. Applied calculus • So, calculus will let us identify the exact price to charge to make profits as large as possible • Take a derivative of the profit function • Solve it for zero (i.e. a flat tangent) • That’s the price to charge given the function

20. Relating this back to what you have learned • We wrote a polynomial function for profits and took its derivative • Our rule: Profits are maximized when marginal profits are equal to zero • Profits = Revenue – Costs • 0 = Marginal Profits = MR – MC • Rewrite this and you have MR = MC

21. Next week • Monday and Wednesday • Lecture on spreadsheet modeling • Tuesday • Lab 12:30 – 1:20 (Lab Guide posted by 12) • Discussion board (details for login Monday) • Respond to questions I post about the assignment during lab time… • Ask any questions on that board you have about the lab work…