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

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

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…

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

Basic Calculus y=f(x)= x 2 -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)

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

An Example Units SoldTotal Revenue Total CostChange in Revenue Change in Cost

Graphical Analysis

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?

Power rule for derivatives Basic rule ◦ Lower the exponent by 1 ◦ Multiply the term by the original exponent If f(x) = ax b Then f’(x) = bax (b-1) E.g. ◦ f(x) = 6x 3 ◦ f’(x) = 18x 2

Examples f(x) = 5x 3 + 3x 2 + 9x – 18 f(x) = 2x 3 + 3y f(x) = √x

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

Profit function p Profits

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

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

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?

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

Graphically the producer’s profit function looks like this

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

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

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…