Lecture 2: Economics and Optimization AGEC 352 Fall 2012 – August 27 R. Keeney

Review Last Wednesday ◦ 2 equations, 3 unknowns ◦ Overcome this problem by making an assumption about the value of one of the unknowns  Assumption: Maximize Revenue  Doesn’t always work but it will for problems you see in this course Today: Similar issue but the equations are more familiar

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 In general, economic decision making focuses on changes in functions… ◦ E.g. The change in revenue vs. the change in cost  If the revenue change is greater than cost, continue expanding production because the next unit will be profitable

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

An Example Units Sold Total Revenue Total Cost Change in Revenue Change in Cost Profit TR-TC

Graphical Analysis

Issue Why is the peak (maximum) of the profit graph not directly above the point where Marginal Revenue = Marginal Cost ◦ Incomplete information used to generate the graph ◦ We are only considering production of whole units

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 (the only rule you need in 352) Basic rule ◦ Lower the exponent by 1 ◦ Multiply the term by the original exponent ◦ Let f’() be the 1 st derivative of f() If f(x) = ax b Then f’(x) = bax (b-1) E.g. ◦ If f(x) = 6x 3 ◦ Then 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 set by the producer 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 these equations into something like the graph below where we search for the price that delivers peak profits?

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

Lab this week Will be posted to ◦ ◦ Consists of Part 1 and Part II ◦ Part I must be completed before the next class meeting ◦ Questions at the end of Part II are due the following Monday  Wednesday this week due to the Labor Day Holiday