Lecture 2: Economics and Optimization

AGEC 352Spring 2011 – January 19

R. Keeney

This week’s assignmentDid 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 labI’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 Calculusy=f(x)= x2 -2x + 4

◦This can be evaluated for any value of xf(1) = 3f(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 ExampleUnits Sold

Total Revenue

Total Cost

Change in

Revenue

Change in Cost

1 5 5.0 -- --2 10 6.5 5.0 1.53 15 9.0 5.0 2.54 20 13.0 5.0 4.05 25 18.5 5.0 5.56 30 26.0 5.0 7.57 35 40 5.0 14.0

Graphical Analysis

-5

0

5

10

15

0 1 2 3 4 5 6 7

Change in Revenue Change in Costs Total Profits

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 derivativesBasic rule

◦Lower the exponent by 1◦Multiply the term by the original

exponentIf f(x) = axb

Then f’(x) = bax(b-1)

E.g.◦f(x) = 6x3

◦f’(x) = 18x2

Examples

f(x) = 5x3 + 3x2 + 9x – 18

f(x) = 2x3 + 3y

f(x) = √x

Applied Calculus: OptimizationIf 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 InformationObjective is to maximize profits

by sales of product represented by Q and sold at a price P that the producer sets

1. Demand is linear2. P and Q are inversely related3. Consumers buy 10 units when

P=04. Consumers buy 5 units when

P=5

More information**Demand must be Q = 10 – P

The producer has fixed costs of 5The constant marginal cost of

producing Q is 3

More informationCost 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 algebraDefinition of ‘Profit’?

How do we simplify this into something like the graph below?

p

Profits

Graphically the producer’s profit function looks like this

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

0 1 2 3 4 5 6 7 8 9 10

Profi

ts

Price Charged

Applied calculusSo, 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 learnedWe wrote a polynomial function

for profits and took its derivativeOur rule: Profits are maximized

when marginal profits are equal to zero

Profits = Revenue – Costs0 = Marginal Profits = MR – MC

◦Rewrite this and you have MR = MC

Next weekMonday and Wednesday

◦Lecture on spreadsheet modelingTuesday

◦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…