Lecture 2: Economics and Optimization

AGEC 352Fall 2012 – August 27

R. Keeney

ReviewLast 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

FunctionsA 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 x

f(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

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 ExampleUnits Sold

Total Revenue

Total Cost

Change in

Revenue

Change in Cost

1 5 5.0 -- --

2 10 6.5 5.0 1.5

3 15 9.0 5.0 2.5

4 20 13.0 5.0 4.0

5 25 18.5 5.0 5.5

6 30 26.0 5.0 7.5

7 35 40 5.0 14.0

An ExampleUnits Sold

Total Reven

ue

Total Cost

Change in Revenue

Change in Cost

ProfitTR-TC

1 5 5.0 -- -- 0

2 10 6.5 5.0 1.5 3.5

3 15 9.0 5.0 2.5 6.0

4 20 13.0 5.0 4.0 7.0

5 25 18.5 5.0 5.5 6.5

6 30 26.0 5.0 7.5 4

7 35 40 5.0 14.0 -5

Graphical Analysis

-5

0

5

10

15

0 1 2 3 4 5 6 7

Change in Revenue Change in Costs Total Profits

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 1st derivative of f()

If f(x) = axb

Then f’(x) = bax(b-1)

E.g.◦ If f(x) = 6x3

◦Then f’(x) = 18x2

Examples

f(x) = 5x3 + 3x2 + 9x – 18

f(x) = 2x3 + 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 InformationObjective 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 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 these

equations into something like the graph below where we search for the price that delivers peak profits?

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

Pro

fits

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

Lab this weekWill be posted to

◦www.agecon.purdue.edu/academic/agec352

◦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