econ optimization technique lecture slides

Chapter 18
Optimization
Techniques
Winter 2014

Agenda…

1) Functional Relationship
2) Marginal Analysis

3) Concept of a Derivative; Rules of Differentiation
4) The Marginal Cost = Marginal Revenue Rule
5) Constrained Optimization

Functional relationships

How an Economic Relationship is
Expressed?

By a Table, a Graph, or an Equation

An Example of Functional Relationships
(Table)
CHERRY CORPORATION
DAILY
SALES
150
100
50
0

PER UNIT
PRICE
10
20
30
40

An Example of Functional Relationships
(Graph)

An Example of Functional Relationships
(Equation)

Q = f (P)
Q=Number of Units sold
P=price

for example Q = 200 - 5P (Figure 2.1)

The number of units sold is a Function of Price

Q is a dependent Value
P is an independent variable

Marginal Value

The marginal value of a dependent variable is the change in this dependent variable (e.g., Q) associated with a 1-unit change in a particular independent variable (e.g., P)

Marginal Values
(An Example)

Relationship between output (independent variable) and profit (dependent) variable
OUTPUT
PER DAY
0
1
2
3
4
5
6
7
8
9
10

TOTAL MARGINAL
PROFIT
PROFIT
0
100
100
250
150
600
350
1000
400
1350
350
1500
150
1550
50
1500
-50
1400
-100
1200
-200

AVERAGE
PROFIT
100.0
125.0
200.0
250.0
270.0
250.0
221.4
187.5
155.6
120.0

The marginal value of a dependent variable is the change in this dependent variable associated with a 1-unit change in a particular independent variable

Central Point
The dependent variable is maximized when its marginal value shifts from positive to negative
OUTPUT
PER DAY
0
1
2
3
4
5
6
7
8
9
10

TOTAL MARGINAL
PROFIT
PROFIT
0
100
100
250
150
600
350
1000
400
1350
350
1500
150
1550
50
1500
-50
1400
-100
1200
-200

AVERAGE
PROFIT
100.0
125.0
200.0
250.0
270.0
250.0
221.4
187.5
155.6
120.0

Marginal Analysis
As mentioned, the