Transcript Managerial Economics & Business Strategy

Managerial Economics &
Business Strategy
Chapter 1
The Fundamentals of Managerial
Economics
Derivative RULES
Y  bX
• Constants

ALWAYS ZERO!!!
• What is the derivative of 10?
– Zero
• Power Functions
Y  10X
3
n
dY
( n 1)
 nbX
dX
dY
2
 30 X
dX
More Rules
• Sums and Differences


The derivative of the sum
(difference) is equal to the sum
(difference) of the derivatives of
the individual terms
If U=g(x) and V=h(x) then…
Y  3X  4 X
2
3
dY dU dV


dX dX dX
dY
2
 6 X  12 X
dX
And more…
• Products

The derivative of the product of
two expressions is equal to the
first term multiplied by the
derivative of the second, PLUS the
second term times the derivative of
the first
Y  5 X (7  X )
2
dY
dV
dU
U
V
dX
dX
dX
dY
2
 10X (7  X )  5 X
dX
dY
 70X  10X 2  5 X 2
dX
dY
 70X  15X 2
dX
And more…
• Quotient

Denominator MULTIPLIED by
derivative of the numerator
MINUS numerator MULTIPLIED
by the derivative of the
denominator ALL DIVIVED BY
the denominator squared
5X  9
Y
2
10 X
dY V (dU dX )  U (dV dX )

2
dX
V
dY 10X (5)  (5 X  9)(20X )

2 2
dX
(10X )
2
dY 50X  100X  180X

4
dX
100X
dY 5 X  10X  18

3
dX
10X
dY 18  5 X
Divide by 10X

3
dX
10X
2
2
Total Revenue
• TR = 7Q – 0.01Q2
• What is the Marginal Revenue function?

MR = 7 - 0.02Q
• TC = 100 – 8Q + 10Q2
• What is the Marginal Cost function?

MC = -8 + 20Q
Partial Derivatives
• When an equation has
MANY independent
variables we may want to
isolate the impact of a
single variable on
dependent variable
• Q = -100P + 5I + Ps + 2N

What is the impact of a change in
the number of customers on Q?
• Holding all other factors
constant
– Remember the derivative of
a constant is ZERO
Q
2
N
Maximum and Minimum Values
• Our goal is to find the
OPTIMAL points

Marginal analysis and derivatives
help us to find these points
• TR = 7Q – 0.01Q2  What
Q will maximize TR?
• Set equal to ZERO to find
revenue-maximizing Q
dTR
 7  0.02Q
dQ
7  0.02Q  0
7  0.02Q
350  Q
• What price and output will maximize profit for the
firm?
• Given : P = 172 – 10Q  demand function
TC = 100 + 65Q + Q2
P = TR – TC
What is TR???
Price * Quantity
TR = 172Q – 10Q2
P = TR – TC = 172Q – 10Q2 – (100 + 65Q + Q2)
= -100 + 107Q – 11Q2
Find the derivative, set equal to zero, and Q*
d
 107  22Q
dQ
0  107  22Q
22Q  107
Q  4.86
*
Is this a Maximum or a Minimum???
How do you know??
• Take the second derivative

Rate of change of the change
• Maximum Value: dY
 0 (first order condit ion)
dX
d2
 0 (secondorder condit ion)
2
dX
• Minimum Value:
dY
 0 (first order condit ion)
dX
d2
 0 (secondorder condit ion)
2
dX
So what was it??
d
 107 22Q
dQ
2
d
 22
2
dX
It is a maximum!!
Quadratic Formula  Yuck
• When you have a quadratic
functional form
• aX2 + bX + c = 0
• You need the quadratic
formula to solve for X
 b  b  4ac
2
x
2a
Let’s try it
• What is the profit-maximizing level of output for a
firm with
TR = 50Q
TC = 100 + 60Q – 3Q2 + 0.1Q3 ???
  50Q  (100 60Q  3Q  0.1Q
2
  100 10Q  3Q  0.1Q
d
2
2
3)
3
 10  6Q  0.3Q
dQ
d
2
 0.3Q  6Q  10
dQ
Plug into the quadratic formula to get Q*
 6  6  4(0.3)(10)
Q* 
2(.3)
2
 6  36  12
Q* 
 0.6
 6  24
*
Q 
 0.6
 6  4.90
*
Q 
 0.6
*
Q  1.833 or 18.166
Which is it??
• Second derivative must be NEGATIVE
• Q* = 18.166 is the maximum

Profit-maximizing level of output
d2
 6  0.6Q
2
dQ
2
d
 6  0.6(1.833)  4.90
2
dQ
d2
 6  0.6(18.166)  4.90
2
dQ
Key Functions we will be using
• Five key functions
Demand

• Linear

Total Revenue
• Quadratic

Production
• Cubic

Total cost
• Cubic

Profit
• Cubic
$
Demand
P=a-bQ
(or Q = a-bP)
Q
$
Total Revenue
TR=a+bQ-cQ2
a=0
Q
$
Production (short run)
TP=a+bL+cL2-dL3
a=0
Q
$
Cost (short run)
TC=a+bQ-cQ2+dQ3
Q
$
Profit
= a-bQ+cQ2-dQ3
a<0
Q
Why is this important?
•
•
•
•
The more data that can be obtained
The more mathematics can be used
The more precise we can be
The closer we can get to maximized profits