CHAPTER 8
OPTIMIZATION
Page
Contents
8.1
The Derivative
131
8.2
Rules of Differentiation
131
8.3
Higher Order in Differentiation
134
8.4
Maxima and Minima of a Function
135
8.5
Point of Inflexion
136
8.6
Uses of the Derivative in Economics
136
Exercise
Objectives:
After working through this chapter, you should be able to:
(i)
explain the term derivative and perform basic differentiation;
(ii)
find the minimum, maximum or inflexion point of a function;
(iii)
understand the use of the derivative in economics;
(iv)
find the optimum point of an economic function.
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Chapter 8: Optimization
8.1
The Derivative
The Derivative measures the instantaneous rate of change of a function. The formal
terminology for the derivative is
f ′ ( x) =
e.g.
dy
∆y
= lim
dx ∆x→0 ∆x
f ( x) = x 2
f ( x + ∆x ) = ( x + ∆x ) 2
= x 2 + 2 x∆x + ( ∆x ) 2
dy
x 2 + 2 x∆x + (∆x) 2 − x 2
= lim
dx ∆x → 0
∆x
= lim 2 x + ∆x
∆x → 0
= 2x
8.2
Rules of Differentiation
Differentiation is the process of determining the derivative of a function.
1.
The Constant Function Rule
f ( x) = a
f ′( x ) = 0
2.
The Linear Function Rule
f ( x ) = a + bx
f ′( x ) = b
3.
The Power Function Rule
f ( x ) = ax n
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Chapter 8: Optimization
f ′( x ) = anx n −1
e.g.
y = 4x 3
dy
= 12 x 2
dx
4.
The Rule for Sums and Difference
f ( x ) = g ( x ) ± h( x )
f ′ ( x) = g ′ ( x) ± h ′ ( x)
e.g.
y = 5x 3 − 3x 2
dy
= 15x 2 − 6 x
dx
5.
The Product Rule
f ( x ) = g ( x ) h( x )
f ′ ( x) = g( x) h ′ ( x) + h ( x ) g ′ ( x)
e.g.
y = 3x 4 (2 x − 5)
g ( x ) = 3x 4
g ′( x ) = 12 x 3
h( x ) = 2 x − 5
h ′( x ) = 2
dy
dx
= (3x 4 )2 + (2 x − 5)12 x 3
= 30 x 3 ( x − 2)
6.
The Quotient Rule
f ( x) =
g( x )
h( x )
f ′( x ) =
h( x ) g ′ ( x ) − g ( x ) h ′ ( x )
[h( x )]2
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Chapter 8: Optimization
e.g.
5x 3
y=
4x + 3
g ( x ) = 5x 3
g ′( x ) = 15x 2
h( x ) = 4 x + 3
h ′( x ) = 4
dy (4 x + 3)(15x 2 ) − (5x 3 )4
=
dx
(4 x + 3) 2
=
7.
40 x 3 + 45x 2
(4 x + 3) 2
The Chain Rule
y = f [ g( x )] where
y = f ( u) &
u = g( x )
dy dy du
=
⋅
dx du dx
e.g.
y = ( x 2 + 3) 3
u = x2 + 3
y = u3
dy
= 3u 2 = 3( x 2 + 3) 2
du
du
= 2x
dx
dy dy du
= 3u 2 . 2 x = 3( x 2 + 3) 2 . 2 x
=
⋅
dx du dx
= 6 x ( x 2 + 3) 2
8.
Special function
(i)
exponential
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Chapter 8: Optimization
f ( x) = e x
f ′( x ) = e x
e.g.
y = e g( x)
dy
= g ′( x )e g ( x )
dx
(ii)
logarithm
f ( x ) = log e x
f ′( x ) =
e.g.
1
x
y = log e g ( x )
dy g ′( x )
=
dx
g( x)
Implicit Function
e.g.
x2 + y2 = 4
d 2
( x + y 2 = 4)
dx
2x + 2 y
dy
=0
dx
dy
x
=−
dx
y
8.3
Higher Order in Differentiation
f ′( x ) =
df ( x )
dx
First derivative
f ′′( x ) =
d
[ f ′( x )]
dx
Second derivative
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Chapter 8: Optimization
e.g.
f ( x ) = x 4 − 3x 3
f ′( x) = 4 x 3 − 9 x 2
1st derivative
f ′′( x ) = 12 x 2 − 18 x
2nd derivative
f ′′′( x ) = 24 x − 18
3st derivative
f ( 4 ) ( x ) = 24
4th derivative
f
8.4
( 5)
( x) = 0
5th derivative
Maxima and Minima of a Function
dy
=0
dx
d2y
<0
dx 2
dy
=0
For a local minimum:
dx
d2y
>0
dx 2
For a local maximum:
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Chapter 8: Optimization
8.5
Point of Inflexion
Where there is an inflexional point, a mere band on the curve, the second derivative has a
zero value (i.e. f″ (x) = 0)
Example 1
f ( x) = x 3
f ′ ( x ) = 3x 2
f ′′( x ) = 6x
when x = 0
f ′( x ) = 0 & f ′′( x ) = 0
∴ Point A is a point of inflexion.
8.6
Uses of the Derivative in Economics
8.6.1
Marginal Concepts
Marginal cost in economics is defined as the change in total cost incurred from the
production of an additional unit. Marginal revenue is defined as the change in total
revenue brought about by the sale of an extra good. Since total cost (TC) and the
total revenue (TR) are both functions of the level of output (Q), marginal cost
(MC) and marginal revenue (MR) can each be expressed mathematically as
derivatives of their respective total functions.
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Chapter 8: Optimization
Example 2
1.
If TR = 75Q − 4Q 2 , then MR =
2.
If TC = Q 2 + 7Q + 23, then MC =
Example 3
Given the total cost function TC = Q 3 − 18Q 2 + 750Q
(a)
Take the first and second derivatives of the total cost function.
(b)
Find the average cost function AC and the relative extrema.
(c)
Do the same thing for the marginal cost function.
Example 4
Given C = 2000 + 0.9Yd, where Yd = Y − T and T = 300 + 0.2Y, use the derivative
dC
to find the Marginal Propensity to Consume MPC, where MPC =
.
dY
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Chapter 8: Optimization
8.6.2
Optimizing Economic Functions
The economist is frequently called upon to help a firm maximize profits and levels
of physical output and productivity, as well as to minimize costs, levels of
pollution, and the use of scarce natural resources.
Example 5
Maximize profits π for a firm, given
total revenue TR = 4000Q − 33Q2, and
total cost
TC = 2Q3 − 3Q2 + 400Q + 5000, assuming Q > 0
Example 6
Prove that marginal cost (MC) must equal marginal revenue (MR) at the profitmaximizing level of output
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Chapter 8: Optimization
Example 7
A producer has the possibility of discriminating between the domestic and foreign
markets for a product where the demands, respectively, are
Q1 = 24 − 0.2P1,
Q2 = 10 − 0.05P2
If TC = 35 + 40Q, what price will the firm charge (a) with discrimination and (b)
without discrimination. Where Q = Q1 + Q2
8.6.3
Price Elasticity of Demand and Supply
In economics, price elasticity of demand (supply) ε measures the percentage
change in quantity demanded (supplied) divided by the percentage change in price.
Mathematically,
ε=
dQ / Q dQ P dQ P
=
=
dP / P
Q dP dP Q
εd < 1
Inelastic Demand is relatively unresponsive to a change of price.
εd = 1
Unitary Demand responds proportionately to a change in price.
εd > 1
Elastic Demand is relatively responsive to a change in price.
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Chapter 8: Optimization
Example 8
The price elasticity of demand at P = 20 is determined below for the demand
function Q = 1400 − P2.
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Chapter 8: Optimization
8.6.4
Relationship Among Total, Marginal, and Average Concepts
A total product (TP) curve of an input is derived from a production function by
allowing the amounts of one input (say, capital) to vary while holding the other
inputs (labour and land) constant.
Example 9
Given TP = 90K2 − K3
(a)
Test the first-order condition to find the critical values.
(b)
Find and maximize the average product of capital APk
(c)
Find and maximize the marginal product of capital MPk
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Chapter 8: Optimization
EXERCISE: OPTIMIZATION
1.
2.
3.
4.
5.
6.
Find (1) the marginal and (2) the average functions for each of the following total
functions. Evaluate them at Q = 3 and Q = 5.
(a)
TC = 3Q2 + 7Q + 12
(b)
TC = 35 + 5Q − 2Q2 + 2Q3
Find the MR functions associated with each of the following supply functions.
Evaluate them at Q = 4 and Q = 10.
(a)
P = Q2 + 2Q + 1
(b)
P = Q2 + 0.5Q + 3
Find the MR functions for each of the following demand functions and evaluate
them at Q = 4 and Q = 10.
(a)
Q = 36 − 2P
(b)
44 − 4P − Q = 0
Maximize the following total revenue TR and total profit π functions by (1) finding
the critical value(s), (2) testing the second-order conditions, and (3) calculating the
maximum TR or π.
(a)
TR = 32Q − Q2
(b)
π = − Q3 − 5Q 2 + 2000Q − 326
1
3
From each of the following total cost TC functions, find (1) the average cost AC
function, (2) the critical value at which AC is minimized, and (3) the minimum
average cost.
(a)
TC = Q3 − 5Q2 + 60Q
(b)
TC = Q3 − 21Q2 + 500Q
Given the following total revenue and total cost functions for different firms,
maximize profit π, π = TR − TC, for the firms.
(a)
TR = 1400Q − 7.5Q2, TC = Q3 − 6Q2 + 140Q + 750
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Chapter 8: Optimization
(b)
7.
TR = 4350Q − 13Q2, TC = Q3 − 5.5Q2 + 150Q + 675
Faced with two distinct demand functions
Q1 = 24 − 0.2P1,
Q2 = 10 − 0.05P2
where TC = 35 + 40Q, what price will the firm charge (a) with discrimination and
(b) without discrimination.
8.
Use the MR = MC method to (a) maximize profit π and (b) check the second-order
conditions, given
TR = 1400Q − 7.5Q2,
TC = Q3 − 6Q2 + 140Q + 750
9.
The demand function is Q = 20 − 5P. (a) Find the inverse function. (b) Estimate
the elasticity at P = 2 and P = 3.
10.
The equation for a production isoquant which depicts the different combinations of
inputs K and L that can be used to produce a specific level of output Q (here 2144
units) is
1
3
16K 4 L4 = 2144
11.
(a)
Find the slope of the isoquant dK/dL which in economics is called the
marginal rate of technical substitution (MRTS)
(b)
Evaluate MRTS at K = 256, L = 108.
Given the demand function
P = 8.25e−0.02Q
12.
(a)
determine the quantity and price at which total revenue will be maximized
and
(b)
test the second-order condition
Land bought for speculation is increasing in value according to the formula
V = 100e
3
t
The discount rate under continuous compounding is 0.09. How long should the
land be held to maximize the present value.
143