Chapter 8: Optimization



Contents8.1The Derivative129

8.2Rules of Differentiation129

8.3Higher Order in Differentiation132

8.4Maxima and Minima of a Function133

8.5Point of Inflexion134

8.6Uses of the Derivative in Economics134


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.

8.1The Derivative

The Derivative measures the instantaneous rate of change of a function. The formal terminology for the derivative is


8.2Rules of Differentiation

Differentiation is the process of determining the derivative of a function.

1.The Constant Function Rule

2.The Linear Function Rule

3.The Power Function Rule


4.The Rule for Sums and Difference


5.The Product Rule


6.The Quotient Rule


7.The Chain Rule


8.Special function





Implicit Function


8.3Higher Order in Differentiation

First derivative

Second derivative


1st derivative

2nd derivative

3st derivative

4th derivative

5th derivative

8.4Maxima and Minima of a Function

For a local maximum:

For a local minimum:

8.5Point 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

 Point A is a point of inflexion.

8.6Uses of the Derivative in Economics

8.6.1Marginal 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.

Example 2



Example 3

Given the total cost function

(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 to find the Marginal Propensity to Consume MPC, where MPC = .

8.6.2Optimizing 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  for a firm, given

total revenueTR = 4000Q 33Q2, and

total costTC = 2Q3 3Q2 + 400Q + 5000, assuming Q > 0

Example 6

Prove that marginal cost (MC) must equal marginal revenue (MR) at the profit-maximizing level of output

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

8.6.3Price Elasticity of Demand and Supply

In economics, price elasticity of demand (supply)  measures the percentage change in quantity demanded (supplied) divided by the percentage change in price. Mathematically,

InelasticDemand is relatively unresponsive to a change of price.

UnitaryDemand responds proportionately to a change in price.

ElasticDemand is relatively responsive to a change in price.

Example 8

The price elasticity of demand at P = 20 is determined below for the demand function Q=1400  P2.

8.6.4Relationship 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


1.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

2.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

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

4.Maximize the following total revenue TR and total profit  functions by (1) finding the critical value(s), (2) testing the second-order conditions, and (3) calculating the maximum TR or .

(a)TR = 32Q  Q2


5.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

6.Given the following total revenue and total cost functions for different firms, maximize profit ,  = TR  TC, for the firms.

(a)TR = 1400Q  7.5Q2,TC = Q3 6Q2 + 140Q + 750

(b)TR = 4350Q  13Q2,TC = Q3 5.5Q2 + 150Q + 675

7.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  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

(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.

11.Given the demand function

P = 8.25e0.02Q

(a)determine the quantity and price at which total revenue will be maximized and

(b)test the second-order condition

12.Land bought for speculation is increasing in value according to the formula

The discount rate under continuous compounding is 0.09. How long should the land be held to maximize the present value.