Basic Mathematics for Economists - Rosser - Chapter 8 doc

• Derive marginal revenue and marginal cost functions using differentiation andrelate them to the slopes of the corresponding total revenue and cost functions.. Differentiation is a meth

8 Introduction to calculus

Learning objectives

After completing this chapter students should be able to:

• Differentiate functions with one unknown variable

• Find the slope of a function using differentiation

• Derive marginal revenue and marginal cost functions using differentiation andrelate them to the slopes of the corresponding total revenue and cost functions

• Calculate point elasticity for non-linear demand functions

• Use calculus to find the sales tax that will maximize tax yield

• Derive the Keynesian multiplier using differentiation

8.1 The differential calculus

This chapter introduces some of the basic techniques of calculus and their application toeconomic problems We shall be concerned here with what is known as the ‘differentialcalculus’

Differentiation is a method used to find the slope of a function at any point Although this

is a useful tool in itself, it also forms the basis for some very powerful techniques for solvingoptimization problems, which are explained in this and the following chapters

The basic technique of differentiation is quite straightforward and easy to apply Considerthe simple function that has only one term

y = 6x2

To derive an expression for the slope of this function for any value of x the basic rules of

differentiation require you to:

(a) multiply the whole term by the value of the power of x, and

(b) deduct 1 from the power of x.

In this example there is a term in x2and so the power of x is reduced from 2 to 1 Using the

above rule the expression for the slope of this function therefore becomes

2× 6x2 −1= 12x

This is known as the derivative of y with respect to x, and is usually written as dy/dx, which

is read as ‘dy by dx’.

y

Figure 8.1

We can check that this is approximately correct by looking at the graph of the function

y = 6x2in Figure 8.1 Any term in x2will rise at an ever increasing rate as x is increased In other words, the slope of this function must increase as x increases The slope is the derivative

of the function with respect to x, which we have just worked out to be 12x As x increases the term 12x will also obviously increase and so we can confirm that the formula derived for

the slope of this function does behave in the expected fashion

To determine the actual value of the slope of the function y = 6x2for any given value

of x, one simply enters the given value of x into the formula

When x = 10, then slope = 180(1,000) = 180,000.

Test Yourself, Exercise 8.1

1 Derive an expression for the slope of the function y = 12x3

2 What is the slope of the function y = 6x4when x= 2?

3 What is the slope of the function y = 0.2x4when x= 3?

4 Derive an expression for the slope of the function y = 52x3

5 Make up your own single-term function and then differentiate it

8.2 Rules for differentiation

The rule for differentiation can be formally stated as:

If y = ax n where a and n are given parameters then

dy

dx = nax n−1

When there are several terms in x added together or subtracted in a function then this rule

for differentiation is applied to each term individually (The special rules for differentiatingfunctions where terms are multiplied or divided are explained inChapter 12.)

Example 8.4

Differentiate the function y = 3x2+ 10x3− 0.2x4

The example above illustrates the point that the derivative of any term in x (to the power of 1)

is simply the value of the parameter that x is multiplied by.

Any constant terms always disappear when a function is differentiated To understand why,

consider a function with one constant such as the function y = 5 This could be written as

y = 5x0 Differentiating this function gives

Even when the power of x in a function is negative or not a whole number, the same rules

for differentiation still apply

Test Yourself, Exercise 8.2

1 Differentiate the function y = x3+ 60x.

2 What is the slope of the function y = 12 + 0.5x4when x= 5?

3 Derive a formula for the slope of the function y = 4 + 4x−1− 4x.

4 What is the slope of the function y = 4x 0.5 when x= 4?

5 Differentiate the function y = 25 − 0.1x−2+ 2x 0.3

6 Make up your own function with at least three different terms in x and then

differentiate it

8.3 Marginal revenue and total revenue

What differentiation actually does is look at the effect of an infinitely small change in the

independent variable x on the dependent variable y in a function y = f(x) This may seem

a strange concept, and the rest of this section tries to explain how it works, but first considerthe following example which shows how a function can be differentiated from first principles

Example 8.14

Differentiate the function y = 6x + 2x2from first principles

the new value of y (i.e y + 5y) can be found by substituting the new value of x (i.e x + 5x)

into the function Thus

This is the same result for dy/dx that would be obtained using the basic rules for differentiation

explained in Section 8.2 It is obviously quicker to use these rules than to differentiate from firstprinciples However, Example 8.14 should now help you to understand how the differentialcalculus can be applied to economics

Up to this point we have been using the usual algebraic notation for a single variable

function, assuming that y is dependent on x Changing the notation so that we can look at

some economic applications does not alter the rule for differentiation as long as functionsare specified in a form where one variable is dependent on another

In introductory economics texts, marginal revenue (MR) is sometimes defined as theincrease in total revenue (TR) received from sales caused by an increase in output by 1 unit.This is not a precise definition though It only gives an approximate value for marginal revenueand it will vary if the units that output is measured in are changed A more precise definition

of marginal revenue is that it is the rate of change of total revenue relative to increases inoutput

T ⬘

T

TR C

BC = the slope of the line AB

which is an approximate value for marginal revenue over this output range

Now suppose that the distance between B and A gets smaller As point B moves along TRtowards A the slope of the line AB gets closer to the value of the slope of TT, which is the

tangent to TR at A (A tangent to a curve at any point is a straight line having the slope atthat point.) Thus for a very small change in output, MR will be almost equal to the slope of

TR at A If the change becomes infinitesimally small, then the slope of AB will exactly equalthe slope of TT Therefore, MR will be equal to the slope of the TR function at any given

q p

You can see that when TR is rising, MR is positive, as one would expect, and when TR isfalling, MR is negative As the rate of increase of TR gets smaller so does the value of MR.When TR is at its maximum, MR is zero

With the function for MR derived above it is very straightforward to find the exact value ofthe output at which TR is a maximum The TR function is horizontal at its maximum pointand its slope is zero and so MR is also zero Thus when TR is at its maximum

MR= 80 − 4q = 0

80= 4q

20= q

One can also see that the MR function has the same intercept on the vertical axis as thisstraight line demand schedule, but twice its slope We can show that this result holds for anylinear downward-sloping demand schedule.

For any linear demand schedule in the format

p = a − bq

TR= pq = (a − bq)q = aq − bq2

MR=dTR

dq = a − 2bq Thus both the demand schedule and the MR function have a as the intercept on the ver- tical axis, and the slope of MR is 2b which is obviously twice the demand schedule’s

slope

It should also be noted that this result does not hold for non-linear demand schedules If

a demand schedule is non-linear then it is best to derive the slope of the MR function fromfirst principles

For those of you who are still not convinced that the idea of looking at an ‘infinitesimally

small’ change can help find the rate of change of a function at a point, Example 8.17 belowshows how a spreadsheet can be used to calculate rates of change for very small increments.This example is for illustrative purposes only though The main reason for using calculus

in the first place is to enable the immediate calculation of rates of change at any point of

a function

Example 8.17

For the total revenue function

TR= 500q − 2q2

find the value of MR when q = 80 (i) using calculus, and (ii) using a spreadsheet that

calculates increments in q above the given value of 80 that get progressively smaller Compare

the two answers

(ii) The spreadsheet shown inTable 8.2can be constructed by following the instructions in

Table 8.1 This spreadsheet shows that as increments in q (relative to the initial given value of 80) become smaller and smaller the value of MR (i.e 5TR/5q) approaches

180 This is consistent with the answer obtained by calculus in (i)

D2 TR = 500q - 2q^2 Label to remind you what function is used

D4 =500*D3-2*D3^2 Calculates TR corresponding to given q value

B7 10 Initial size of increment in q.

B8 =B7/10 Calculates an increment in q that is only 10% of

the value of the one in cell above

B9 to

B13

Copy cell B8 formula

down column B

Calculates a series of increments in q that get

smaller and smaller each time

A7 =B7+D$3 Calculates new value of q by adding the

increment in cell A7 to the given value of 80

A8 to

A13

Copy cell A7 formula

down column A

Calculates a series of values of q that increase by

smaller and smaller increments each time

C7 =500*A7-2*A7^2 Calculates TR corresponding to value of q in cell

D7 =C7-D$4 Calculates the change in TR relative to the initial

given value in cell D4

Copy cell E7 formula

down column E decreasing increments in q and TR

Test Yourself, Exercise 8.3

1 Given the demand schedule p = 120 − 3q derive a function for MR and find the

output at which TR is a maximum

2 For the demand schedule p = 40 − 0.5q find the value of MR when q = 15.

3 Find the output at which MR is zero when p = 720 − 4q 0.5describes the demandschedule

4 A firm knows that the demand function for its output is p = 400 − 0.5q What

price should it charge to maximize sales revenue?

5 Make up your own demand function and then derive the corresponding MRfunction and find the output level which corresponds to zero marginal revenue

8.4 Marginal cost and total cost

Just as MR can be shown to be the rate of change of the TR function, so marginal cost (MC)

is the rate of change of the total cost (TC) function In fact, in nearly all situations where one

is dealing with the concept of a marginal increase, the marginal function is equal to the rate

of change of the original function, i.e to derive the marginal function one just differentiatesthe original function

Figure 8.4

The example above is somewhat unrealistic in that it assumes an MC function that is a straightline This is because the TC function is given as a simple quadratic function, whereas onenormally expects a TC function to have a shape similar to that shown in Figure 8.4 Thisrepresents a cubic function with certain properties to ensure that:

(a) the rate of change of TC first falls and then rises, and

(b) TC never actually falls as output increases, i.e MC is never negative (Although it is

quite common to find economies of scale causing average costs to fall, no firm is going

to find the total cost of production falling when output increases.)

The flattest point of this TC schedule is at M, which corresponds to the minimum value

It is obvious from this TC function that total fixed costs TFC= 40 and total variable costsTVC= 82q − 6q2+ 0.2q3 Therefore,

at the minimum point of AVC

(When you have covered the analysis of maximization and minimization in the next chapter,come back to this example and see if you can think of another way of solving it.)

Test Yourself, Exercise 8.4

1 If TC= 65 + q 1.5 what is MC when q= 25?

2 Derive a formula for MC if TC= 4q3− 20q2+ 60q + 40.

3 If TC = 0.5q3− 3q2+ 25q + 20 derive functions for: (a) MC, (b) AC, (c) the

slope of AC

4 What is special about MC if TC= 25 + 0.8q?

5 Make up your own TC function and then derive the corresponding MC function

8.5Profit maximization

We are now ready to see how calculus can help a firm to maximize profits, as the followingexamples illustrate At this stage we shall just use the MC = MR rule for profit maxi-mization The second condition (MC cuts MR from below) will be dealt with in the nextchapter

Example 8.21

A monopoly faces the demand schedule p = 460 − 2q

and the cost schedule TC= 20 + 0.5q2

How much should it sell to maximize profit and what will this maximum profit be? (All costsand prices are in £.)

Example 8.22

A firm faces the demand schedule p = 184 − 4q

and the TC function TC= q3− 21q2+ 160q + 40

What output will maximize profit?

Test Yourself, Exercise 8.5

1 A monopoly faces the following TR and TC schedules:

TR= 300q − 2q2

TC= 12q3− 44q2+ 60q + 30

What output should it sell to maximize profit?

2 A firm faces the demand function p = 190 − 0.6q

and the total cost function TC= 40 + 30q + 0.4q2

(a) What output will maximize profit?

(b) What output will maximize total revenue?

(c) What will the output be if the firm makes a profit of £4,760?

3 A firm’s total revenue and total cost functions are

although, as was explained inChapter 4, economic theory normally defines a demand function

in the format q = f(p), with q being the dependent variable rather than p However, because the usual convention is to have p on the vertical axis in supply and demand graphical analysis, and also because cost functions have q as the independent variable, it usually helps to work with the inverse demand function p = f(q) The examples below show how to derive the relationship between MR and q by finding the inverse demand function.

A firm faces the demand schedule q = 200 − 4p

and the cost schedule TC= 0.1q3− 0.5q2+ 2q + 8

What price will maximize profit?

√

57.85 0.6

Disregarding the negative solution as output cannot be negative

Test Yourself, Exercise 8.6

1 Given the demand function q = 150 − 3p, derive a function for MR.

2 A firm faces the demand schedule q = 40 − p 0.5 (where p 0.5 ≥ 0, q ≤ 40) and

the cost schedule TC= q3− 2.5q2+ 50q + 16 What price should it charge to

maximize profit?

3 Find the MR function corresponding to the demand schedule q = (60 − 2.5p) 0.5

A

T ⬘ B

8.7 Point elasticity of demand

Price elasticity of demand is defined as

e = (−1)percentage change in quantity

percentage change in price

However, looking at the changes in price and quantity between points A and B on the demandschedule D in Figure 8.5, the question you may ask is ‘percentage of what’? Clearly the

change in quantity 5q is a much larger percentage of q1 than of the larger quantity q2.Although arc elasticity gives an approximate ‘average’ measure, a more precise measure can

be obtained by finding the elasticity of demand at a single point on the demand schedule In

Chapter 4some simple examples of point elasticity based on linear demand schedules wereconsidered With the aid of calculus we can now also derive point elasticity for non-lineardemand schedules

If the movement along D from A to B in Figure 8.5 is very small then we can assume

As B gets nearer to A the value of 5p/5q, which is the slope of the straight line AB, gets

closer to the slope of the tangent TTat A (Note that, as price falls in this example, 5p is

negative, giving a negative value for the relevant slopes.) Thus for an infinitesimally smallmovement from A

5p

dq = slope of D at A