Chapter 5: Univariable Calculus

UNIVERSITY OF BRISTOL : DEPARTMENT OF ECONOMICS

MATHEMATICS FOR ECONOMISTS - COURSE 11122

Chapter 4 : Univariate calculus

1.Concept of Differentiability

Suppose f = f (x) with x and any two points on the x-axis. The straight line through

(, )and (x,) has slope b where:

If is fixed then b is a function of x ( ) so as we change x ,b changes.

For the function graphed above, as x gets closer and closer to , the straight line gets

closer and closer to the tangent to the graph of at , and the slope b gets closer and

closer to the slope of the tangent at (which is ()). We write it mathematically as

follows:

As x converges to i.e as , b converges to 

i.e. ()

If exists we say that f is differentiable at .

We call the limit the derivative ofat. If f is differentiable at all x then we say that

is differentiable and we denote the derivative by (x).

Alternatively let and , then .

at or ()

Derivatives of standard functions

(i) () = n for all values of n (iii) (ax) = axln a

(ii) (iv) (ln )) =

The derivative of a function is itself a function of x. The derivative may or may not be

differentiable itself but if it is then we can differentiate to obtain the second derivative of the

original function. We denote the second derivative of by (x) or .Similarly,

the nth derivative of is denoted (n) or .

2. Properties of differentiable functions

Let f = f (x), g =g (x), u= u (x) and v = v (x) where  and  are constants; then

(i) (u + v) =  + 

(ii) ( u v ) = u + v

(iii)

(iv)If h(x) = g ( f (x)) then (x) = g'( f (x )).(x)(function of a function rule).

Result 4.1 If f (x) is differentiable at , then it is continuous at .

Example 4.1 Differentiate(i) (ii)

3. Applications to economic functions

Example 4.2 Price elasticity of demand

The price elasticity of demand (denoted by ) for a demand function , is defined

as the ratio of the percentage change in the quantity demanded i.e. x 100%

to the percentage change in price i.e. x 100% .

Hence = = .

When the changes in p are infinitesimal, the above can be written as

= .

Similarly the price elasticity of supply is defined as

= .

Notes

(1) Since and have been defined in terms of percentage changes, they are therefore

independent of units of measurement.

(2) and are evaluated at a point on the demand function (curve) and

supply function (curve) respectively so they are known as point elasticities.

Find the price elasticity of demand and the price elasticity of supply for the demand and supply

functions given below at the point p = 4 and comment on them:

(a) (b)

Solution (a) and

= . = = -0.178

so inelastic demand since demand is weakly responsive to changes in price i.e the percentage

change in demand is less than the percentage change in price.

(b) = 4 + 8 = 12 and = 20.

= . = = 2.4

so elastic supply since supply is strongly responsive to changes in price i.e the percentage change in supply is greater then the percentage change in price.

Example 4.3 Marginal utility

Given a utility function where x is the quantity of a good that a consumer consumes,

the change in total utility resulting from some (infinitesimally small) change in the quantity x is

given by the derivative or which is known as the marginal utility (MU).

In turn the change in MU resulting from some (infinitesimally small) change in x is given by

the derivative = or . Clearly ifMU declines as x increases infinitesimally,

then , indicating the operation of the law of diminishing utility.

A consumer has a utility function where and x. Does this

utility function possess the property of diminishing marginal utility?

Example 4.4 Marginal Cost

In the theory of cost, the firm is assumed to have a total cost function which relates total cost

the change in total cost given some (infinitesimally small) increase in the firm’s output is given

by the derivative or , which is known as the marginal cost.

The marginal cost can be interpreted as the extra cost incurred by the firm as an extra unit

of output is produced.

Given the total cost function , state whether the marginal

cost is an increasing or decreasing function of the quantity produced.

Example 4.5 Marginal propensity to consume and to save

In macroeconomics we frequently encounter the concept of a consumption function, which

relates aggregate consumption expenditure (C) to the level of national income (Y). Thus given

a consumption function , the change in consumption for some (infinitesimally small)

change in income is called the marginal propensity to consume, MPC, so MPC = or .

Moreover if saving (S) is defined as the amount of national income not devoted to consumption,

then a saving function can be written as . This permits us to define the marginal

propensity to save (MPS) as MPS = or .

(a)Sketch the general shape of the consumption function for .

(b) Find the marginal propensity to consume (MPC) when Y = 91.

(d) Determine whether or not MPC and MPS change in the same direction as Y changes.

( for the solution see WorkedExample 4.5) .

Example 4.6 Marginal product

In the short-run analysis of production , the production process is assumed to involve

fixed inputs and only one variable input, labour services (l). In this case the production function,

which relates the level of output Q or y to the quantity of labour input, can be written as

. The derivative or is known as the marginal product of labour (MP).

If MP declines as l increases, then or , indicating the operation of the law of

diminishing marginal product.

Example 4.7 Marginal revenue and marginal revenue product

If a firm has a total revenue function where Q denotes output, then the

marginal revenue (MR) is the derivative or . It can be interpreted as the extra

revenue earned by a firm from producing and selling an extra unit of output.

Suppose the firm has a production function , then we can express R as a function of l,

namely . The change in total revenue for some (infinitesimally small) change in

quantity of labour input (l) is given by the derivative or . This derivative is known as

the marginal revenue product (MRP) of labour.

To obtain , we can use the function of a function rule or chain rule.

i.e. MRP = = . = MR . MP where

A firm’s production and revenue functions are respectively, and

. Find :

(a) the marginal revenue product of labour when l = 16,

(b) the price at which marginal revenue is zero

4.The differential of f at

Suppose we want to approximate a non-linear function f (x) by a linear function at a

particular point , graphically we are approximating the curve f (x) by the tangent to the curve

at the point .

The tangent to the curve f (x) at has slope (). The equation of a tangent to the curve

f (x) is given by

Hence (1)

The differential of at is denoted by (x) and is defined as:

(x)=()

Rearranging (1) gives

i.e. the change in as we move from to x is approximately equal to (x).

Hence the differential is a linear approximation to the change in the value of in going from

to x . It will be a good approximation when x is close to .

Alternative (more general) form of the differential.

Let = then

(x) =  () or (x) =

More briefly still =

Important note

Example 4.8

The profit function of a firm is given by where x is the

output. Find the differential of at the point x = 3 and use it to estimate the change in

profit when the output increases from 3 units to 3.5 units. Compare this estimated change in profit

with the actual change in profit.

A derivative as the ratio of two differentials

From above = .

Divide both sides of the equation by dx , then i.e.

the derivative of .

So that the derivative which has previously been regarded as a single entity can

now be reinterpreted as the ratio of the two differentials df and dx .

Application to elasticities

From above = so suppose that f(x) = ln x, then

So the differential of ln x which is d(ln x) =

From Example 4.2, = . = .

i.e the price elasticity of demand is the ratio of the differentials of and .

Similarly the price elasticity of supply is is the ratio of the differentials of and

so that =.

Example 4.9 Given , find the price elasticity of demand using logs.

5. Unconstrained Optimisation

Let (a , b) be an open interval and choose any  (a , b) and x (a , b).

We say that f has a local maximum at if f () f (x) for all x  in the interval (a , b).

If f () f (x) for all x , we say f has a strict local maximum at .

We can illustrate these definitions as follows;

Notice that these definitions are local i.e. they refer only to some interval around .

A stronger condition is:

f has a globalmaximum at if f ()  f (x) for allx

f has a strictglobal max. at if f () > f (x) for allx .

NOTES

(a) Note that any global maximum is also a local maximum.

(b) If we consider the tangent to a graph at a local maximum we observe that the tangent is

horizontal. An equivalent (and important) way of restating this is contained in the next result

(for which no proof is given).

Result 4.2 If f (x) is a differentiable function and has a local maximum at , then

We also say that satisfies the first order condition for a maximum.

Example 4.10 Suppose f (x) = . Ithas a maximum at x = 1 and .

(d)Now is a necessary condition for a local maximum but it is not

a sufficient condition.

Example 4.11 Suppose f (x) = . The point x = 0 is a stationary point but it is not a

maximum but a point of inflection.

(e) Having derived a necessary condition for a maximum we wish to know if we can find a (simple) condition that is also sufficient.

It can be shown that a sufficient condition for a function f (x) to

have a local strict maximum at is that f () < 0.

We normally refer to this as the second order condition for a maximum so we can say that

satisfies the second order for a maximum if f () < 0.

6. Points of inflection.

Given a function f (x) , whose second derivative (x) at the point x = is = 0,

then the point , will be

(i) a stationary point of inflection if and ;

(ii) a non-stationary point of inflection if and .

Example 4.12

Show that the cost function where x is the output, has a

non-stationary point of inflection when x = 3.

7. More economic examples

Example 4.13

A firm has a total revenue function and a total cost function given by

where q is the level of output.

Find : (a) the firm’s profit-maximising output level ,and the corresponding values of profit, price

and total revenue at this output level.

(b) the revenue-maximising output level, and the corresponding values of profit, price and

total revenue at this output level.

the revenue-maximising output level.

Example 4.14 (for the solution see Worked Example 4.14 )

The demand curve for a product is given by, where b > 0. A monopolist can supply q at

cost cq where c is a constant. Determine the monopolist’s optimal policies for b < 1, b = 1, and

b > 1 if he wishes to maximise his profit. Interpret b in terms of the elasticity of demand.

7. Further economic applications of differentiation

The contents of Chapter 4 are useful in a great deal of economics. You will hopefully have seen

that there is a direct connection between the derivative and the economic concept of

marginality. The derivative can be interpreted as the marginal change in the economic

variable y given some (infinitesimally small) change in x and hence is directly related to

the marginal concepts so useful in economics.

Economic analysis is full of such marginal changes : marginal propensity to consume, marginal

cost, marginal propensity to import, marginal rate of substitution, marginal product, marginal

utility and so on.

Below are further economic applications of calculus.

Example 4.15

Consider an economic variable which grows from an initial value of according to the

exponential function :

The proportional rate of growth at time t of y is defined as

Since , the proportional rate of growth of y is .

Since , the slope of the graph of the natural log of a variable with respect to time

indicates the rate of growth of the variable over time.

The constancy of the slope indicates how close the growth process is to exponential growth

at a constant rate.

Example 4.16

A classic application of the optimisation technique involves a firm that has output q.

This is produced at a cost C(q) and sold with resulting revenue R(q). The firm's profit from

producing output q is given by:

A profit maximising firm seeks a (global) maximum of this function. We have

so that at a maximum

Thus at a profit maximising point marginal revenue is set equal to marginal cost

To ensure that we have a maximum we need to check our second order condition:

Example 4.17

Total differentials are widely used in macroeconomics.

Suppose, for example, that we have a (consumption) function f(Yt) that gives consumption in

period , as a function of income in period t, . A natural question to ask is: how much

does consumption change if income changes from to ? If the change in income is small

and the consumption function is differentiable the (approximate) answer is the total differential

of f at

i.e. .

The expression is the marginal propensity to consume.

Reading for Chapter 4 : Chapter 6, Bradley and PATTON

C.Osborne November 2001