Differentiation in Economics – Objectives 1

Differentiation in Economics – Objectives 1

• Understand that differentiation lets us identify marginal relationships in economics

• Measure the rate of change along a line or curve

• Find dy/dx for power functions and practise the basic rules of differentiation

• Apply differentiation notation to economics examples

Differentiation in Economics – Objectives 2

• Differentiate a total utility function to find marginal utility

• Obtain a marginal revenue function as the derivative of the total revenue function

• Differentiate a short-run production function to find the marginal product of labour

Differentiation in Economics – Objectives 3

• Understand the relationship between total cost and marginal cost

• Measure point elasticity of demand and supply

• Find the investment multiplier in a simple macroeconomic model

Differentiation

• Differentiation provides a technique of measuring the rate at which one variable alters in response to changes in another

Changes for a Linear Function

• For a linear function• The rate of change of y with

respect to x is measured by • The slope of the line =

right) the to (distance up) (distance

xy

Differentiation Terminology

• Differentiation: finding the derivative of a function

• Tangent: a line that just touches a curve at a point

• Derivative of a function: the rate at which a function is changing with respect to an independent variable, measured at any point on the function by the slope of the tangent to the function at that point

Derivatives

• The derivative of y with respect to x is

denoted

• The expression

should be regarded as a single symbol and you should not try to work separately with parts of it

xy

dd

xy

dd

Using Derivatives

• The derivative

is an expression that measures the slope of the tangent to the curve at any point on the function y = f(x)

• A derivative measures the rate of change of y with respect to x and can only be found for smooth curves

• To be differentiable, a function must be continuous in the relevant range

xy

dd

Tangents at points A, B and C

The slope of the tangent at A is steeper than that at B; the tangent at C has a

negative slope

0

50

100

150

200

0 1 2 3 4 5 6 7 8 9Output, x

Total Revenue, y y = 56x - 4x2

A

B

C

Working with Derivatives

• The derivative

is itself a function of x

• If we wish we can evaluate

for any particular x value by substituting that value of x

xy

dd

xy

dd

Small Increments Formula

• For small changes x it is approximately true that

y = x.

• We can use this formula to predict the effect on y, y, of a small change in x, x

• This method is approximate and is valid only for small changes in x

xy

dd

Rules of Differentiation for Functions of the Form y =

f(x)

• The Constant Rule Constants differentiate to zero, i.e. if y = c where c is a constant

= 0xy

dd

• If y = axn where a and n are constants

= n.axn –1

• Multiply by the power, then subtract 1 from the power

Power-Function Rule

xy

dd

• Another way of handling the constant a in the function y = a.f(x) is to write it down as you begin differentiating and multiply it by the derivative of f(x)

=

• The derivative of a constant times a function is the constant times the derivative of the function

xx

dad n

xx

dd

an

.

Constant Times a Function Rule

Indices in Differentiation

• When differentiating power functions, remember the following from the rules of indices

x1 = xx0 = 1

= x – n

x = x0.5 = x1/2

nx1

• If y = f(x) + g(x)

=

• If y = f(x) – g(x)

=

• The derivative of a sum (difference) is the sum (difference) of the derivatives

xx

xx

dgd

dfd

xx

xx

dgd

dfd

Sum – Difference Rule

xy

dd

xy

dd

• If y = c + mx

= m

• The derivative of a linear function is the slope of the line 

Linear – Function Rule 1

xy

dd

• If y = mx

= m

• The derivative of a constant times the variable with respect to which we are differentiating is the constant

Linear – Function Rule 2

xy

dd

• To find dy/dx, we may obtain dx/dy and turn it upside down, i.e.

=

• There must be just one y value corresponding to each x value so that the inverse function exists

yx dd /

1

Inverse Function Rule

xy

dd

When Differentiating

• Ascertain which letters represent constants

• Identify the variable with respect to which you are differentiating and use it as x in the rules

Utility Functions

• To find an expression for marginal utility, differentiate the total utility function

• If total utility is given by U = f(x)

• MU =xU

dd

Revenue Functions

• To find marginal revenue, MR, differentiate total revenue, TR, with respect to quantity, Q

• If TR = f(Q)

• MR = QTR

dd

Short-run Production Functions

• The marginal product of labour is found by differentiating the production function with respect to labour

• If output produced, Q, is a function of the quantity of labour employed, L, then

• Q = f(L)

• MPL = LQ

dd

Total and Marginal Cost

• Marginal cost is the derivative of total cost, TC, with respect to Q, the quantity of output, i.e.

• MC =

• When MC is falling, TC bends downwards When MC is rising, TC bends upwards

QddTC

Variable and Marginal Cost

• Marginal cost is also the derivative of variable cost, VC, with respect to Q, i.e.

• MC =Qd

dVC

Point Elasticity of Demand and of Supply

• Point price elasticity =• For price elasticity of demand

use the equation for the demand curve

• Differentiate it to find dQ/dP then substitute as appropriate

• Supply elasticity is found from the supply equation in a similar way

QP

PQ

dd

Finding Point Elasticities

• Point price elasticity =

• If the demand or supply function is given in the form P = f(Q), use the inverse function rule

• =

• For downward sloping demand curves, dQ/dP is negative, so point elasticity is negative as price falls the quantity demanded

increases

QP

PQ

dd

QPd

d1

PQ

dd

Elasticity Values

• Demand elasticities are negative, but we ignore the negative sign in discussion of their size

• As you move along a demand or supply curve, elasticity usually changes

• Functions with constant elasticity: Demand: Q = k/P where k is a constant

has E = – 1 at all prices Supply: Q = kP where k is a constant

has E = 1 at all prices

Elasticity at Different Points on Linear Demand

Curves• Elasticity varies from – to 0 as

you move down a linear demand curve

• Two demand curves with the same intercept on the P axis have the same elasticity at every price

• For two demand curves with different intercepts on the P axis, the one with the lower intercept has the greater elasticity at every price

Finding the Investment Multiplier 1

1. Write down the equilibrium condition for the economy

Y = ADIncome = Aggregate Demand

2. Write an expression for ADAD = C + I + G + X – Z

Substitute into this, but do not substitute a numerical value for the autonomous expenditure I so

AD = f(Y, I)

Finding the Investment Multiplier 2

Substituting AD in the equilibrium condition gives an equation where Y occurs on both sides

3. Collect terms in Y on the left-hand side and solve for Y

4. Now differentiateIf Y = income and I = investment

dY/dI is the investment multiplier

Maximum and Minimum Values – Objectives 1

• Appreciate that economic objectives involve optimization

• Identify maximum and minimum turning points by differentiating and then finding the second derivative

• Find maximum revenue• Show which output maximizes profit

and whether it changes if taxation is imposed

Maximum and Minimum Values – Objectives 2

• Identify minimum turning points on cost curves

• Find the level of employment at which the average product of labour is maximized

• Choose the per unit tax which maximizes tax revenue

• Identify the economic order quantity which minimizes total inventory costs

Derivatives and Turning Points

• Sign of around a turning point:

before at critical value after

• Maximum + 0 –• Minimum – 0

+

xy

dd

Second Derivative of a Function

• After obtaining the first derivative

of the function we differentiate that and the result is called the second derivative of the original function

=

• Second derivative: is obtained by differentiating a derivative

2

2

xy

dd

xy

x dd

dd

xy

dd

To Identify Possible Turning Points:

• Differentiate, set equal to zero and solve for x

• Find and look at its sign to distinguish

a maximum from a minimum• The first and second order conditions are:

Maximum Minimum 0 0

– ve +ve

2

2

xy

dd

xy

dd

xy

dd

2

2

xy

dd

Point of Inflexion

• There is also the possibility that d2y/dx2 may be zero

• In this case we have neither a maximum nor a minimum

• Here the curve changes its shape, bending in the opposite direction

• This is called a point of inflexion

Maximum Total Revenue

• For maximum total revenue• Differentiate the TR function

with respect to output, Q• Set the derivative equal to

zero and solve for Q• Find the second derivative

and check that it is negative 2

2

QdTRd

Maximum Profit

• For maximum profit, = TR – TC• Substitute the expressions for TR

and TC in the profit function so = f(Q)

• Differentiate the profit function with respect to output, Q

• Set the derivative equal to zero and solve for Q

• Find the second derivative and check that it is negative

2

2

Qdd

Indirect taxation 1

• A lump sum tax, T, increases fixed cost but does not affect marginal cost or average variable cost

• Price and quantity are unchanged• Profit falls by the amount of the

lump sum tax • The effect of the tax falls on the

producer

Indirect taxation 2

• A per unit tax, t, shifts the average and marginal cost curves up by the amount of the tax and total cost increases by t.Q, where Q is the quantity of output sold

• Price rises and quantity falls• Profit is reduced • The effect of a per unit tax is

shared between the producer and buyers of the good

Minimum Average Cost

• At the minimum point of ACAC = MC

• Marginal Cost intersects Average Cost at the minimum point of the AC curve

Average and Marginal Product of Labour

• When average product is maximized, APL=MPL

• The MPL curve intersects the APL curve at that point

• MPL reaches a maximum at a lower value of L than that where APL is a maximum

• After the maximum of MPL there are diminishing marginal returns, since the marginal product of labour is falling

Tax Rate which Maximizes Tax Revenue

• To find the per unit rate of tax, t, which maximizes tax revenue

• Write the supply and demand equations in the form P = f(Q)

• Equate these and solve for Q in terms of t, finding an equilibrium expression for Q

• Multiply by t to find tax revenue tQ• Differentiate with respect to t and set

= 0 for a maximum

Minimizing Total Inventory Costs

• To find economic order quantity EOQ, choose Q to minimizeTotal Inventory Cost =

• Differentiate with respect to Q and set = 0 for a minimum

2Q

CQD

C HO

Further Rules of Differentiation –

Mathematics Objectives• Appreciate when further rules

of differentiation are needed• Differentiate composite

functions using the chain rule• Use the product rule of

differentiation• Apply the quotient rule

Further Rules of Differentiation – Economics

Objectives• Show the relationship between

marginal revenue, elasticity and maximum total revenue

• Analyse optimal production and cost relationships

• Differentiate natural logarithmic and exponential functions

• Use logarithmic and exponential relationships in economic analysis

Chain Rule

• If y = f(u) where u = g(x)

• =

• Chain rule: multiply the derivative of the outer function by the derivative of the inner function

xu

uy

dd

dd.

xy

dd

Product Rule

• If y = f(x)g(x)• u = f(x), v = g(x)

• = v. + u.

• Product rule: the derivative of the first term times the second plus the derivative of the second term times the first

xu

dd

xv

dd

xy

dd

Quotient rule

• If y = f(x)/g(x)• u = f(x), v = g(x)

• Quotient rule: the derivative of the first term times the second minus the derivative of the second term times the first, all divided by the square of the second term

2

..

vxv

uxu

v

xy d

ddd

dd

Marginal Revenue, Price Elasticity and Maximum

Total Revenue

• For any demand curve, given that E is point price elasticity of demand and is negative

and maximum total revenue occurs when E = – 1

EP

11MR

Optimal Production and Cost Relationships

• Maximum output occurs where dQ/dL = 0

• A firm operating in perfectly competitive product and labour markets:has short-run marginal cost curve

MC = W/MPL where MPL is the marginal product of labour and W is the wage rate

to maximize profits, it employs labour until MVP = W

where P is the price of its product and MVP = P.MPL is the marginal value product of labour

Marginal and average cost

• MC is below AC before a minimum turning point of AC

• At the turning point of AC, MC intersects AC from below

Exponential Functions

• For the exponential function y = ex

• = ex

• More generally we can write the rule as shown below:

• For the exponential function y = aemx

• = maemx

xy

dd

xy

dd

Natural Logarithmic Functions 1

• If y = loge x

= x –1

xy

dd

Natural Logarithmic Functions 2

• More generally: if y = loge mx

= x –1

• and if y = loge axm

xxy 1

dd

xxy m

dd