Differentiation in Economics – Objectives 1 - PowerPoint PPT Presentation

• 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

• 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

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

• For a linear function
• The rate of change of y with respect to x is measured by
• The slope of the line =

• 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

• 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

• 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

• 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

• For small changes Dx it is approximately true that

D y = Dx.

• We can use this formula to predict the effect on y,Dy, of a small change in x, Dx
• This method is approximate and is valid only for small changes in x

• The Constant Rule

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

= 0

Power-Function Rule

• If y = axn where a and n are constants

= n.axn–1

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

Constant Times a Function Rule

• 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

• When differentiating power functions, remember the following from the rules of indicesx1 = x x0 = 1

= x – n

x = x0.5 = x1/2

Sum – Difference Rule

• 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

Linear – Function Rule 1

• If y = c + mx

= m

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

Linear – Function Rule 2

• If y = mx

= m

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

Inverse Function Rule

• 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

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

• To find an expression for marginal utility, differentiate the total utility function
• If total utility is given by U = f(x)
• MU =

• To find marginal revenue, MR, differentiate total revenue, TR, with respect to quantity, Q
• If TR = f(Q)
• MR =

• 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 =

• 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

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

• 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

• 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

• 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 = kPwhere k is a constant has E = 1 at all prices

• 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

1. Write down the equilibrium condition for the economy

Y = AD

Income = Aggregate Demand

2. Write an expression for AD

AD = 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)

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

• 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

• 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

• Sign of around a turning point:

before at critical value after

• Maximum + 0 –
• Minimum – 0 +

• 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

• 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

• 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

• 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

• For maximum profit, p = TR – TC
• Substitute the expressions for TR and TC in the profit function so p = 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

• 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

• 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

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

• 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

• 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

• To find economic order quantity EOQ, choose Q to minimize

Total Inventory Cost =

• Differentiate with respect to Q and

set = 0 for a minimum

• 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

• 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

• If y = f(u) where u = g(x)
• =
• Chain rule: multiply the derivative of the outer function by the derivative of the inner function

• 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

• 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

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

and maximum total revenue occurs when E = – 1

• 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 = Wwhere P is the price of its product and

MVP = P.MPL is the marginal value product of labour

• MC is below AC before a minimum turning point of AC
• At the turning point of AC, MC intersects AC from below

• For the exponential function y = ex
• = ex
• More generally we can write the rule as shown below:
• For the exponential function y = aemx
• = maemx

• If y = logex

= x –1

• More generally: if y = loge mx

= x –1

• and if y = loge axm