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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 d y /d x for power functions and practise the basic rules of differentiation - PowerPoint PPT Presentation
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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
