Differentiation in Economics – Objectives 1

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

2. 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

3. 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

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

5. 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 =

6. 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

7. 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

8. 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

9. Tangents at points A, B and CThe slope of the tangent at A is steeper than that at B; the tangent at C has a negative slope

10. 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

11. Small Increments Formula • 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

12. 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 = 0

13. 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

14. 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

15. Indices in Differentiation • When differentiating power functions, remember the following from the rules of indicesx1 = x x0 = 1 = x – n x = x0.5 = x1/2

16. 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

17. Linear – Function Rule 1 • If y = c + mx = m • The derivative of a linear function is the slope of the line 

18. 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

19. 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

20. 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

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

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

23. 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 =

24. 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

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

26. 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

27. 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

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

29. 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

30. Finding the Investment Multiplier 1 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)

31. 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

32. 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

33. 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

34. Derivatives and Turning Points • Sign of around a turning point: before at critical value after • Maximum + 0 – • Minimum – 0 +

35. 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

36. 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

37. 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

38. 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

39. Maximum Profit • 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

40. 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

41. 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

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

43. 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

44. 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

45. Minimizing Total Inventory Costs • To find economic order quantity EOQ, choose Q to minimize Total Inventory Cost = • Differentiate with respect to Q and set = 0 for a minimum

46. 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

47. 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

48. 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

49. 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

50. 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