DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIUES 1 LECTURE 4 DIFFERENTIATION 1 Differentiation Managers are often concerned with the way that a variable changes over time Prices, for example, tend to increase with inflation; sales vary from month to month; the number of people employed changes each week and so on It would be useful to have a means to monitor such change Differential calculus is a branch of mathematics concerned with the notion of rate of change We earlier defined the slope of a straight line as the change in function value ( Δy ) per unit change in x ( Δ x) It was also noticed that the slope of a straight line is the gradient when measured between any two points on the line The situation is, however, different for a non-linear function As the slope varies along the curve for Δy non-linear functions the difference quotient ( ) will generally provide a poor Δ x approximation of the slope of the curve for any interval y 1
Δ y Δ x y = f( x) Figure 1 Measuring the Slope of a Non-linear Function x Consider the function y = f( x) : y T C f ( x) Δy C T Δ x x x + Δ x x Figure At x = x, f ( x) = f( x ) Suppose x is changed to ( x + Δ x ), the value of the function is: y +Δ y = f( x +Δ x) (1)
The change in the functional value is: Δ y= f( x +) y = () The rate of change of f ( x ) is: Geometrically, f ( x +Δ x) ) Δy Δ y = f ( x + x) f( x ) (3) is the slope of the chord CC (passing through f ( x ) and As Δ x becomes smaller, the slope of CC approaches the slope of the line TT The slope of this line (if it exists) is written as Assuming the limit exists, f ( x + Δ x ) f ( x ) (4) Equation (4) is known as the limiting condition The limit of Δ y Δ x (the difference quotient) is called the derivative of f ( x ) at x = x It represents the slope of the curve at x EXAMPLE 1 Consider the profit function: π = 115 + 5 5, (5) 3
where π is profit and is output To find the derivative of (5), the difference quotient at any value of is: Clearly, Δπ d lim ( ) = 5 1, hence π = 5 1 Δ d Δ 3 Rules for Differentiation The method for finding the derivative of a function is imprecise Fortunately simple rules exist for differentiating the kinds of function frequently used in economics But first consider the general linear equation: y = a+ bx (6) the derivative of the function is b Proof: f( x + ) f( x) a+ b( x+) ( a+ bx) a+ bx+ b a bx b b = b This is the derivative for any linear function: y = a+ bx or b = For a quadratic function: y = a+ bx+ cx the derivative is 4
For a cubic function: the derivative is b c = + x 3 y = a+ bx+ cx + b cx 3 = + + Note the derivatives are obtained if the following rule is applied to each term in the equations: y = ax n is EXAMPLE n 1 = nax (7) π = 115 + 5 5 = + 1 115 5 5 dπ = 5 1 d The rule can be proven by applying the derivate formula to the general quadratic, cubic and higher-order equations The following derivative rules are provided without proof Expression Function Derivative y = f( x) = f ( x) Constant Function y = a = Power Function y = ax (Polynomial) Exponential Function y = e (to base e) n f ( x) = nax = n 1 f ( xe ) f x ( ) 5
Natural Log Function (to base e) Sum of Functions Multiplication by a Constant Product of Functions y = ln f( x) uotient of Functions f ( x) y gx ( ) f ( x) = f ( x) y = f( x) + g( x) = f ( x) + g ( x) y = k f( x) = kf ( x) y = f( x) g( x) = f ( xgx ) ( ) + f( xg ) ( x) f ( x) g( x) f ( x) g ( x) ( g( x)) = = Inverse Function x = f 1 ( y) 1 = f ( x) Table1 Derivative Rules 7 Derivative of a Composite Function (Chain Rule) Many functions can be considered composite functions of several functions Let y = f( x) and z = g( y) then z = h( x) = g( f( x)) is a composite function The derivative dz is obtained by applying dz dz = or chain rule 8 More Definitions 6
d ( TR) Marginal Revenue (MR) = d d ( TC) Marginal Cost (MC) = d Average Revenue (AR) = Average Cost (AC) = (TC) (TR) Average Cost (AC) = AFC + AVC EXAMPLE 3 15 The demand function for a good is P = 15 (a) Find expressions for TR, MR and AR (b) Evaluate TR, MR and AR at = 1 and = 5 (c) Calculate the value of for which MR = (d) Calculate the value of for which AR = (e) Graph TR, MR and AR on the same diagram EXAMPLE 4 15 A firm has average cost function AC = 9 + + 75 (a) Find an expression for TC and calculate TC when =15 (b) Write down the equations for FC and TVC (c) Find the MC function (d) Show that MC > for all Plot the MC and AC curves on the same diagram 7