DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 10 LECTURE NON-LINEAR FUNCTIONS 0. Preliminaries The following functions will be discussed briefly first: Quadratic functions and their solutions Cubic functions Other Polynomial Functions 1. Non-linear Functions Linear functions are of limited applicability to real-life situations, hence the need for more versatile non-linear models. For eample, a revenue function of the type TR = PQ (1) is reasonable for a perfectively competitive firm that faces a constant price. Should the firm be a monopolist then (1) is not appropriate, as total revenue depends on Q. In that case, TR = P( Q) Q. () Further, with an inverse demand curve, P = 0 Q, () the corresponding revenue function is: TR = (0 Q) Q (4) = 0 Q Q which is non-linear. In particular, the revenue function is quadratic in Q, and has the shape: 1
TR = 0 Q
Equation (4) is a member of the class of equations y = a + b + c () where = Q, a =, b= 0 and c= 0. The roots (or solutions) of a quadratic equation are the values of for which the quadratic function equals zero. The roots to () are found using the formula: EXAMPLE 1, 1 b± b 4ac =. (6) a A firm's total cost function is given by the equation TC = 00 + Q, while the demand function is given by the equation P =107 Q. (a) Write down the equation for total revenue (TR) function. (b) Graph the TR function for 0<Q<60. Hence, estimate the output Q, and total revenue when TR is maimum. (c) Plot the total cost function on the diagram in (b). Estimate the break-even point from the graph. Confirm your answer algebraically. (d) State the range of values for which the company makes a profit. EXAMPLE The demand and supply functions for a good are given by the equations: ( 4) P d = Q + + 100, P s = ( Q + ). (a) Sketch each function on the same diagram; hence, estimate the equilibrium price and quantity. (b) Confirm the equilibrium using algebra.
. Polynomial Functions Quadratic functions are members of a larger class of functions called polynomial (multi-term) functions. Their general form is: y = a + a + a + + a 0 1 0 1... n n (7) although (7) is more commonly written, y = a + a + a + + a 0 1... n n (8) as = 1 and =. 0 1 Specific subclasses of polynomials include: Constant Function: y = a 0 (9) where a1 = a =... = a n = 0. Linear Function: where a = a =... = a n = 0. y = a0 + a1 (10) Quadratic Function: y = a + a + a 0 1 (11) where a = a4 =... = a n = 0. Cubic Function: y = a + a + a + a 0 1 (1) should a4 = a =... = a n = 0. 4
Quartic Function: y= a + a+ a + a + a 0 1 4 4 (1) where a = a6 =... = a n = 0. 4. Rational Functions Rational functions are defined as the ratio of one (or more) polynomial functions. For eample, y = 1 + + 4 (14) is a rational function. An important rational function is the rectangular hyperbola, which has the mathematical form, α y = α > 0 (1) is used in economics to represent demand and supply curves. Two other classes of functions important in eplaining growth and decay are logarithmic and eponential functions.. Logarithm to the Base b The logarithm to the base b of a number to yield. It is denoted log b., is the power to which b must be raised EXAMPLE When b = the following values are defined:
= = log1 as 1 = = log as = = log as = = 0 log1 as 1 1 log = as = 1 1 log = as = It should be noted that: (a) the log of < 0 does not eist and (b) the log of 0< < 1 is always negative. 6. Log Rules Several rules for manipulating logarithms are useful. They are: (a) log of a product log( y) = log + log y log ( ) = = = (17) (b) log of a ratio log( ) = log log y y log ( ) = = = (18) (c) lo g of an eponent 6
k log = klog log = = = (19) The definition of logarithm is applicable for any b > 0. The natural (or ln) logarithm has the natural constant e (=.718.) as its base. The natural log is frequently used by economists because it is easy to differentiate. It is denoted: y = log ln. (0) e 7. Eponential Functions Another important class of functions has the form: f ( ) = b (1) where b > 0. Note here the inde () is changing while the base remains constant. The corresponding graph will pass through the ordered pairs: f ( ) 1 0-1 - 8 4 1 1/ 1/4 7
As > 0 increases, f ( ) increases very rapidly. However, as < 0 becomes more negative f ( ) declines slowly. When b= e, the base of the natural logarithm, the function is the natural eponential function. Any economic variable that ehibits a constant proportional growth will behave like an eponential function, e.g., compound interest. EXAMPLE 4 The demand and supply functions for a brand of tennis shoes are: 00 P d = and P s = 16 + Q, where P is the price per pair, Q is the quantity in Q + 1 thousands of pairs. (a) Calculate the equilibrium price and quantity. (b) Graph the supply and demand functions; hence confirm your answer graphically. 8