ECON0702: Mathematical Methods in Economics Yulei Luo SEF of HKU January 18, 2008 Luo, Y. (SEF of HKU) MME January 18, 2008 1 / 44 Comparative Statics and The Concept of Derivative Comparative Statics is concerned with the comparison of di¤erent equilibrium states that are associated with di¤erent sets of values of parameters and exogenous variables. When the value of some parameter or exogenous variable that is associated with an initial equilibrium changes, we can get a new equilibrium. The question posted in the Comparative Statics analysis is: How would the new equilibrium compare with the old one? Note that in the CS analysis, we don’tconcern with the process of adjustment of the variables; we merely compare the initial equilibrium state with the …nal equilibrium. Luo, Y. (SEF of HKU) MME January 18, 2008 2 / 44 (Conti.) The problem under consideration is essentially one of …nding a rate of change: the rate of change of the equilibrium value of an endogenous variable with respect to the change in a particular parameter or exogenous variable. Hence, the concept of derivative is the key factor in comparative statics analysis. We will study the rate of change of any variable y in response to a change in another variable x : y = f (x). (1) Note that in the CS analysis context, y represents the equilibrium value of an endogenous variable, and x represents some parameter or exogenous variable. The di¤erence quotient. We use the symbol ∆ to denote the change from one point, say x0, to another point, say x1. Thus ∆x = x1 x0. When x changes from x0 to x0 + ∆x, the value of the function y = f (x) changes from f (x0) to f (x0 + ∆x). The change in y per unit of change in x can be expressed by the di¤erence quotient: ∆y f (x0 + ∆x) f (x0) = (2) ∆x ∆x Luo, Y. (SEF of HKU) MME January 18, 2008 3 / 44 Quick Review of Derivative, Di¤erentiation, and Partial Di¤erentiation The derivative of the function y = f (x) is the limit of the di¤erence quotient ∆y exists as ∆x 0. The derivative is denoted by ∆x ! dy ∆y = y 0 = f 0(x) = lim (3) dx ∆x 0 ∆x ! Note that (1) a derivative is also a function; (2) it is also a measure of some rate of change since it is merely a limit of the di¤erence quotient; since ∆x 0, the rate measured by the derivative is an instantaneous rate! of change; and (3) the concept of the slope of a curve is merely the geometric counterpart of the concept of derivative. Example: If y = 3x2 4, dy = y = 6x. (4) dx 0 Luo, Y. (SEF of HKU) MME January 18, 2008 4 / 44 The concept of limit. For a given function q = g(v), if, as v N (it can be any number) from the left side (from values less than !N), q approaches a …nite number L, we call L the left-side limit of q. Similarly, we call L the right-side limit of q. The left-side limit and right-side limit of q are denoted by lim q and lim + q, v N v N respectively. The limit of q at N is said! to exist if ! lim q = lim q = L (5) v N v N + ! ! and is denoted by limv N q = L. Note that L must be a …nite number. ! The concept of continuity. A function q = g(v) is said to be continuous at N if limv N q exists and limv N g(v) = g(N). Thus continuity involves the following! requirements:! (1) the point N must be in the domain of the function; (2) limv N g(v) exists; and (3) ! limv N g(v) = g(N). ! Luo, Y. (SEF of HKU) MME January 18, 2008 5 / 44 The concept of di¤erentiability. By the de…nition of the derivative of a function y = f (x), at x0, we know that f 0(x0) exists if and only if ∆y the limit of the di¤erence quotient exists at x = x0 as ∆x 0, ∆x ! that is, ∆y f 0 (∆x) = lim ∆x 0 ∆x ! f (x0 + ∆x) f (x0) = lim (Di¤erentiation condition).(6) ∆x 0 ∆x ! dy Di¤erentiation is the process of obtaining the derivative dx . Note that the function y = f (x) is continuous at x0 if and only if lim f (x) = f (x0) (Continuity condition). (7) x x0 ! Luo, Y. (SEF of HKU) MME January 18, 2008 6 / 44 Continuity and di¤erentiability are very closely related to each other. Continuity of a function is a necessary condition for its di¤erentiability, but this condition is not su¢ cient. Consider example: f (x) = x , which is clearly continuous at x = 0, but is not di¤erentiablej j at x = 0. Continuity rules out the presence of a gap, whereas di¤erentiability rules out sharpness as well. Therefore, di¤erentiability calls for smoothness of the function as well as its continuity. Most of the speci…c functions used in economics are di¤erentiable everywhere. Luo, Y. (SEF of HKU) MME January 18, 2008 7 / 44 Rules of Di¤erentiation The central problem of comparative-static analysis, that of …nding a rate of change, can be identi…ed with the problem of …nding the derivative of some function f (x), provided only a small change in x is being considered. Constant function rule: for y = f (x) = c, where c is a constant, then dy = f (x) = 0. (8) dx 0 Power function rule: if y = f (x) = x α where α ( ∞, ∞) is any real number, then 2 dy = αx α 1. (9) dx Power function rule generalized: if y = f (x) = cx α, then dy = cαx α 1. (10) dx Luo, Y. (SEF of HKU) MME January 18, 2008 8 / 44 (continue. For two or more functions of the same variable) Sum-di¤erence rule: d d d [f (x) g (x)] = f (x) g (x) = f 0 (x) + g 0 (x) , (11) dx dx dx which can easily extend to more functions d n n d n ∑ fi (x) = ∑ fi (x) = ∑ fi0 (x) . (12) dx "i=1 # i=1 dx i=1 Product rule: d d d [f (x) g (x)] = f (x) g (x) + g (x) f (x) (13) dx dx dx = f (x) g 0 (x) + g (x) f 0 (x) (14) Quotient rule: d f (x) f (x) g (x) g (x) f (x) = 0 0 . (15) dx g (x) g 2 (x) Luo, Y. (SEF of HKU) MME January 18, 2008 9 / 44 Marginal-revenue Function and Average-revenue Function Suppose that the average-revenue function (AR) is speci…ed by AR = 15 Q,then the total revenue function (TR) is TR = AR Q = 15Q Q2, (16) which means that the marginal revenue (MR) function is given by d(TR) MR = = 15 2Q. (17) dQ In general, if AR = f (Q) , then TR = f (Q) Q and MR = f (Q) + f 0 (Q) Q, (18) which means that MR AR = f 0 (Q) Q. TR PQ Note that since AR = Q = Q = P, we can view AR as the inverse demand function for the product of the …rm. If the market is perfect competition, that is, the …rm takes the price as given, then P = f (Q) =constant, which means that f 0 (Q) = 0 and thus MR = AR. Luo, Y. (SEF of HKU) MME January18,2008 10/44 Marginal-cost Function and Average-cost Function Suppose that a total cost function is C = C (Q) , (19) the average cost (AC) function and the marginal cost (MC) function are given by C (Q) AC = and MC = C (Q) . (20) Q 0 The rate of change of AC with respect to Q is > 0 MC > AC d C (Q) 1 C (Q) 2 3 = 2C 0 (Q) 3 = = 0 i¤ MC = AC dQ Q Q Q 8 8 < 0 MC < AC 6 7 6 MC 7 < < 6 AC 7 6 AC 7 4 5 4| {z } 5 : : (21) | {z } | {z } Luo, Y. (SEF of HKU) MME January18,2008 11/44 Rules of Di¤erentiation Involving Functions of Di¤erent Variables Consider cases where there are two or more di¤erentiable functions, each of which has a distinct independent variables, Chain rule: If we have a function z = f (y), where y is in turn a function of another variable x, say, y = g(x), then the derivative of z with respect to x gives by dz dz dy = = f (y) g (x) . (22) dx dy dx 0 0 Intuition: Given a ∆x, there must result a corresponding ∆y via the function y = g(x), but this ∆y will in turn being about a ∆z via the function z = f (y). Example: Suppose that total revenue TR = f (Q) , where output Q is a function of labor input L, Q = g (L) . By the chain rule, the marginal revenue of labor is dTR dTR dQ MRL = = = f (Q) g (L) . (23) dL dQ dL 0 0 Luo, Y. (SEF of HKU) MME January18,2008 12/44 Inverse function rule: If the function y = f (x) represents a one-to-one mapping, i.e., if the function is such that a di¤erent value of x will always yield a di¤erent value of y, then function f will have an inverse function 1 x = f (y) , (24) 1 note that here the symbol f doesn’tmean the reciprocal of the function f (x).