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ECON2285: Mathematical Economics ECON2285: Mathematical Economics Yulei Luo FBE, HKU September 2, 2018 Luo, Y. (FBE, HKU) ME September2,2018 1/44 Comparative Statics and The Concept of Derivative Comparative Statics is concerned with the comparison of different equilibrium states that are associated with different 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 final equilibrium. Luo, Y. (FBE, HKU) ME September2,2018 2/44 (Continued.) The problem under consideration is essentially one of finding 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 difference quotient. We use the symbol D to denote the change from one point, say x0, to another point, say x1. Thus Dx = x1 x0. When x changes from x0 to x0 + Dx, the value of the function y = f (x) changes from f (x0) to f (x0 + Dx). The change in y per unit of change in x can be expressed by the difference quotient: Dy f (x0 + Dx) f (x0) = (2) Dx Dx Luo, Y. (FBE, HKU) ME September2,2018 3/44 Quick Review of Derivative, Differentiation, and Partial Differentiation The derivative of the function y = f (x) is the limit of the difference quotient Dy exists as Dx 0. The derivative is denoted by Dx ! dy Dy = y 0 = f 0(x) = lim (3) dx Dx 0 Dx ! 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 difference quotient; since Dx 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. dx 0 Luo, Y. (FBE, HKU) ME September2,2018 4/44 (Continued.) 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 finite 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 (4) v N v N + ! ! and is denoted by limv N q = L. Note that L must be a finite 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. (FBE, HKU) ME September2,2018 5/44 (Continued.) The concept of differentiability. By the definition of the derivative of a function y = f (x), at x0, we know that f 0(x0) exists if Dy and only if the limit of the difference quotient Dx exists at x = x0 as Dx 0, that is, ! Dy f 0 (x0) = lim Dx 0 Dx ! f (x0 + Dx) f (x0) = lim (Differentiation condition). Dx 0 Dx ! dy Differentiation 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). x x0 ! Luo, Y. (FBE, HKU) ME September2,2018 6/44 (Continued.) Continuity and differentiability are very closely related to each other. Continuity of a function is a necessary condition for its differentiability, but this condition is not suffi cient. Consider example: f (x) = x , which is clearly continuous at x = 0, but is not differentiablej j at x = 0. Continuity rules out the presence of a gap, whereas differentiability rules out sharpness as well. Therefore, differentiability calls for smoothness of the function as well as its continuity. Most of the specific functions used in economics are differentiable everywhere. Luo, Y. (FBE, HKU) ME September2,2018 7/44 Rules of Differentiation The central problem of comparative-static analysis, that of finding a rate of change, can be identified with the problem of finding 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. dx 0 Power function rule: if y = f (x) = x a where a ( ¥, ¥) is any real number, then 2 dy = ax a 1. dx Power function rule generalized: if y = f (x) = cx a, then dy = cax a 1. dx Luo, Y. (FBE, HKU) ME September2,2018 8/44 (Continued. For two or more functions of the same variable) Sum-difference rule: d d d [f (x) g (x)] = f (x) g (x) = f 0 (x) + g 0 (x) , dx ± dx ± dx which can easily extend to more functions d n n d n ∑ fi (x) = ∑ fi (x) = ∑ fi0 (x) . 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) dx dx dx = f (x) g 0 (x) + g (x) f 0 (x) Quotient rule: d f (x) f (x) g (x) g (x) f (x) = 0 0 . dx g (x) g 2 (x) Luo, Y. (FBE, HKU) ME September2,2018 9/44 Marginal-revenue Function and Average-revenue Function Suppose that the average-revenue function (AR) is specified by AR = 15 Q,then the total revenue function (TR) is TR = AR Q = 15Q Q2, (5) · which means that the marginal revenue (MR) function is given by d(TR) MR = = 15 2Q. (6) dQ In general, if AR = f (Q) , then TR = f (Q) Q and MR = f (Q) + f 0 (Q) Q, 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 firm. If the market is perfect competition, that is, the firm takes the price as given, then P = f (Q) =constant, which means that f 0 (Q) = 0 and thus MR = AR. Luo, Y. (FBE, HKU) ME September2,2018 10/44 Marginal-cost Function and Average-cost Function Suppose that a total cost function is C = C (Q) , (7) the average cost (AC) function and the marginal cost (MC) function are given by C (Q) AC = and MC = C (Q) . 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 iff 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 : : | {z } | {z } Luo, Y. (FBE, HKU) ME September2,2018 11/44 Rules of Differentiation Involving Functions of Different Variables Consider cases where there are two or more differentiable 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) . (8) dx dy dx 0 0 Intuition: Given a Dx, there must result a corresponding Dy via the function y = g(x), but this Dy will in turn being about a Dz 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) . (9) dL dQ dL 0 0 Luo, Y. (FBE, HKU) ME September2,2018 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 different value of x will always yield a different value of y, then function f will have an inverse function 1 x = f (y) , (10) 1 note that here the symbol f doesn’tmean the reciprocal of the function f (x). For monotonic functions, the corresponding inverse functions exist. Generally speaking, if an inverse function exists, the original and the inverse functions must be both monotonic. For inverse functions, the rule of differentiation is dx 1 = . (11) dy dy dx 5 4 Examples: Suppose that y = x + x, y 0 = 5x + 1 and dx 1 = . dy 5x4 + 1 Luo, Y. (FBE, HKU) ME September2,2018 13/44 Partial Differentiation In CD analysis, several parameters appear in a model, so that the equilibrium value of each endogenous variable may be a function of more than one parameter.
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