ECON2285: Mathematical Economics
                                                       Yulei Luo
                                                       SEF of HKU
                                                September 9, 2017
          Luo, Y. (SEF of HKU)                               ME                                September 9, 2017     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’t concern with the process of
            adjustment of the variables; we merely compare the initial equilibrium
            state with the …nal equilibrium.
          Luo, Y. (SEF of HKU)                               ME                                September 9, 2017     2 / 44
            (Continued.) 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.
            Wewill 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 x , to another point, say x . Thus ∆x = x x .
                                             0                                      1                        1       0
            When x changes from x to x +∆x, the value of the function
                                                 0        0
            y = f (x) changes from f (x ) to f (x + ∆x). The change in y per
                                                       0             0
            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)                               ME                                September 9, 2017     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 = y0 = f0(x) = lim ∆y                                                (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 = y0 = 6x.
                                                       dx
          Luo, Y. (SEF of HKU)                               ME                                September 9, 2017     4 / 44