Comparative statics, as the name suggests, is concerned with the comparison of different equilibrium states that are associated with different sets of values of parameters and exogenous variables. For purposes of such a comparison, we always start by assuming a given initial equilibrium state. In the isolated-market model, for example, such an initial equilibrium will be represented by a determinate price P* and a corresponding quantity Q*. Similarly, in the simple national-income model of (3.23), the initial equilibrium will be specified by a determinate F and a corresponding C*\ Now if we let a disequilibrating change occur in the model in the form of a change in the value of some parameter or exogenous variable the initial equilibrium will, of course, be upset. As a result, the various endogenous variables musr undergo certain adjustments. If it is assumed that a new equilibrium state relevant to the new values of the data can be defined and attained, the question posed in the comparative-static analysis is: How would the new equilibrium compare with the old?
It should be noted that in comparative statics we still disregard the process of adjustment of the variables; we merely compare the initial (/^change) equilibrium state with the final (/josichange) equilibrium state. Also, we still preclude the possibility of instability of equilibrium, for we assume the new equilibrium to be attainable, just as we do for the old.
A comparative-static analysis can be either qualitative or quantitative in nature, [fwc are interested only in the question of, say, whether an increase in investment Aj will increase or decrease the equilibrium income the analysis will be qualitative because the direction of change is the only matter considered. But if wc are concerned with the magnitude of the change in Y* resulting from a given change in I(i (that is. the size of the investment multiplier), the analysis will obviously be quantitative. By obtaining a quantitative answer, however, we can automatically tell the direction of change from its algebraic sign. Hence the quantitative analysis always embraces the qualitative.
It should be clear that 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. For this reason, the mathematical concept of derivative takes on preponderant significance in comparative statics, because that concept the most fundamental one in the branch of mathematics known as differential calculus -is directly concerned with the notion of rate of changc! Later on, moreover, we shall find the concept of derivative to be of extreme importance Cor optimi7ation problems as well.
Even though our present context is concerned only with the rates of change of the equilibrium values of the variables in a model, we may carry on the discussion in a more general manner by considering the rate of change of any variable y in response to a change in another variable x, where the two variables are related to each other by the function
Applied to the comparative-static context, the variable y will represent the equilibrium value of an endogenous variable, and* will be some parameter. Note that, for a start, we are restricting ourselves to the simple case where there is only a single parameter or exogenous variable in the model. Once we have mastered this simplified case, however, the extension to the ease of more parameters will prove relatively easy.
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