Constructing the Model

Since only one commodity is being considered, it is necessary to include only three variables in the model: the quantity demanded of the commodity (Qd\ the quantity supplied of the commodity (gT), and its price {P). The quantity is measured, say, in pounds per week, and the price in dollars. Having chosen the variables, our next order of business is to make certain assumptions regarding the working of the market. First, we must specify an equilibrium condition- something indispensable in an equilibrium model. The standard assumption is that equilibrium occurs in the market if and only if the excess demand is zero {Qd - Qs ā€” 0), that is, if and only if the market is cleared. But this immediately raises the question of how Qd and Qs themselves are determined. To answer this, we assume that Q(f is a decreasing linear function of P (as P increases, Qd decreases). On the other hand, Qs is postulated to be an increasing linear function of P (as P increases, so does X w'th the proviso that no quantity is supplied unless the price cxceeds a particular positive level. In all, then, the model will contain one equilibrium condition plus two behavioral equations which govern the demand and supply sides of the market, respectively.

FIGURE 3,1

Translated into mathematical statements, the model can be written as

Four parameters, a, b, c, and d, appear in the two linear functions, and all of them are specified to be positive. When the demand function is graphed, as in Fig. 3.1, its vertical intercept is at a and its slope is -A, which is negative, as required. The supply function also has the required type of slope, d being positive, but its vertical intercept is seen to be negative, at -c. Why did wc want to specify such a negative vertical intercept? The answer is that, in so doing, we force the supply curve to have a positive horizontal intercept at P]: thereby satisfying the proviso slated earlier that supply will not be forthcoming unless the price is positive and sufficiently high.

The reader should observe that, contrary to the usual practice, quantity rather than price has been plotted vertically in Fig. 3.1. This, however, is in line with the mathematical convention of placing the dependent variable on the vertical axis. In a different context in which the demand curve is viewed from the standpoint of a business firm as describing the average-revenue curve, AR = Pā€” j\Q(i)? we shall reverse the axes and plot P vertically.

With the model thus constructed, the next step is to solve it, i.e., to obtain the solution values of the three endogenous variables, Qd? QJf and P. The solution values are those values that satisfy the three equations in (3.1) simultaneously; i.e., they are the values which, when substituted into the three equations, make the latter a set of true statements. In the context of an equilibrium model, those values may also be referred to as the equilibrium values of the said variables.

Many writers employ no special symbols to denote the solution values of the endogenous variables. Thus, Qd is used to represent cither the quantity-demanded variable (with a whole range of values) or its solution value (a specific value); and similarly for the symbols

Qs andP. Unfortunately, this practice can give rise to possible confusions, especially in the context of comparative-static analysis (e.g., Sec. 7,5). To avoid such a source of confusion, we shall denote the solution value of an endogenous variable with an asterisk. Thus, the solution values of Qd, Op and P, are denoted by Q*d, Q*y and P% respectively. Since Qd = Qt> however, they can even be replaced by a single symbol Q*. Hence, an equilibrium solution of the model may simply be denoted by an ordered pair (P*, Q*), In case the solution is not unique, several ordered pairs may each satisfy the system of simultaneous equations; there will then be a solution set with more than one element in it. However, the multiple-equilibrium situation cannot arise in a linear model such as the present one.

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