012

Then (3.13) becomes—after transposing the c0 and y0 terms to the right-hand side of the equals sign:

which may be solved by further elimination of variables. From the first equation, it can be found that P2 = -(c0 + c]P])/c2. Substituting this into the second equation and solving, we get

Note that P, is entirely expressed, as a solution value should be, in terms of the data (parameters) of the model. By a similar process, the equilibrium price of the second commodity is found to be

For these two values to make sense, however, certain restrictions shoukl be imposed on the model. First, since division by zero is undefined, we must require the common denominator of (3.14) and (3.15) to be nonzero, that is, c,y2 c'iYi-Second, to assure positivity, the numerator must have the same sign as the denominator.

The equilibrium prices having been found, the equilibrium quantities Q\ and Q1 can readily be calculated by substituting (3.14) and (3.15) into the second (or third) equation and the fifth (or sixth) equation of (3.12). These solution values will naturally also be expressed in terms of the parameters. (Their actual calculation is left to you as an exercise.)

Numerical Example

Suppose that the demand and supply functions are numerically as follows:

What will be the equilibrium solution?

Before answering the question, let us take a look at the numerical coefficients. For each commodity, Qsj is seen to depend on P; alone, but Qdl is shown as a function of both prices. Note that while P, has a negative coefficient in Qdl, as we would expect, the coefficient of P2 is positive. The fact that a rise in P2 tends to raise Qd] suggests that the two commodities are substitutes for each other. The role of P| in the Qd2 function has a similar interpretation.

With these coefficients, the shorthand symbols c, and y, will take the following values:

By direct substitution of these into (3.14) and (3.15), we obtain

And the further substitution of P} and P2 into (3.16) will yield

Thus all the equilibrium values turn out positive, as required. In order to preserve the exact values of P, and P2 to be used in the further calculation of Q, and Q2, it is advisable to express them as fractions rather than decimals.

Could we have obtained the equilibrium prices graphically? The answer is yes. From (3.13), it is clear that a two-commodity model can be summarized by two equations in two variables Pt and P2. With known numerical coefficients, both equations can be plotted in the PtP2 coordinate plane, and the intersection of the two curves will then pinpoint P] and P2.

«-Commodity Case

The above discussion of the multicommodity market has been limited to the case of two commodities, but it should be apparent that we are already moving from partial-equilibrium analysis in the direction of general-equilibrium analysis. As more commodities enter into a model, there will be more variables and more equations, and the equations will get longer and more complicated. If all the commodities in an economy are included in a comprehensive market model, the result will be a Walrasian type of general-equilibrium model, in which the excess demand for every commodity is considered to be a function of the prices of all the commodities in the economy.

Some of the prices may, of course, carry zero coefficients when they play no role in the determination of the excess demand of a particular commodity; e.g., in the excess-demand function of pianos the price of popcorn may well have a zero coefficient. In general, however, with n commodities in all, we may express the demand and supply functions as follows (using qdi and qst as function symbols in place of / and g):

In view of the index subscript, these two equations represent the totality of the 2n functions which the model contains. (These functions are not necessarily linear.) Moreover, the equilibrium condition is itself composed of a set of n equations,

When (3.18) is added to (3.17), the model becomes complete. You should therefore count a total of 3n equations.

Upon substitution of (3.17) into (3.18), however, the model can be reduced to a set of n simultaneous equations only:

qdi(pi,p2,...,pn)-qsl(pl.p2 ,pn) = 0 (/= 1,2,...,«)

Besides, inasmuch as £, = qdl — qsr where ei is necessarily also a function of all the n prices, the above set of equations may be written alternatively as ei(pl,p2,...,pn) = 0 (/= 1,2,...,«)

Solved simultaneously, these « equations will determine the « equilibrium prices pl—if a solution does indeed exist. And then the qt may be derived from the demand or supply functions.

Solution of a General-Equation System

If a model comes equipped with numerical coefficients, as in (3.16), the equilibrium values of the variables will be in numerical terms, too. On a more general level, if a model is expressed in terms of parametric constants, as in (3.12), the equilibrium values will also involve parameters and will hence appear as "for mulas," as exemplified by (3.14) and (3.15). If, for greater generality, even the function forms are left unspecified in a model, however, as in (3.17), the manner of expressing the solution values will of necessity be exceedingly general as well.

Drawing upon our experience in parametric models, we know that a solution value is always an expression in terms of the parameters. For a general-function model containing, say, a total of m parameters (a,, a2,..., am)—where m is not necessarily equal to n—the n equilibrium prices can therefore be expected to take the general analytical form of

This is a symbolic statement to the effect that the solution value of each variable (here, price) is a function of the set of all parameters of the model. As this is a very general statement, it really does not give much detailed information about the solution. But in the general analytical treatment of some types of problem, even this seemingly uninformative way of expressing a solution will prove of use, as will be seen in a later chapter.

Writing such a solution is an easy task. But an important catch exists: the expression in (3.19) can be justified if and only if a unique solution does indeed exist, for then and only then can we map the ordered w-tuple (a,, a2,..., am) into a determinate value for each price Pr Yet, unfortunately for us, there is no a priori reason to presume that every model will automatically yield a unique solution. In this connection, it needs to be emphasized that the process of "counting equations and unknowns" does not suffice as a test. Some very simple examples should convince us that an equal number of equations and unknowns (endogenous variables) does not necessarily guarantee the existence of a unique solution.

Consider the three simultaneous-equation systems

In (3.20), despite the fact that two unknowns are linked together by exactly two equations, there is nevertheless no solution. These two equations happen to be inconsistent, for if the sum of x and y is 8, it cannot possibly be 9 at the same time. In (3.21), another case of two equations in two variables, the two equations are functionally dependent, which means that one can be derived from (and is implied by) the other. (Here, the second equation is equal to two times the first equation). Consequently, one equation is redundant and may be dropped from the system, leaving in effect only one equation in two unknowns. The solution will then be the equation^ = 12 — 2x, which yields not a unique ordered pair (3c, y) but an infinite number of them, including (0,12), (1,10), (2, 8), etc., all of which satisfy that equation. Lastly, the case of (3.22) involves more equations than unknowns, yet the ordered pair (2,18) does constitute the unique solution to it. The reason is that, in view of the existence of functional dependence among the equations (the first is equal to the second plus twice the third), we have in effect only two independent, consistent equations in two variables.

These simple examples should suffice to convey the importance of consistency and functional independence as the two prerequisites for application of the process of counting equations and unknowns. In general, in order to apply that process, make sure that (1) the satisfaction of any one equation in the model will not preclude the satisfaction of another and (2) no equation is redundant. In (3.17), for example, the n demand and n supply functions may safely be assumed to be independent of one another, each being derived from a different source—each demand from the decisions of a group of consumers, and each supply from the decisions of a group of firms. Thus each function serves to describe one facet of the market situation, and none is redundant. Mutual consistency may perhaps also be assumed. In addition, the equilibrium-condition equations in (3.18) are also independent and presumably consistent. Therefore the analytical solution as written in (3.19) can in general be considered justifiable.*

For simultaneous-equation models, there exist systematic methods of testing the existence of a unique (or determinate) solution. These would involve, for linear models, an application of the concept of determinants, to be introduced in Chap. 5. In the case of nonlinear models, such a test would also require a knowledge of so-called "partial derivatives" and a special type of determinant called the Jacobian determinant, which will be discussed in Chaps. 7 and 8.

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