P2

Substituting the equilibrium price into the demand or supply equation, we obtain the corresponding equilibrium quantity:

where the reduced-form coefficients are a1Po - aoP1

The demand-and-supply model given in Eqs. (19.2.12) and (19.2.22) contain six structural coefficients—a0, a1, a2, p0, p1, and p2—and there are six reduced-form coefficients—n0, n1, n2, n3, n4, and n5—to estimate them. Thus, we have six equations in six unknowns, and normally we should be able to obtain unique estimates. Therefore, the parameters of both the demand-and-supply equations can be identified, and the system as a whole can be identified. (In exercise 19.2 the reader is asked to express the six structural coefficients in terms of the six reduced-form coefficients given previously to show that unique estimation of the model is possible.)

To check that the preceding demand-and-supply functions are identified, we can also resort to the device of multiplying the demand equation (19.2.12)

746 PART FOUR: SIMULTANEOUS-EQUATION MODELS

by X (0 < X < 1) and the supply equation (19.2.22) by 1 — X and add them to obtain a mongrel equation. This mongrel equation will contain both the predetermined variables It and Pt—i; hence, it will be observationally different from the demand as well as the supply equation because the former does not contain Pt—1 and the latter does not contain It.

Overidentification

For certain goods and services, income as well as wealth of the consumer is an important determinant of demand. Therefore, let us modify the demand function (19.2.12) as follows, keeping the supply function as before:

Demand function: Qt = a0 + a1 Pt + a2It + a3Rt + u1t (19.2.28)

Supply function: Qt = Po + P1 Pt + ft Pt—1 + u2t (19.2.22)

where in addition to the variables already defined, R represents wealth; for most goods and services, wealth, like income, is expected to have a positive effect on consumption.

Equating demand to supply, we obtain the following equilibrium price and quantity:

where

Pt = no + n It + nRt + nPt-i + v Qt = n + ns It + n6 Rt + ny Pt-1 + wt no ft - ao ai - ft ni a2

The preceding demand-and-supply model contains seven structural coefficients, but there are eight equations to estimate them—the eight reduced-form coefficients given in (i9.2.31); that is, the number of equations is greater than the number of unknowns. As a result, unique estimation of all

Models

Problem

CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 747

the parameters of our model is not possible, which can be shown easily. From the preceding reduced-form coefficients, we can obtain that is, there are two estimates of the price coefficient in the supply function, and there is no guarantee that these two values or solutions will be identi-cal.4 Moreover, since p1 appears in the denominators of all the reduced-form coefficients, the ambiguity in the estimation of p1 will be transmitted to other estimates too.

Why was the supply function identified in the system (19.2.12) and (19.2.22) but not in the system (19.2.28) and (19.2.22), although in both cases the supply function remains the same? The answer is that we have "too much," or an oversufficiency of information, to identify the supply curve. This situation is the opposite of the case of underidentification, where there is too little information. The oversufficiency of the information results from the fact that in the model (19.2.12) and (19.2.22) the exclusion of the income variable from the supply function was enough to identify it, but in the model (19.2.28) and (19.2.22) the supply function excludes not only the income variable but also the wealth variable. In other words, in the latter model we put "too many" restrictions on the supply function by requiring it to exclude more variables than necessary to identify it. However, this situation does not imply that overidentification is necessarily bad because we shall see in Chapter 20 how we can handle the problem of too much information, or too many restrictions.

We have now exhausted all the cases. As the preceding discussion shows, an equation in a simultaneous-equation model may be underidentified or identified (either over- or just). The model as a whole is identified if each equation in it is identified. To secure identification, we resort to the reduced-form equations. But in Section 19.3, we consider an alternative and perhaps less time-consuming method of determining whether or not an equation in a simultaneous-equation model is identified.

As the examples in Section 19.2 show, in principle it is possible to resort to the reduced-form equations to determine the identification of an equation

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