2 ,

and if CT„y = 0, then

pHm a = a + (P - a)

The bias is now negative since (p - a) is expected to be negative. Again, suppose that the supply elasticity is estimated to be 0.6; then we know that the true price elasticity is > 0.6.

In practice, when we regress g on p we do not know whether we are estimating the demand function or the supply function. However, if the regression coefficient is, say, +0.3, we know that the supply elasticity is 0.3 + a positive number and the demand elasticity is 0.3 + a negative number. Thus the practically useful conclusion is that the supply elasticity is > 0.3. On the other hand, if the regression coefficient is -0.9, we know that the supply elasticity is -0.9 -b a positive number and the demand elasticity is -0.9 + a negative number. Thus the practically useful conclusion is that the demand elasticity < -0.9 (or greater than 0.9 in absolute value).

In the example above we also note that

/2 p,gA

plim pas crO

a as or? -> 0

Since a;, is the variance of the random shift in the demand function, a], 0 means that the demand function does not shift or is stable. Thus the regression of 9 on p will estimate the demand elasticity if the demand function is stable and the supply elasticity if the supply function is stable.

Figure 9.2 gives the case of a supply curve shifting and the demand curve not shifting. What we observe in practice are the points of intersection of the demand curve and the supply curve. As can be seen, the locus of these points of intersection traces out the demand curve. In Figure 9.3, where the demand

This expression is not very useful, but if „, = 0 (i.e., the shifts in the demand and supply functions are unrelated), we get

plim p = + 0 - a) (-)

Now p is expected to be negative and a positive. Thus the bias term is expected to be positive. Hence if we find a price elasticity of demand of -0.8, the true price elasticity is < -0.8.

We can show by a similar reasoning that if we estimate the supply function by OLS we get

Figure 9.2. Stable demand and shifting supply.

curve shifts and the supply curve does not, this locus determines the supply curve.*

Workings Concept of Identification

After concluding that if the demand curve did not shift much, but the supply curve did, then the locus of the points of intersection would come close to tracing a demand curve. Working added that by "correcting" for the influence of an additional important determining variable (such as income) in the demand curve, we could reduce its shifts and hence get a better approximation of the price elasticity of demand. If we add income to the demand function, we get the equation system

Qt = PPi + 7 + «r demand function

q, = ap, + V,

supply function

What Working said was that by introducing y, in the demand function, we can get a better estimate of p. What he had in mind was that the introduction of y, reduced the variance of u,.

If we use the results on identification discussed in Sections 9.3 and 9.4, we come to the conclusion that the supply equation is identified (it has one missing variable) but the demand function is not. This is exactly the opposite of what Working said. The problem here is that the concept of identification we have

*This was the main conclusion of the article by E. J. Working, "What Do Statistical Demand Curves Show?" Quarterly Journal of Economics, February 1927.

Figure 9.3. Stable supply and shifting demand.

discussed is in terms of our ability to get consistent estimates. What Working was concerned about is the least squares bias. His argument is that if al is somehow reduced, we get good estimates by OLS (i.e., the bias would be negligible). Consider two estimates p, and Pj for p. Suppose that

plim p, = p and plim pj = 0.999p

then p, is consistent but is not. But for all practical purposes 02 is also consistent.

Suppose that we include all the relevant explanatory variables in the demand function so that al is very small, whereas the supply function includes only price as the "explanatory" variable. Then, even if the demand equation is not identified by the rules we discussed earlier, we are still justified in estimating the equation by OLS. On the other hand, even if the supply function is over-identified, if al is very large, we get a poor estimate of the supply elasticity a. We can get a consistent estimator for a, but its variance will be very high.

In summary, it is not true that if an equation is under-identified, we have to give up all hopes of estimating it. Nor does it follow that an equation that is over-identified can be better estimated than one that is exactly-identified or under-identified. Further discussion of the demand supply model considered here can be found in Maddala* and Leamer.

»G. S. Maddala, Econometrics (New York: McGraw-Hill, 1977), pp. 244-249. "E. E. Leamer, "Is It a Demand Curve or Is It a Supply Curve: Partial Identification Through Inequality Constraints," The Review of Economics and Statistics, Vol. 63. No. 3, August 1981, pp. 319-327.