Info

(with all the derivatives evaluated at the critical values x and y), then the stationary value of U will assuredly be a maximum. The presence of the derivatives Uxx, Uyy, and Uxy in (12.32) clearly suggests that meeting this condition would entail certain restrictions on the utility function and, hence, on the shape of the indifference curves. What are these restrictions?

Considering first the shape of the indifference curves, we can show that a positive | H | means the strict convexity of the (downward-sloping) indifference curve at the point of tangency E. Just as the downward slope of an indifference curve is guaranteed by a negative dy/dx( = - Ux/Uy), its strict convexity would be ensured by a positive d2y/dx2. To get the expression for d2y/dx2, we can differentiate — Ux/Uy with respect to x; but in doing so, we should bear in mind not only that both Ux and Uy (being derivatives) are functions of x and y but also that, along a given indifference curve, y is itself a function of x. Accordingly, Ux and U can both be considered as functions of x alone; therefore, we can get a total derivative d2y d I U'\ 1 I dUx dUv

Since x can affect Ux and Uy not only directly but also indirectly, via the intermediary of y, we have dUx dy dU dy

where dy/dx refers to the slope of the indifference curve. Now, at the point of tangency E—the only point relevant to the discussion of the second-order condition—this slope is identical with that of the budget constraint; that is, dy/dx = - Px/Pv. Thus we can rewrite (12.34) as dU P dUv P

(12 34') —= U - U — —1 = U - U — 1 ; dx u« dx x>' u>yp

Substituting (12.34') into (12.33) and utilizing the information that uz p.

and then factoring out Uy/P, , we can finally transform (12.33) into d2v 2PPUxv-P2Uxx-P2Uvl 1/71 (12 33') y — * 1 >l — I I

It is clear that when the second-order sufficient condition (12.32) is satisfied, the second derivative in (12.33') is positive, and the relevant indifference curve is strictly convex at the point of tangency. In the present context, it is also true that the strict convexity of the indifference curve at the tangency implies the satisfaction of the sufficient condition (12.32). This is because, given that the indifference curves are negatively sloped, with no stationary points anywhere, the possibility of a zero d2y/dx2 value on a strictly convex curve is ruled out. Thus strict convexity can now result only in a positive d2y/dx2, and hence a positive \H |, by (12.33'). _ .

Recall, however, that the derivatives in \H\ are to be evaluated at the critical values x and y only. Thus the strict convexity of the indifference curve, as a sufficient condition, pertains only to the point of tangency, and it is not inconceivable for the curve to contain a concave segment away from point E, as illustrated by the broken curve segment in Fig. 12.7a. On the other hand, if the utility function is known to be a smooth, increasing, strictly quasiconcave function, then every indifference curve will be everywhere strictly convex. Such a utility function has a surface like the one in Fig. 12.4b. When such a surface is cut with a plane parallel to the xy plane, we obtain for each of such cuts a cross section which, when projected onto the xy plane, becomes a strictly convex, downward-sloping indifference curve. In that event, no matter where the point of tangency may occur, the second-order sufficient condition will always be satisfied. Besides, there can exist only one point of tangency, one that yields the unique absolute maximum level of utility attainable on the given linear budget. This result, of course, conforms perfectly to what the diamond on the right of Fig. 12.6 states.

You have been repeatedly reminded that the second-order sufficient condition is not necessary. Let us illustrate here the maximization of utility while (12.32) fails to hold. Suppose that, as illustrated in Fig. 12.7b, the relevant indifference curve contains a linear segment that coincides with a portion of the budget line. Then clearly we have multiple maxima, since the first-order condition UX/UY =

Px/Py is now satisfied at every point on the linear segment of the indifference curve, including E2, and E,. In fact, these are absolute constrained maxima. But since on a line segment d2y/dx2 is zero, we have j H \ = 0 by (12.33')- Thus maximization is achieved in this case even though the second-order sufficient condition (12.32) is violated.

The fact that a linear segment appears on the indifference curve suggests the presence of a slanted plane segment on the utility surface. This occurs when the utility function is explicitly quasiconcave rather than strictly quasiconcave. As Fig. 12.7ft shows, points £,, E2, and E3, all of which are located on the same (highest attainable) indifference curve, yield the same absolute maximum utility under the given linear budget constraint. Referring to Fig. 12.6 again, we note that this result is perfectly consistent with the message conveyed by the diamond on the left.

Comparative-Static Analysis

In our consumer model, the prices Px and Pv are exogenous, as is the amount of the budget, B. If we assume the satisfaction of the second-order sufficient condition, we can analyze the comparative-static properties of the model on the basis of the first-order condition (12.31), viewed as a set of equations FJ = 0 0'= 1,2,3), where each FJ function has continuous partial derivatives. As pointed out in (12.19), the endogenous-variable Jacobian of this set of equations must have the same value as the bordered Hessian; that is, |/| = |//|. Thus, when the second-order condition (12.32) is met, |J| must be positive and it does not vanish at the initial optimum. Consequently, the implicit-function theorem is applicable, and we may express the optimal values of the endogenous variables as implicit functions of the exogenous variables:

These are known to possess continuous derivatives that give comparative-static information. In particular, the derivatives of the last two functions x and y, which are descriptive of the consumer's demand behavior, can tell us how the consumer will react to changes in prices and in the budget. To find these derivatives, however, we must first convert (12.31) into a set of equilibrium identities as follows:

By taking the total differential of each identity in turn (allowing every variable to change), and noting that Uxy = Uyx, we then arrive at the linear system

- Px dx - Py dy = x dPx + y dPy - dB (12.37) - Px dX + Uxx dx + Uxy dy = X dPx

To study the effect of a change in the budget size (also referred to as the income of the consumer), let dPx = dPy = 0, but keep dB + 0. Then, after dividing (12.37) through by dB, and interpreting each ratio of differentials as a partial derivative, we can write the matrix equation*

0 0

Post a comment