169: 634/973 - BAGH EXAMPLE 15.1.4 As in Example 14.9.2, let u(x, y) denote the total well-being of a society, where x denotes GDP and y denotes a measure of the pollution level. As in that example, assume that u'x (x, y) > 0 and u, (x, y) 0. Then total well-being becomes a function U(x) = u(x, h(x)) of x alone. Find a necessary condition for U(x) to have a maximum at x = x* > 0, and give this condition an economic interpretation. Solution: By Theorem 9.1.1, a necessary condition for U(x) to have a maximum at x* > 0 is that U'(x*) = 0. Now we use the chain rule (15.1.1) to obtain: U'(x) = us(x, h(x)) . 1 + u, (x, h(x)) . h(x) ( * ) 616 CHAPTER 15 / PARTIAL DERIVATIVES IN USE IMG290.jp eg ti... So U'(x*) = 0 requires that in... u'x ( x* , h(x* ) ) = -u, (x*, h(x*) )h(x*) ( * * ) d... To interpret this condition, consider increasing x* by a small amount d, which can be pos- MG291.jpe itive or negative. By Eq. (14.2.6), the gain due to the increase in GDP is approximately g ati... u'x(x*, h(x*))8. On the other hand, the level of pollution increases by about h'(x*) & units. But we lose -uy(x*, h(x*)) in well-being per unit increase in pollution. So all in all we lose atI... about -uy(x*, h(x*))h'(x*)8 from the extra pollution that results from this increase in x*. Equation (**) just states that the direct gain from increasing x* by any small amount & must IMG292.jp We sys... equal the indirect loss from the increased pollution that results from this increase. Other- eg wise a slight increase in well-being could be achieved by a small change & in x whose sign matches the sign of U'(x* ) given by Eq. (*). "tv JJ Q S Cali 4 TALK( a ) J ( x , y ) = x t y; ( b ) J ( X, y ) = X Y; (c) f ( x, y) = x . y; (d) f (x, y) = x/ y; (e) f (x, y) = G(x). SM 7. [HARDER] Consider Example 15.1.4, where u(x, y) = In(x* +y*) - alny and y = h(x) = Vaxt + b with positive constant parameters a, a, and b. Find the optimal x* in this case