Question 11.7.2: Let x be the GDP of a country, and let y be a measure of its......
Let x be the GDP of a country, and let y be a measure of its level of pollution. If the function u(x, y) purports to measure the total well-being of the society, what signs do you expect u^{\prime}_{x}(x, y) and u^{\prime}_{y}(x, y) to have? Can you guess what economists usually assume about the sign of u^{\prime\prime}_{xy}(x, y)?
It is reasonable to expect that well-being increases as GDP increases, but decreases as the level of pollution increases. Hence, we will usually have u^{\prime}_{x}(x, y) > 0 and u^{\prime}_{y}(x, y) < 0. According to (11.6.2),
f^{\prime\prime}(x)=\left ( \begin{matrix} f^{\prime\prime}_{11}(x) & f^{\prime\prime}_{12}(x) & … & f^{\prime\prime}_{1n}(x) \\ f^{\prime\prime}_{21}(x) & f^{\prime\prime}_{22}(x) & … & f^{\prime\prime}_{2n}(x) \\ . & . & & . & \\. & . & & . \\ . & . & & . \\ f^{\prime\prime}_{n1}(x) & f^{\prime\prime}_{n2}(x) & … & f^{\prime\prime}_{nn}(x) \end{matrix} \right ) (11.6.2)
u^{\prime\prime}_{xy }= (∂/∂y)(u^{\prime}_{x}) is approximately equal to the change in u^{\prime}_{x} per unit increase in y, the level of pollution. Moreover, u^{\prime}_{x} is, approximately, the increase in welfare per unit increase in x. It is often assumed that u^{\prime\prime}_{xy} < 0. This implies that the increase in welfare obtained by an extra unit of x will decrease when the level of pollution increases.^{14} Because of Young’s Theorem, 11.6.1, the inequality u^{\prime\prime}_{xy} < 0 also implies that u^{\prime\prime}_{yx} < 0. Thus the increase in welfare obtained from being exposed to one unit less pollution, which is approximately −u^{\prime}_{y}, increases with consumption x. This accords with the controversial view that as people can afford to consume more, they also become less tolerant of pollution.
^{14} An analogy: When a confirmed nonsmoker sits in a room filled with tobacco smoke, the extra satisfaction from one more piece of cake will decrease if the concentration of smoke increases too much.