91Ó°ÊÓ

The productivity of a country is given by \(f(x, y)=300 x^{2 / 3} y^{1 / 3}\), where \(x\) and \(y\) are the amount of labor and capital. (a) Compute the marginal productivities of labor and capital when \(x=125\) and \(y=64\). (b) Use part (a) to determine the approximate effect on productivity of increasing capital from 64 to 66 units. while keeping labor fixed at 125 units. (c) What would be the approximate effect of decreasing labor from 125 to 124 units while keeping capital fixed at 64 units?

In a certain suburban community, commuters have the choice of getting into the city by bus or train. The demand for these modes of transportation varies with their cost. Let \(f\left(p_{1}, p_{2}\right)\) be the number of people who will take the bus when \(p_{1}\) is the price of the bus ride and \(p_{2}\) is the price of the train ride. For example, if \(f(4.50,6)=7000\), then 7000 commuters will take the bus when the price of a bus ticket is $$\$ 4.50$$ and the price of a train ticket is $$\$ 6.00$$. Explain why \(\frac{\partial f}{\partial p_{1}}<0\) and \(\frac{\partial f}{\partial p_{2}}>0\).

A company manufactures and sells two products, I and II, that sell for \(\$ p_{\mathrm{I}}\) and \(\$ p_{\mathrm{II}}\) per unit, respectively. Let \(C(x, y)\) be the cost of producing \(x\) units of product \(\mathrm{I}\) and \(y\) units of product II. Show that if the company's profit is maximized when \(x=a, y=b\), then $$ \frac{\partial C}{\partial x}(a, b)=p_{\mathrm{I}} \quad \text { and } \quad \frac{\partial C}{\partial y}(a, b)=p_{\mathrm{II}} $$.

A company manufactures and sells two competing products, I and II, that cost \(\$ p_{\mathrm{I}}\) and \(\$ p_{\text {II per unit, respectively, to produce. Let }} R(x, y)\) be the revenue from marketing \(x\) units of product \(\mathrm{I}\) and \(y\) units of product II. Show that if the company's profit is maximized when \(x=a, y=b\), then $$ \frac{\partial R}{\partial x}(a, b)=p_{\mathrm{I}} \quad \text { and } \quad \frac{\partial R}{\partial y}(a, b)=p_{\mathrm{II}} . $$.