Homework 1 - Math 105, Section 204 Due at the beginning of lecture on January 20, 2012 Name: SID: 1. Consider the production function 3 1 f (x, y) = 60x 4 y 4 , which gives the number of units of goods produced when utilizing x units of labor and y units of capital. (a) Compute ∂f and ∂f . These quantities are referred to as the marginal ∂x ∂y productivities of labor and of capital respectively. (b) A firm with this production function currently operates with 16 units of capital and 81 units of labor. If the amount of capital is held fixed at y = 16 and the amount of labor increases by 1 unit, estimate the increase in the quantity of goods produced. (c) Suppose instead that the amount of labor is held fixed at x = 81 and the amount of capital increases by 1 unit. Estimate the increase in the quantity of goods produced. 1 2 2. A monopolist markets a product in two countries and can charge different amounts in each country. Let x be the number of units to be sold in the first country and y the number of units to be sold in the second country. Due to the laws of demand, the monopolist must set the price at 97 − (x/10) dollars in the first country and 83 − (y/20) dollars in the second country to sell all units. The cost of producing these units is 20, 000 + 3(x + y). Find the values of x and y that maximize the profit. 3 3. Does there exist a function F (x, y) such that Fx (x, y) = xy, Fy (x, y) = y 2 ? If yes, find such a function. If not, explain why not.