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.