The production function is f(x1, x2) = x11/2 x21/2 . If the price of factor 1...

The production function is f(x₁, x₂) = x₁^1/2 x₂^1/2 . If the price of factor 1 is $12 and the price of factor 2 is $24, in what proportions should the firm use factors 1 and 2 if it wants to maximize profits?
(a) x₁ = x₂

(b) x₁ = 0.50x₂

(d) x₁ = 24x₂

(e) We can't tell without knowing the price of output.

Solutions

Expert Solution

(c) x₁ = 2x₂

At max: MP(x1)/P(x1) = MP(x2)/P(x2)

=> (0.5x1^-0.5x2^0.5)/12 = (0.5x1^0.5x2^-0.5)/24

=> x2/12 = x1/24

=> x2 = x1/2

=> 2x2 = x1 (or x1 = 2x2)

Rahul Sunny answered 2 years ago

Next > < Previous

Related Solutions

If a firm’s production is f(x1,x2) = 2x11/2 x21/4. If the price of factor 1 is...

If a firm’s production is f(x1,x2) = 2x11/2 x21/4. If the price of factor 1 is $5 and the price of factor 2 is $10, in what proportions should the firm use factors 1 and 2 if it wants to maximize profits? Please show all work clearly.

The production function of a competitive firm is given by f(x1, x2) = x1^1/3 (x2 −...

The production function of a competitive firm is given by f(x1, x2) = x1^1/3 (x2 − 1)^1/3 The prices of inputs are w1 = w2 = 1. (a) Compute the firm’s cost function c(y). (b) Compute the marginal and the average cost functions of the firm. (c) Compute the firm’s supply S(p). What is the smallest price at which the firm will produce? 2 (d) Suppose that in the short run, factor 2 is fixed at ¯x2 = 28. Compute...

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a)...

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2. (a) The marginal product of factor 1 (increases, decreases, stays constant) ------------ as the amount of factor 1 increases. The marginal product of factor 2 (increases, decreases, stays constant) ----------- as the amount of factor 2 increases. (b) This production function does not satisfy the definition of increasing returns to scale, constant returns to scale, or decreasing returns to scale. How can this be? (c)Find...

A firm has two variable factors and a production function f(x1; x2) = (2x1 + 4x2)^1/2....

A firm has two variable factors and a production function f(x1; x2) = (2x1 + 4x2)^1/2. On a graph, plot three input combinations and draw production isoquants corresponding to an output of 3 and to an output of 4. Also, mention the technical rate of substitution(s) for the isoquants. Show all working.

Linda sells an android app. Her firm’s production function is f (x1, x2) = x1 +...

Linda sells an android app. Her firm’s production function is f (x1, x2) = x1 + 2x2 where x1 is the amount of unskilled labor and x2 is the amount of skilled labor that she employs. Draw the isoquant at 20 units of output and another at 40 units of output. Does this production function exhibit increasing, decreasing, or constant return to scale? If Linda faces factor prices (w1, w2) = (1, 1), what is the cheapest way to produce...

An industry has 1000 firms, each with the production function f(x1; x2 ) x1^.5 x2^.5. Theprice...

An industry has 1000 firms, each with the production function f(x1; x2 ) x1^.5 x2^.5. Theprice of factor 1 is 1 and the price of factor 2 is 1. In the long run, both factors are variable, but inthe short run, each firm is stuck with using 100 units of factor 2.The long run industry supply curve:Can somebody explain how to solve?

. Consider a firm that has a production function y = f(x1, x2) = 2x 1/4...

. Consider a firm that has a production function y = f(x1, x2) = 2x 1/4 1 x 1/4 2 facing input prices w1 = 2 and w2 = 4. Assume that the output price p = 8. • What will be the profit maximizing output level? -What will be the profit? • -If this firm is divided up into two equal-size smaller firms, what would happen to its overall profits? Why

Sara’s utility function is u(x1, x2) = (x1 + 2)(x2 + 1). a. Write an equation...

Sara’s utility function is u(x1, x2) = (x1 + 2)(x2 + 1). a. Write an equation for Sara’s indifference curve that goes through the point (2,8). b. Suppose that the price of each good is 1 and that Clara has an income of 11. Draw her budget line. Can Sara achieve a utility of 36 with this budget? Why or why not? c. Evaluate the marginal rate of substitution MRS at (x1, x2) = (1, 5). Provide an economic interpretation...

Let X1 and X2 have the joint pdf f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere (a) Find the...

Let X1 and X2 have the joint pdf f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere (a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1 = x1. (b) Find the conditional expectation and variance of X1|X2 = x2 and X2|X1 = x1. (c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and P(0 < X1 < 1/2). (d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that var(Y ) ≤ var(X2).

T/F 1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear...

T/F 1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear function, where x = (x1,...,xn). 2) The function f(x1,x2,x3,x4) = (x2,x1,x4,x3) is linear. 3) For a given matrix A and vector b, equation Ax = b always has a solution if A is wide

Subjects