Question

In: Economics

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 the short run total, average and marginal cost functions and the firm’s supply. What is the smallest price at which the firm will produce? (e) Suppose that the market demand is given by D(p) = 240/p. Compute, for the long run, the equilibrium: price, output of each firm, and number of firms. (The output of a firm does not have to be an integer number, but the number of firms must be an integer.)

Solutions

Expert Solution

The production function is given as . The cost of production is or .

(a) The MRTS would be as or or . The optimal input combination would be where or or .

Putting this in the production function, we have or or or , and since , we have or . These are the conditional input demand. Putting these in the cost of production, we have the cost function as or or .

(b) The marginal cost would be as or or . The average cost would be as ro or .

(c) The VC would be as , and the AVC would be . The supply curve for the firm would be where MC=p, sucht that MC>AVC. As can be verified, since , we have MC>AVC for all y>0. The supply function would be as or . The minimum price at which the firm can produce would be the minimum of AVC, which is where MC=AVC. We have and , and MC=AVC would only be where y=0. At y=0, we have the minimum of AVC where . Hence, the firm would produce for p>0.

(d) For x2 fixed at 28 in the short run, we have the production function as or or , and we have or as the short run conditional input demand. The short run total cost function would be as or .

The marginal cost would be as , and the average cost would be as . The short run supply would be where MC=p, provided MC>AVC. We have AVC as , and MC>AVC for all y>0. Hence, the short run supply would be as or or . The smallest price would be where MC=AVC, and since and , we have MC=AVC where y=0. Hence, the minimum AVC would be , meaning that for p>0, the firm produces.

(e) In the long run, the firms produces at minimum of AC, which would be where or or or or or , and the minimum of AC would hence be . Hence, the long run equilibrium price would be p=3. The output of each firm would be y=1.

The long run quantity demanded would be units. For each firm produces y units, we have the total quantity supplied would be , and since quantity supplied would be equal to quantity demanded, we have or , ie there would be 80 firms.

Rahul Sunny answered 2 years ago
Next > < Previous

Related Solutions

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...
. 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
3. Given is the function f : Df → R with F(x1, x2, x3) = x...
3. Given is the function f : Df → R with F(x1, x2, x3) = x 2 1 + 2x 2 2 + x 3 3 + x1 x3 − x2 + x2 √ x3 . (a) Determine the gradient of function F at the point x 0 = (x 0 1 , x0 2 , x0 3 ) = (8, 2, 4). (b) Determine the directional derivative of function F at the point x 0 in the direction given...
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.
Consider a perfectly competitive firm that produces output using a wellbehaved production function y = f(x1,x2)....
Consider a perfectly competitive firm that produces output using a wellbehaved production function y = f(x1,x2). Let the price of output be denoted by p, the price of input 1 be denoted by w1, and the price of input 2 be denoted by w2. (a) State the firm’s cost-minimization problem, and draw the solution to the cost minimization problem on a graph. (b) What are the solutions to the cost-minimization problem? What are these solutions called? Which variables are these...
Consider a perfectly competitive firm that produces output using a wellbehaved production function y = f(x1,x2)....
Consider a perfectly competitive firm that produces output using a wellbehaved production function y = f(x1,x2). Let the price of output be denoted by p, the price of input 1 be denoted by w1, and the price of input 2 be denoted by w2. j)  Suppose x2 is fixed at some level x2. Graph the solution to the profit maximization problem where profit is maximized by choosing x1. [Note that this profit maximization problem is the short-run version of the profit...
If the joint probability distribution of X1 and X2 is given by: f(X1, X2) = (X1*X2)/36...
If the joint probability distribution of X1 and X2 is given by: f(X1, X2) = (X1*X2)/36 for X1 = 1, 2, 3 and X2 = 1, 2, 3, find the joint probability distribution of X1*X2 and the joint probability distribution of X1/X2.
The prices of inputs (x1,x2,x3,x4) are (4,1,3,2): (a) If the production function is given by f(x3,x4)...
The prices of inputs (x1,x2,x3,x4) are (4,1,3,2): (a) If the production function is given by f(x3,x4) =min⁡{x1+x2,x3+x4} what is the minimum cost of producing one unit of output? (b) If the production function is given by f(x3,x4)=x1+x2 +min⁡{x3+x4} what is the minimum cost of producing one unit of output?
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?
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT