Question

In: Economics

2. A firm uses labor and machines to produce output according to the production function f...

2. A firm uses labor and machines to produce output according to the production function f (L,K) = 2 L K ,where L is the number of units of labor used and K is the number of machines. The cost of labor is $40 per unit and the cost of using a machine is $10. The Long Run Decision of the Firm

(a) Write down the equation for an isocost line for this firm. What is the slope of these isocost lines?

(b) Find , and . Using TRS and the slope of isocost lines, write down the tangency condition (the optimal choice condition) for this firm.

(c) When the desired level of output of this firm is , what is the equation for the isoquant of this firm? Use the tangency condition (b) and the equation for the isoquant, find the optimal choice of input 1 and the optimal choice input 2. That is, solve the simultaneous equation system: the tangency condition and the equation for the isoquant.

(d) Use the optimal choices of input 1 and input 2, find the long run total cost, the long run average cost, and the long run marginal cost.

Solutions

Expert Solution

2. (a) The isocost line would be as , which is basically the cost of production. For a given cost, the slope would be as or or or . The slope in this case would be .

(b) The production function is . The TRS for a given Q would be as or or or .

Equating the slope of isocost and TRS, we have the required optimal combination of inputs as .

(c) For the required output be Q, the equation of isoquant would be . For the tangency condition , we have or .

Putting this in the isoquant, we have or or or , and since , we have or or . These are the optimal input choices (conditional demands of the inputs).

(d) The long run cost function would be as or or or or . The average cost would be as or or . The marginal cost would be as or or or .

Rahul Sunny answered 8 months ago
Next > < Previous

Related Solutions

A firm called AstraZeneca uses labor and machines to produce output according to theproduction function ?(?,?)=4?1/2?1/2...
A firm called AstraZeneca uses labor and machines to produce output according to theproduction function ?(?,?)=4?1/2?1/2 where ? is the number of units of labor used and ? is the number of machines. The cost of labor is $40 per unit and the cost of using a machine is $10. (a) On a graph, draw an isocostline for this firm, showing combinations of machines and labor that cost $400 and another isocost line showing combinations that cost $200. What is...
A firm uses a single input to produce a good according to the production function q...
A firm uses a single input to produce a good according to the production function q = f(x) = 4x(1/2), where x is the input. The good sells for $4 per unit, and the cost of input x is w. a. Create a function showing the firm’s profit as a function of x and w. b. Find the profit maximizing amount of x as a function of w. What is the profit maximizing amount of output? c. Will this firm...
A firm produces output according to the production function: Q= F(K, L) = 4K + 8L.
A firm produces output according to the production function: Q= F(K, L) = 4K + 8L. a. How much output is produced when K= 2 and L = 3? b. If the wage rate is $60 per hour and the rental rate on capital is $20 per hour, what is the cost-minimizing input mix for producing 32 units of output? c. If the wage rate decreases to $20 per hour but the rental rate on capital remains at $20 per hour, what is the...
Suppose your firm uses 2 inputs to produce its output: K (capital) and L (labor). the...
Suppose your firm uses 2 inputs to produce its output: K (capital) and L (labor). the production function is q = 50K^(1/2)L^(1/2). prices of capital and labor are given as r = 2 and w = 8 a) does the production function display increasing, constant, or decreasing returns to scale? how do you know and what does this mean? b) draw the isoquants for your firms production function using L for the x axis and K for y. how are...
A firm produces output according to the following function: q = f (L, K) = L^1/2...
A firm produces output according to the following function: q = f (L, K) = L^1/2 K^1/4 . The cost of labor is $8 per hour and the rental cost of capital is $2 per hour. a. Determine the returns to scale for this function. b. Suppose the firm wishes to produce at cost $96. How much capital and how much labor does the firm employ? c. Derive the short-run cost function with optimal amount of K from part b....
The following table shows the production function of a firm Input (Number of machines) Total output...
The following table shows the production function of a firm Input (Number of machines) Total output Marginal product 0 0 1 10 2 35 3 70 4 120 5 165 6 175 7 170 8 155 Calculate the marginal product of the firm at each level of input. Where does diminishing marginal product start?
A firm has the following production function: q=KL The firm will produce 64 units of output...
A firm has the following production function: q=KL The firm will produce 64 units of output and faces prices for labor and capital of $4 and $1 respectively. What is the optimal quantity of labor and capital the firm should employ in order to minimize the cost of producing 64 units of output? What is the minimum cost of producing 64 units of output? Show the firms optimal production decision on a graph.
2. A firm is required to produce 234 of output using quantities of labor and capital...
2. A firm is required to produce 234 of output using quantities of labor and capital (L, K) = (21, 6) . For each of the following production functions, state whether it is possible to produce the required output with the given input combination. If it is possible, state whether producing Q = 234 with input combination is technically efficient or inefficient. (a) Q = 8L + 7K (b) Q=3KL+L-4 (c) Q=9√KL (d) Q= min{2L, 25K}
Consider a firm which produces according to the following production function by using labor and capital:...
Consider a firm which produces according to the following production function by using labor and capital: f(l,k) = k1/4 / 1/2 (a) Solve the cost minimization problem of this firm for the given wage rate, w and the rental rate of capital,v. (b) Derive the cost function of the firm. (c) Solve the profit maximization of the firm for a given price p, and derive the supply function. (d) Derive the profit function of the firm. (e) Suppose demand for...
Consider a firm which produces according to the following production function by using labor and capital:...
Consider a firm which produces according to the following production function by using labor and capital: f(l,k) = K1/2 L1/2 (a) Solve the cost minimization problem of this firm for the given wage rate, w and the rental rate of capital,v. Derive the long-run total cost function of the firm. (b) Derive the short-run total cost, the short-run average cost, the short- run marginal cost function of the firm under the assumption that capital is the fixed input. (c) What...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT