Problem 3* A competitive firm has the following production function: f(L,K) = min {L, K}. Suppose the wage rate is 2, and the rental rate of capital is 4. (a) Derive the long-run cost function for the firm. (b) Calculate the long-run marginal cost curve and the average cost curve for the firm. (c) Write down the long-run supply function and graph the long-run supply curve for the firm.
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- The long-rung production function of a competitive firm is f(L,K)= L1/3K 1/2 , where L is the amount of labor and K is the amount of capital. The cost per unit of labor is w and the cost of capital is r, which is the interest rate. The price per unit of output is p. (A. What are the returns to scale of this production function? (B. Find the long-run profit maximizing levels of capital, labor and output.A competitive firm has a production function described as follows. Q = f(K, L) = 100K}L where K and L are levels of capital and labor in production respectively. (a) Find the optimal bundle of capital and labor to produce 400000 units of output, if the wage is w=$8 and the rental price of capital is r=$1. Also, find cost of producing 400000 units of output. (b) Suppose that in the short run this firm must use 40000 units of capital but can vary its amount of labor freely. How does optimal number of labor and cost of production changes? (consider prices and quantity given part a) (c) Write down a formula that describes the marginal product of labor in the short run as a function of the amount of labor used. (d) If the wage is w=$10 and the price of output is p=$2, how much labor will the firm demand in the short run? (e) Write down an equation for the firms short-run demand for labor as a function of w and p. (f) Write down an equation for the firms short-run cost function as a function…Month (m): 4 Day (d): 1 Use the two numbers above, m and d, to complete the cost function for a perfectly competitive firm: Cost (q) = m q2 + d = (30) For a cost function like yours, Marginal Cost (MC) = 2 m q . Specifically, what are the following for the cost function you wrote out above? Fixed Cost = Average Total Cost = Cost (q)/q = Variable Cost = Average Fixed Cost = FC/q = Marginal Cost = 2 m q = Average Variable Cost = VC/q = (15) Fill in the table with your values from your cost function. q Total Cost AVC AFC ATC MC 0 0 0 -- -- -- 1 2 3 4 5 6
- Suppose that a firm has a short-run cost function C(q) = q2 +1. Part a Find the short-run supply function of this firm. Part b Given this production function, at what output is average cost (AC) minimized?Suppose the long-run production function for a competitive firm is f(x1,x2)= min {3x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. A: Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. B: Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? C: Write down the formulas and draw the graphs of the average cost and marginal cost functions, as functions of y.The table above shows the total cost function for a typical firm producing hats in a perfectly competitive market. The market price for hats is $9 per hat. (a) Calculate the average variable cost of the fifth unit. Show your work. (b) What is the firm’s profit-maximizing quantity of hats? Explain using marginal analysis. (c) If the rent of the building the firm occupies increases, what will happen to the firm’s profit-maximizing quantity of hats in the short run? Explain. (d) Draw a correctly labeled graph showing the firm’s demand and marginal cost curves, and show the profit-maximizing quantity of hats, labeled Q*.
- A firm producing hockey sticks has a production function given by q = 2Vkl In the short run, the firm's amount of capital equipment is fixed at k = 100. The rental rate for k is r = 5, and the wage rate for r = $1, and the wage rate for l is w = $4. (a) Calculate the firm's short-run total cost curve. Calculate the short-run average cost curve. (b) What is the firm's short-run marginal cost function? What are the SC, SAC, and SMC for the firm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks? (c) Graph the SAC and the SMC curves for the firm. Indicate the points found in part (b). (d) Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point Suppose now that capital used for producing hockey sticks is fixed at ki in the short run. (e) Calculate the firm's total costs as a function of q, w, r, and k1. (f) Given q,w,and r, how should the capital stock be…The cost function of a UC Irvine donut shop is: C(q)=10+ 10q + q?, so the marginal cost function is: MC= 10+ 2q. In these equations, q is the output in terms of boxes of donuts. (a) What is the firm's average cost curve? (Note: just write the equation, no graph necessary) What is the firm's average variable cost curve? (Note: just write the equation, no graph (b) necessary) (c) If the price of a box of donuts is $20, what is the optimal output for this firm?Consider a firm with the following cost function: C (q) = 4q^2 + 100 Find the long-run supply and the short-run supply of the firm, under the assumptions that the total cost function is the same in the long and in the short run, but the xed cost is sunk in the short run.
- Suppose the production function for a competitive firm is y=f(x1,x2)= x1 1/4 x2 1/4 . The cost per unit of the first input is w1 and the cost of the second input is w2. A: What are the returns to scale of this production function? B: Find the cheapest input bundle, x1 and x2, that yields the given output level of y. C: Write down the formula of the firm’s total costs as a function of y. D: Are the average costs increasing, constant or decreasing in y? Are the marginal costs increasing, constant or decreasing in y?A competitive firm uses two variable factors to produce its output, with a production function y = min{ x1, x2 }.The price of x1 is w1 = $8 and the price of x2 is w2 = $5. Due to a lack of warehouse space, the company cannot use more than 10 units of x1. The firm must pay a fixed cost of $80 if it produces any positive amount but doesn't have to pay this cost if it produces no output. What is the smallest integer price that would make a firm willing to produce a positive amount? please solve asap?A competitive firm uses two variable factors to produce its output, with a production function y = min{ x1, x2 }.The price of x1 is w1 = $8 and the price of x2 is w2 = $5. Due to a lack of warehouse space, the company cannot use more than 10 units of x1. The firm must pay a fixed cost of $80 if it produces any positive amount but doesn't have to pay this cost if it produces no output. What is the smallest integer price that would make a firm willing to produce a positive amount?