Using the production function q = (z," + z2)2/a for a manufacturing firm, where z; denotes labor, zz denotes capital and a denotes a constant parameter, derive the ) conditional factor demand functions and (i) cost function. If a/la-1) - +1 then show that the cost function will take a very simple form. Briefly explain how market regulators can make use of the cost function for this firm.
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- Using the production function q = (z1a + z2ª)1/a for a manufacturing firm, where zı denotes labor, z2 denotes capital and a denotes a constant parameter, derive the (i) conditional factor demand functions and (ii) cost function. If a/(a-1) = +1 then show that the cost function will take a very simple form. Briefly explain how market regulators can make use of the cost function for this firm.Using the production function q = (z1ª + z2a)1/a for a manufacturing firm, where z1 denotes labor, z2 denotes capital and a denotes a constant parameter, derive the (i) conditional factor demand functions and (ii) cost function. If a/(a-1) = +1 then show that the cost function will take a very simple form. Briefly explain how market regulators can make use of the cost function for this firm.For a firm with a Total Cost function TC C(Q), recall that the definitions of Marginal Cost (MC) and AverageTotal Cost (ATC) are given as follows:Marginal Cost (MC) = C'(0)Average Total Cost (ATC) = C(Q)/QProve that the marginal cost is equal to the average total cost (MC = ATC) when the average total cost is at itsminimum value.(Hint: By using the definition of average cost function, find the quantity that minimizes it by applying unconstraint optimization.Show that the FOC of the minimization problem implies that MC=ATC has to be satisfied at the minimum level of average cost.)
- 2. A firm has the following Cobb-Douglas production function: q = LªK!-a, where 0 < a <1 is a parameter. Suppose that in the short run K = 1. The rental rate of a unit of K is $10, and the wage rate of a unit of L is $20. (a) Derive the marginal cost of the firm (expressed in terms of a and q). (b) Derive the average variable cost of the firm (expressed in terms of a and q). (c) Derive the cost function of the firm.A firm has the following information on production and costs from past data: Output (Y) 6 12 18 Total Cost (TC) 2775 5361 8199 If the total cost function is known to be TC = aY'+ bY² + kY + ƒ , and the demand for the product of the firm is Y = 320- (1/2). P answer the following: • Determine the coefficients of the cubic cost function. • Derive all cost and revenue curves and the profit function. • Show that the MC cuts the AVC when AVC is at its minimum point. Plot the relevant graph indicating all points. • Calculate the break even and profit maximizing levels of output and price. • What is the relationship between price, marginal revenue and own price elasticity of demand at the profit maximization point.The following data show the total output for a firm when different amounts of labour are combined with a fixed amount of capital. Assume the wage per unit of labour is $10 and the cost of the capital is $50. Labour per period 0 1 2 3 4 5 TABLE 7-3 Total output per period 0 10 30 90 132 150 Refer to Table 7-3. The average variable cost when this firm is producing 90 units of output is Select one: OA 33 cents. OB. 17 cents. OC 68 cents. OD. 98 cents. OE 89 cents.
- Given a firm's cost function C = F(x) = ax² +bx + d: Find the firm's Average cost funetion (A(x) and Marginal cost function M(x). O Is C(x) more appropriatc as a long-run or short-run cost function? Explain. Determine whether the Marginal cost function is strictly monotonic for a> 0, b>0, and d> 0. Does this finction have an inverse for economically meaningful values of x? If yes, find dx/dM. If not, show/explain what restriction would be necessary for inverse to exist. a. b. c. d.Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.The Acme Anvil Company's output is given by the Cobb-Douglas Production function P = 60L2/3 K1/3, where P is the number of anvils produced when Lis the amount spent on labor and K is the amount spent on capital. a. What is the production if L = 150 and K = 150? b. Find the marginal productivities. C. Evaluate the marginal productivities with L = 150 and K = 150. d. Interpret the meanings of the marginal productivities found in part c. e. If their budget is $300 then there is a constraint L+ K = 300. Use Lagrange multipliers (2) to find the values of L and K that will maximize production and find the maximum production f. Find 2. 12| is called the marginal productivity of money and will give the number additional units produced for each dollar increase in the budget. Interpret 2 for this problem. MacBook Pro urses/34419/files/5014304?wrap=1
- Suppose that the Acme Gumball Company has a fixed proportions production function that requires it to use two gumball presses and one worker to produce 1,000 gumballs per hour. a. Explain why the cost per hour of producing 1,000 gumballs is 2v + w (where v is the hourly rent for gumball presses and w is the hourly wage). b. Assume Acme can produce any number of gumballs they want using this technology. Explain why the cost function in this case would be TC = q(2v +w), where q is output of gumballs per hour, measured in thousands of gumballs. c. What is the average and marginal cost of gumball production (again, measure output in thousands of gumballs)? (show the complete formula) Draw the graph for the average and marginal cost curves for gumballs assuming v=3, w-5 (show working) Now draw the graph for these curves for v=6, w=5.( show working) Explain why these curves have shifted.Suppose that the Acme Gumball Company has a fixed proportions production function that requires it to use two gumball presses and one worker to produce 1,000 gumballs per hour. a. Explain why the cost per hour of producing 1,000 gumballs is 2v + w (where v is the hourly rent for gumball presses and w is the hourly wage). b. Assume Acme can produce any number of gumballs they want using this technology. Explain why the cost function in this case would be TC = q(2v +w), where q is output of gumballs per hour, measured in thousands of gumballs. c. What is the average and marginal cost of gumball production (again, measure output in thousands of gumballs)? Graph the average and marginal cost curves for gumballs assuming v=3, w-5 Now graph these curves for v=6, w=5. Explain why these curves have shifted.3. If MPL = 5L¹/³ is a firm's marginal-product function, where L is the single input labor and output is zero when L = 0, find the production function for the firm.