Suppose we have one consumer with a Cobb Douglas utility function for consumption, x, and leisure, R: u(x, R) = a ln r+(1-a) ln R. The consumer is endowed with one unit of labor/leisure and there is one firm with a constant-returns to scale technology: x = aL. Determine r(p, w) ane R(p, w).
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- Suppose that a food manufacturing company produces two goods, A and B, using the same plant and labor. The total cost of producing A and B is give by: (in the picture) Is this total cost function consistent with the present of increasing or decreasing returnsto scaleProblem 5 The production function of COVID vaccines for firm O is given by v° = VKL. and requires at least one unit of labor L and one unit of capital K, i.e. L>1 und K > 1. a) After a year and some thorough research, the production function changes to VO = V4KL. (I) Compute the marginal products with respect to labor L and capital K for both production functions ! (II) Which company's the marginal products is/are larger ? What happened to the original level of production after a year ? (III) Do the production functions exhibit increasing, constant or decreasing returns to scale ? b) Suppose now, we compare production functions of company O that discovered the vaccine to a second company M. The production function of M is given by VM = K0.6 L0.4 (I) Compute the marginal products with respect to labor L and capital K for this pro- duction function ! (II) If both companies used the same equal amount of labor L and capital K, which of them will generate more output ? (III) If the…A firm uses labor and machines to produce outputaccording to the production function f(L, M) = 4L1/2M1/2, where L is the number of units of laborused and M is the number of machines. The cost of labor is $40 per unit and the cost of using amachine is $10.1. Draw an isocost line for this firm, showing combinations of machines and labor that cost $400and another isocost line showing combinations that cost $200.2. Suppose that the firm wants to produce its output in the cheapest possible way. Find thenumber of machines it would use per worker. (Hint: The firm will produce at a point wherethe slope of the production isoquant equals the slope of the isocost line.)
- The production function for the automotive and parts industry is q=L0 25 K0.16 M0.59 where M is energy and materials (based loosely on Klein, 2003). What kind of returns to scale does this production function exhibit? This production function exhibits constant returns to scale. The marginal product of materials, MPM, is O A. MPM = 0.59(q/M) O B. MPM = 0.59L0 25 K0.16M - 0.41 MPM =L0 25K0. 16MO.59 O D. both a and b. O E. all of the above.For a firm to maximize profit, it must minimize the cost of producing whatever quantity it produces. Use the isocost and isoquant tools to present a firm that is choosing the optimal levels of labor and capital (i.e., tools) to produce a certain quantity and a certain cost. Then, show in your diagram how this firm would respond if it were to expand and spend more on its inputs, assuming it is best for the firm to become more “capital intensive” as it grows. Comment on WHY a firm might best become more capital intensive as it expands, even when the relative prices of labor and capital remain unchanged.Suppose the production function for high quality brandy is given by : Q = √KL Where q is the output of brandy per week and L is labor hours per week ., in the short run K is fixed at 100, so the short run production function is Q = 10√L a. Ifthecapitalrentsfor10$andthewageare5$perhour.,writetheshortruntotalcost function. b. How much will the firm produce at a price of 20$ per bottles of brandy ? c. Howmanylaborhourswillbehiredperweek?
- (a) Suppose we have the following production function: Q=K²/³L2/³ Does the technology have constant returns to scale (CRS)? If not, what? Show your work and explain what it means.< (b) What is the firm's cost minimi on problem in the long-run? Explain and illustrate in a figure using isocost and isoquants how termine optimal inputs K and L. ← (c) Now, consider the short-run where K is fixed at 8 (so that K²/³ = 4). Let r = $20 and w = $10. Show then discuss how short-run average costs change as Qrises. You can use math or a table/figure to do this. (d) Suppose P = $15. Show that optimal Q = 64 in the short-run. You can use math or a spreadsheet. Show details. < (e) Now suppose P = $5. Should the firm shut-down and set Q=0? Explain. <Let f: Rn → R be increasing returns to scale. Recall, thismeans that f(αx) ≥ αf(x) for all x ∈ Rn and all α ≥ 1. Further, assume for some vector of inputs, profits are strictly positive. Argue that there is no profit maximizing vector of inputs.Let the production function be given by q=2(square root of L)(square root of K)} and Price of K be 5 and the Price of L be 2. Then, the slope of the isoquant is given by: Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a: -L/K b: -K/L c: -(K+L) d: -(KL) e: none of the above
- Fill the missing values on the table below by using the following cost formulas: Let Q be the amount of the output a firm produces. The essential cost equation: TC = FC+VC Mathematically, if Q +0, can I do as following equation or not? TC FC VC A=Average, then ATC = AFC + AVC TC ATC= : AFC = FC ; AVC = VC ATC TC,-TC: Oj - Q. AQ Output (Q) (Instructional Moddules per Average Fixed Cost (AFC) Average Total Marginal Cost Cost (ATC) Fixed Variable Total Average Costs Costs Costs Variable (MC) (FC) (VC) (TC) Cost (AVC) Month) XXXXXXXXXXX XXXXXXXXXX XXXXXXXXXXX XXXXXXXXXXX D4 B A S38 B $13 D1 C1 D2 D3 C2 C3 2 B H2 H1 H3 S33 H4 3 B L2 L1 L3 L4 54 C4 516 J3 J4 1 S72 4 B. 12 B. E1 C5 E2 E3 E4 M2 M1 C6 M3 M4 S36 S19 G3 7. B. 12 1 C7 13 14 G1 5185 C8 G2 G4Exercise A.5 A company operates with production function q-K¹/2 [1/2 a) What type of returns to scale does this production function present? As production varies, how would you expect long-term average costs to evolve? b) Assume that the firm's capital equipment is fixed and equal to K = 100 units. Find the firm's short-run production function and the marginal productivity of labour and graph these functions. As production varies, how would you expect marginal costs to evolve in the short term? c) If the prices of capital and labour are r = 8€ and w = 2€ respectively, obtain the expression of the short-run total cost function of the firm. b) Obtain the expressions of the functions of short-term average total costs, average variable costs and marginal costs of this company. Find the production level for which the average total cost is minimized and the corresponding average total cost level. d) Indicate the value of the average total cost, the average variable cost and the marginal cost…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