The technology for making nails is described by the Cobb-Douglas production function Q(L,K) = L*K, where L is the number of workers and K is the number of machines. Each worker is two times as expensive as a machine. The MRTS between Labor and Capital at the cost-minimizing output combination is
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- Suppose a Cobb-Douglas Production function is given by the function: P(L,K)=8L^0.7K^0.3Furthemore, the cost function for a facility is given by the function:C(L,K)=300L+500KSuppose the monthly production goal of this facility is to produce 20,000 items. In this problem, we will assume LL represents units of labor invested and KK represents units of capital invested, and that you can invest in tenths of units for each of these. What allocation of labor and capital will minimize total production Costs?Units of Labor LL = (Show your answer is exactly 1 decimal place)Units of Capital KK = Incorrect (Show your answer is exactly 1 decimal place)Also, what is the minimal cost to produce 20,000 units? (Use your rounded values for LL and KK from above to answer this question.)The minimal cost to produce 20,000 units is $ Hint: Your constraint equation involves the Cobb Douglas Production function, not the Cost function. When finding a relationship between LL and KK in your system of…A lot of companies use the marginal cost function for their business models to determine the production level to minimize total costs. As we all know, if C (x) represents the total cost of producing x units of a product, then the average cost per unit at this production Let assume C (x) is differentiable on all x > 0 . This implies that the C(x) level is average cost function is also differentiable for all x > 0 since C(x) can be written as C (x) · - , and both C (x) and are differentiable for all x > 0. Suppose you go to an interview and the chairman of the interview committee gives you the following questions: In our company , we use marginal cost to determine the production level to minimize the costs. However, there is another useful model in business, the average cost function, that we also track. 1. Is it true or not that the marginal cost C '(x) is equal to the average cost at the critical points of the average cost function? Please elaborate and explain to us. 2. This year, the…Skylar owns a firm that produces holiday ornaments. Her production function is given by: f (L, K) = (min{x1 , 2x2})1/2 where x1 is the amount of plastic used, and x2 is the amount of labour. A) Does this production function have increasing, decreasing or constant returns to scale? B) If the cost of plastic is w1, and the cost of labour is w2, what is the cost function of producing "y" holiday ornaments with this technology?
- Hannah and Sam run Moretown Makeovers, a home remodeling business. The number of square feet they can remodel in a week is described by the Cobb-Douglas production function Q=F(L,K) Q=10L^0.5K^0.5,where L is their number of workers and K is units of capital. The wage rate is $250 per week and a unit of capital costs $250 per week. Suppose that when initially producing 100 square feet a week, they use 10 units of capital.a. What is their short-run cost of remodeling 1,000 square feet per week? Instructions: Enter your answer as a whole number. $ b. What is their long-run cost of remodeling 1,000 square feet per week? Instructions: Enter your answer as a whole number. $Suppose that firms face the following production function: Q = LAK^. Suppose also that price of labor (L)=w=3 and price of capital (K)=r%3D9. If the firm wants to produce 120 units of output, what is the optimum amount of labor, capital and associated cost respectively? 60, 60, 120 30, 30, 360 240, 240, 2880 120, 120, 144OA firm produces plastic bins using labor (measured in man-hours) and capital (measured in machine-hours), according to the production function Q = f(L,K) = LK, where Q is the number of plastic bins produced. Suppose that the cost of labor is $20 per worker-hour and the cost of capital is $10 per machine-hour. What is the cost minimizing input combination if the firm wants to produce 28,800 plastic bins? Hint: The marginal products are MP, = K and MPg = L.
- You manage a store that mass-produces candies by teams of workers using assembly candy machines. The technology is summarized by the production function q = 3 LK where q is the number of candies per week in thousands, K is the number of assembly candy machines, and L is the number of labor teams. Each candy assembly machine rents for r = per week, and each team costs w = $1000 per week. Candy costs are given by the cost of labor teams and machines, plus $600 per thousand of candies for ingredients. Your store has a fixed installation of 2 candy assembly machines as part of its design. = $6,000 What is the cost function for your store – namely, how much would it cost to produce q candies? What are average and marginal costs for producing q candies? How do average costs vary with output? а. b. How many teams are required to produce 3 (thousands) candies? What is the average cost per thousand candies? You are asked to make recommendations for the design of a new production facility in the…Suppose a Cobb-Douglas Production function is given by the function: P(L, K) = 8L0.' K0.9 Furthemore, the cost function for a facility is given by the function:C(L, K) = 200L + 100K Suppose the monthly production goal of this facility is to produce 8,000 items. In this problem, we will assume L represents units of labor invested and K represents units of capital invested, and that you can invest in tenths of units for each of these. What allocation of labor and capital will minimize total production Costs? Units of Labor L = (Show your answer is exactly 1 decimal place) Units of CapitalK = (Show your answer is exactly 1 decimal place) Also, what is the minimal cost to produce 8,000 units? (Use your rounded values for L and K from above to answer this question.) The minimal cost to produce 8,000 units is $ Hint: 1. Your constraint equation involves the Cobb Douglas Production function, not the Cost function. 2. When finding a relationship between L and K in your system of equations,…The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C) = c₂LC1C²2 C2 are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co 5, C₁ = 0.25, and c2₂ 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $90,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. = = where co, C₁, and Max s.t. L, C ≥ 0 A ≤ 90,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in dollars)? Hint: Put bound constraints on the variables based on the budget constraint. Use L ≤ 3,000 and C ≤ 1,000 and use the Multistart option as described in Appendix 8.1. (Round your answers to the nearest integer when…