1) Assume the production process can be represented by the following production function: Q=4K¹/2¹/2 We are operating in the short-run and thus K is fixed. We currently have use of 16 units of K. Note that the wage is $200 per worker, the rental rate of K is $150 per unit, and our output sells for $100 per unit. art run
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- Suppose a brewery uses a Cobb-Douglas production function for his production. He studies the production process and finds the following. An additional machine-hour of fermentation capacity would increase output by 600 bottles per day (i. e. MPK = 600). An additional man-hour of labor would increase output by 1200 bottles per day (i. e. MP₁ = 1200). The price of a man-hour of labor is $40 per hour. The price of a machine-hour of fermentation capacity is $8 per hour. 1. Is the brewery currently minimizing its cost of production? Check using the minimization condition. 2. It turns out, the brewery is not optimally choosing the factors of production. To lower its production cost, which factor of production should the brewery increase and which factor should he decrease? 3. Suppose that the price of a machine-hour of fermentation capacity rises to $20 per hour. How does this change the answer from part 1?Suppose that the production function for Hannah and Sam's home remodeling business is Q = F(L,K) Q = 10L0.1K0.4.Assume the wage rate is $8,000 per week and the cost of renting a unit of capital is $1,000 per week.a. What is the least-cost input combination for remodeling 400 square feet each week? Instructions: Round your answers to 2 decimal places. units of labor and units of capital. b. What is the total cost? Instructions: Round your answer to 2 decimal places. $ .revised jrl 08-11-2011Consider the production function Q = (0.5K1/3 + 0.5L1/3)3 with a corresponding MRTS = K2/3/L2/3. a. Prove that this production function exhibits constant returns to scale. b. Suppose the firms want to minimize the cost of producing 8 units of output. It pays a wage rate W of $40 and capital cost R of $10. How many units of labor and capital should the firm use?
- 2) Assume a firm has the following production function: 1 = K³L3¸ Assume w= 2, v = 16 and that K = 27 in the short run Further assume the firm faces a market price of 32. a) What is the firm's level of production and labor that can maximize the profit in the short run? b) If you were their economic consultant, what would you advise them to do regarding short run production and future adjustments to K and L? ExplainGiven the following data on input and output levels. Suppose the output price is $5 and input price is $10. Find the values of APP and MPP when X = 6: X 0 2 4 6 8 10 12 Y 0 100 250 450 600 700 750 50 and 150 6 and 200 100 and 75 75 and 100Suppose a firm has the following production function Q = f(K,L) = (1/2)L(2/3)K(1/3) and a total cost function TC = wL + rK The rental cost of capital is 2 euros per hour and the labor wage is 4 euros per hour. Suppose also that in the short-run, the firm uses 8 machines in the production process. (a) Show that the firm’s short-run average cost (SRAC) is SRACK=8 = (16/Q)+ 4√? (b) Draw the SRAC curve(c) By solving the first order conditions that the global minimum of production is 4(d) Show that the firm’s long-run average cost (LRAC) is 12. Hint: Start by obtaining the firm’s long-run expansion path(e) Draw both the SRAC and the LRAC and explain why the SRAC is always equal or above the LRAC.
- Suppose that the production function for Hannah and Sam's home remodeling business is Q=f(L,K) Q=1040.2K0.3 Assume the wage rate is $8,000 per week and the cost of renting a unit of capital is $2,000 per week. a. What is the least-cost input combination for remodeling 100 square feet each week? Instructions: Round your answers to 2 decimal places. units of labor and units of capital. b. What is the total cost? Instructions: Round your answer to 2 decimal places.Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is: q= 4K¾L where q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. (K For this production, MP, L and MP = 3 K K The weekly cost function for the paper plant is C= 10K +2L where C is the total weekly cost What ratio of capital to labor minimizes Miguel and Jake's total costs? (Hint: Find the Marginal Rate of Technical Substitution (MRTS) for capital and labor.)A farmer uses K units of machinery and L hours of labor to produce C tons of corn, with the production function, C = 2 L0.5 K. The farmer currently uses L = 20 and K = 10. Suppose the input price ratio is w/r = 1. What would you tell the farmer to do in order to minimize his cost of production given the current level of output?
- a) Determine whether the following production functions exhibit increasing, constant, or decreasing returns to scale. • F(K, L) = 4 K0.$L0.6 • F(K, L) = AK0³L04 b) Calculate the marginal product of labor (MPL) for the production function of the following form: Q = 10L0.75K©25(A) Suppose the production function for T-shirts can be represented as q = L0.5 KO.5. Show that the marginal productivity of labor diminishes in the short run. What is the cost minimizing bundles of labor and capital in the long run for T-shirts where w = and r = 10 when q=100 units?Consider the production function f (x1, x2)=min{x1, x2}. Assume the price of the inputs are w1 and w2. Assuming that to increase the quantity of input 1 by one unitwhile holding constant the quantity of input 2 1) show that the technology has constant returns to scale. 2) Write the cost minimization problem of thev firm