1. Given the following cost and production functions, find where production is optimized. C = 25x₁ + 5x2 Q = 5x₁²5x75 2. A firm wants to minimize the cost of producing 100 units of output given that the cost of labor (w) is $25 and the cost of capital (r) is $5. Output is produced according to the Cobb-Douglas production function: Q = 5K-5L.5.
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- A manufacturer named Kjell is facing the following product function.1 x = f(N, K) = 100Nº.8K0,2 The price for labor is NOK 500 per unit. The price for capital is NOK 125 per unit. a) Does the product function provide constant, increasing, or decreasing scale benefits? b) Kjell wants to produce 400 cakes at the lowest possible cost. How much labor and does he need capital? What will be the cost? c) The bank is not impressed by Kjell's production technology. He therefore receives a loan of only NOK 1,875. How many cakes can Kjell produce a maximum of NOK 1,875. d) Make a factor diagram (N, K). Note Kjell's adaptation from problem b) and c). Draw the isocost lines and the substituent template. e) A short-term strike leads to the price of labor rising to NOK 750. Kjell still has only NOK 1,875. How will Kjell change his adaptation from c)?How many cakes does he produce now? f) Kjell eventually starts making money, and wants to expand production to 650 cakes. What will be Kjell's new…1 1 2. Suppose a producer's production function is F(K, L) = 3KZL² The cost of K is 20 (r = 20) and the cost of L is 80 (w = 80) The firm wants to produce 30 units of output. a. What is the cost-minimizing way for the firm to produce 30 units of output? How many units of capital and how many units of labor should be used?A firm's cost function is given by C(x,y)=2x²+y²+2, where two outputs are produced, x and y. What are the outputs levels that minimize the costs to the firm? What is that minimal cost? Using hessian determinant show that cost is indeed minimized
- An accountant for a car rental company was recently asked to report the firm's costs of producing various levels of output. The accountant knows that the most recent estimate available of the firm's cost function is C(Q) = 100 + 10Q + Q^2, where costs are measured in thousands of dollars and output is measured in thousands of hours rented. a. What is the average fixed cost of producing 2 units of output? b. What is the average variable cost of producing 2 units of output? c. What is the average total cost of producing 2 units of output? d. What is the marginal cost of producing 2 units of output? e. What is the relation between the answers to (a), (b), and (c) above? Is this a general property of average cost curves?4) A firm faces a production function of twittle-twaps: Q(K,Lp,Ln) = 5*K(2/5)*LP(1/3)*LN(1/5) per hour, where capital (K), production labor (LP), and non-production labor (LN) are input factors used in production. The firm operates in a competitive market, where they are a price taker within the capital & labor markets and its own price (r = 40, wP = 25, wN = 50, P = 20). Answer the following.a. If capital and non-production labor are fixed at K = 32 and LN = 243, what is the general form MPLP and graph Q wrt to LP changing [you do not need to solve for LP yet].b. Is this production function decreasing, constant, or increasing returns to scale and why.c. Given the wage of production workers and the price of twittle-twaps, what is the optimal number of LP to employ to maximize profits and the quantity produced (VMPLP = wP).d. If the firm can control both K and LP, what does the Isoquant curve look like and its slope in relative terms if LN is fixed at 243 units [IQ slope =…4) A firm faces a production function of twittle-twaps: Q(K,Lp,Ln) = 5*K(2/5)*LP(1/3)*LN(1/5) per hour, where capital (K), production labor (LP), and non-production labor (LN) are input factors used in production. The firm operates in a competitive market, where they are a price taker within the capital & labor markets and its own price (r = 40, wP = 25, wN = 50, P = 20). Answer the following.a. If capital and non-production labor are fixed at K = 32 and LN = 243, what is the general form MPLP and graph Q wrt to LP changing [you do not need to solve for LP yet].b. Is this production function decreasing, constant, or increasing returns to scale and why.c. Given the wage of production workers and the price of twittle-twaps, what is the optimal number of LP to employ to maximize profits and the quantity produced (VMPLP = wP).d. If the firm can control both K and LP, what does the Isoquant curve look like and its slope in relative terms if LN is fixed at 243 units [IQ slope =…
- Suppose 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.The weekly marginal cost of producing x pairs of tennis shoes is given by the following function, where C(x) is cost in dollars. C′(x)=15+500x+1 If the fixed costs are $4,000 per week, find the cost function. What is the average cost per pair of shoes if 1,000 pairs of shoes are produced each week?A company manufactures mountain bikes. The research department produced the marginal cost function C'(x)=500-3, 0≤x≤ 900, where C'(x) is in dollars and x is the number of bikes produced per month. Compute the increase in cost going from a production level of 450 bikes per month to 600 bikes per month. Set up a definite integral and evaluate it. ← The increase in cost is $
- A firm’s production is represented by the following function: Q = L1/4 K3/4 . The rental rate of capital (r) is $40 and the wage rate (w) is $12.a. For a given level of output, what should be the ratio of capital to labor in order to minimize costs?b. How much capital and labor should be used to produce 400 units of output? What is the total cost?c. What is the short run total cost if output is decreased to 300 units?d. How would the capital labor choice and total cost change in the long run?e. Would the firm prefer to relocate if input prices are r = 60 and w = 8 at an alternativelocation assuming relocation is costless?Consider the following production function of DVDS: Q = K0.5L 0.5, where Q represents DVDS (boxes per hour), K denotes capital input (units per hour), and L denotes labour input (units of worker time per hour). The unit cost of capital and labour are $40 and $20, respectively. a) What is the optimal input ratio of labour and capital for the production? b) Determine the appropriate input mix to produce 800 boxes of DVDS. Compute the cost of production. c) Suppose the government decided to offer a subsidy that would make the cost of labour $15. What is the optimal input mix to produce the same level of output, and the new cost of production? Also, compute the substitution effect in the employment of labour.If C(x) the total cost function of producing x items, then the marginal cost at x = a, given by C'(a), is the approximate cost to produce the (a + 1)st item. If the cost of producing x golf carts is C(x) = 0.001x³ + 0.2x2 + 39x + 200 what is the cost of producing 20 golf carts? C(20) = $ What the marginal cost when x = 20? C'(20) = $