Suppose a firm has production function F(L, K) = L6K.6. Following the steps in the slides, derive this firm's long run cost function C(q) when w = 15 and r = 8. Also derive the firm's long run marginal cost C'(Q). Why is the firm's long run marginal cost decreasing?
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- 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.K Suppose that a firm's production function is q=5x in the short run, where there are fixed costs of $1,000, and x is the variable input whose cost is $625 per unit. What is the total cost of producing a level of output q? In other words, identify the total cost function C(q). The total cost of producing ya level of output q is A. C(q)= 1,000. OB. OC. OD. OE C(q)= 1,000+ C(q)= 1,000+ C(q) = 1,000+ 625q² C(q)= 25 q² 25 625q² 25 0.5 q B Dollars per unit of output 2000- 1800- 1600- 1400- 1200- 1000- 800- 600 400- 200- a 12 14 10 Quantity nchack 18 (1) IncorrectK A cereal factory has weekly fixed costs of $22,000. It costs $1.26 to produce each box of cereal. A box of cereal sells for $4.06. Find the rule of the cost function c(x) that gives the total weekly cost of producing x boxes of cereal. OA. c(x)=1.26x OB. c(x)=22,000+ 1.26x OC. c(x)=22,000 +4.06x OD. c(x)=22,000+ 2.8x
- Suppose that a firm has increasing marginal cost, and a positive fixed cost. What happens to minimum effcient scale if the xed cost increasesSuppose the production function for high quality brandy is given by : Q = √KLWhere q is the output of brandy per week and L is labor hours per week ., in the short run K is fixed at 100, so the short run production function is Q = 10√La. If the capital rents for 10$ and the wage are 5$per hour ., write the short run total cost function. b. How much will the firm produce at a price of 20$ per bottles of brandy ?c. How many labor hours will be hired per week?The''Sad Student'' Company finds that it costs a total of (q)=9720+500q-1.5q^(2)+0.005q^(3) Dollars to produce q lots of its latest board game named ''Calculus Teacher Shooter'' each lot of 100 games. D)Interpret the effect of the constant term in the expression of total cost function on marginal cost function. E)Analyze the behavior average cost funtion. F)Analyze interaction of marginal cost and average cost numerically; and derive the formula of this relationship with respect to relevant mathematical economics theory and interpret your formula.
- The table below shows the long run total cost function for a firm. The firm's cost function exhibits: Quantity of Output 0 2 3 4 5 Odiseconomies of scale Total Cost ($) O constant returns to scale O economies of scale O decreasing marginal cost O increasing marginal cost 10 20 30 40 50 K1. A firm has the production function y = 4LK. The marginal products are given by MPL = 4K and MPK = 4L. (a) Provide an expression for the long run total cost function. (b) Now suppose that w = r = 25. Write out the expression for the long run total cost curve, and plot it on a graph. 1 (c) With w = r = 25, derive the long run average cost curve, and plot it on a graph. Does this firm experience increasing or decreasing average cost, as the output increases?production function: ƒ (k, 1) = 9 function. k Derive the firm's cost
- Cost A company buys and retails baseball caps, andthe total cost function is linear. The total cost for200 caps is $2690, and the total cost for 500 caps is$3530. What is the marginal cost, which is the rate ofchange of the function?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.)9 Question 9 The long run cost function for the production function F(L, K) = ALªK³ is c(q,w,r) = (4) a‡ß watßrats ((2)a+³ + (2)a‡³). 1. Under what conditions on A, a, and ß does the cost function exhibit economies of scale? 2. Under what conditions on A, a, and ß is c(\q, w, r) 1? 3. What is the relationship between your answers for the previous parts? What do your results say about the relationship between a cost function exhibiting economies of scale and a cost function exhibiting increasing returns to scale? 4. Prove the relationship in general: that a cost function exhibits increasing returns to scale if and only if it exhibits economies of scale. To do this, you should show the following two statements: • If c(Ag) q (hint: remember that λq > q since λ > 1). q • If there exists some › and q such that c(\q) > Ac(q), then there exists some q and q' such that AC(q) q.
