Let input prices be (wK, WL) and output price be p. The production function is given by ƒ (K, L) = min [√K, L]. (a) (b) Derive the cost function. Use the cost function to derive the output supply function.
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![1. Let input prices be (wK, WL) and output price be p. The production function is given
by f (K, L) = min [√K, L].
(b)
Derive the cost function.
Use the cost function to derive the output supply function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F655e5d67-ab19-404c-b883-3aa3c693f6a1%2Fa4cfa809-06f1-4205-bea9-a3f1358eae5c%2Fvqozdzf_processed.jpeg&w=3840&q=75)
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- Juan Valdez owns a coffee farm in Colombia. His production function is: f(x1,x2)=(x1−1)^0.25 x2^0.5 Assume the price of input 1 is r and the price of input 2 is w. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm?The cost function for producing x items is C(x)=x2-3x+625. a.Find the average cost function. b.What is the minimum average cost? c.Find the marginal cost function. d.Is the point of intersection of the average cost and marginal cost function the same which produced the minimum the average cost? 2.Determine where the function f(x)=4x3-3x2+6 is concave and where it is convex. 3.An efficiency study conducted for Spektra Electronics Showed that the number of Base Commander handsets assembled by the average worker t hours after starting work at 8 a.m is given by N(t)=-t2+6t2+15t (0t4) At what time during the morning shift is the average worker performing at peak efficiency?b Now suppose Q = 2L +3K. Let the market price of L be w = 5 and the price of K be r = 4. Let both L and K can vary with production. Compute the input demand functions as a function of Q. (4 Points) c Calculate the marginal cost and average cost of the above function in subpart (b). Show them graphically. At what prices of textile will the producer shut down production. (3 Points) d Now suppose Q = 10LK. The market prices of inputs are as in subpart (b) above. Compute the input demand functions as a function of Q. Find the optimal production when the price of textile is $10 per yarn. (5 Points)
- 6 A production function is given by f(x1,x2)=(max{x1,x2})^0.5. Prices of inputs are p1=10 and p2=20. What is the lowest cost of producing 10 units of output?a= min[x,y]^(1/2) a)Draw isoquants for a = 1, a = 4 and a = 9. b)What kind of returns to scale does the production function have? c)The price of x is p_x = 1 and of y p_y = 2. Calculate the cost function. d)Rent is $ 400, calculate average and marginal costs.X and X2 are the two factors used in production A firm's production function: fcx₁, x₂) = max {x₁, X₂} the price of X, is W₁=8, and the price of X₂ is W₂=10 The firm paproduce 100 total units of output, what is the total cost?
- Suppose that cost function is C(Q) = mQ 2 + nQ + p. Find the point thataverage cost function A(Q) has a minimum point.The production function for a product is given by q= 10K^(1/2)L^(1/2) where K is capital, and Lis labor and q is output. a) Find the marginal product of labor and the marginal product of capital. b) Find the marginal rate of technical substitution between labor and capital. c) Denote the wage of labor by w and the rental of capital by r. What is the cost minimization condition for a firm? Show it diagrammatically. d) Now suppose w =30 and r= 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) Part A, B, CThe production function for a product is given by q= 10K^(1/2)L^(1/2) where K is capital, and L is labor and q is output. a) Find the marginal product of labor and the marginal product of capital. b) Find the marginal rate of technical substitution between labor and capital. c) Denote the wage of labor by w and the rental of capital by r. What is the cost minimization condition for a firm? Show it diagrammatically. d) Now suppose w =30 and r = 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) Part A, B, C
- 1. Let a firm’s cost function be c(y1, y2) = F + αy1y2 + y21 + y22, where α is some constant.a) What restriction on α is required to guarantee that this cost function exhibits economies ofscope?b) What restriction on α is required to guarantee that this cost function exhibits cost complementarities in both goods? Note that you’ll need to verify cost complementarities for y1 andy2.2. Given the industry demand function X(p) = 100 − 2p, consider the following scenarios:• The market is a perfectly competitive market. Assume there are identical firms with marginalcost of 12 in this perfectly competitive market.• The market is dominated by one monopolist with a marginal cost of 12. This monopolist isable to achieve 1st degree pricing.• The market is dominated by one monopolist with a marginal cost of 12, but the monopolistis able to achieve only 2nd degree pricing. Assume the menu offers only 2 choices:(Q∗1 = 30, p∗1 = 35), and (Q∗2 = 60, p∗2 = 20).• The market is dominated by one monopolist…Given the production function f(x,x2) x,ax,, calculate the profit maximizing demand and supply functions, and the profit function. ii. %3DConsider 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