Suppose that a firm\'s production function is given by Q = L K and that its cost
ID: 1191661 • Letter: S
Question
Suppose that a firm's production function is given by Q = L K and that its costs are given by C = w L + r K, where w = $2 and r = $8. Assuming the firm wants to produce 40,000 units of output (and minimize its cost in doing so). how much capital and labor should be used? what is this minimum cost to produce the 40,000 units? Suppose that a firm's production function is given by Q = 10 L^1/2 K and that its costs are given by C = w L + r K, where w = $15 and r = $30. If the firm attempts to maximize output subject to C = $450,000, then how much capital and labor should it use? what is the maximum output that can be produced? Farmer Jones has two apple orchards (i. e. , the North orchard and the South orchard), and farmer Jones has a total of 50 farm laborers (L_T = 50) that he can assign to pick apples in either the North orchard or in the South orchard. L_T = L_N + L_s = 50, where L_N is the number of laborers assigned to pick apples in the North orchard, and Ls_ is the number of laborers assigned to pick apples in the South orchard.Explanation / Answer
Q.1 Production Function: Q = LK
Cost Function: wL + rK and w=2 and r=8
(i) We know that at cost minimization level wage rate/capital rate = Marginal product of labor/marginal product of capital
that is W/r = MPl/MPk
And MPl = dQ/dL = K and MPk = dQ/dK = L
Thus w/r = K/L
or 2/8 = K/L
or K = L/4 .... eq i
Firms wants to produce 40,000 units that is Q = 40,000
Thus 40,000 = KL
from equation i we get 40,000 = L*L/4
L2 = 160,000
L = 400 units
And from equation 1, K = 400/4 = 100 units
Thus to produce 40,000 units of out at minimum cost the firm should employ 100 units of capital and 400 units of labor.
(ii) Cost = wL + rK
putting L = 400, K = 100, w =2 and r = 8 in above equation, we get
Cost = 2*400 + 8*100 = 800 + 800 = 1600
Thus to produce at minimum cost, the frm will have to incurr $1600 as total cost of production.
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