Consider the following production function q=f(k, l)= kl where k is capital and
ID: 1168684 • Letter: C
Question
Consider the following production function q=f(k, l)= kl where k is capital and 1 is labor used. Both inputs are variable. Rental rate of capital is given by r and wage rate of labor if given by w. Find Conditional (Factor) Demand for k and 1 Find the Cost function for this firm in terms of q, w and r only. Find an expression for Marginal Cost (MC) and Average Total Cost (ATC) What is the total cost if this firm produces 10 units of output if rental rate is 1 and wage rate is 2? What is the fixed cost? What is the Variable cost? Find Marginal Cost and Average Total Cost at 10 units of output if rental rate is 1 and wage rate is 2. Is this firm experiencing Economies of Scale or Diseconomies of Scale at the output of 10 units?Explanation / Answer
a) Conditional factor demand is the cost-minimizing level of an input (factor of production) such as labor or capital, required to produce a given level of output, for given unit input costs (wage rate and rental rate) of the input factors.
Minimize wL + rK subject to KL = q
It can be written as minimize wL + r(q/L) w.r.t L
w + rq(-1/L2) = 0
L = Squareroot(rq/w)
K = q/L = Squareroot(qw/r)
b) Cost function = wL + rK = Squareroot(wrq) + Squareroot(wrq) = 2squareroot(wrq) = C
C) ATC = C/q = 2squareroot(wr/q)
MC = Differentiate C w.r.t L(the variable factor) = w
d) q=10 r=1 w=2
Fixed cost = 1* squareroot(qw/r) = 2squareroot(5)
variable cost = 2* squareroot(qr/w) =2 squareroot(5)
e) MC = w = 2
ATC = 2squareroot(wr/q) = 2squareroot(0.2)
f) Economies of scale is when AC>MC as in that part AC is decreasing and diseconomies of scale is when AC<MC.
Here AC<MC at 10 units so the firm is experiencing diseconomies of scale.