Question

Consider the production function q=aK + bL.

a. Show that the cost-minimizing choice of K and L may not be unique. (The cost-minimizing K and L levels are those used at a firm’s cost-minimizing point; the levels are not unique if there is more than one optimal combination of K and L for any one isoquant.)

b. Show on a diagram that, if the cost-minimizing choice of inputs is unique, it will generally entail the use of only K or only L. Under what conditions would the firm choose only K (and no L)? That is, what is the mathematical relationship between w/r and b/a when the firm chooses only K?

Answer #1

A firm’s production function is Q = min(K , 2L), where Q
is the number of units of output produced using K units of capital
and L units of labor. The factor prices are w = 4 (for labor) and r
= 1 (for capital). On an optimal choice diagram with L on the
horizontal axis and K on the vertical axis, draw the isoquant for Q
= 12, indicate the optimal choices of K and L on that isoquant,...

a. A cost minimizing firm’s production is given by Q=L1/2K1/2.
Suppose the desired output is Q=10. Let w=12 and r=4. What is this
firm’s cost minimizing combination of K & L? What it the total
cost of producing this output?
b. Suppose the firm wishes to increase its output to Q=12. In
the short run, the firm’s K is fixed at the amount found in (a),
but L is variable. How much labor will the firm use? What will the...

a. A cost minimizing firm’s production is given by
Q=L^(1/2)K^(1/2)
. Suppose the desired output is
Q=10. Let w=12 and r=4. What is this firm’s cost minimizing
combination of K & L? What it the
total cost of producing this output?
b. Suppose the firm wishes to increase its output to Q=12. In
the short run, the firm’s K is fixed
at the amount found in (a), but L is variable. How much labor
will the firm use? What will...

3. A firm’s production function is Q=min(K, 3L ). Input prices a
re as follows: w=$ 2 and r=$1.
On the optimal choice diagram below, draw the isoquant for Q=12.
Calculate the optimal choice of K and L for this level of output as
well as the total cost.
Then, draw in (with a dotted line) the isocost line consistent
with your Total Cost value.
It won't let me copy the graph template but it is a simple graph
with...

1. Consider the production
function q=K2L0.5
a) Find the cost minimizing quantities of K and L for q = 100, r
as the price of K and w as the price of L.
b) Find the cost minimizing quantities of K and L for q = 1000,
r as the price of K and w as the price of L. Explain whether or not
the output expansion [change from part a) to part b)] is labor
saving or capital saving.

There are two inputs labor, L, and capital, K. Their cost
minimizing levels are given by K(y) = 2y and L(y) =y^2.L and K are
respectively priced w=1/2 and r= 3
.a) Find the firm’s cost curve.
b) What is the firm’s exit price
c) Graphically show how the long run supply curve is derived
from cost curves (make sure to label the axes, the curves, the
intercepts, and the slope).
d) A tp= $12, what is the profit-maximizing level...

The firms production function is: Q=2L^2/3 K^1/3
A) Suppose the firm wants to determine the cost minimizing
combination for L and K for any given values of q, w, and r. Solve
for the the firms factor demand functions for L and K (i.e. express
the optimal quantity of L and K in terms of W, r and Q)
B) Using these factor demand functions, solve for the firm's
long run cost function.

A firm’s production process is represented by y= L^2/3 K^1/3.
The price of Labor, w is $2 and the price of capital, r, is
$27.
(a) Write down the firm’s cost minimization problem
(b) What is the firm’s MRTS?
(c) What are the firm’s cost minimizing levels of labor and
capital (these will both be functions of y)?
(d) What is the firm’s cost curve (ie, derive C(y))?
(e) If the firm chooses output y= 450, what are the firms...

A firm’s production technology is given by the production
function q = 0.25 LK where L represents labor hours, K machine
hours and q the amount of output. The market wage and rental rates
are, w= $16 and r = $256. The firm is operating in the long run
where it can adjust both inputs.
(b) Suppose that the firm wants to produce 100 units of output.
Determine the cost minimizing combination of L and K. Calculate the
resulting long...

A cost-minimizing firm has the following production function:
Q=LK+2M. Where L denotes Labor, K denotes Capital, and M denotes
Materials. The prices for the inputs are as follows: w=$4, r=$8,
and m=$1. The firm is operating in the long run. Answer the
following questions as you solve for the total cost of producing
400 units of output. Assume an interior solution (i.e. positive
values of all inputs).
a) Set up constrained optimization problem of the firm:
b) Write out the...

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