Question

(2) Consider the production function f(L, K) = 2K √ L. The marginal products of labor and capital for this function are given by MPL = K √ L , MPK = 2√ L. Prices of inputs are w = 1 per hour of labor and r = 4 per machine hour. For the following questions suppose that the firm currently uses K = 2 machine hours, and that this can’t be changed in the short–run.

(e) What is the (short–run) efficient scale of production? What is the (short–run) average cost at the efficient scale of production?

(f) Assume that in the short run the r ∗ K = 4 ∗ 2 = 8 dollars that the firm pays for its capital are sunk. What is the short–run profit maximizing quantity as a function of the market price p? Draw the short–run supply curve of the firm. What are the firm’s profits (as a function of p)?

(g) How would the short–run supply curve change if the fixed cost (of 8 dollars) is avoidable instead of sunk?

Answer #1

Consider the production function Q = f(L,K) = 10KL / K+L. The
marginal products of labor and capital for this function are given
by
MPL = 10K^2 / (K +L)^2, MPK = 10L^2 / (K +L)^2.
(a) In the short run, assume that capital is fixed at K = 4.
What is the production function for the firm (quantity as a
function of labor only)? What are the average and marginal products
of labor? Draw APL and MPL on one...

A firm produces output according to the production function.
Q=sqrt(L*K) The
associated marginal products are MPL = .5*sqrt(K/L) and MPK =
.5*sqrt(L/K)
(a) Does this production function have increasing, decreasing, or
constant marginal
returns to labor?
(b) Does this production function have increasing, decreasing or
constant returns to
scale?
(c) Find the firm's short-run total cost function when K=16. The
price of labor is w and
the price of capital is r.
(d) Find the firm's long-run total cost function...

3. Consider the production function, Q = [L0.5 +
K0.5] 2 . The marginal products are given as
follows: MPL = [L0.5 + K0.5] L-0.5
and MPK = [L0.5 + K0.5] K-0.5 and
w = 2, r = 1.
A). what is the value of lambda
B). Does this production function exhibit increasing, decreasing
or constant returns to scale?
C).Determine the cost minimizing value of L
D).Determine the cost minimizing value of K
E).Determine the total cost function
F).Determine the...

A firm has the production function:
Q = L 1 2 K 1 2
Find the marginal product of labor (MPL), marginal
product of capital (MPK), and marginal rate of technical
substitution (MRTS).
Note: Finding the MRTS is analogous to finding the
MRS from a utility function:
MRTS=-MPL/MPK. Be sure to simplify your
answer as we did with MRS.
A firm has the production function:
Q = L 1 2 K 3 4
Find the marginal product of labor (MPL),...

a firm produces a product with labor and capital as inputs. The
production function is described by Q=LK. the marginal products
associated with this production function are MPL=K and MPK=L. let
w=1 and r=1 be the prices of labor and capital, respectively
a) find the equation for the firms long-run total cost curve
curve as a function of quantity Q
b) solve the firms short-run cost-minimization problem when
capital is fixed at a quantity of 5 units (ie.,K=5). derive the...

(a) Show that the following Cobb-Douglas production function,
f(K,L) = KαL1−α, has constant returns to scale.
(b) Derive the marginal products of labor and capital. Show
that you the MPL is decreasing on L and that the MPK is decreasing
in K.

Suppose a firm’s production function is given by Q = L1/2*K1/2.
The Marginal Product of Labor and the Marginal Product of Capital
are given by:
MPL = (K^1/2)/2L^1/2 & MPK = (L^1/2)/2K^1/2)
a) (12 points) If the price of labor is w = 48, and the price of
capital is r = 12, how much labor and capital should the firm hire
in order to minimize the cost of production if the firm wants to
produce output Q = 10?...

Consider a firm which has the following production function
Q=f(L,K)=4?LK
(MPL=2?(K/L) and MPK=2?(L/K).
(a) If the wage w= $4 and the rent of capital r=$1, what is the
least expensive way to produce 16 units of output? (That is, what
is the cost-minimizing input bundle (combination) given that
Q=16?)
(b) What is the minimum cost of producing 16 units?
(c) Show that for any level of output Q, the minimum cost of
producing Q is $Q.

2. Consider the following production functions, to be used in
this week’s assignment:
(A) F(L, K) = 20L^2 + 20K^2
(B) F(L, K) = [L^1/2 + K^1/2]^2
a (i) Neatly draw the Q = 2,000 isoquant for a firm with
production function (A) given above, putting L on the horizontal
axis and K on the vertical axis. As part of your answer, calculate
three input bundles on this isoquant. (ii) Neatly draw the Q = 10
isoquant for a firm...

The production of sunglasses is characterized by the production
function Q(L,K)= 4L1/2K 1/2 . Suppose that the price of labor is
$10 per unit and the price of capital is $90 per unit. In the
short-run, capital is fixed at 2,500. The firm must produce 36,000
sunglasses. How much money is it sacrificing by not having the
ability to choose its level of capital optimally? That is, how much
more does it cost to produce 36,000 sunglasses the short-run
compared...

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