Question

Consider the Cobb-Douglas production function F (L, K) =
(A)(L^α)(K^1/2) , where α > 0 and A > 0.

1. The Cobb-Douglas function can be either increasing, decreasing
or constant returns to scale depending on the values of the
exponents on L and K. Prove your answers to the following three
cases.

(a) For what value(s) of α is F(L,K) decreasing returns to
scale?

(b) For what value(s) of α is F(L,K) increasing returns to scale?

(c) For what value(s) of α is F(L,K) constant returns to scale?

Answer #1

the Cobb-Douglas production function F (L, K) = (A)(L^α)(K^1/2) , where α > 0 and A > 0.

1. The Cobb-Douglas function can be either increasing, decreasing or constant returns to scale depending on the values of the exponents on L and K.

(a)

It will show decreasing return to scale if the sum of (a+ b)<1, the b is 1/2 given, so the value of a must be less than 1/2, for making the Cobb Douglas production function decreasing.

(b)

It will show increasing return to scale if the sum of (a+ b)>1, the b is 1/2 given, so the value of a must be greater than 1/2, for making the Cobb Douglas production function increasing.

(c)

It will show constant return to scale if the sum of (a+ b)=1, the b is 1/2 given, so the value of a must be 1/2, for making the Cobb Douglas production function constant.

(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.

for a firm with Cobb-Douglas production function
q = f (k, L) = k ^ (1/2) L ^ (1/2)
calculate the total, average and marginal cost.

1. Using the Cobb-Douglas production function:
Yt =
AtKt1/3Lt2/3
If K = 27, L = 8 A = 2, and α = 1/3, what is the value of Y?
(For K and L, round to the nearest whole number) ______
2. If Y = 300, L = 10, and α = 1/3, what is the marginal product
of labor? ______
3. Using the values for Y and α above, if K = 900, what is the
marginal product of capital?...

A? Cobb-Douglas production function
A. exhibits constant returns to scale.
B. exhibits decreasing returns to scale.
C. exhibits increasing returns to scale.
D. can exhibit? constant, increasing, or decreasing returns to
scale.

For each part of this question write down a Cobb-Douglas
production function with the returns to scale called for and
perform a proof for each that shows the production function has the
correct returns to scale.
Constant returns to scale
Decreasing returns to scale
Increasing returns to scale
Increasing returns to scale

Suppose that a firm has the Cobb-Douglas production function Q =
12K ^ (0.75) L^ (0.25). Because this function exhibits (constant,
decreasing, increasing) returns to scale, the long-run average cost
curve is (upward-sloping, downward-sloping, horizontal), whereas
the long-run total cost curve is upward-sloping, with (an
increasing, a declining, a constant) slope.
Now suppose that the firm’s production function is Q =
KL. Because this function exhibits (constant, decreasing,
increasing) returns to scale, the long-run average cost curve is
(upward-sloping, downward-sloping,...

Which is/are incorrect about the Cobb-Douglas production
function: Y equals K to the power of alpha L to the power of 1
minus alpha end exponent (0 < alpha < 1 )? All are correct it
increases in both K and L the share of total income that goes to
capital and labor depend on the amount of K and L it exhibits
diminishing marginal returns to both K and L it is constant returns
to scale

Consider the production function Y = F (K, L) = Ka *
L1-a, where 0 < α < 1. The national saving rate is
s, the labor force grows at a rate n, and capital depreciates at
rate δ.
(a) Show that F has constant returns to scale.
(b) What is the per-worker production function, y = f(k)?
(c) Solve for the steady-state level of capital per worker (in
terms of the parameters of the model).
(d) Solve for the...

2. Consider a Cobb-Douglas production function Q = A . L^a . K^b
. Answer the following in terms of L, K, a, b
(a) What is the marginal product of labour ?
(b) What is the marginal product of capital ?
(c) What is the rate of technical substitution (RTS L for
K)?
(d) From the above what is the relation between K L and RT
SL,K?
(e) What is the relation between ∆ K L ∆RT SL,K (f)...

1. Consider the following production function:
Y=F(A,L,K)=A(K^α)(L^(1-α))
where α < 1.
a. Derive the Marginal Product of Labor(MPL).
b. Show that this production function
exhibit diminishing MPL.
c. Derive the Marginal Production of Technology (MPA).
d. Does this production function exhibit diminishing MPA? Prove
or disprove

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