Question

Consider the following production function: Y = A ̄K2 L1 , where Y is production, A ̄ is productivity, K is capital, and L is labor. Let w denote the wage rate and r denote the rental rate of capital. 21 Suppose you solve the profit maximization problem of the firm: max A ̄K L wL rK. What is K,L the expression for wL ? Y (a) wL =1↵. Y (b) wL =↵. Y (c) wL = 1. Y3 (d) wL = 2. Y3 (e) None of the above.

Answer #1

The production function is given by y=L1/2, where y
is the output, and L is the amount of labor input. Assume the wage
rate is w so that the cost of using L unit of labor input is wL.
Let p denote the unit price of the output. Note that w and p are
exogenously given.
(1) Find the function for the profit (in terms of p, w and
L).
(2) Find the optimal choice of labor input and the...

A closed economy has the following Cobb-Douglas production
function: F(KL) = K2/5 (EL)3/5, where the notation is as in class.
The depreciation rate is 1.5% and the saving rate is 20%. The
economy is in steady state, where the population decreases at a
rate 1% and capital K increases at a rate 1%. (a) Find the growth
rates of the following variables (i) labor efficiency, E (ii) the
number of workers per machine, L/K (iii) the average productivity
of capital,...

Let a production function for a country, Eastasia, be defined
as: Y=5*L1/3*K2/3.
Suppose L=27 and K=64. Find the level of GDP.Now, suppose L=13.5
and K=32. Find the level of output in the economy.
Does this function have constant
returns to scale? Explain.
Now, let
Y=5*L1/3*K2/3+L. Does this function have
constant, increasing, or decreasing returns to scale? Explain.

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...

Consider the following Cobb-Douglas production function: y(K,L)
= 2K^(0.4)*L^(0.6), where K denotes the amount of capital and L
denotes the amount of labour employed in the production
process.
a) Compute the marginal productivity of capital, the marginal
productivity of labour, and the MRTS (marginal rate of technical
substitution) between capital and labour. Let input prices be r for
capital and w for labour. A representative firm seeks to minimize
its cost of producing 100 units of output.
b) By applying...

Suppose that a firm has production function F(L, K) = L1/4 K3/4
for producing widgets, the wage rate for labor is w = $32, and the
rental rate of capital is r = $6. Suppose the firm has an order to
produce 40 units of output.
a) Carefully write out the firm’s cost minimization problem,
using information specific to this problem.
b) Express two equations—specific to this problem—that the
optimal solution satisfies.
c) Solve these two equations for L* and...

Consider the following production function: x = f(l,k) =
Albkbwhere x is the output, l is the labour
input, k is the capital input, and A, b are positive constants.
(a) Set up the cost minimization problem and solve for the first
order conditions using the Lagrange Method. Let w be the wage rate
and r the rental rate of capital.
(b) Using your answer in (a), find how much labour and capital
would the firm use to produce x...

Consider a firm the faces the following production
technology:
Y = 500K − K2
where Y denotes output, and K the capital stock.
The price of capital, pk, is 500, the real interest
rate, r, is 5%, and the depreciation rate, d, is 15%.
(a) Derive the expected future marginal product of capital, MPKf
. How does MPKf vary with K?
(b) What is the user cost of capital and the firm’s desired
capital stock? If the initial capital stock...

A firm produces output (y), using capital (K) and labor (L). The
per-unit price of capital is r, and the per-unit price of labor is
w. The firm’s production function is given by, y=Af(L,K), where A
> 0 is a parameter reflecting the firm’s efficiency.
(a) Let p denote the price of output. In the short run, the
level of capital is fixed at K. Assume that the marginal product of
labor is diminishing. Using comparative statics analysis, show that...

Production Function: Y = AKx L1-x = (100)
K1/2 L1/2
Find the minimum cost technique and total equilibrium output
when the following hold:
TC = 18$ : pk = 1 $/K : w0 = 3 $/L
and the minimum cost technique and total equilibrium output when
the wage decreases:
TC = 18$ : pk = 1 $/K : wt = 2 $/L
Including the chosen technique, show 2 different techniques on
each respective isoquanT

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