Question

The production function is given by y=L^{1/2}, 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 corresponding maximized profit (in terms of p and w).

(3) Suppose that both wage rate, w, and the price, p, double. How will this affect the firm's optimal decision? Explain.

Answer #1

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)...

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...

The production function is Y=K0.5L0.5 where K is capital, L is
labor and Y is output. The price of L is 1 and the price of K is
2.
a) Find the optimal levels of K and L that should be employed to
produce 100 units of output. What is
the cost of producing this level of output?
b) Will the optimal capital-labor ratio change if the price of
labor goes up to 2 and the price of K goes...

1. A firm production function is given by q(l,k) =
l0.5·k0.5, where q is number of units of
output produced, l the number of units of labor input used and k
the number of units of capital input used. This firm profit
function is π = p·q(l,k) – w·l – v·k, where p is the price of
output, w the wage rate of labor and v the rental rate of capital.
In the short-run, k = 100. This firm hires...

Consider a firm using quantities L1 and L2 of two kinds of
labour as its only inputs in order to produce output Q=L1+L2. Thus,
each unit of labour produces one unit of
output. Suppose that we also have two segmented labor markets,
with the following inverse labor supply functions.
w1=α1+β1L1
w2=α2+β2L2
which shows the wage that must be paid to attract a given labor
supply. Assume that the firm is competitive and take price of
output P as given. (α1,...

Suppose that you have estimated the following output function
where L is labor and K is capital: Y = K1/4 L1/2.
-You know that the current price of labor is $10 and capital
cost is $150 per machine (capital). You currently use 81 units of
capital. For #1a, the output (Y) is 100.
1a: How many employees (L) do you need to hire to achieve your
output goal?
b. Given a fixed level of capital (K=81) and a price of...

Suppose soft drink production at bottling plant is described by
the production function:
? =??^.?
where y is the number of cases of soft drinks produced and x is
the number of hours of labor hired. We also assume the wage for
labor is r, price for soft drink is p and fixed costs is
$1,000.
k. Find the optimal input of x that maximizes the profit. (Hint:
this is the input-side
optimization, you should derive optimal x as a...

ABC Corp produces widgets with labour as the only variable
input. Its production function is y = z2, where y is output and z
is labour input. The maximum output possible with its plant is 100
units of output. Denote the price of output by p > 0 and the
wage rate by w > 0. Does the Extreme Value Theorem guarantee an
answer to ABC’s profit maximization problem? Defend your
answer.

ABC Corp produces widgets with labour as the only variable
input. Its production function is y = z2, where y is output and z
is labour input. The maximum output possible with its plant is 100
units of output. Denote the price of output by p > 0 and the
wage rate by w > 0. Does the Extreme Value Theorem guarantee an
answer to ABC’s profit maximization problem? Defend your
answer.

Suppose a competitive firm’s production function is Y= 20
L1/2 K1/3. L is Labor , K is capital and Y is
output.
a) (4) Find the marginal product of labor and capital.
b) (4) What is Marginal Rate of technical Substitution of Labor
for Capital?
c) (2) Does this production function exhibit increasing,
decreasing or constant returns to scale? Show your work.

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