Question
In: Economics
Consider a firm that operates using only a single variable input, labor (denoted ℓ). The production...
Consider a firm that operates using only a single variable
input, labor (denoted ℓ). The production function of this firm is
given by:
Φ(ℓ)=?=2√ℓ
where ? denotes output. The firm is a price-taker in both the
output market and the input market for labor. Each unit of labor
costs the firm ?>0. The market price for the firm’s output is
given by ?>0.
Show that the firm’s profit function is ?∗(?,?)=?2?. Show that ?∗(?,?) is homogeneous of degree one and interpret this result.
Derive the comparative statics terms ??∗/??, ??∗/??, ?ℓ∗/??, and ?ℓ∗/?? and interpret these results.
Solutions
Rahul Sunny answered 1 year agoRelated Solutions
Consider a firm that operates using only a single variable input, labor (denoted ℓ). The production...
Consider a firm that operates using only a single variable
input, labor (denoted ℓ). The production function of this firm is
given by:
Φ(ℓ)=?=2√ℓ
where ? denotes output. The firm is a price-taker in both the
output market and the input market for labor. Each unit of labor
costs the firm ?>0. The market price for the firm’s output is
given by ?>0.
(a) Show that the firm’s total cost function is ??(?;?)=
(?/4)?2 and Show that the firm’s profit
function...
A firm produces a single output using two variable inputs (denoted ?1 and ?2). The firmproduction...
A firm produces a single output using two variable inputs
(denoted ?1 and ?2). The firmproduction function is given by ? =
Φ(?1, ?2) = (?1?2) ^0.5. The firm can employ as much of either
input it desires by incurring constant (and respectively denoted)
per-unit input costs of ?1 and ?2. Assume throughout that all
prices and quantities are positive and infinitely divisible.
Finally, let ?0 denote the “target” level of output that the firm
envisions producing when deciding how...
Consider a firm which produces according to the following production function by using labor and capital:...
Consider a firm which produces according to the following
production function by using labor and capital:
f(l,k) = k1/4 / 1/2
(a) Solve the cost minimization problem of this firm for the
given wage rate, w and the rental rate of capital,v.
(b) Derive the cost function of the firm.
(c) Solve the profit maximization of the firm for a given price
p, and derive the supply function.
(d) Derive the profit function of the firm.
(e) Suppose demand for...
Consider a firm which produces according to the following production function by using labor and capital:...
Consider a firm which produces according to the following
production function by using labor and capital: f(l,k) =
K1/2 L1/2
(a) Solve the cost minimization problem of this firm for the
given wage rate, w and the rental rate of capital,v. Derive the
long-run total cost function of the firm.
(b) Derive the short-run total cost, the short-run average cost,
the short- run marginal cost function of the firm under the
assumption that capital is the fixed input.
(c) What...
A firm uses a single input to produce a good according to the production function q...
A firm uses a single input to produce a good according to the
production function q = f(x) = 4x(1/2), where x is the input. The
good sells for $4 per unit, and the cost of input x is w.
a. Create a function showing the firm’s profit as a function of
x and w.
b. Find the profit maximizing amount of x as a function of w. What
is the profit maximizing amount of output?
c. Will this firm...
(a) Consider a one input production technology utilizing labor that is represented by the function, x...
(a) Consider a one input production technology utilizing labor
that is represented by the function, x = Aln(L) with A>0. Is
the producer’s choice set convex? Show your working.
(b) Find the total cost function associated with this production
technology.
Note: Show all necessary working and steps so that understanding
the solution is easy.
If you owned a business and labour is the only variable input, explain: a) using numerical...
If you owned a business and labour is the only variable input,
explain:
a) using numerical examples, how decreasing marginal product of
labour will affect your marginal returns and marginal cost.
2 Marks
b) how a change in marginal cost can affect your average total
cost and your profit.
Isoquant Analysis. A firm with production function q = K1/4L1/4 operates with variable labour and variable...
Isoquant Analysis. A firm with production function q =
K1/4L1/4 operates with variable labour and
variable capital. The firm sells output at a competitive price p =
80 and hires labour at w = 2 and capital at r = 0.5.
This firm
will produce q =
will hire L =
will lease capital K =
will incur total cost C =
will earn profits π =
Now the price rises to p = 120. This firm
will produce q...
Consider the following cost information for a firm that operates in a perfectly competitive market. Labor...
Consider the following cost information for a firm that operates
in a perfectly competitive market. Labor is a variable input.
Q (quantity of output)
Total cost ($)
0
15
1
25
2
45
3
75
4
110
5
165
6
225
OUTCOME #1: Suppose that the market price is
$20. State the condition for the optimal quantity of
output. Using the condition, find the quantity of
output that the firm should produce in the short run.
OUTCOME #2: In...
Consider the following cost information for a firm that operates in a perfectly competitive market. Labor...
Consider the following cost information for a firm that operates
in a perfectly competitive market. Labor is a variable
input.
Q (quantity of output)
Total cost ($)
0
15
1
25
2
45
3
75
4
110
5
165
6
225
(1) As the firm increase the output from 1 unit to 2 units, does
the marginal product of labor rise or fall?
Explain.
(2) Suppose that the market price is $30. Find the optimal
quantity of output that...
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