Question

1. A firm production function is given by q(l,k) =
l^{0.5}·k^{0.5}, 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 its profit maximizing
level of labor input. In the short-run, this firm demand equation
for labor l is a factor a of (p/w)^{2}: l = a
(p/w)^{2}. In this specific case, factor a is equal to
**[a]**. (HINT: This is unconstrained profit
maximization problem. Set your first order condition by taking the
derivative of your profit function with respect to l equal to zero,
assume the second order condition is satisfied, and solve for l.
NOTE: Write your answer in number format, with 2 decimal places of
precision level; do not write your answer as a fraction. Add a
leading zero and trailing zeros when needed.)

2. An individual utility function is given by U(c,h) = c·h,
where c represents consumption during a typical day and h hours of
leisure enjoyed during that day. Let l be the hours of work during
a day, then l + h = 24. The real hourly market wage rate the
individual can earn is w = $20. This individual receives daily
government transfer benefits equal to n = $100. For the graphical
analysis of this individual’s utility maximization problem,
consumption c is plotted on the vertical axis and hours of leisure
h is plotted on the horizontal axis. The y-intercept for this
individual’s full income constraint is **[y]**. (NOTE:
Write your answer in number format, with 2 decimal places of
precision level; do not write your answer as a fraction. Add a
leading zero and trailing zeros when needed.)

Answer #1

2. An individual utility function is given by U(c,h) = c·h,
where c represents consumption during a typical day and h hours of
leisure enjoyed during that day. Let l be the hours of work during
a day, then l + h = 24. The real hourly market wage rate the
individual can earn is w = $20. This individual receives daily
government transfer benefits equal to n = $100. For the graphical
analysis of this individual’s utility maximization problem,...

A firm’s production function is Q(L,K) = K^1/2 + L. The firm
faces a price of labor, w, and a price of capital services, r.
a. Derive the long-run input demand functions for L and K,
assuming an interior solution. If the firm must produce 100 units
of output, what must be true of the relative price of labor in
terms of capital (i.e. w/r) in order for the firm to use a positive
amount of labor? Graphically depict this...

An individual utility function is given by U(x,y) = x·y1/2. This
individual demand equation for x is a factor a of I/px: x* = a
(I/px). In this specific case, factor a is equal to [a]. (NOTE:
Write your answer in number format, with 2 decimal places of
precision level; do not write your answer as a fraction. Add a
leading zero and trailing zeros when needed.)

Consider a firm using the production technology given by q =
f(K, L) = ln(L^K)
If capital is fixed at K = 2 units in the short run, then what
is the profit maximizing allocation of output if the price of
output and respective input prices of labor and capital are given
by (p, w, r) = (2, 1, 5)?

Suppose an agricultural firm has the production function:
f(l; k; a) = l^(1/4) * k^(1/4) * a^(1/4)
where the price of labor is w, the price of capital is r and
acreage (a) has price s.
(a) Verify that this is a valid production function.
(b) Solve the rm's cost minimization problem for the conditional
input demands,
cost function, average cost function, and marginal cost
function.
(c) Suppose that there was a tax on one or more inputs. For each...

A firm’s production function is given by Q = 5K1/3 +
10L1/3, where K and L denote quantities of capital and
labor, respectively.
Derive expressions (formulas) for the marginal product of each
input.
Does more of each input increase output?
Does each input exhibit diminishing marginal returns?
Prove.
Derive an expression for the marginal rate of technical
substitution (MRTS) of labor for capital.
Suppose the price of capital, r = 1, and the price of labor, w
= 1. The...

Suppose a firm has a production function given by q = 3L +
K.
The firm can purchase labor, L at a price w = 24, and capital, K
at a price of r = 5.
What is the firm’s total cost function?

Given production function: Q=L3/5K1/5.
Where L is labor, K is capital, w is wage rate, and r is rental
rate.
What kinds of returns to scale does your firm face?
Find cost minimizing level of L and K, and long run cost
function.

Firm A’s production function and cost line are given by:Q=Q(L,K)=2 L^(1/2) K^(1/2) (The production function)?L+?=30000 (The cost line of firm A):?L is the amount of labor hired.?K is the amount of capital hired.??p_L or the price of labor is 1 dollar per
unit.??p_K or the price of capital is 1 dollars per
unit.C or cost (think of it as the firm’s budget) is 30000
dollars.How much labor and capital should this firm optimally hire?

Suppose the final goods production function is fixed-proportion,
Q = f(E, L) = min{E,L}, where Q is output level, E is energy input
and L is the labor in- put. Let m be the marginal cost of energy
per unit and w be the price of labor per unit. Suppose the demand
function for final good is P = 1 - Q
a). (10) Suppose energy and final good are produced by two
different firms. Derive the cost function of...

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