Please show steps to resolve these problems please use Lagrangian Multiplier if possible.
1. A manufacturer has the following production function:
Q=100 K^. 2 L^. 9
If the price per unit of labor is $20 and the price per unit of capital is $10,
a) What is the optimal combination of labor and capital to use in order to maximize output for a total cost of $2,200?
b) How much output will be produced?
c) What is the equation for the expansion path?
d) What is the meaning and value of λ (lambda)?
e) What is the value of the marginal rate of technical substitution for labor and capital at the optimal point?
f) Is there diminishing marginal productivity for labor and capital? Explain
g) Are there increasing returns to scale? Explain
2. Given the following short-run production function, Q = 1500L + 60L^2 - L^3
where Q is output and L is variable input, find
a. The point of diminishing marginal returns.
b. The point where the elasticity of production is equal to one.
c. The boundary between stage II and stage III.
d. Show that marginal product is equal to average product when average product is at its maximum.
Lagrangian Multiplier is utilized in the problems.
Gabriella Inc. is a business that helps individuals on job with local employers. Gabriella does not hire individuals; rather it provides the service of calling individuals for employers who then hire them. A key feature of Gabriella's success is its ability to reach a large number of individuals in a short period of time by means of the telephone. In making these telephone calls, the company can use telephone operators; each using a single phone, and it can use computer operators, each using a computerized process with automatic dialing capability. A production function for the calling operation is as follows:
Y = 400p + 800q + 4pq
Where: Y = number of call made per day
p = number of telephone operators per day
q = number of computer operators using phone-computer combinations
Each phone operator using a single phone costs the firm an average of $90 per day, and each computer operator matched with a phone computer combination costs $180 per day. The total expense budget allocated to this project has been limited to $3600 per day.
A. Use the Lagrangian multiplier technique to determine the optimal levels of p and q to maximize the total number of calls that can be made per day.
B. Calculate and interpret the value of the Lagrangian multiplier in this case.