2. (i) The production function for a firm is given by
Q = LK
where Q denotes output; Land K labor and capital inputs.
Wage rate and rental rate are given by w and r respectively.
(a) Show whether or not the above production function exhibits
diminishing marginal productivity of labor.
(b) Determine the nature of the Return to Scale as exhibited by the above
production function
(c) Using the Lagrangean Multiplier method, calculate the least cost
combinations of labor and capital and the resulting long run total cost
function for the above production function. Explain the economic
significance of the Lagrangean Multiplier and calculate its value.
3(d) Using the above cost function, calculate the numerical value of long run
total cost when Q =225, w = 16 and r = 144.
(e) Using Excel- Solver verify your answer to (d) above.
(ii) As the manager of an 80-unit motel you know that all units are occupied
when you charge $60 a day per unit. Each occupied room costs $4 for
service and maintenance a day. You have also observed that for every x
dollars increase in the daily rate above $60, there are x units vacant.
Determine the daily price that you should charge in order to maximize
profit.
I can not help with the first part, do not have formulas or understand notation.
ii. ) fully occupied profit = 80*60 -80*4 = 4480
price/unit = 60 + x
no of units occupied = 80 - x
cost = 4 (80-x)
so profit = (80-x)(60+x) - 4(80-x) = P
P = (80-x)(56+x)
P = 4480 + 24 x - x^2
max where dP/dx = 0 (or find vertex of parabola)
2 x = 24
x = 12
price = 60 + 12 = 72
(i) To determine whether the production function exhibits diminishing marginal productivity of labor, we need to calculate the marginal product of labor (MPL).
The marginal product of labor is the additional output gained from employing one more unit of labor while keeping the capital input constant. It can be calculated as the derivative of the production function with respect to labor.
Differentiating the production function with respect to labor, we get:
∂Q/∂L = K
Since the derivative with respect to L is simply the capital input (K), it does not depend on the level of labor input (L). Therefore, the marginal product of labor is constant and not diminishing. Hence, the production function does not exhibit diminishing marginal productivity of labor.
(b) The nature of the return to scale is determined by how the output changes when all inputs are proportionately increased. Let's consider the case when both labor and capital inputs are increased by the same factor (λ). The new production function can be represented as:
Q(λL, λK) = (λL)(λK) = λ^2LK
If Q(λL, λK) = λ^2Q(L, K), then the production function exhibits constant returns to scale. If Q(λL, λK) > λ^2Q(L, K), then it exhibits increasing returns to scale. If Q(λL, λK) < λ^2Q(L, K), then it exhibits decreasing returns to scale.
From the given production function, we can observe that Q(λL, λK) = (λL)(λK) = λ^2LK = λ^2Q(L, K). Therefore, the production function exhibits constant returns to scale.
(c) To calculate the least cost combinations of labor and capital and the resulting long-run total cost function using the Lagrangean Multiplier method, we need to set up the Lagrangean equation.
The Lagrangean equation for this problem can be set up as:
L = Q - wL - rK + λ(Q - LK)
where λ is the Lagrangean multiplier.
To find the least cost combinations, we need to take partial derivatives of the Lagrangean equation with respect to L, K, and λ, and set them equal to zero. Solving these equations will give us the optimal values of L and K.
To find the resulting long-run total cost function, we substitute the optimal values of L and K back into the production function (Q = LK) and solve for Q.
The economic significance of the Lagrangean multiplier is that it represents the rate at which the objective function (in this case, total cost) changes with respect to the constraint (in this case, output level). The Lagrangean multiplier provides the optimal trade-off between increasing output and minimizing costs. Its value can be calculated by solving the Lagrangean equations.
(d) To calculate the numerical value of long-run total cost when Q = 225, w = 16, and r = 144, we need to use the long-run total cost function obtained from part (c).
By substituting the given values into the long-run total cost function, we can calculate the numerical value of the total cost.
(e) To verify the answer obtained in part (d) using Excel Solver, you can set up a spreadsheet with the necessary formulas and constraints. By setting the objective function to minimize the total cost and setting the constraints according to the given values, you can use the Solver tool in Excel to find the optimal values of labor and capital inputs, as well as the resulting total cost. Make sure to test different values and ranges to ensure the accuracy of the solution.
(ii) To determine the daily price that maximizes profit, we need to consider the revenue and cost functions.
The revenue function can be calculated as the product of the daily rate (p) and the number of occupied units (80 - x), where x represents the number of vacant units when the daily rate is increased.
The cost function can be calculated as the sum of the per-unit service and maintenance cost ($4) multiplied by the number of occupied units (80 - x).
The profit function can be calculated as the difference between the revenue and cost functions.
To maximize profit, we need to find the value of x that maximizes the profit function. One way to do this is by setting up a table or graph to analyze the relationship between the daily rate (p), vacant units (x), and profit. By testing different values of x and calculating the corresponding profit, we can identify the daily price that maximizes profit.