4 (ii) You manage Dirt Diggers, an excavating firm that excavates
roadside ditches for laying drainpipes. Its output follows the
production function:
Q = 10L -.1L2
where L denotes labor hours and Q the length of the ditch in meters. You
can hire labor at the going wage rate of $12 per hour. As the manager of
DD you want to earn as high a profit as possible.
(a) You have received an offer to excavate 250 meters for a lump sum
payment of $500. Should you accept the offer ? Explain with
appropriate calculations.
(b) Suppose that instead of the previous offer, you are now offered as
much or as little excavation work at a price of $2.00 per meter dug.
Should you accept the offer ? Why or why not ? If you accept the
offer calculate DD’s resulting profit. Also, calculate the optimal level
of output (meter dug) and the level of labor usage.
(iii) As a manager of a firm you find the marginal cost of the firm to be $10 and
the fixed cost $100. For the range of prices that you are planning to charge,
own price elasticity of demand is believed to be –1.5. Calculate the optimal
(profit maximizing) price that you should charge. Show all calculations.
(a) To determine whether you should accept the offer to excavate 250 meters for a lump sum payment of $500, you need to compare the profit earned from accepting the offer to the profit earned from excavating 250 meters using the production function.
First, let's calculate the revenue from accepting the offer:
Revenue = Lump sum payment = $500
Next, let's calculate the cost of excavating 250 meters:
Q = 250 (given)
Labor Hours (L) = Q/Output per labor hour = 250/(10L - 0.1L^2)
Labor Cost = Labor Hours * Wage rate = (250/(10L - 0.1L^2)) * $12
Total Cost = Labor Cost
Finally, let's calculate the profit by subtracting the total cost from the revenue:
Profit = Revenue - Total Cost
Compare the profit obtained from accepting the offer to the profit you would earn without accepting the offer. If the profit from accepting the offer is higher, then you should accept it.
(b) To determine whether you should accept the offer of as much or as little excavation work at $2.00 per meter dug, we need to compare the profit earned from this offer to the profit earned from different levels of output using the production function.
Let's start by calculating the profit for different levels of output (meters dug) using the production function:
Profit = Revenue - Total Cost
Revenue = Price per meter * Output (Q)
Total Cost = Labor Cost
To find the optimal level of output and labor usage, calculate the profit for different levels of output and labor usage, and choose the level that maximizes profit. This can be done by trying different values of L and calculating the corresponding Q, Revenue, Total Cost, and Profit for each combination.
Once you have determined the optimal level of output (meter dug) and labor usage, you can calculate the resulting profit by using the above formula.
(iii) To calculate the optimal (profit-maximizing) price, we need to use the formula for optimal price in relation to own price elasticity of demand.
Optimal price = Marginal cost / (1 + (1/elasticity of demand))
Given:
Marginal cost = $10
Fixed cost = $100
Elasticity of demand = -1.5
Substituting the values into the formula:
Optimal price = $10 / (1 + (1/-1.5))
Simplify the calculation by multiplying the denominator with -1:
Optimal price = $10 / (1 - (2/3))
Further simplify the denominator:
Optimal price = $10 / (1/3)
To divide by a fraction, we can multiply by its reciprocal:
Optimal price = $10 * (3/1)
Solve the multiplication:
Optimal price = $30
Therefore, the optimal (profit-maximizing) price that you should charge is $30.