Consider the problem of a firm that needs to decide how much output, denoted by x, to sell.
The marginal revenue function of the firm is given by 10−x. It's marginal cost function is goven by x.
In addition, the firm faces a legal constrain: it is allowed to sell a minimum of 0 units, and a maximum of 5 units.
QUESTION. What would be the impact on the profits of the firm of a change in the law that allows it to sell up to a maximum of 6 units?
Please help me to answer this que...
To answer this question, we need to analyze the impact of the change in the law on the firm's profits. Here's how we can approach the problem:
1. Determine the optimal level of output: The firm's profit is maximized when its marginal revenue equals its marginal cost. The MR (marginal revenue) function given is 10 - x, and the MC (marginal cost) function is x. To find the optimal output level, set MR equal to MC and solve for x:
10 - x = x
Simplifying the equation, we get:
2x = 10
x = 5
So, the optimal output level for the firm is 5 units.
2. Calculate profits before the law change: With the maximum limit on sales set at 5 units, the firm's profits can be calculated by multiplying the difference between the revenue (price per unit multiplied by the number of units sold) and the total cost (marginal cost multiplied by the number of units sold) at the optimal output.
Profit = (Price * Quantity) - (MC * Quantity)
In this case, the price per unit is the same as the marginal revenue, which is (10 - x). Substituting the optimal output value, we have:
Profit = (10 - 5) * 5 - 5 * 5
Profit = 25 - 25
Profit = 0
Hence, before the law change, the firm's profit is 0.
3. Calculate profits after the law change: With the maximum limit on sales increased to 6 units, we calculate the profits in a similar manner. However, we need to consider two scenarios:
a. If the optimal output level is still 5 units (meaning the firm's marginal revenue is greater than the cost of producing the 6th unit):
In this case, the profit calculation remains the same as before since the firm will continue to produce and sell only up to the optimal output level.
b. If the optimal output level increases to 6 units (meaning the firm's marginal revenue is equal to the cost of producing the 6th unit or greater):
In this scenario, the firm can produce and sell 6 units. Therefore, the profit calculation will be:
Profit = (10 - 6) * 6 - 6 * 6
Profit = 4 * 6 - 36
Profit = 24 - 36
Profit = -12
Hence, if the optimal output level increases to 6 units, the firm's profit would be -12.
In summary, the impact on the firm's profits of a change in the law that allows it to sell up to a maximum of 6 units depends on whether the optimal output level remains at 5 units or increases to 6 units. If the optimal output level remains at 5 units, the profit remains at 0. However, if the optimal output level increases to 6 units, the profit would be -12.
To determine the impact on the profits of the firm, we need to compare the revenue and cost of selling 5 units with the revenue and cost of selling 6 units.
Step 1: Calculate the revenue and cost functions.
The marginal revenue function, given as 10 - x, represents the additional revenue the firm receives when it sells one additional unit. We can calculate the total revenue function by integrating the marginal revenue function:
Revenue = ∫(10 - x) dx
Revenue = 10x - (x^2)/2 + C
The marginal cost function, given as x, represents the additional cost incurred when producing one additional unit.
Cost = ∫x dx
Cost = (x^2)/2 + C'
Note: C and C' are constants of integration.
Step 2: Calculate the profits.
Profits = Revenue - Cost
For selling 5 units:
Profits at 5 units = (10*5 - (5^2)/2 + C) - ((5^2)/2 + C')
For selling 6 units:
Profits at 6 units = (10*6 - (6^2)/2 + C) - ((6^2)/2 + C')
Step 3: Compare the two profit values.
By comparing the profits at selling 5 units and 6 units, you can determine the impact on the firm's profits of being allowed to sell up to a maximum of 6 units. If the profits at 6 units are higher, then the change in the law would likely result in increased profits for the firm. If the profits at 6 units are lower, then the change in the law would likely result in decreased profits for the firm.