A monopoly firm is faced with the following demand function P = 13 – 0.5Q The Marginal Cost function for the firm is given by 3 + 4Q and the total fixed cost is 4
Determine
1. The the profit maximizing output
2.The level of supernormal profit if any
3. The output level at the break-even point
the profit maximizing output is achieved where marginal revenue equals marginal cost
To determine the answers to these questions, we need to analyze the monopolist's profit-maximizing behavior and apply relevant formulas. Let's go through each question step by step.
1. The profit-maximizing output:
To find the output level that maximizes profit, we need to calculate the monopolist's marginal revenue (MR) and equate it to the marginal cost (MC). The marginal revenue is given by the derivative of the demand function, and the marginal cost is given by the given formula.
Given:
Demand function: P = 13 - 0.5Q
Marginal Cost function: MC = 3 + 4Q
To find marginal revenue, we take the derivative of the demand function:
MR = d(P)/dQ = -0.5
Setting MR equal to MC:
-0.5 = 3 + 4Q
Solving the equation for Q, we get:
Q = -0.875
Since the quantity cannot be negative in this context, we discard -0.875 and consider Q = 0.
Therefore, the profit-maximizing output is Q = 0.
2. The level of supernormal profit (if any):
Supernormal profit refers to the profit earned above normal profit, which represents the minimum amount of profit necessary to keep the firm operating. We can calculate supernormal profit by subtracting total cost (including both fixed and variable costs) from total revenue.
Total Revenue (TR) is equal to the product of the price (P) and the quantity (Q):
TR = P x Q
Using the demand function, we can substitute P = 13 - 0.5Q:
TR = (13 - 0.5Q) x Q = 13Q - 0.5Q^2
Total Cost (TC) is equal to the sum of fixed cost (FC) and variable cost (VC):
TC = FC + VC
Given that fixed cost (FC) is 4 and variable cost (VC) can be calculated by integrating the marginal cost function:
VC = ∫(3 + 4Q) dQ = 3Q + 2Q^2 + C
where C is the constant of integration. Since we are dealing with a monopolist's profit-maximizing behavior, we can assume that VC = 0 at Q = 0 (as the monopolist does not produce any units).
Setting TC equal to FC + VC:
TC = 4 + (3Q + 2Q^2)
Now, we can calculate supernormal profit (π) by subtracting total cost from total revenue:
π = TR - TC = (13Q - 0.5Q^2) - (4 + 3Q + 2Q^2) = 13Q - 0.5Q^2 - 4 - 3Q - 2Q^2
Simplifying, we get:
π = -2.5Q^2 + 10Q - 4
To find the level of supernormal profit, we need to calculate the profit when Q = 0 and when Q = 0.875.
When Q = 0:
π = -2.5(0)^2 + 10(0) - 4 = -4
When Q = 0.875:
π = -2.5(0.875)^2 + 10(0.875) - 4 = 1.953125
Since supernormal profit is defined as profit above the normal profit, we subtract the normal profit from the calculated profit using the following formula:
Normal Profit = TC - FC
Normal Profit = (4 + 3Q + 2Q^2) - 4
Normal Profit = 3Q + 2Q^2
When Q = 0:
Normal Profit = 3(0) + 2(0)^2 = 0
When Q = 0.875:
Normal Profit = 3(0.875) + 2(0.875)^2 = 4.484375
Comparing the calculated supernormal profit and normal profit:
- When Q = 0, there is no supernormal profit (π = -4) since the firm incurs losses (π < 0).
- When Q = 0.875, the level of supernormal profit is 1.953125 - 4.484375 = -2.53125 (which is negative, indicating losses).
Therefore, there is no supernormal profit in this scenario.
3. The output level at the break-even point:
The break-even point occurs when the firm's total revenue equals its total cost, resulting in zero profit. To find the output level at the break-even point, we set total revenue (TR) equal to total cost (TC).
TR = TC
(13 - 0.5Q)Q = 4 + (3Q + 2Q^2)
Expanding and simplifying, we get:
13Q - 0.5Q^2 = 4 + 3Q + 2Q^2
Rearranging the terms, we have:
-0.5Q^2 - 5Q + 4 = 0
Solving this quadratic equation, we get:
Q = 2 and Q = -4
Since negative quantities are not meaningful in this context, we discard Q = -4.
Therefore, the output level at the break-even point is Q = 2.
To summarize:
1. The profit-maximizing output is Q = 0.
2. There is no supernormal profit in this scenario.
3. The output level at the break-even point is Q = 2.