A monopoly firm is faced with the following demand function P = 26 – 0.5Q. The Marginal Cost function for the firm is given by 6 + 6Q and the total fixed cost is 4.

Determine
a) The profit maximizing output.
b) The level of supernormal profit if any.
c) The output level at the break-even point.
A firm operating in a perfectly competitive market has to sell all its output at the price of $10 per unit. Its marginal cost function is given by Q + 4 and the total fixed cost is 1.
Determine
a) The profit maximizing output level.
b) The level of supernormal profit if any.

Mr = mc

To answer the questions, we need to understand the concepts of monopolistic competition and perfect competition. In monopolistic competition, there is only one firm in the market, while in perfect competition, there are many firms.

a) The profit-maximizing output for a monopolistic firm can be found by setting marginal revenue (MR) equal to marginal cost (MC). MR is calculated by differentiating the demand function with respect to quantity and multiplying it by the price.

Given that P = 26 - 0.5Q, we can differentiate it to find MR:

MR = d(P)/dQ = -0.5.

To find MC, we use the given marginal cost function: MC = 6 + 6Q.

Setting MR equal to MC, we have:

-0.5 = 6 + 6Q.

Solving for Q, we get:

Q = (-0.5 - 6) / 6 = -1.083.

However, since quantity cannot be negative, we take the positive value:

Q = 1.083.

Therefore, the profit-maximizing output for the monopolistic firm is approximately 1.083 units.

b) To find the level of supernormal profit, we need to compare total revenue (TR) and total cost (TC) at the profit-maximizing output.

TR is calculated by multiplying the price (P) by the quantity (Q):

TR = P * Q = (26 - 0.5Q) * Q.

Using Q = 1.083, we find:

TR = (26 - 0.5 * 1.083) * 1.083.

Calculating TR, we get:

TR ≈ 27.06.

TC is calculated by summing the marginal cost and the fixed cost:

TC = MC * Q + fixed cost.

Using Q = 1.083, the marginal cost function MC = 6 + 6Q, and the fixed cost of 4, we find:

TC = (6 + 6 * 1.083) * 1.083 + 4.

Calculating TC, we get:

TC ≈ 16.87.

Profit (π) is calculated as the difference between TR and TC:

π = TR - TC = 27.06 - 16.87.

Calculating π, we find:

π ≈ 10.19.

If the profit is positive, it indicates that the firm is earning supernormal profit. In this case, the level of supernormal profit for the monopolistic firm is approximately 10.19.

c) The output level at the break-even point is where the firm generates zero profit. Since profit is determined by subtracting total cost (TC) from total revenue (TR), the break-even point occurs when TR is equal to TC.

In this case, we need to find the output level (Q) where TR = TC. TR is calculated as before: TR = P * Q.

Using the given demand function P = 26 - 0.5Q, we have:

TR = (26 - 0.5Q) * Q.

TC is calculated as: TC = MC * Q + fixed cost.

Given the marginal cost function MC = 6 + 6Q and fixed cost of 4, we have:

TC = (6 + 6Q) * Q + 4.

Setting TR equal to TC, we get:

(26 - 0.5Q) * Q = (6 + 6Q) * Q + 4.

Solving for Q will give us the output level at the break-even point.

For the second question:

a) In a perfectly competitive market, the profit-maximizing output is determined by setting marginal cost (MC) equal to the market price.

Given the marginal cost function Q + 4 and the market price of $10, we have:

MC = Q + 4 = price.

Setting MC equal to the price, we get:

Q + 4 = 10.

Solving for Q, we find:

Q = 6.

Therefore, the profit-maximizing output for the firm operating in a perfectly competitive market is 6 units.

b) In perfect competition, there is no supernormal profit in the long run. If firms are earning positive profit, it will attract new firms to enter the market, which will increase supply and drive down prices. As a result, firms will only earn normal profit, covering their total cost but not earning any additional profit.

Therefore, the level of supernormal profit for the firm operating in a perfectly competitive market is zero.