To find the inverse demand function for your firm’s product, you can start by solving the demand function for P (price). The demand function is given as:

Q = 36 – 4P

First, subtract Q from both sides:

4P = 36 – Q

Now, divide both sides by 4 to solve for P:

P = (36 – Q)/4

Now, simplify:

P = 9 – 0.25Q

So, the inverse demand function for your firm’s product is:

P = 9 – 0.25Q

b. To determine the profit-maximizing price and level of production, you need to find the level of output where marginal cost (MC) equals marginal revenue (MR) and then use the demand function to find the corresponding price.

The total cost function is given as:

C(Q) = 4 + 4Q + Q^2

To find the marginal cost (MC), take the derivative of the cost function with respect to Q:

MC = dC/dQ = 4 + 2Q

The demand function, as you found earlier, is:

P = 9 – 0.25Q

To find marginal revenue (MR), take the derivative of the revenue function with respect to Q:

MR = d(9Q – 0.25Q^2)/dQ = 9 – 0.5Q

Now, set MC equal to MR and solve for Q:

4 + 2Q = 9 – 0.5Q

2.5Q = 5

Q = 2

Now that you’ve found the level of production (Q = 2), you can use the demand function to find the price:

P = 9 – 0.25Q P = 9 – 0.25(2) P = 9 – 0.5 P = 8.5

So, the profit-maximizing price is $8.50, and the level of production is 2 units.

c. To calculate your firm’s maximum profits, you need to use the profit function, which is the difference between total revenue and total cost:

Profit = Total Revenue – Total Cost

Total Revenue = P * Q Total Cost = C(Q)

Using the values of P and Q you found in part b:

Total Revenue = $8.50 * 2 = $17

Total Cost = C(2) = 4 + 4(2) + (2)^2 = 4 + 8 + 4 = $16

Now, calculate the profit:

Profit = $17 – $16 = $1

So, your firm’s maximum profits are $1.

d. In the long run, in a monopolistically competitive market, firms enter or exit until they achieve zero economic profits. Economic profits are zero when total revenue equals total cost. In this case, your firm is making a small profit of $1.

The long-run adjustments you should expect are as follows:

Exit will occur until profits rise sufficiently high.

Since your firm is making a profit, other firms may enter the market to capture some of these profits. This increased competition will eventually drive down demand and prices, leading to lower profits. As a result, some firms may exit the market until profits reach a level close to zero.

In the long run, monopolistically competitive markets tend to reach a state where firms earn normal profits, meaning they cover all their costs but do not make any additional economic profit. This is because in such markets, product differentiation and consumer preferences play a significant role, leading to a degree of market power for each firm but also intense competition.