A monopolist faces the following demand curve

A monopolist faces the following demand curve:

Q = 144/P2 where Q is the quantity demanded and P is price.

Its average variable cost is AVC = Q1/2 and its fixed cost is 5.

What are its profit-maximizing price and quantity? What is the resulting profit?

Suppose the government regulates the price to be no greater than $4 per unit. How much will the monopolist produce, and what will its profit be?

Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. What price will do this?

The correct answer and explanation is :

Step 1: Profit Maximization for the Monopolist

Given:

Demand curve: ( Q = \frac{144}{P^2} )
Average Variable Cost (AVC): ( AVC = Q^{1/2} )
Fixed Cost (FC): ( FC = 5 )

1.1: Inverse Demand Function

We can express the price ( P ) as a function of quantity ( Q ) by rearranging the demand curve:

[
Q = \frac{144}{P^2} \quad \Rightarrow \quad P = \sqrt{\frac{144}{Q}}
]

1.2: Total Revenue (TR)

Total Revenue is the price times the quantity, ( TR = P \times Q ). Substituting the inverse demand function:

[
TR = \sqrt{\frac{144}{Q}} \times Q = 12 \sqrt{Q}
]

1.3: Total Cost (TC)

Total Cost is the sum of fixed cost and variable cost. The variable cost is derived from the Average Variable Cost (AVC):

[
VC = AVC \times Q = Q^{1/2} \times Q = Q^{3/2}
]

Thus, the Total Cost (TC) is:

[
TC = FC + VC = 5 + Q^{3/2}
]

1.4: Profit Function

Profit (( \pi )) is Total Revenue minus Total Cost:

[
\pi = TR – TC = 12\sqrt{Q} – (5 + Q^{3/2})
]

1.5: Maximizing Profit

To find the profit-maximizing quantity, we take the first derivative of the profit function with respect to ( Q ) and set it equal to zero:

[
\frac{d\pi}{dQ} = \frac{d}{dQ}\left( 12\sqrt{Q} – 5 – Q^{3/2} \right)
]

[
\frac{d\pi}{dQ} = \frac{6}{\sqrt{Q}} – \frac{3}{2} Q^{1/2}
]

Set the derivative equal to zero:

[
\frac{6}{\sqrt{Q}} = \frac{3}{2} Q^{1/2}
]

Solve for ( Q ):

[
12 = 3 Q \quad \Rightarrow \quad Q = 4
]

1.6: Finding the Profit-Maximizing Price

Substitute ( Q = 4 ) into the inverse demand function to find the price:

[
P = \sqrt{\frac{144}{Q}} = \sqrt{\frac{144}{4}} = \sqrt{36} = 6
]

Thus, the monopolist’s profit-maximizing price is ( P = 6 ), and the quantity produced is ( Q = 4 ).

1.7: Profit

To find the profit, we calculate Total Revenue and Total Cost at ( Q = 4 ).

Total Revenue: ( TR = 12 \sqrt{4} = 12 \times 2 = 24 )
Total Cost: ( TC = 5 + 4^{3/2} = 5 + 8 = 13 )

Profit:

[
\pi = TR – TC = 24 – 13 = 11
]

So, the monopolist’s profit-maximizing quantity is 4, the price is 6, and the profit is 11.

Step 2: Government Price Regulation

Suppose the government regulates the price to no greater than ( P = 4 ).

At ( P = 4 ), we can find the quantity demanded using the demand curve:

[
Q = \frac{144}{P^2} = \frac{144}{4^2} = \frac{144}{16} = 9
]

So, the monopolist will produce 9 units.

2.1: Profit under Price Regulation

Now, we calculate the monopolist’s profit at ( Q = 9 ) and ( P = 4 ).

Total Revenue: ( TR = P \times Q = 4 \times 9 = 36 )
Total Cost: ( TC = 5 + 9^{3/2} = 5 + 27 = 32 )

Profit:

[
\pi = TR – TC = 36 – 32 = 4
]

So, under the price regulation, the monopolist will produce 9 units and make a profit of 4.

Step 3: Ceiling Price for Maximum Output

To induce the monopolist to produce the largest possible output, the government needs to set a price that causes the monopolist to produce the highest quantity.

The monopolist maximizes output when price is equal to marginal cost (MC), since production will occur as long as MC is less than or equal to the price.

3.1: Marginal Cost (MC)

Marginal Cost is the derivative of Total Cost with respect to ( Q ):

[
MC = \frac{d}{dQ} (5 + Q^{3/2}) = \frac{3}{2} Q^{1/2}
]

To find the price that leads to the largest quantity, set price equal to marginal cost:

[
P = MC = \frac{3}{2} Q^{1/2}
]

From the inverse demand function:

[
P = \sqrt{\frac{144}{Q}}
]

Set the two expressions for price equal:

[
\sqrt{\frac{144}{Q}} = \frac{3}{2} Q^{1/2}
]

Squaring both sides:

[
\frac{144}{Q} = \frac{9}{4} Q
]