# A monopolist faces a demand curve given by Q = 200 - 2p and has constant marginal (and average...

## Question:

A monopolist faces a demand curve given by {eq}Q = 200 - 2p {/eq} and has constant marginal (and average total cost) of 20. What is the economic profit made by this profit-maximising monopolist if they engage in perfect price discrimination?

A) 0

B) 800

C) 3200

D) 6400

E) None of the above

## Perfect Price Discrimination

Perfect price discrimination, also known as first-degree price discrimination, is a situation that occurs when the seller charges each buyer exactly the price they are willing to pay.

## Answer and Explanation: 1

Under perfect price discrimination, the monopolist will charge the price that equals the demand. Thus, let's find the inverse demand function (make P the subject of the formula),

{eq}\begin{align*} Q&=200-2P \\ Q-200 &= -2P \\ P&=-0.5Q + 100 \end{align*} {/eq}

Now, finding the Profit function,

{eq}\begin{align*} Profit&=Total \ Revenue - Total \ Cost \\ Profit &= PQ - (ATC)Q \\ Profit&=(-0.5Q + 100)Q - 20Q \\ Profit&=-0.5Q^2 + 100Q -20Q \\ Profit&=-0.5Q^2+80Q \end{align*} {/eq}

To maximize profit,

{eq}\begin{align*} \frac{\partial Profit}{\partial Q}&= -Q + 80 = 0 \\ Q&= 80 \\ \end{align*} {/eq}

Thus, the quantity that maximizes profit is {eq}Q=80 {/eq}. The associated maximum price will be: {eq}P =-0.5Q + 100 = -0.5(80) + 100 =60 {/eq}

The maximum profit will therefore be computed as:

{eq}\begin{align*} Profit &=-0.5Q^2+80Q \\ Profit &= -0.5(80)^2 + 80(80) \\ Profit &= -0.5(80)^2 + 80(80) \\ Profit &= 3200 \end{align*} {/eq}

Thus, the correct answer is **C. 3200**

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