Question

The monopolist faces a demand curve given by \(D(p)=100-2 p .\) Its cost function is \(c(y)=2 y .\) What is its optimal level of output and price?

Short answer

The optimal output is 48 units, and the optimal price is 26.

Step-by-step solution

Determine the Revenue Function

The revenue function for a monopolist is given by the product of price and quantity, which is the inverse of the demand function. Start with the demand equation: \[ D(p) = 100 - 2p \]To express quantity in terms of price, solve for \(p\): \[ p = 50 - 0.5Q \]The revenue function \(R(Q)\) is then:\[ R(Q) = p \cdot Q = (50 - 0.5Q) \cdot Q = 50Q - 0.5Q^2 \]

Derive the Marginal Revenue Function

The marginal revenue function \(MR(Q)\) is the derivative of the revenue function with respect to quantity:\[ R(Q) = 50Q - 0.5Q^2 \]Taking the derivative, we have:\[ MR(Q) = \frac{d}{dQ}(50Q - 0.5Q^2) = 50 - Q \]

Derive the Marginal Cost Function

The cost function is given as \( c(y) = 2y \). Hence, the marginal cost function \( MC(Q) \) is the derivative of the cost function with respect to \(y\):\[ MC(Q) = \frac{d}{dQ}(2Q) = 2 \]

Set Marginal Revenue Equal to Marginal Cost

To find the optimal level of output, set the marginal revenue equal to the marginal cost:\[ MR(Q) = MC(Q) \]\[ 50 - Q = 2 \]Solve for \(Q\):\[ Q = 48 \]

Determine the Optimal Price

Find the price corresponding to the optimal output using the demand curve equation:\[ p = 50 - 0.5Q \]Substitute \(Q = 48\):\[ p = 50 - 0.5(48) = 50 - 24 = 26 \]

Conclusion

The optimal level of output is 48 units, and the optimal price is 26 currency units per unit.

Key concepts

Revenue Function

In economics, the revenue function is crucial for understanding how a monopolist makes money.
To find the revenue function, we need to know the demand curve, which shows the relationship between price and quantity demanded.
For a monopolist, revenue is defined as price (\( p \)) times quantity (\( Q \)). Given the demand curve \( D(p) = 100 - 2p \), we need it in terms of quantity.
This demand curve can be manipulated to express price in terms of quantity: solving yields \( p = 50 - 0.5Q \).
Now, the revenue function is derived as:

• Revenue (\( R(Q) \)) = Price \( \times \) Quantity = \((50 - 0.5Q) \times Q \)
• This simplifies to \( R(Q) = 50Q - 0.5Q^2 \).

This function helps determine how revenue changes with different levels of output.

Marginal Revenue

Marginal Revenue (MR) reflects the additional revenue gained from selling one more unit of output.
It is derived from the revenue function, highlighting how the monopolist’s decisions about pricing affect total revenue.
From the revenue function \( R(Q) = 50Q - 0.5Q^2 \), we find the marginal revenue by taking the derivative with respect to \( Q \).
This yields:

• MR(\( Q \)) = \( \frac{d}{dQ}(50Q - 0.5Q^2) = 50 - Q \).

Understanding marginal revenue is crucial as a monopolist will continue to produce as long as this is greater than marginal cost.

Marginal Cost

The Marginal Cost (MC) denotes how much it costs to produce one additional unit of output.
For a monopolist, knowing the marginal cost is key to determining the optimal level of production.
The given cost function is \( c(y) = 2y \). This tells us that costs increase linearly with output.
To find the marginal cost, we take the derivative with respect to \( y \) (or \( Q \)).

• MC(\( Q \)) = \( \frac{d}{dQ}(2Q) = 2 \).

In simpler terms, each additional unit costs 2 currency units to produce.

Demand Curve

The demand curve represents consumer willingness to buy different quantities at varying price levels.
In our scenario, the monopolist faces a demand curve given by \( D(p) = 100 - 2p \).
This tells us that for every currency unit increase in price, the quantity demanded drops by 2 units.

• The inverse function, or the way price depends on quantity, is \( p = 50 - 0.5Q \).

This relationship is pivotal for determining the optimal price and output level, as it guides the monopolist in balancing revenue with cost.
By analyzing where MR equals MC, the monopolist finds the point on the demand curve representing optimal pricing and production.
Understanding the demand curve allows a monopolist to maximize profit by predicting consumer reaction to price changes.