Question

The monopolist faces a demand curve given by \(D(p)=10 p^{-3}\). Its cost function is \(c(y)=2 y .\) What is its optimal level of output and price?

Short answer

The optimal output is \(\frac{10}{27}\) and the optimal price is 3.

Step-by-step solution

Understanding the Demand Curve

The demand function is given by \(D(p) = 10p^{-3}\). This equation applies for price \(p\) and represents the relationship between the price and demand.

Derive the Inverse Demand Function

To find the inverse demand function, solve for \(p\) in terms of quantity \(y\). We know that quantity \(y = D(p) = 10p^{-3}\). So, rearranging gives \(p = (10/y)^{1/3}\).

Calculate Revenue

Revenue \(R\) is given by \(R = p imes y\). Substitute the inverse demand function: \(R = ((10/y)^{1/3}) imes y = 10^{1/3} y^{2/3}\).

Calculate the Derivative of Revenue (Marginal Revenue)

To find marginal revenue, take the derivative of \(R\) with respect to \(y\): \(MR = \frac{d}{dy}(10^{1/3} y^{2/3}) = \frac{2}{3}10^{1/3}y^{-1/3}\).

Calculate Costs and Marginal Cost

The cost function is \(c(y) = 2y\), thus the marginal cost \(MC\) is given by the derivative: \(MC = \frac{d}{dy}(2y) = 2\).

Set Marginal Revenue Equal to Marginal Cost

Set the marginal revenue equal to the marginal cost for profit maximization:\[ \frac{2}{3}10^{1/3}y^{-1/3} = 2 \].

Solve for the Optimal Output Level (y)

Multiply both sides of the equation by \(y^{1/3}\) and solve for \(y\):\[ \frac{2}{3}10^{1/3} = 2y^{1/3} \]\[ y^{1/3} = \frac{2}{3 \times 2}10^{1/3} = \frac{1}{3}10^{1/3} \]Cubbing both sides results in:\[ y = (\frac{1}{3}10^{1/3})^3 = \frac{10}{27} \].

Calculate the Optimal Price

Substitute \(y = \frac{10}{27}\) back into the inverse demand function \(p = (10/y)^{1/3}\):\[ p = (\frac{10}{\frac{10}{27}})^{1/3} = 3^{1/3} = 3 \].

Key concepts

Understanding the Demand Curve

The demand curve is a fundamental concept in economics, especially in monopoly optimization. It illustrates the relationship between the price of a product and the quantity demanded by consumers. In our exercise, the demand curve is represented by the function \(D(p) = 10p^{-3}\). This means that as the price \(p\) changes, the quantity demanded by consumers also changes accordingly.
For instance, when the price is high, the quantity demanded is low, and vice versa. This inverse relationship helps monopolists predict how much consumers will purchase at different price levels. Furthermore, understanding the demand curve allows businesses to plan pricing strategies effectively.

• The demand curve can be seen as a tool to gauge consumer behavior.
• It helps monopolists to decide at which price and quantity to operate to maximize profit.

By carefully analyzing the demand curve, a monopolist can make informed decisions about output and pricing.

Inverse Demand Function

The inverse demand function is key in determining optimal pricing in a monopoly setting. It reconfigures the demand function to express price \(p\) as a function of quantity \(y\). From the given demand function \(D(p) = 10p^{-3}\), we can derive the inverse demand function: \(p = (10/y)^{1/3}\).
This function helps the monopolist see how changes in output levels affect the price they can charge. It's crucial for setting prices systematically to balance supply with consumer demand.

• Expresses price directly in terms of quantity.
• Assists in assessing how to adjust prices with changes in output levels.

Having the inverse demand function allows a monopolist to control pricing more strategically, optimizing for maximum profitability.

Marginal Revenue and Marginal Cost

Marginal revenue (MR) and marginal cost (MC) are pivotal in optimizing a monopoly's production and pricing. Marginal revenue is the additional revenue from selling one more unit. In our problem, MR is found by differentiating the revenue function \(R = 10^{1/3} y^{2/3}\) with respect to \(y\), resulting in \(MR = \frac{2}{3}10^{1/3}y^{-1/3}\).
Marginal cost, on the other hand, is the cost of producing one more unit and is derived from the cost function \(c(y) = 2y\), which results in \(MC = 2\).

• MR: Shows revenue change with additional output.
• MC: Indicates cost change with additional output.

Aligning MR and MC is critical for determining the optimal production level where profit is maximized. In essence, producing where MR equals MC ensures that the cost of producing an extra unit is justified by the revenue it generates.

Profit Maximization

Profit maximization is a central objective for monopolies. It involves determining the level of output and price where profits are the highest. This is achieved by equating marginal revenue with marginal cost: \(\frac{2}{3}10^{1/3}y^{-1/3} = 2\). Solving this equation gives the monopolist the quantity \(y\) at which profit is maximized, which in our case is computed to be \(\frac{10}{27}\).
Once the optimal quantity is established, the corresponding optimal price is calculated by substituting the quantity back into the inverse demand function. Here, we find the optimal price as \(3\).

• Achieved when MR equals MC.
• Ensures maximum profitability for monopolists.

Profit maximization strategy ensures that resources are used efficiently and financial outcomes are optimized, reinforcing sustainable growth and competitive advantage in the market.