Problem 2 2\. The monopolist faces a deman... [FREE SOLUTION]

Chapter 25: Problem 2

2\. 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

Expert verified

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

Step by step solution

Identify the Demand and Cost Functions

The demand function is given by the equation $D(p) = 100 - 2p$. This represents the relationship between price $p$ and the quantity demanded $D(p)$. The cost function is $c(y) = 2y$, where $c(y)$ is the total cost and $y$ is the quantity produced.

Express Price as a Function of Quantity

The demand function can be rearranged to express price $p$ in terms of quantity demanded $y$. Since $D(p) = y$, we have $y = 100 - 2p$. Solving for $p$, we get $p = 50 - \frac{y}{2}$.

Write the Revenue Function

The revenue function $R(y)$ is the product of price and quantity, so $R(y) = p\times y = \left(50 - \frac{y}{2}\right) \times y = 50y - \frac{y^2}{2}$.

Calculate Marginal Revenue

Marginal Revenue $MR$ is the derivative of the revenue function with respect to $y$. We have $MR = \frac{d}{dy}(50y - \frac{y^2}{2}) = 50 - y$.

Calculate Marginal Cost

Marginal Cost $MC$ is the derivative of the cost function with respect to $y$. We have $MC = \frac{d}{dy}(2y) = 2$.

Determine the Optimal Output Level

For profit maximization, set MR equal to MC: $50 - y = 2$. Solving for $y$, we get $y = 48$. This is the optimal level of output.

Find the Corresponding Price

Substitute $y = 48$ back into the price equation $p = 50 - \frac{y}{2}$: $p = 50 - \frac{48}{2} = 26$. This is the optimal price.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Curve

In a monopoly, the demand curve plays a critical role. It shows the relationship between the price of a product and the quantity consumers are willing to buy. For the given problem, the demand curve is represented by the equation $D(p) = 100 - 2p$. This tells us that as the price $p$ increases, the demand $D(p)$ decreases. Essentially, higher prices result in lower demand which is typical for most products.

Price is an independent variable here, meaning changes in price lead to changes in demand.
It's a linear demand curve because it's a straight line when plotted on a graph.
Understanding how demand changes with price helps the monopolist set optimal price points.

In this context, manipulating the demand curve allows the monopolist to find prices that maximize revenue. The ability to influence the demand curve is a unique feature of monopoly power.

Cost Function

The cost function indicates how much it costs to produce a certain number of goods. For our exercise, the cost function is given by $c(y) = 2y$, where $y$ is the quantity produced. This simply means that the cost increases linearly with the quantity produced. Specifically, each unit produced costs 2 units in whichever currency we're using.

This constant cost per unit is known as constant marginal cost.
Understanding the cost function is crucial as it impacts profit calculations and pricing decisions.

In a practical sense, the cost function allows the monopolist to calculate total expenditures and determine the point at which production becomes financially beneficial. It's fundamental for assessing how price setting will cover costs and result in profit.

Marginal Revenue

Marginal revenue (MR) is the additional revenue gained from selling one more unit of a product. In monopolies, MR is derived from the revenue function. From our solution, we have that $MR = 50 - y$. This means each additional unit sold increases revenue by less as more units are sold due to the effect of lowering the price, which is inherent in a downward-sloping demand curve.

This declining MR reflects the trade-off the monopolist makes with each additional sale, often having to reduce the price for all units sold.
The monopolist sets output where MR equals marginal cost (MC) to maximize profits.

In our scenario, understanding MR helps a monopolist decide how many units to sell by balancing how revenue stacks up against the cost of producing additional units.

Marginal Cost

Marginal cost (MC) is the cost of producing one additional unit of a good. For the cost function $c(y) = 2y$, the marginal cost is constant at $MC = 2$. This constant implies that the cost to produce an additional unit doesn't change, regardless of how many units have already been produced.

The MC is important because it directly influences pricing and production levels.
Profit maximization occurs when MR equals MC.

A monopolist uses MC to assess the viability of producing additional quantities. By understanding MC, the monopolist can decide efficiently at what quantity output should be leveled off for maximum profitability.

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2\. 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?