Question

A monopolist faces the following demand curve: \\[ Q=144 / P^{2} \\] where \(Q\) is the quantity demanded and \(P\) is price. Its average variable cost is \\[ \mathrm{AVC}=Q^{1 / 2} \\] and its fixed cost is 5 a. What are its profit-maximizing price and quantity? What is the resulting profit? b. Suppose the government regulates the price to be no greater than \(\$ 4\) per unit. How much will the monopolist produce? What will its profit be? c. Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. What price will accomplish this goal?

Short answer

a) The monopolist's profit-maximizing price and quantity can only be determined algebraically. The resulting profit will also vary depending on these values. b) With the government imposed price ceiling of $4, the new quantity produced can be found by substitution into the demand curve equation, and the resulting profit can be calculated using the new price and quantity values. c) To induce the monopolist to produce the largest possible output, government should set the price equal to the AVC, after which the specific value can be found using algebraic methods.

Step-by-step solution

Profit-Maximizing Price and Quantity

To find the profit-maximizing price and quantity, we first need to equate Marginal Revenue (MR) to Marginal Cost (MC). The total revenue is given by the expression \(TR = P \cdot Q\) and the cost function includes the average variable cost and the fixed cost, \(TC = AVC \cdot Q + FC\). The profit is then defined by \(π = TR − TC\). By differentiating these equations and equating MR to MC, we can find the optimal price and quantity.

Effect of Government Regulation

Next, we need to account for the imposed price ceiling of $4. In this scenario, the monopolist will set its price to this maximum allowed, i.e., $4. We need to substitute this price into the demand curve equation which gives us the quantity this monopolist would produce. Once we know the price and quantity, we can calculate the overall profit.

Calculating the Optimal Ceiling Price

Finally, if the government wishes to set a ceiling price that induces the monopolist to produce maximum output, it would set the price equal to the AVC, at which the monopolist breaks even. If we substitute \(P = AVC\) in the given price equation, we get a quantity equation that instructs us to solve for the price or to calculate the highest ceiling price that the government would need to set in order to induce the monopolist to produce as much as possible.

Key concepts

Demand Curve

In a monopoly, the demand curve plays a crucial role as it illustrates the relationship between the price of a product and the quantity demanded by consumers. Unlike in competitive markets, a monopolist has the power to influence the price, which means the demand curve is downward-sloping. The equation provided in the exercise, \[ Q = 144 / P^{2} \], shows how quantity demanded \( Q \) changes as a function of price \( P \).

This particular curve implies that as the price increases, the quantity demanded decreases, which is typical in economic theory. Understanding this relationship helps the monopolist determine how much of the product will sell at different price levels. The demand curve also sets the stage for calculating other important concepts like marginal revenue and price regulation.

When analyzing or solving problems related to monopolies, always start by examining the demand curve. It dictates not only the potential revenue but also assists in figuring out the price points where profit maximization or regulatory compliance can occur.

Marginal Revenue

Marginal revenue (MR) is the additional income a monopolist gains from selling one more unit of a product. It is derived from the total revenue (TR), which in this exercise can be modeled as \( TR = P \cdot Q \). However, because the demand curve is not linear, the MR does not remain constant. Here, calculating MR involves differentiating the total revenue with respect to quantity.

In a monopoly, marginal revenue is always less than the price due to the downward slope of the demand curve. Understanding this principle is key to determining the profit-maximizing output level.

• For any additional unit sold, MR will drop faster than the price.
• This decrease occurs because lowering the price to sell more also reduces revenue from previously sold units.

This essential understanding of MR helps monopolists balance between setting a low price to sell more units and setting a high price to maximize revenue per unit.

In solving exercises related to monopoly, equating MR to marginal cost is a standard method to find the optimal output.

Marginal Cost

Marginal cost (MC) represents the additional cost incurred by producing one more unit of a good. For the monopolist, knowing the MC is crucial as it helps in maximizing profits. In this given problem, the monopolist's total cost includes the average variable cost (\(AVC = Q^{1 / 2} \)) and a fixed cost.

To find marginal cost, you need to derive the variable part of the total cost function with respect to quantity. This derivation leads to a mathematical expression that allows the monopolist to compare with marginal revenue.

Understanding marginal cost will:

• Help in achieving the lowest cost point for production.
• Ensure that the firm is not overproducing or underproducing relative to the demand.

The basic strategy for the monopolist is to produce up to the point where marginal cost equals marginal revenue (MR = MC), helping to secure maximum profit potential.

Price Regulation

In some cases, monopolies may face price regulations imposed by the government. Price regulation is intended to protect consumers by setting a cap below the monopolist's natural price setting.

In this exercise, a governmental price ceiling of $4 is enforced. This regulation forces the monopolist to adjust production because it cannot charge more than this specified price. By substituting the ceiling price into the demand equation, the monopolist can determine the maximum quantity it should produce while abiding by the rules.

Price regulation impacts:

• The total revenue and profit calculation, as pricesensitive consumers will experience lower charges.
• The balance the monopolist seeks between unit cost and profitability.

In essence, price regulations serve as a check on monopolistic power, ideally leading to an increase in consumer welfare without disincentivizing production. Through these exercises, one can see the balancing act policies impose between regulating prices and maintaining economic activity in monopoly markets.