Question

Michelle's Monopoly Mutant Turtles (MMMT) has the exclusive right to sell Mutant Turtle t-shirts in the United States. The demand for these t-shirts is \(Q=10,000 / P^{2} .\) The firm's short-run cost is \(\mathrm{SRTC}=\) \(2000+5 Q,\) and its long-run cost is \(\mathrm{LRTC}=6 Q\) a. What price should MMMT charge to maximize profit in the short run? What quantity does it sell, and how much profit does it make? Would it be better off shutting down in the short run? b. What price should MMMT charge in the long run? What quantity does it sell and how much profit does it make? Would it be better off shutting down in the long run? c. Can we expect MMMT to have lower marginal cost in the short run than in the long run? Explain why.

Short answer

a. In the short run, MMMT should charge $10 per t-shirt and sell 1000 t-shirts to make a profit of $5000. It is not better off shutting down. b. In the long run, MMMT should charge $0.12 per t-shirt and sell 694,444 t-shirts to make a loss of $556,347.68. It is better off shutting down. c. MMMT has a lower marginal cost in the short run as compared to in the long run due to economies of scale.

Step-by-step solution

Calculate the Revenue

Using the demand function \(Q=10000 / P^{2}\), we can find the price as a function of quantity, i.e., \(P= \sqrt{10000 / Q }\). So, the revenue (R) as a function of quantity (Q) is defined as \(R=P*Q=Q*\sqrt{10000 / Q} = \sqrt{10000*Q}\).

Find the Marginal Revenue

The marginal revenue(MR) is determined by differentiating the revenue function w.r.t the quantity \(Q\). The derivative of \(\sqrt{10000*Q}\) w.r.t \( Q \) is \( MR = \frac{5000}{\sqrt{Q}}\).

Find the Optimal Quantity and Price for the Short-Run

Setting marginal cost equal to marginal revenue will give the optimal quantity for the short run. With \(\mathrm{SRTC}= 2000 + 5Q\), the marginal cost(MC) is the derivative w.r.t \(Q\), which is 5. Equating this to MR, we get \(5 = \frac{5000}{\sqrt{Q}} => Q = 1000\). Substituting this back into the demand function will give the short-run price \(P = \sqrt{10000/1000} = 10\).

Calculate the Short-Run Profit

The short-run profit can be calculated by subtracting the total cost from the total revenue. It's calculated as Profit = \( total revenue - total cost = 1000*10 - (2000 + 5*1000) = 5000\). While the firm makes a positive profit, it does not need to consider shutting down in the short run.

Find the Optimal Quantity and Price for the Long-Run

Setting marginal cost equal to marginal revenue for the long-run. With \(\mathrm{LRTC}= 6Q\), the marginal cost(MC) is 6. Equating this to MR we find \(\frac{5000}{\sqrt{Q}} = 6 => Q = 694,444\). Substituting this back into the demand function gives the long-run price \(P = \sqrt{10000/694,444} = 0.12\).

Calculate the Long-Run Profit

The long-run profit can be calculated the same way as the short-run profit. It's given by Profit = \( total revenue - total cost = 694,444*0.12 - 6*694,444 = -556,347.68 \). Due to negative profit, the firm would be better off shutting down in the long run.

Analyze the Marginal Cost in the Short- and Long-Run

Comparing the marginal costs in the short run and in the long run (5 vs 6), it can be observed that the marginal cost is lower in the short run. This could be due to economies of scale. In the short run, fixed costs have already been paid and cannot be recovered, which leads to a lower marginal cost. However, in the long run, all costs are variable and can increase, leading to a higher marginal cost.

Key concepts

Short-Run Marginal Cost (SMC)

In microeconomics, understanding the behavior of costs is crucial for businesses to make informed decisions. The short-run marginal cost (SMC) refers to the additional cost a company incurs to produce one more unit of a product within a limited time frame, where at least one input, typically capital, is fixed.

The SMC can be represented mathematically by taking the derivative of the short-run total cost (SRTC), which includes both fixed and variable costs, with respect to quantity. For example, consider Michelle's Monopoly Mutant Turtles (MMMT) - their SRTC is given by the equation \(2000 + 5Q\). Here, \(2000\) represents fixed costs, and \(5Q\) are the variable costs that depend on the quantity produced. To find the SMC, we find the derivative of SRTC with respect to \(Q\), which yields a constant \(5\) for MMMT.

In a perfect competition scenario, a firm maximizes its profit by producing up to the point where SMC equals marginal revenue (MR). However, in the case of MMMT, since they are a monopoly, they face no competition and can directly equate their MR to SMC to find the profit-maximizing output in the short run, which in their case happens at producing \(1000\) units.

Long-Run Marginal Cost (LRMC)

When considering a time horizon that allows all factors of production to be variable, we examine the long-run marginal cost (LRMC). The LRMC indicates the cost to produce one additional unit when a firm can adjust all inputs, and there are no fixed costs - all costs are variable.

For MMMT, the long-run total cost (LRTC) is expressed as \(6Q\). In this scenario, the LRMC is also \(6\), since it's simply the coefficient of \(Q\) in the LRTC function. This cost does not change with the quantity because the firm does not have any fixed costs to dilute over a larger number of units, a situation typical for the long-run where the company can scale all inputs up or down.

Determining the profit-maximizing output level in the long run involves finding where LRMC equals MR. MMMT's calculations show a different optimal output and price in the long run, as these will now reflect economies or diseconomies of scale and the flexibility in varying all inputs to production.

Demand Function

The demand function is a mathematical representation that shows the relationship between the quantity of a good that consumers are willing to purchase and the price of the good. It typically illustrates how quantity demanded diminishes with an increase in price, assuming all other factors remain constant (ceteris paribus).

For MMMT, the demand function is given as \(Q=10,000 / P^{2}\). This inverse relationship indicates that when prices rise, the quantity demanded falls, as per the law of demand. For MMMT's product, this function can be used to determine the price at which a specific quantity of t-shirts will be demanded. By adjusting the equation, MMMT can calculate the maximum revenue it can earn for different prices and quantities.

Understanding the demand function allows businesses to set prices strategically to maximize revenue based on how much consumers are willing to buy at different price levels. MMMT's demand function is particularly crucial for determining its pricing and production strategies as a monopoly.

Marginal Revenue (MR)

Marginal revenue (MR) is the increase in total revenue that results from selling one additional unit of a product. It's a fundamental concept in microeconomics for profit maximization, as it helps firms understand the benefits of increasing production.

In a monopolistic setting like MMMT's, MR is calculated by differentiating the revenue function with respect to the quantity. The revenue function itself is found by multiplying the price function (the inverse of the demand function) by the quantity. For MMMT, the revenue function is \(R = Q * \sqrt{10000/Q}\), from which we identify MR as \(MR = \frac{5000}{\sqrt{Q}}\).

Profit maximization occurs at the quantity for which MR equals MC. However, due to the unique downward-sloping demand curve of a monopoly, MR does not equal price. Therefore, MMMT's MR is essential for determining the optimal quantity and price for profit maximization in both the short run and the long run, as MR guides the monopoly on how much to produce to extract maximum profits.