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 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

For short-run profit maximization, MMMT should calculate the price and quantity by setting MR equal to SRMC and calculate the profit. If the profit is negative, it should shut down. The same principle applies to long-run profit maximization. Comparing SRMC and LRMC shows that MMMT's marginal cost in the short run is lower than in the long run.

Step-by-step solution

Find the marginal revenue

Marginal revenue (MR) can be derived from the demand function. The demand function is \(Q=10,000 / P^{2}\). Therefore, the revenue function \(R=P*Q=P*(10,000 / P^{2})=10000 / P\). Taking its derivative with respect to Q obtains the MR: \(MR=dR/dQ=-10000 / P^{2}\)

Short-run profit maximization

In the short-run, a firm maximizes its profit by setting MR equal to marginal cost (MC). The short-run total-cost (SRTC) function is \(SRTC=2000+5Q\). So, the short-run marginal-cost (SRMC) can be obtained by taking the derivative: \(SRMC=dSRTC/dQ=5\). Setting MR equal to SRMC yields the following equation: -10000 / P^{2}=5. Solving this equation gets the price P and substitute P into demand function gets the quantity Q. Substituting Q into the short-run profit equation \(𝜋 = TR - TC\) where \(TR = Q * P\) and \(TC = SRTC\), the short-run profit can be calculated. If the profit is less than zero, it would be better to shut down.

Long-run profit maximization

In the long-run, the firm also sets MR equal to MC to maximize its profit. The long-run total-cost (LRTC) is \(LRTC=6Q\). The LRM, which is the derivative of LRTC, is also 6. So, the equation -10000 / P^{2}=6 can be solved the same way as in step 2 to find the price P and quantity Q in the long run. Substituting Q into the long-run profit equation \(𝜋 = TR - TC\) where \(TR = Q * P\) and \(TC = LRTC\), the long-run profit can be calculated. If the profit is less than zero, it would be better to shut down in the long run.

Compare Marginal Cost in Short-run and Long-run

Just by looking at the two MC functions (SRMC = 5 and LRMC = 6), it's concluded that LRMC is larger than SRMC, which means MMMT would have higher marginal cost in the long run.

Key concepts

Marginal Revenue

Understanding the concept of marginal revenue (MR) is vital for any firm striving to maximize profits, including monopoly businesses like Michelle's Monopoly Mutant Turtles (MMMT). Marginal revenue represents the additional income received from selling one more unit of a product. In a monopoly, where the business controls the market, marginal revenue decreases with each additional unit sold due to the downward-sloping demand curve.

For MMMT, to calculate MR, we differentiate the revenue function with respect to quantity, which, from the demand function, is derived to be \(MR = -10000 / P^{2}\). This implies that as more t-shirts are sold, and the price decreases, the marginal revenue diminishes. It is crucial for MMMT to understand their MR as it guides them on setting the price and quantity to maximize profits, as seen in the exercise.

Short-run Cost

Short-run costs are expenses that affect a company's operations in the immediate time frame. These costs typically include fixed and variable elements. For MMMT, the short-run total cost (SRTC) function is given as \(SRTC = 2000 + 5Q\), where the fixed cost amounts to 2000, and the variable cost is proportionate to the quantity \(Q\) of t-shirts produced, at 5 dollars per shirt.

Understanding the SRTC enables MMMT to determine their minimum production costs in the short run and helps in calculating marginal costs, which are essential in decision-making processes related to pricing and maximizing profits in the short-term period. MMMT must produce where its marginal revenue equals its short-run marginal cost to ensure profit is maximized, without which, shutting down might be the more prudent decision.

Long-run Cost

In contrast to short-run costs, long-run costs do not have fixed expenses because, over a lengthier period, all costs become variable. For MMMT, the long-run total cost (LRTC) is given as \(LRTC = 6Q\), suggesting that each t-shirt's cost is 6 dollars, regardless of the quantity produced. This simplifies the analysis as there's no fixed cost to consider.

For long-term profit maximization, MMMT needs to focus on investing in processes or technologies that reduce long-run marginal costs. If the revenue generated from selling t-shirts does not exceed these costs, then continuing operation may not be sustainable in the long-run, and MMMT might benefit from considering shutting down.

Demand Function

The demand function illustrates the relationship between the price of a good and the quantity demanded by consumers. For MMMT, the demand for their exclusive t-shirts is given by \(Q = 10,000 / P^{2}\), demonstrating an inverse relationship between price \(P\) and quantity \(Q\). As the price goes up, the quantity demanded goes down, which is typical in most markets.

Known as price elasticity of demand, this measure can indicate how sensitive consumers are to price changes. A monopoly like MMMT can leverage their demand function to set optimal prices that balance their desire to maximize profits with the market's willingness to purchase their t-shirts.

Marginal Cost

Marginal cost (MC) is the extra expense incurred from producing one additional unit. It's a crucial factor for businesses like MMMT when it comes to setting prices and maximizing profits. In the short run, MMMT's marginal cost is \(dSRTC/dQ = 5\), indicating a constant marginal cost for each additional t-shirt made. However, in the long run, the marginal cost, found by differentiating the LRTC, is \(dLRTC/dQ = 6\), which is slightly higher than the short run.

This discrepancy between short-run and long-run marginal costs can occur due to factors such as economies of scale or investment in more efficient production techniques over time. MMMT would need to ensure that their pricing strategy covers the marginal cost in both the short and long run to maintain profitability.