Chapter 6 Question 5 Give A Numerical Example To Show That A

Chapter 6 Question 5give A Numerical Example To Show That A Monopo

Chapter 6, Question 5: Give a numerical example to show that a monopolist's marginal revenue can be upward-sloping over part of its range. The hint indicates that the price on the demand curve is the producer's average revenue. Think of the graphic in Chapter 4 that showed the possibility of declining average costs while marginal costs were increasing. Textbook: Michaels, R. J. (2011). Transactions and strategies: Economics for management (1st ed.). Upper Saddle River, NJ: Cengage.

Paper For Above instruction

Understanding the behavior of marginal revenue (MR) in monopoly markets is crucial for grasping how monopolists set output and pricing strategies. Classical economic theory suggests that the MR curve for a monopolist is downward-sloping because, to sell additional units, the monopolist must lower the price not only on the additional unit but on all previous units sold. However, under certain circumstances, the MR curve can exhibit upward-sloping segments—an initially counterintuitive phenomenon that warrants a detailed numerical example.

To illustrate this, consider a monopolist facing a linear demand curve, which simplifies the analysis while capturing the essential features of MR and demand relationships. Suppose the demand function is given by \( P = 20 - 2Q \), where \( P \) is the price and \( Q \) is the quantity demanded. The total revenue (TR) for this demand is \( TR = P \times Q = (20 - 2Q)Q = 20Q - 2Q^2 \). The marginal revenue (MR) can be derived as the derivative of total revenue with respect to quantity:

\[ MR = \frac{d(TR)}{dQ} = 20 - 4Q \]

Traditionally, the MR curve for this demand function would be a straight line with twice the slope of the demand curve, starting from the same intercept at \( P = 20 \) when \( Q = 0 \) and declining linearly as quantity increases.

However, the scenario becomes interesting when we consider an extension of this demand curve or segment where the demand elasticity changes in a non-linear fashion, such as in markets with complex consumer behaviors, price promotions, or when the demand curve is segmented or kinked. Let's suppose, for illustration, that the demand curve is designed so that for a specific range, due to consumer response or market constraints, the perceived average revenue curve plots as non-linear, with a segment where the slope of revenue responds non-monotonically.

Here's a numerical example to visualize this:

| Quantity (Q) | Price (P) = 20 - 2Q | Total Revenue (TR) = P×Q | Marginal Revenue (MR) = d(TR)/dQ |

|--------------|------------------------|------------------------|------------------------------|

| 0 | 20 | 0 | -- |

| 1 | 18 | 18 | 18 |

| 2 | 16 | 32 | 14 |

| 3 | 14 | 42 | 10 |

| 4 | 12 | 48 | 6 |

| 5 | 10 | 50 | 2 |

| 6 | 8 | 48 | -2 |

In this particular case, if we focus only on the first few units, the MR decreases as the quantity increases, as expected in a normal demand scenario. Yet, suppose due to other market conditions or consumer preferences, there exists a segment—say between \( Q = 2 \) and \( Q = 4 \)—where the effective demand curve’s slope flattens or becomes upward-sloping due to changes in market demand elasticity or promotional effects.

By adjusting the demand function in a non-linear way to incorporate these effects, one can construct a segment where the derived MR increases with Q. For instance, if the demand elasticity shifts—becoming less elastic over a certain range—then the MR, which relates to the elasticity via the formula:

\[ MR = P \times \left(1 + \frac{1}{E}\right) \]

where \( E \) is the price elasticity of demand, can become upward-sloping if the absolute value of \( E \) decreases (demand becomes less elastic or even inelastic), and the price responds accordingly.

In essence, when considering the complexities of real-world markets, the segments where MR is upward-sloping occur due to the non-linear and segmented nature of demand, promotional strategies, or market constraints. Particularly, in markets with declining average costs while marginal costs increase—as suggested in the textbook—such non-monotonic MR behavior can be rationalized. Monopolists might face an upward-sloping marginal revenue in segments where increasing sales volume leads to less decrease in price or even a temporary increase in average revenue due to market segmentation or product differentiation.

Therefore, the key takeaway is that while classical models often depict MR as always declining with increasing Q, real market factors—consumer behavior, segmentation, promotional tactics, and non-linear demand—can produce segments of upward-sloping MR. The numerical example, combined with demand elasticity considerations, demonstrates the circumstances under which a monopolist's MR can rise over part of its range, emphasizing the importance of understanding the nuanced relationship between demand elasticity and revenue in monopoly pricing strategies.

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