Question

Use the following demand schedule to calculate total revenue and marginal revenue at each quantity. Plot the demand, totalrevenue, and marginal-revenue curves, and explain the relationships between them. Explain why the marginal revenue of the fourth unit of output is \(\$ 3.50,\) even though its price is \(\$ 5 .\) Use Chapter 6 's total-revenue test for price elasticity to designate the elastic and inelastic segments of your graphed demand curve. What generalization can you make as to the relationship between marginal revenue and elasticity of demand? Suppose the marginal cost of successive units of output was zero. What output would the profit-seeking firm produce? Finally, use your analysis to explain why a monopolist would never produce in the inelastic region of demand. LO12.3 $$\begin{array}{|c|c|c|c|} \hline \text { Price (P) } & \begin{array}{c} \text { Quantity } \\ \text { Demanded (Q) } \end{array} & \text { Price (P) } & \begin{array}{c} \text { Quantity } \\ \text { Demanded (O) } \end{array} \\ \hline \$ 7.00 & 0 & \$ 4.50 & 5 \\ 6.50 & 1 & 4.00 & 6 \\ 6.00 & 2 & 3.50 & 7 \\ 5.50 & 3 & 3.00 & 8 \\ 5.00 & 4 & 2.50 & 9 \\ \hline \end{array}$$

Short answer

For the fourth unit, MR is lower than price due to diminishing returns. Elasticity dictates MR behavior, impacting monopolist output decisions.

Step-by-step solution

Calculate Total Revenue (TR) for Each Quantity

Total Revenue is calculated as the product of Price and Quantity Demanded. For each quantity level, compute TR using the formula: \( TR = P \times Q \). For instance, when quantity demanded \( Q = 1 \), the price \( P = 6.50 \), so \( TR = 6.50 \times 1 = 6.50 \).

Calculate Marginal Revenue (MR)

Marginal Revenue is the change in Total Revenue when one more unit is sold. Compute it using: \( MR = \Delta TR/\Delta Q \). For the second unit, where \( Q = 2 \) and \( TR = 12 \), \( MR = (12 - 6.50)/1 = 5.50 \). Apply this for each increment in quantity.

Plot the Curves

Create graphs for the Demand Curve (Price vs. Quantity), Total Revenue Curve (TR vs. Quantity), and Marginal Revenue Curve (MR vs. Quantity). Start by plotting points for each of them and join these points with smooth lines. Check that the MR curve lies below the Demand curve due to diminishing returns.

Analyze Relationship Between Curves

Observe that the MR curve is below the Demand curve, showing that each additional unit adds less to TR than the price due to decreasing MR. The Total Revenue curve increases at a decreasing rate and peaks as Marginal Revenue becomes zero.

Marginal Revenue for Fourth Unit

Even though the price of the fourth unit is \\(5, the MR is lower at \\)3.50 because MR reflects changes in TR, impacted by previous levels of sales and diminishing returns.

Identify Elastic and Inelastic Demand Segments

Total-revenue test: if TR increases with a fall in price, demand is elastic; if TR decreases, demand is inelastic. Analyze the TR curve to identify the segments, usually elastic at high prices and less quantity, inelastic at low prices and higher quantity.

Generalization on Marginal Revenue and Elasticity

Marginal Revenue is positive when demand is elastic, reaches zero when demand is unit elastic, and becomes negative when demand is inelastic. This means MR impacts elasticity directly.

Determine Output for Zero Marginal Cost

If marginal cost is zero, produce all units where MR is non-negative, maximizing profit. Thus, produce where MR is zero, typically the maximum TR point.

Monopolist Production Decisions

A monopolist won't produce where demand is inelastic because additional units decrease total revenue, despite increasing costs. This means moving past the unit elastic point reduces total profits.

Key concepts

Demand Curve

The demand curve is a fundamental concept in economics. It represents the relationship between the price of a good and the quantity demanded by consumers over a certain period. On a graph, the demand curve slopes downward from left to right, showing that as the price decreases, the quantity demanded generally increases.

This negative slope reflects the law of demand, which states that consumers will buy more of a good as its price falls, and less when the price increases.

In our exercise, we plot the demand curve using data points from the given demand schedule. By placing price on the vertical axis and quantity on the horizontal, we can visualize how changes in price influence demand behavior.

This graphical representation helps in understanding consumer purchasing patterns and is crucial for businesses in pricing strategies and forecasting sales.

Elasticity of Demand

Elasticity of demand measures how sensitive the quantity demanded of a good is to changes in its price. It is a critical concept for understanding how changes in price can affect a firm's revenue and market strategy.

When demand is elastic, a small decrease in price leads to a large increase in quantity demanded. In contrast, if demand is inelastic, a price decrease does not significantly affect the quantity demanded. This sensitivity is computed using the formula:
\[ E_d = \frac{\% \text{ change in quantity demanded}}{\% \text{ change in price}}\]

Elasticity provides insight into whether a firm should raise or lower prices to increase total revenue. Elastic segments of the demand curve indicate potential for revenue growth through price reductions, whereas inelastic segments suggest less benefit from lowering prices.

Total Revenue

Total Revenue (TR) represents the total income a firm receives from selling its goods and is calculated as the product of the price of the good and the quantity sold:
\[ TR = P \times Q\]

In our exercise, we calculate the total revenue for each quantity level provided in the demand schedule. By doing this, we can plot the Total Revenue Curve.

The shape of the total revenue curve gives insights into the market conditions. Initially, as quantity sold increases, TR rises, reflecting elastic demand. Eventually, TR reaches a peak as sales slow, indicating the point of unit elasticity. Beyond this point, increasing quantities decrease TR, signaling inelastic demand.

This curve is crucial for businesses to understand how their revenue changes with price adjustments and consumer purchasing behavior.

Price Elasticity

Price elasticity of demand is a concept that quantifies the responsiveness of the quantity demanded to a change in price. It relates directly to revenue maximizing strategies. If the price elasticity of demand (often denoted as E) is greater than one, demand is considered elastic, implying a price reduction could potentially increase total revenue.

In our exercise, by applying the total-revenue test, we can classify demand as elastic or inelastic. When a price drop leads to increased TR, the demand is elastic; conversely, if TR decreases, the demand is inelastic.

Importantly, understanding this relationship helps firms in decision-making. By pricing goods optimally, they can align with consumer responsiveness, avoiding losses and enhancing profitability. Simplifying, price elasticity impacts how pricing changes will affect total revenue and overall market performance.

Monopolist Decisions

Monopolists face unique challenges compared to firms in competitive markets because they control the market for their product or service. Their pricing and output decisions are influenced heavily by the demand curve and elasticity considerations.

In our scenario, a monopolist assesses the inelastic and elastic segments of their demand curve to make decisions. A monopolist ideally produces up to where the marginal revenue (MR) is zero, which corresponds to the peak of the total revenue curve. Producing beyond this point, where demand turns inelastic, would decrease total revenue as additional units cause revenue losses more than cost savings.

This strategic decision ensures profit maximization. Understanding the demand curve's elasticity helps the monopolist avoid expanding production into regions where total costs outweigh revenues, which is crucial for sustained profitability. These decisions underline the importance of comprehending demand behavior to optimize output.