Profit Comparison: Price Discrimination vs. Single Price

The Uniform Pricing Alternative

Before a firm decides to price discriminate, it must consider the simpler alternative: charging all customers the same price. To calculate the profit in this scenario, the firm no longer looks at its market segments separately. Instead, it must aggregate them into a single total market demand.

The process for finding the profit-maximizing uniform price is as follows:

1. Aggregate Market Demand: Horizontally sum the demand equations of all market segments. This means adding the quantities demanded from each group at every given price: $Q_{Total}(P)=Q_1(P)+Q_2(P)+...$
2. Find Inverse Demand and Marginal Revenue: From the total market demand function, derive the inverse demand function $P(Q_{Total})$ and the corresponding marginal revenue curve, $M{R}_{Total}$.
3. Apply the Profit-Maximization Rule: Set the total marginal revenue equal to marginal cost ($M{R}_{Total}=MC$) to find the optimal total quantity, $Q^{\ast }$.
4. Determine the Uniform Price: Plug $Q^{\ast }$ back into the total inverse demand function to find the single profit-maximizing price, $P^{\ast }$.
5. Calculate Profit: Compute total profit as $\pi =(P^{\ast }\times Q^{\ast })-TC(Q^{\ast })$.

A crucial point to consider, given your background in functions and domains, is that aggregate demand curves are often "kinked." If one group of consumers has a much lower maximum willingness to pay than another, there will be a price range where only the high-demand group purchases the product. This creates different segments in the total demand curve, and you must be careful to identify the correct one when solving for the optimal price and quantity.

The following video provides an excellent walkthrough of this calculation, including how to handle the complication of one market segment being priced out.

Worked Example: A Side-by-Side Comparison

Let's apply this method to the cinema example from our previous lesson and compare the outcomes directly.

Market Data:

• Adult Demand: $Q_A=200-P_A$
• Student Demand: $Q_S=160-2P_S$
• Total Cost: $TC=1000+20Q$, which gives a constant Marginal Cost ($MC$) of £20.

Scenario 1: Third-Degree Price Discrimination (Recap)

In the last lesson, we treated each group as a separate monopoly and found:

• Adults: $Q_A=90$, $P_A=\pounds 110$
• Students: $Q_S=60$, $P_S=\pounds 50$
• Total Profit: ${\pi }_{PD}=[(110\times 90)+(50\times 60)]-[1000+20(150)]=\pounds 12,900-\pounds 4,000=\text{£8,900}$

Scenario 2: Single Uniform Price (New Calculation)

Now, let's find the profit if the cinema must charge a single price, $P$.

1. Aggregate Market Demand
We first express quantity as a function of price for both groups:

• $Q_A=200-P$
• $Q_S=160-2P$

Students will only buy if the price is below their maximum willingness to pay ($P<80$). Adults will buy if $P<200$. This creates a kink in the total demand curve at $P=80$.

• If $\pounds 80\le P<\pounds 200$: Only adults buy. The total demand is $Q_T=Q_A=200-P$.
• If $P<\pounds 80$: Both groups buy. The total demand is $Q_T=Q_A+Q_S=(200-P)+(160-2P)=360-3P$.

2. Find Inverse Demand and Marginal Revenue for Each Segment

• Segment 1 (High Price/Low Quantity): For $P\ge 80$, the inverse demand is $P=200-Q_T$. The corresponding marginal revenue is $M{R}_1=200-2Q_T$. This segment is valid for quantities up to where the price hits £80, which is $Q_T=200-80=120$. So, this applies for $0\le Q_T\le 120$.

• Segment 2 (Low Price/High Quantity): For $P<80$, the inverse demand is $P=120-\frac{1}{3}{Q}_T$. The corresponding marginal revenue is $M{R}_2=120-\frac{2}{3}{Q}_T$. This applies for $Q_T>120$.

3. Set MR = MC in Both Potential Segments
Our $MC$ is £20. We must check which segment's MR curve intersects the MC curve.

• Check Segment 1: Set $M{R}_1=MC$:
  $200-2Q_T=20⟹2Q_T=180⟹Q_T=90$
  This quantity $Q_T=90$ is within the valid range for this segment ($\le 120$). This is a valid solution. Let's find the profit.

  • Price: $P=200-90=\pounds 110$. (Note: at this price, students buy zero, as expected).
  • Profit: ${\pi }_1=(110\times 90)-(1000+20\times 90)=9900-2800=\pounds 7,100$.

• Check Segment 2: Set $M{R}_2=MC$:
  $120-\frac{2}{3}{Q}_T=20⟹100=\frac{2}{3}{Q}_T⟹Q_T=150$
  This quantity $Q_T=150$ is within the valid range for this segment ($>120$). This is also a valid solution.

  • Price: $P=120-\frac{1}{3}(150)=120-50=\pounds 70$.
  • Profit: ${\pi }_2=(70\times 150)-(1000+20\times 150)=10500-4000=\pounds 6,500$.

4. Choose the Optimal Uniform Price
The firm compares the profits from the two potential strategies. ${\pi }_1=\pounds 7,100$ is greater than ${\pi }_2=\pounds 6,500$. Therefore, the profit-maximizing uniform price strategy is to charge £110. At this price, the firm sells 90 tickets exclusively to adults.

3. The Comparison

Now we can quantify the value of price discrimination.

By segmenting the market, the cinema increases its profit by £1,800 (£8,900 - £7,100). It does this by keeping the adult price the same but also capturing the student market with a lower price, a market it would have otherwise excluded.

Visualizing the Profit Gain

The image below compares a single-price monopoly with a quantity discrimination scheme. While the mechanics differ slightly from third-degree discrimination, the principle is identical: a discriminating monopolist captures more surplus than a single-price monopolist.

In both second- and third-degree price discrimination, the firm's goal is to find a way to charge higher prices to customers with a high willingness to pay and lower prices to customers with a lower willingness to pay, thereby converting what would have been consumer surplus (like area E) or deadweight loss (like area G) into additional profit.

Conclusion

You are now able to perform a complete analysis comparing the profitability of third-degree price discrimination against a uniform monopoly price. This is a powerful tool for evaluating business strategy and a key skill for your exam.

Key Takeaways:

• To find the optimal uniform price, you must first horizontally sum the demand curves of all market segments.
• Be mindful of kinks in the aggregate demand curve, which require you to check potential profit-maximizing points in each segment of the curve.
• A firm will only adopt price discrimination if the profit from doing so is greater than the profit from the best uniform-price strategy.
• Price discrimination can increase profits by allowing a firm to sell to lower-demand groups that would be priced out of the market by a high uniform price.

In our next lesson, we will explore another pricing strategy used by firms that face capacity constraints and fluctuating demand over time: peak-load pricing.