What you should have done

Solutions to last week's pricing riddles

If you didn’t see last week’s post, click here before reading further. Now, here are the solutions to each problem.

Problem 1: Competing for market share

The way to solve this problem is for firm A to put itself in firm B’s shoes. Once firm A sets its price, firm B will try to maximize its revenue.¹ Since firm A’s price (p_A) is fixed at that point, firm B’s revenue is a function of p_B alone:

\(\begin{aligned} R_B &=p_B\cdot m_B\\[3pt] &=p_B\cdot(50+1.5(p_A-p_B))\\[3pt] &=-1.5(p_B)^2+(1.5p_A+50)p_B. \end{aligned}\)

Technically, I should use quantity instead of market share. However, using share won’t change the price decision because quantity is a constant multiple of share. Next, being a student of calculus, the CEO of firm B will differentiate R with respect to B’s price to find the price(s) that might maximize revenue:

\(\begin{aligned} 0&=\frac{dR_B}{dp_B}\\[3pt] &=-3p_B+(1.5p_A+50)\\[3pt] \implies p_B&=\frac{1.5p_A+50}{3}. \end{aligned}\)

In principle, this price may minimize revenue. However, the revenue function is a downward-opening parabola in p_B (cf., the first formula above), so we can quickly rule that out. Thus, the value of p_B just derived is the the revenue-maximizing price that firm B will set for its product. Moreover, firm A knows that firm B will pick this price. This allows firm A to rewrite its revenue equation in terms of p_A alone:

\(\begin{aligned} R_A &=p_A\cdot m_A\\[3pt] &=p_A\cdot(50+1.5(p_B-p_A))\\[3pt] &=p_A\cdot\left(50-1.5p_A+1.5\left(\frac{1.5p_A+50}{3}\right)\right)\\[3pt] &=p_A\cdot(75-.75p_A)\\[3pt] &=.75p_A\cdot(100-p_A).\end{aligned}\)

This is also a downward-opening parabola, with zeros at p_A = 0 and 100. The vertex of a parabola is half-way between the vertices, so firm A can maximize revenue with a price of p_A = 50. At this price, firm A’s revenue is

\(\begin{aligned} R_A&=p_A\cdot m_A\\[3pt] &= .75p_A\cdot(100-p_A)\\[3pt] &=.75\cdot50\cdot(100-50)=1875. \end{aligned}\)

What about firm B? If p_A = 50, then

\(\begin{aligned} p_B&=\frac{1.5p_A+50}{3}\\[3pt] &=\frac{1.5\cdot(50)+50}{3}\\[3pt] &=\frac{125}{3}\approx 41.67. \end{aligned}\)

Evidently, firm B’s best play is to undercut firm A’s price to steal market share. At this price, their revenue would be

\(\begin{aligned} R_B &=-1.5(p_B)^2+(1.5p_A+50)p_B\\[3pt] &= -1.5\left(\frac{125}{3}\right)^2+\left(1.5\cdot50+50\right)\cdot\left(\frac{125}{3}\right)\\[3pt] &\approx 2604.17. \end{aligned}\)

Despite getting to set its price first, firm A ends up with a smaller market share (37.5%) and less revenue than firm B.

Problem 2: Pricing a bundle

Let me remind you of each customer type’s willingness to pay (WTP) for each service.

To solve this problem, the first thing to realize is that you should always be pricing at (someone’s) willingness to pay. Take ESPN+, for example. It makes no sense to charge $14. Why? Only the sports junkie would subscribe for $14, but they would be willing to pay $15. So if you decide to set the price in excess of $12, you may as well go for $15. Having established that, there are four potential prices for each of Disney+ and ESPN+.

Total revenue for each subscription service at each of the four rational price points. For an example calculation, take Disney+ priced at $5. Each customer’s WTP is at least $5, so all four customers will buy, yielding revenue of 4*5 = $20. At $8 per subscription, three of four will buy, so revenue is 3*8 = $24.

Without bundling, we see that the total revenue Papa Disney can capture is $54, which is obtained by charging $10 for ESPN+ and either $8 or $12 for Disney+.

To investigate the potential for bundling, we need the combined WTP for each customer for both subscriptions.

If you want to offer only the bundle, then the right price is $20. At that level, all four customers will purchase for total revenue of $80. (If you priced it at $22, only the Renaissance Family would buy the bundle, thus total revenue would be $22.)

There’s one more strategy to investigate, which is mixed bundling. This is where you offer both the bundle and the individual options, aiming for high prices for both. The only conceivable way to make more money in this case is to charge $22 for the bundle. If you do so, you’ll get $2 extra from one customer (Renaissance Family). However, you’ll lose at least $2 from each of the other three, so there’s no way this will work. This tells us that the optimal pricing strategy is to offer the Disney+/ESPN+ bundle (only) at a price of $20.²

1. I was a little loose in phrasing this problem and didn’t mention costs in the consideration of profit. Let’s ignore that issue and just focus on maximizing revenue. (For instance, maybe each firm has already produced enough of their respective products to serve the entire market.) Incorporating cost would only change the equations, not the strategy.

2. Mixed bundling is often a good strategy. The problem here is that there is very little variability in WTP for the bundle. Mixed bundling is more effective when the people who would pay the most for the individual services don’t value the bundle that highly. For example, if the two fanatics each had a WTP of $1 for their non-preferred service, then the optimal pure bundle play is (15 + 1) = $16, which would generate revenue of $64. In that case, you would be better off charging $20 for the bundle (x2 = $40 in revenue) and $15 for ESPN+ and Disney+ separately (one each at $15), yielding $70 in total revenue.