IBM Research | Ponder This | September 2007 solutions
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September 2007

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Ponder This Solution:

1. The expected revenue (per customer per unit time) as a function of price is p*(1-p) for each channel (since a fraction 1-p of the potential customers will value a channel more than p and will purchase it at price p). It is easy to see p*(1-p) is maximized when p=.5. So the expected revenue per channel is .25 or .50 for both channels.

2. Let the bundled price be p. For p < 1 a fraction (1-p*p/2) of the customers will buy the channel. So the expected revenue is p*(1-p*p/2). By elementary calculus this is maximized for p=sqrt(2/3) =.81649658+ and has value .544331+. For 1 < p < 2 a fraction (2-p)*(2-p)/2 of the customers will buy the channel for expected revenue p*(2-p)*(2-p)/2 which is maximized for p=1 and has value .5. So the optimal choice is p=sqrt(2/3)=.8164958+ for revenue .544331+.

3.1 The average value to consumers willing to pay p is (1+p)/2 so the consumer surplus for price p is (1-p)*(.5*(1+p)-p)=.5*(1-p)**2= .125 (for p=.5) per channel or .25 for both channels.

3.2 Assuming p<1 the consumer surplus for price p is Integral(p to 1){(t-p)*t*dt} + Integral(1 to 2){(t-p)*(2-t)*dt} = {(1/3)-p/2+p*p*p/6} + {(2/3)-p/2} = 1-p+p*p*p/6 = .274225+ for p=sqrt(2/3).

So for this particular example a bundled pricing plan is better for both the company and its potential customers (on average).

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