the analytics of constrained optimal decisions microeco nomics spring 2016 the monopoly model (II): further pricing analysis ………….1platform and consumables ………….3 the “razor/blade” setup: one buyer session five ………….6the “razor/blade” setup: several buyers

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |1 platform and consumables ► We discussed a model of price discrimination based on an “access fee plus price per download”  an initial fee is required to gain access to a website from where songs can be downloaded  once the access fee is paid the buyer can download songs at a pre-specified price per each download ► In fact this is just a simple/stylized example of a variety of situations encountered in real life  Disney Park Access/Rides model: an initial “entry” fee and per-ride fees  Gillette Razor/Blade model: a moderately low price for the razor and somewhat high price for blades  Printer/Ink model: lower price on the device, higher price for accessories  Keurig Brewer/K-Cups model: a machine and a proprietary system for packaging single-serve coffee ► The common theme is a “two-part tariff” strategy: (i) an initial fixed fee is required to gain access to a platform (ii) a reoccurring fee is required for obtaining the complementary consumable Remark. Without the initial fee the consumer would retain the consumer surplus (the “triangle” above the price paid per consumable and below the demand curve). The access fee is designed to “extract” this surplus and transfer it to the platform/consumable provider. “razor/blade” models

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |2 platform and consumables “razor/blade” models ► A simple example: the price is P = $0.6 and marginal cost is MC = $0.2. There are 20 units sold and consumer surplus is calculated as the gray area in the left diagram - that area is $6. The producer’s surplus in this case is the blue rectangle in the left diagram - that area is $8. ► If the producer is able to charge an “entry” fee of $6 and then charge $0.6 per unit then the consumer would end up with no surplus at all and still buys 20 units. The whole surplus is now “captured” by the producer – the blue area in the right diagram. PP MC demand CS PS

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |3 pricing in the “razor/blade” setup ► The razor/blade model with one buyer. We will keep the setup as simple as possible and we will consider:  a razor/blade producer (call it Gillette) that incurs a cost F > 0 for each razor it manufactures. The marginal cost of producing the blades is 0 (zero);  a buyer who is interested in using Gillette products, has a demand for blades B = a – P, with B the number of blades it buys and P the price of one blade. To use the blades one razor should be acquired at a price R. Gillette is the only producer (a monopolist) of this innovative razor/blade system. Notice how the problem can be set in the following terms:  the producer sets a pair of prices ( R, P ) for razor and per blade  the buyer decides whether it is worth “investing” R in the razor, and if yes, it acquires the razor and then the number of blades based on its demand curve and the price P per blade  being aware of buyer’s behavior, the producer has to find the pair of prices ( R, P ) that maximizes its profit P = Remark. Is it possible for the buyer to find Gillette’s offer not so attractive? Consider a case in which R = $6 and P = $2 and buyer demand is B = 5 – P. By investing $6 in the razor the buyer can now buy and use Gillette blades at $2 per blade. At this price the buyer would like to have 3 blades.  The consumer surplus derived from buying and using the blades (ignore for the moment the price paid for the razor) is $4.5 - the area of the shaded triangle. However, with the $6 paid for the razor the net consumer surplus becomes –$1.5. Of course the buyer can simply ignore the offer and end up with a zero net consumer surplus – a better choice perhaps… demand CS

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |4 pricing in the “razor/blade” setup ► The razor/blade model with one buyer. The previous example, as simple as it might seem, suggests a very powerful restriction on producer’s choice of R and P. Specifically, for any price level P for the blade the maximum razor price that the producer can set and have the buyer taking the offer is given by the consumer surplus that the buyer would enjoy when the price of the razor is zero – that is the area above the price level P and below the demand curve up to the number of blades that the buyer is willing to buy. For a demand curve B = a – P the maximum razor price is R max = 1/2  ( a – P ) 2 P demand B = a – P CS a a – P Remark. It is fairly straightforward to calculate the consumer surplus: for a blade price of P the number of blades is B = a – P. The vertical difference on the y-axis is also a – P thus the area of the shaded triangle is simply 1/2  ( a – P ) 2.  If the producers charges a razor price equal to 1/2  ( a – P ) 2 the net consumer surplus in this case is zero as the shaded area is now “transferred” to the producer. You can think of the buyer as “breaking-even”. We are closer now to answering the central question here:  being aware of buyer’s behavior, the producer has to find the pair of prices ( R, P ) that maximizes its profit as we already figured out that the optimal (profit maximizing) razor price should be R * = 1/2  ( a – P ) 2. It remains to find out the optimal blade price P *. a

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |5 pricing in the “razor/blade” setup ► The razor/blade model with one buyer. With a cost of F > 0 per razor, producer’s profit for the pair ( R, P ) is given by  = [ R – F ] + [ P  B ] Above we know that R * = 1/2  ( a – P ) 2 and B = a – P thus the profit is written as  = [ 1/2  ( a – P ) 2 – F ] + [ P  ( a – P ) ] After some simple but tedious computations, the profit as a function of the blade price is  = –1/2  P 2 + 1/2  a 2 – F Obvious optimal solution here for the blade price P * = 0. The complete solution is thus:  razor price R * = 1/2  a 2, blade price P * = 0 and producer’s profit  * = 1/2  a 2 – F  number of blades bought is exactly a at a price per blade of 0 and net consumer surplus is 0 profit from razor profit from blades P * = 0 demand B = a – P R* = 1/2  a 2 a a Remark. The result we obtained (zero price for blades) is at odds with the evidence that actually the consumables are prices relatively high comparative to the fee necessary to access the platform… What are we missing ?

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |6 pricing in the “razor/blade” setup ► The razor/blade model with several buyers. The previous model was quite simplistic in that it assumes there is only one consumer that could buy the razor and blades. Here we consider an extension of that setup with several potential buyers:  there are N buyers who are interested in using Gillette products, buyer i demand for blades is given by B i = i – P, with B i the number of blades consumer i buys and P the price of one blade. To use the blades one razor should be acquired at a price R. All buyers pay the same price for razor and blades.  the producer sets a pair of prices ( R, P ) for razor and per blade  buyer i decides whether it is worth “investing” R in the razor, and if yes, it acquires the razor and then the number of blades based on its demand curve and the price P per blade  being aware of buyers’ behavior, the producer has to find the pair of prices ( R, P ) that maximizes its profit Remark. With N = 10 there are ten parallel demand curves, each with slope –1. Buyer i “reservation price” is exactly i. Of course the world is somewhat more complex than this but keeping simple assumptions helps understanding the logic of the model (not to say that algebra will be simple at all even in this simple model)

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |7 pricing in the “razor/blade” setup ► The razor/blade model with several buyers. 1 1 Step 1. Given the pair of prices ( R, P ) which buyers will take the offer?  Buyer i ’s demand curve is given by B i = i – P. There are two situations in which buyer i will refuse the offer ( R, P ): (a) The blade price is above buyer i ’s reservation price, that is P > i (b) The net consumer surplus is less than zero, that is (– R ) + 1/2  ( i – P ) 2 1/2  ( i – P ) 2  An easy conclusion from the above discussion is that buyer i will take the offer ( R, P ) if: P  i and R  1/2  ( i – P ) 2  Notice that if buyer k is willing to take an offer ( R, P ) then so are all buyer k + 1, k + 2, …, N. Why? If buyer k finds offer (R,P) acceptable then it must be the case that P  k and R  1/2  ( k – P ) 2 but this implies that these two inequalities also hold for all buyers with a higher reservation than k, i.e. for buyers k + 1, …, N.  Notice that once a price P is set then the optimal razor price is R* = 1/2  ( k – P ) 2 where k is such that k – 1 < P  k.  We are now in the position to answer the main question “Given the pair of prices ( R, P ) which buyers will take the offer?”: For a pair of prices ( R*, P ), with R* = 1/2  ( k – P ) 2 buyers k, k + 1, …, N will take the offer, where k is such that k – 1 < P  k.

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |8 pricing in the “razor/blade” setup ► The razor/blade model with several buyers. 2 2 Step 2. Given the pair of prices ( R*, P ) what is producer’s profit?  We saw that setting the pair of prices ( R*, P ), with R* = 1/2  ( k – P ) 2 buyers k to N will take the offer, with k such that k – 1 < P  k, that is a total of N – k + 1 buyers.  How many razors ? Each of the N – k + 1 buyers that accept the offer will buy one razor, thus in total N RZS = N – k + 1  How many blades ? Buyer i from the N – k + 1 buyers that accept the offer will buy i – P blades thus in total N BLD = ( k – P ) + ( k + 1 – P ) + ( k + 2 – P ) + … + ( N – 1 – P ) + ( N – P ) = 1/2  ( N – k +1)  ( N + k ) – ( N – k +1)  P  Producer’s profit is thus  = [ R * – F ]  N RZS + [ P  N BLD ] = [ R * – F ]  ( N – k + 1) + P  [ 1/2  ( N – k +1)  ( N + k ) – ( N – k +1)  P ]  With a bit of algebra  = ( N – k + 1)  [ R * – F + 1/2  ( N + k )  P – P 2 ] buyer k buyer k + 1 buyer k + 2 buyer N – 1 buyer N

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |9 pricing in the “razor/blade” setup ► The razor/blade model with several buyers. 3 3 Step 3. What pair of prices ( R*, P ) will maximize producer’s profit?  We saw that setting the pair of prices ( R*, P ), with R* = 1/2  ( k – P ) 2 buyers k to N will take the offer, with k such that k – 1 < P  k, that is a total of N – k + 1 buyers and producer’s profit is  = ( N – k + 1)  [ R * – F + 1/2  ( N + k )  P – P 2 ]  There are three variables in the above expression, namely R *, k and P. However, there is an additional relation between these three variables: R* = 1/2  ( k – P ) 2. Plug this into the profit function to get now an expression of R * and P :  = ( N – k + 1)  [1/2  ( k – P ) 2 – F + 1/2  ( N + k )  P – P 2 ]  This is a function of two variables and so with calculus techniques one would calculate the partial derivatives with respect to k and P and set these derivatives equal to zero. The solution of the resulting system of two equations with two unknowns would be the solution to optimizing (maximizing) the profit. Once k and P are found these are used to find R *.  An alternative way to find the pair ( R *, P ) that maximize the profit above is to use Excel. For example, setting F = 1 the solution is found as k = 7, P = 1.5 and R * = There are thus four buyers that take the offer and they all pay $ for the razor and an additional $1.5 per blade. Each buyer will buy a different number of blades corresponding to their demand curves.

microeconomic s the analytics of constrained optimal decisions lecture 5 the monopoly model (II): further pricing analysis  2016 Kellogg School of Management lecture 5 page |10 pricing in the “razor/blade” setup ► The razor/blade model with several buyers. 4 4 Step 4. How do we visualize the solution?  Buyers i = 7, 8, 9 and 10 are taking the offer and the number of razors they buy is 4. The producer receives 4  $ = $ In addition, the four buyers are buying 5.5, 6.5, 7.5 and 8.5 respectively for a total of 28 blades for a price of P = 1.5 each. The producer receives thus $1.5  28 = $42.00 for the blades.  The producer incurs $1.00 per razor produced thus his profit is $ $42.00 – $4.00 = $  Notice that Buyer 7 has zero net consumer surplus, however Buyers 8, 9 and 10 each has a positive net consumer surplus (the gray area). For all four buyers the blue triangle represents the surplus that the producer is able to extract through the razor price. Buyer 7 Buyer 8 Buyer 9Buyer 10 PS CS