the analytics of constrained optimal decisions microeco nomics spring 2016 dynamic pricing (III) ………….1uber as a platform ………….7 digital markets session ten

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |1 uber (dynamic) pricing uber as a platform Main issues to consider: ► what determines demand (required transport) and supply (offer riding) for uber cars? ► how should Uber determine the optimal pricing policy?

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |2 uber (dynamic) pricing uber: a simple model We will make a few assumptions : ► CLIENTS care only about the price they pay per mile and each client i has a reservation price P max ( i ) thus anytime the price per mile is above P max ( i ) client i will not take an Uber drive. As a result, the demand curve for Uber rides is a downward slopping curve as a horizontal summation of individual demand for one ride per client. To simplify further the algebra let’s assume the market demand for Uber rides is # Clients = P MAX – bP ride (note: here P MAX is the highest reservation price among all potential Uber clients) ► DRIVERS care only about the price per mile they receive from Uber and each driver j has a cost per mile C ( j ) thus anytime the price per mile received by the driver is below C ( j ) driver j will not provide an Uber drive. As a result, the supply curve for Uber rides is an upward slopping curve as a horizontal summation of individual supply for one ride per driver. To simplify further the algebra let’s assume that the most efficient driver has a cost per mile of zero thus the supply curve as it depends on the price received by the driver is # Drivers = P driver ► UBER cares (only) about the profit per mile as a difference between (i) the price charged to clients and (ii) the price paid to drivers, thus Uber’s profit is written as  Uber = ( P ride – P driver )  # Drives Uber’s policy is to pay the driver a fraction f (about 80%) of the price charged to the client. The remaining fraction is retained by Uber, thus Uber’s profit as a function of the price per ride is  Uber = (1 – f )  P ride  # Drives ► For a given f, what price per mile ( P ride ) should Uber set in order to maximize its profit?

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |3 uber (dynamic) pricing uber: a simple model demand(P ride ) supply(P ride ) drivers’ cost function ► Below is a simple illustration of two possible price per mile levels (Left) P ride = 100 and (Right) P ride = 40. P MAX = 140 and f = 80%, thus  demand curve:# Clients ( P ride ) = 140 – 2 P ride  supply curve:# Drivers ( P ride ) = 0.8 P ride rides “idle” cars demand(P ride ) supply(P ride ) drivers’ cost function rides clients waiting Uber’s profit = (100 – 80)  20 = 400 Uber’s profit = (40 – 32)  32 = 256

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |4 uber (dynamic) pricing uber: a simple model demand(P ride ) supply(P ride ) drivers’ cost function P ride P MAX P driver P MAX - P ride P driver ► Let’s try to solve Uber’s general problem of solving for the optimal price to charge clients in order to maximize its profit, taking f as given for now. Thus  demand curve:# Clients ( P ride ) = P MAX – b  P ride  supply curve:# Drivers ( P ride ) = f  P ride rides “idle” cars ► Uber’s profit is  Uber = (1 – f )  P ride  # Drives with #Drives = #Clients = P MAX – b  P ride thus  Uber = (1 – f )  P ride  ( P MAX – b  P ride ) ► This can be written as  Uber = (1 – f )  [ P MAX  P ride – b  ( P ride ) 2 ] ► The optimal price is Opt.P ride = P MAX / (2 b) and the maximum profit is max  Uber = (1 – f )  P MAX / (4 b)  Uber ► In this simple model there are two ways to create an optimal “surge in price”: wither the reservation prices increases or b decreases (assuming f is constant all the time). In both cases the demand curve shifts to the right.

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |5 uber (dynamic) pricing uber: a more realistic model We will alter a bit some of our assumptions : ► CLIENTS care not only about the price they pay per mile but also about their estimate time of waiting ( ETW ) which is calculated in a simplistic way as (with k a proportionality constant) ETW = k  # Drivers / # Clients ► Let’s go back to the simple example with P MAX = 140 and f = 80%, thus  demand curve:# Clients ( P ride ) = 140 – 2 P ride  supply curve:# Drivers ( P ride ) = 0.8 P ride and assume that P ride = 100. In that case there were 20 clients soliciting a ride and 80 cars available thus ETW = k  # Drivers / # Clients = k  80 / 20 = 4 k ► Obviously the client cares also about the price is has to pay for the ride thus we have to introduce a new way to model client’s demand for an Uber ride. Now we have to introduce a demand function that combines P ride and ETW. Obviously the client prefers both to be as small as possible, however it might be “flexible” in the sense that for a smaller ETW perhaps is willing to accept a higher P ride and vice-versa. ► We will model the demand now as an “acceptance area”: client i with the reservation Pmax ( i ) will solicit an Uber ride if the requested price P ride satisfies P ride  P max ( i ) – ETW P ride ETW acceptance area P ride  P max ( i ) – ETW rejection area P ride > P max ( i ) – ETW

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |6 uber (dynamic) pricing uber: a more realistic model ► Uber’s problem remains the same: find the price that maximizes it’s profit. But know the problem it’s a bit more complicated as: - choosing a too high price might get the client into the rejection area (thus no Uber ride solicited from that client) - choosing a too low price might get the client into the acceptance area but perhaps some profit is lost The problem is further complicated because the ETW for one client in fact depends on all the clients that might solicit an Uber drive. ► Proposed solution: simulate the number of clients soliciting an Uber drive for different levels of P ride and choose the one that provides the highest expected profit.

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |7 learning and pricing in the digital markets market description ► … long story short … ● … you are a monopolist that sells online digital content (music, movies, live-race event, etc.) … ● … that could be accessed only through an app … ● … that can be downloaded for free … ● … by anybody who becomes aware of this app initial pool of possible buyers ● It is time zero and there are N potential buyers that might be interested in buying from you “exposure” to the app ● You “expose” the time zero pool to your app; ● All N potential buyers become aware of what they have to do in case they intend to make a purchase ● Out of the N potential buyers (exposed to the app) a fraction f will download the app download the app final pool of possible buyers final pool of possible buyers purchase decision ● The final pool of potential buyers is n = f  N ● The price for the digital content is P ● Any potential buyer has an unknown value v to you, it is uniformly distributed between 0 and V max ● The purchase will be initiated if P  v

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |8 learning and pricing in the digital markets market description ► The size of the initial pool of potential buyers that get exposed to the app is N. Of this pool a fraction f will go ahead and download the app thus the pool of potential buyers that downloaded the app is n = f  N. ► We assumed above that the price of the content is not known or that the decision to download the app is independent/random with respect to the final price ► The fact that the value of the digital content to the buyer is uniformly distributed in the interval [0, V max ] then given a price P, the probability that the buyer will make the purchase is q ( P, V max ) = Pr[purchase| P ] = Pr[ P  v ] = 1 – P / V max ► You are facing a pool of n potential buyers and, for a price P, each of these buyers will buy the content with probability q ( P, V max ). ► What is the probability that you’ll make k sales ? The answer is given by the classic “binomial distribution” result: Pr[# of sales = k | n, P ] = C( n, k )  q k  (1 – q) n – k ► What is the average/expected number of sales that you make? This is the average of the classic “binomial distribution” result: E[# of sales| n, P ] = n  q = n  [1 – P / V max ] ► What is the average/expected profit that you make? Assuming the marginal cost is zero, the expected profit, given a price level P is E[  | n, P ] = P  E[# of sales| n, P ] = n  P  [1 – P / V max ] ► Profit maximizing price: P * = V max /2

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |9 learning and pricing in the digital markets from initial pool to making a purchasing decision V max The initial (  ) pool of potential buyers V max P* = V max /2 The final (  ) pool of potential buyers value potential buyers N individuals n individuals ► The optimal solution is strikingly similar to considering the demand curve P = V max – Q and MC = 0, that is Q * = V max /2 and P * = V max /2.

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |10 learning and pricing in the digital markets dynamic pricing: two period model ► … most of the story still valid but … ● … you sell the content in two periods, say there are two race-events (Formula 1) ● … from the initial pool of N individuals you get the pool of n 1 = f  N potential buyers that after downloading the app will face a price P 1 and decide whether to buy or not ● … those who downloaded the app and made a purchase in the first period will not return in the second period ● … for the second period your pool of final potential buyers come from two sources: (i) individuals that did not download the app in period 1 (a total of N – n 1 ) (ii) individuals that downloaded the app in period 1 (a total of n 1 – K 1 ) but have not make a purchase yet ● … from pool (i) you’ll get f  ( N – n 1 ) = f  (1 – f )  N into the final pool ● … in pool (ii) first let’s identify K 1 : it is the average number of individuals that made the purchase given a price P 1, but this is equal to n 1  [1 – P 1 / V max ]; thus from pool (ii) you’ll have a total of f  N  P 1 / V max What are the possible values for those moving forward into the final pool? quiz

microeconomic s the analytics of constrained optimal decisions lecture 10 dynamic pricing (III)  2016 Kellogg School of Management lecture 10 page |11 learning and pricing in the digital markets dynamic pricing: two period model ► … your final pool: ● … from pool (i) you’ll get f  ( N – n 1 ) = f  (1 – f )  N with values (uniform) between 0 and V max ● … in pool (ii) get f  N  P 1 / V max with values (uniform) between 0 and P 1 ► What is the average/expected number of sales that you make? This is the average of the classic “binomial distribution” result: ● … from pool (i) you’ll get: f  (1 – f )  N  [1 – P 2 / V max ] ● … from pool (ii) you’ll get: f  N  P 1 / V max  [1 – P 2 / P 1 ] = f  N  P 1 / V max – f  N  P 2 / V max ► What is the average/expected profit that you make? Assuming the marginal cost is zero, the expected profit, given price levels ( P 1, P 2 ) is Period 1 :  1 ( P 1 ) = f  N  P 1  [1 – P 1 / V max ] Period 2 :  2 ( P 1, P 2 ) = f  (1 – f )  N  P 2  [1 – P 2 / V max ] + f  N  P 2  P 1 / V max – f  N  P 2  P 2 / V max ► What are the pricing strategies? ► Uniform P 1 = P 2 = Pmaximize the sum of profits ► Myopic P 1, P 2 maximize  1 ( P 1 ) then maximize  2 ( P 1, P 2 ) ► Dynamic P 1, P 2 maximize  1 ( P 1 ) +  2 ( P 1, P 2 ) ► Dynamic P 1, P 2(i), P 2(ii), with P 2(i) - the price of individuals coming from (i) P 2(ii) - the price of individuals coming from (ii)