Transcript Microeconomics MECN 430 Spring 2016

session ten
dynamic pricing (III)
uber as a platform ………….1
digital markets ………….7
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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?
 2016 Kellogg School of Management
lecture 10
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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 Pmax(i) thus anytime the price per mile
is above Pmax(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 = PMAX – bPride (note: here PMAX 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 = Pdriver
► 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 = (Pride – Pdriver)  #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) Pride  #Drives
► For a given f, what price per mile (Pride) should Uber set in order to maximize its profit?
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lecture 10
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
uber (dynamic) pricing
uber: a simple model
► Below is a simple illustration of two possible price per mile levels (Left) Pride = 100 and (Right) Pride = 40. PMAX = 140 and f = 80%,
thus
 demand curve:
#Clients(Pride) = 140 – 2Pride
 supply curve:
#Drivers(Pride) = 0.8Pride
supply(Pride)
supply(Pride)
140
drivers’ cost
function
100
140
drivers’ cost
function
80
40
32
demand(Pride)
demand(Pride)
rides
“idle” cars
Uber’s profit = (100 – 80)20 = 400
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32
80
20
rides
60
clients waiting
Uber’s profit = (40 – 32)32 = 256
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
uber (dynamic) pricing
uber: a simple model
► 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:
 supply curve:
#Clients(Pride) = PMAX – b Pride
#Drivers(Pride) = f Pride
supply(Pride)
► Uber’s profit is
Uber = (1 – f) Pride  #Drives
PMAX
with #Drives = #Clients = PMAX – b Pride thus
drivers’ cost
function
Pride
Uber = (1 – f) Pride  (PMAX – b Pride)
Uber
► This can be written as
Pdriver
Uber = (1 – f)  [PMAX  Pride – b (Pride)2]
► The optimal price is
Opt.Pride = PMAX / (2b)
demand(Pride)
and the maximum profit is
PMAX - Pride
maxUber = (1 – f)  PMAX / (4b)
rides
Pdriver
“idle” cars
►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.
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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 PMAX = 140 and f = 80%, thus
 demand curve:
 supply curve:
#Clients(Pride) = 140 – 2Pride
#Drivers(Pride) = 0.8Pride
and assume that Pride = 100. In that case there were 20 clients soliciting a ride and 80 cars available thus
ETW = k  #Drivers / #Clients = k  80 / 20 = 4k
► 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 Pride 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 Pride and vice-versa.
Pride
rejection area
Pride > Pmax(i) – ETW
► 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 Pride satisfies
acceptance area
Pride  Pmax(i) – ETW
Pride  Pmax(i) – ETW
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ETW
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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 Pride and choose the one that
provides the highest expected profit.
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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
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“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
download the app
● Out of the N potential
buyers (exposed to the
app) a fraction f will
download the app
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final pool of
possible buyers
● The final pool of
potential buyers is
n = fN
● The price for the
digital content is P
purchase decision
● Any potential buyer
has an unknown value
v to you, it is uniformly
distributed between 0
and Vmax
● The purchase will be
initiated if P  v
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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,Vmax] then given a price P, the
probability that the buyer will make the purchase is
q(P,Vmax) = Pr[purchase|P] = Pr[P  v] = 1 – P/Vmax
► 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,Vmax).
► 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)qk(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/Vmax]
► 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/Vmax]
► Profit maximizing price:
P* = Vmax/2
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lecture 10
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
from initial pool to making a purchasing decision
learning and pricing in the digital markets
The final () pool of potential buyers
The initial () pool of potential buyers
Vmax
value
Vmax
P* = Vmax/2
potential buyers
► The optimal solution is strikingly similar to considering the demand curve P = Vmax – Q and MC = 0, that is Q* = Vmax/2 and
P* = Vmax/2.
 2016 Kellogg School of Management
lecture 10
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
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 n1 = f N potential buyers that after downloading the app
will face a price P1 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 – n1)
(ii) individuals that downloaded the app in period 1 (a total of n1 – K1) but have not make a
purchase yet
● … from pool (i) you’ll get f(N – n1) = f(1 – f)N into the final pool
● … in pool (ii) first let’s identify K1: it is the average number of individuals that made the purchase given a price P1,
but this is equal to n1[1 – P1/Vmax]; thus from pool (ii) you’ll have a total of fNP1/Vmax
quiz
What are the possible values for those moving forward into the final pool?
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lecture 10
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microeconomics
lecture 10
dynamic pricing (III)
the analytics of constrained optimal
decisions
learning and pricing in the digital markets
dynamic pricing: two period model
► … your final pool:
● … from pool (i) you’ll get f(N – n1) = f(1 – f)N with values (uniform) between 0 and Vmax
● … in pool (ii) get fNP1/Vmax with values (uniform) between 0 and P1
► 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 – P2/Vmax]
● … from pool (ii) you’ll get: fNP1/Vmax[1 – P2/P1] = fNP1/Vmax – fNP2/Vmax
► What is the average/expected profit that you make? Assuming the marginal cost is zero, the expected profit, given price levels
(P1,P2) is
Period 1: 1(P1) = fNP1[1 – P1/Vmax]
Period 2: 2(P1,P2) = f(1 – f)N P2[1 – P2/Vmax] + fNP2P1/Vmax – fNP2P2/Vmax
► What are the pricing strategies?
► Uniform P1 = P2 = P
maximize the sum of profits
► Myopic P1 , P2
maximize 1(P1) then maximize 2(P1,P2)
► Dynamic P1, P2
maximize 1(P1) + 2(P1,P2)
► Dynamic P1, P2(i), P2(ii) , with P2(i) - the price of individuals coming from (i)
P2(ii) - the price of individuals coming from (ii)
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