assignment two
preferences, utility maximization & demand curve
cost functions ………….1
cost curves ………….3
firms’ and market supply curves ………….4
market exit and entry ………….7
long-run supply curve ………….9
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
the snob effect
► Separate visits
► Different expectations
► Identical expectations
 As long as only one of the two friends is visiting the club each is willing to pay up to $1,000 for access.
 If each expects that the other one will show up too, both decrease their willingness to pay to $500.
 Suppose they end up visiting the club in different days within a given week.
 Under the assumption that the two friends are visiting the club in different nights each is willing to pay up to $1,000 to enter the club.
For any price below $1,000 each will demand one night, i.e. they will visit the club one night as long as the price is no more than $1,000.
 For the week in which the two friends visit the club in different nights the club will face a demand for two nights whenever the price is at
most $1,000. For any price above $1,000 none of the two friends will visit the club, i.e. demand will be zero.
Friend 1
Friend 2
Market
Price
Price
Price
$1,000
$1,000
$1,000
$500
$500
$500
0
1
 2016 Kellogg School of Management
Nights
0
Nights
1
assignment 2
0
1
2
Nights
page | 1
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
the snob effect
► Separate visits
► Different expectations
► Identical expectations
 As long as only one of the two friends is visiting the club each is willing to pay up to $1,000 for access.
 If each expects that the other one will show up too, both decrease their willingness to pay to $500.
 Suppose that within a given week Friend 1 expects Friend 2 will show up in the same night while Friend 2 expects that Friend 1 will
show up in a different night.
 Friend 1 expects that the night he will choose to visit the club his friend, Friend 2, will show up too. Thus Friend 1 is willing to visit the
club only if the price is at most $500. If the price is above $500 Friend 1 prefers not to show up, i.e. demand is zero.
 Friend 2 expects that the night he will choose to visit the club his friend, Friend 1, will not show up. Thus Friend 2 is willing to visit the
club only if the price is at most $1,000. If the price is above $1,000 Friend 2 prefers not to show up, i.e. demand is zero.
Friend 1
Friend 2
Market
Price
Price
Price
$1,000
$1,000
$1,000
$500
$500
$500
0
1
 2016 Kellogg School of Management
Nights
0
Nights
1
assignment 2
0
1
2
Nights
page | 2
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
the snob effect
► Separate visits
► Different expectations
► Identical expectations
 As long as only one of the two friends is visiting the club each is willing to pay up to $1,000 for access.
 If each expects that the other one will show up too, both decrease their willingness to pay to $500.
 Suppose that within a given week both Friend 1 and Friend 2 expect that the other will show up in the same night.
 Friend 1 expects that the night he will choose to visit the club his friend, Friend 2, will show up too. Thus Friend 1 is willing to visit the
club only if the price is at most $500. If the price is above $500 Friend 1 prefers not to show up, i.e. demand is zero.
 Friend 2 expects that the night he will choose to visit the club his friend, Friend 2, will show up too. Thus Friend 2 is willing to visit the
club only if the price is at most $500. If the price is above $500 Friend 2 prefers not to show up, i.e. demand is zero.
Friend 1
Friend 2
Market
Price
Price
Price
$1,000
$1,000
$1,000
$500
$500
$500
0
1
 2016 Kellogg School of Management
Nights
0
Nights
1
assignment 2
0
1
2
Nights
page | 3
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
demand for a pay-per-view event
► Setup and intuition
► The binomial distribution
► Expected demand
 Let consider one individual subscriber labeled “Subscriber 1”. From the TV provider’s perspective Subscriber 1’s reservation price RP1
can be any number between 0 and 1 with equal probability.
 Let’s say the TV provider sets a price P between 0 and 1. Subscriber 1 will pay for the “pay-per-view” (PPV) event as long as P  RP1.
What is the probability that this will happen? Since RP1 is uniformly distributed between 0 and 1:
Pr[ RP1  P ] = 1 – P
 We can represent Subscriber 1’s decision (from TV provider’s perspective) as a random variable that takes value 0 (no pay for the
PPV event) with probability P and takes value 1 (pay for the PPV event) with probability 1 – P:
no pay for PPV event
if RP1 < P
pay for PPV event
if RP1  P
1 
0

S1 : 
 P 1 P 
Pr[ RP1  P ] = 1 – P
Pr[ RP1 < P ] = P
 2016 Kellogg School of Management
assignment 2
page | 4
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
demand for a pay-per-view event
► Setup and intuition
► The binomial distribution
► Expected demand
 If we add a second subscriber, call her “Subscriber 2”, from TV provider’s perspective there are now two random variables
representing payment for the PPV event:
1 
1 
0
0


S1 : 
S2 : 
P
1

P
P
1

P




 Of course, the number of payers for the PPV event can be 0 (none of the two subscribers pay), 1 (either the first subscriber or the
second subscriber pays) or 2 (both subscribers pay). This can be represented as the sum of the two random variables above:
 0
S (2)  S1  S2 :  2
P
1
2

2 
2P (1  P ) (1  P ) 
 Calculation of probabilities for S(2):
S(2) = 0 : Subscriber 1 does not pay AND Subscriber 2 does not pay  probability P2
probability P
probability P
S(2) = 1 : Subscriber 1 does pay AND Subscriber 2 does not pay OR Subscriber 1 does not pay AND Subscriber 2 does pay  probability 2P(1 – P)
probability 1 – P
probability P
probability P
probability 1 – P
S(2) = 2 : Subscriber 1 does pay AND Subscriber 2 does pay  probability (1 – P)2
probability 1 – P
 2016 Kellogg School of Management
probability 1 – P
assignment 2
page | 5
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
demand for a pay-per-view event
► Setup and intuition
► The binomial distribution
► Expected demand
 The setup so far “reminds” us of the binomial distribution:
The binomial distribution with parameters N and Psuccess is the discrete probability distribution of the number of
successes in a sequence of N independent “yes”/”no” experiments, each of which yields success with probability Psuccess
 The probability that there are exactly k success when the experiment is repeated N times is given by
N 
Pr[# successes  k ]     (Psuccess )k  (1  Psuccess )N k
k 
N 
N!
where   
 k  k !(N  k )!
 Note that the number of success when the experiment is repeated N times can be 0, 1, 2, …, N. Thus the random variable X giving
the number of success when the experiment is repeated N times has the distribution (Ps stands for Psuccess)
 0

X :  ( P )N
 s

1
1
N  (Ps )  (1  Ps )
...
N 1
k
N 
   (Ps )k  (1  Ps )N k
k 
...
N


(1  Ps ) 

N
 Finally, for such a binomial distribution:
Expected Value = E[X] = NPsuccess
Variance = Var[X] = NPsuccess(1 – Psuccess)
 2016 Kellogg School of Management
assignment 2
page | 6
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
demand for a pay-per-view event
► Setup and intuition
► The binomial distribution
► Expected demand
 The pay-per-view event situation can be modeled using the binomial distribution. Why?
 The binomial distribution assumes an experiment is repeated N times and each experiment is successful with probability Psuccess:
- think of “a subscriber pays for the PPV event” as the experiment, there are N subscribers
- the TV provider having these N subscribers each potentially paying for the PPV event
- this means repeating N times the experiment “a subscriber pays for the PPV event”
 We already saw that the probability that a subscriber pays for the PPV event (this is the probability of success for an experiment) is
actually 1 – P with P the price set by the TV provider.
 The probability to have exactly k subscribers paying for the PPV event is thus:
N 
Pr[# PPV event payments  k ]     (1  P )k  (P )N k
k 
where
N 
N!
  
 k  k !(N  k )!
 The expected number of subscribers paying for the PPV event is
E[# PPV event payments] = N(1 – P)
 The variance of number of subscribers paying for the PPV event is
Var[# PPV event payments] = NP(1 – P)
 2016 Kellogg School of Management
assignment 2
page | 7
microeconomics
assignment 2
preferences, utility maximization & demand curve
the analytics of constrained optimal
decisions
demand for a pay-per-view event
► Setup and intuition
► The binomial distribution
► Expected demand
 The expected demand when the price is P is thus given by N(1 – P) and this is represented in the diagram on the left. Since we know
the variance of the number of subscribers paying for the PPV event we can represent this as a confidence interval of size +/- three
standard deviation around the mean as shown in the diagram on the right (this is an approximately 99.7% confidence interval)
price P
Expected demand
price P
expected # of payments
expected # of payments
 2016 Kellogg School of Management
Expected demand and confidence interval
assignment 2
page | 8