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Price Discrimination
Price discrimination is always good for firms as they can increase revenue. However the impact on consumers is not clear. For example, if firms are uncertain
about the demand curve, they may discover that it is more elastic than they
expected. This may lead them to reduce prices which may be be beneficial
for consumers. On the other hand, extracting more surplus from consumers
through price discrimination is clearly worse for the consumers.
In the paper, Price discrimination in Broadway theatre (Rand 2004), Leslie
wants to resolve the ambiguity on welfare effects. This is done by first estimating
a structural model that includes both second degree and third degree price
discrimination. Once the structural parameters are estimated, counterfactuals
using alternative pricing policies (for example uniform pricing) are generated.
1.1
Context: Broadway theatre pricing
The data consists of price and quantity sold for all 17 different ticket categories
for all 199 performance of Seven Guitars, a play that ran on Broadway in 1996.
Figure 1:
Figure 1 shows the layout of a typical broadway theatre. Ticket prices do not
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Figure 2:
vary by cost-the marginal cost of every ticket sold for a given performance is zero.
As table 1 (below) indicates, there is different kind of price variation. Prices vary
across different ticket categories: depending on quality of seating arrangement
(orchestra, mezzanine etc. etc.). The mean price for orchestra ticket is the
highest. All full price options are available to all potential costumers and are
sold by telephone. Discount tickets are also available who receive coupon in
mail (or in restaurants etc.). Another kind of ticket, which is available to all
potential consumers, requires consumers to incur a non pecuniary cost of having
to wait in line at a discount booth. For discount price tickets, the buyers are
seated in the high quality region of the theatre, though generally not in the best
seats.
What explains the observed variation in prices? The price of full price ticket
depends on quality. Hence ignoring quality will make prices endogenous. Therefore seat quality has to be explicitly factored in. However note that while even
seats within the high quality area are of different quality, their prices are the
same. People who buy full ticket price may get assigned to bad seats within the
high price region.
There is also time series variation in prices (See standard deviation of prices).
Prices vary by times in the week (matinee cheaper than evening). Therefore this
has to be also factored in while estimating demand.
A third source of variation is that though every seat in theatre may differ in
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quality, the firm used only seat quality categories for the purpose of setting the
different prices. But for most performances, only two quality categories were
used. The medium quality region was offered at a different price in only 50 of
the 199 performances and introduced in 133rd performance. Hence for a given
seat quality, for given time of week, there is variation in ticket price. While the
introduction of the medium quality seat is correlated with time (as a response
to falling demand over time). the time trend is smooth, while the variation is
discrete.
There is also price variation in discount ticket categories: coupon and booth.
Coupon ticket prices were not in response to fluctuating demand but reflected
usual market practices. So they are plausibly exogenous. Booth ticket prices
are 50 percent of full tickets and have the same issues are full price tickets.
However, for a given performance, people who get discount tickets also get to
sit in the best quality section (Orchestra). Hence there is price variation within
the high quality area.
The author observes price and quantity as well as variables that help capture shifts in demand for the show (advertising, Tony awards, number of other
broadway shows).
1.2
Structural Model:
Individuals are differentiated along two dimensions: income and the taste for
the play relative to the outside alternative. Let yi ≥ 0 denote the income of
consumer i. Let ξ ≥ 0 denote consumer’s i’s relative taste. The pair (yi , ξ i ) are
known to individuals but not to the firm. Let yi ∼ F (y) and ξ i ∼ G(ξ). Both
distributions are known to the firm. F and G are independent.
There are M potential consumers who come in a random sequence: {(y1 , ξ 1 ) , (y2 , ξ 2 ) , ..., (yM , ξ M )} .
Let qih denote the quality of seat option in the high quality region. Let qim = qm
and qil = ql .Subject to availability, the net utlity to individual i from choosing
a full price ticket for seat quality j ∈ {l, m, h} is
Uij = qij [B (yi ) − pj ]η
in which B is the budget for entertainment. Consumer’s marginal utility
from seat quality depends on level of income, leading to self selection process
in which high income individuals choose high quality seats and low income
individuals choose low quality seats. Moreover,
B (yi ) = δ 1 yiδ2
where δ 1 > 0 and δ 2 ∈ (0, 1] .Wealthier people spend a greater absolute
amount of income on entertainment, but a lower proportion of their total income
than less wealthy people.
In addition to above full price ticket options, with probability λ(yi |γ) consumer i receives a coupon that can be used to purchase a ticket for a high quality
seat at price pc < ph and obtain a utility:
Uic = qih [B (yi ) − pc ]η
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The density λ(yi |γ) is the outcome of some coupon technology available to
the firm where γ is the efficiency of the coupon technology (how accurately
coupons are targeted to low income individuals).
Consumers can also go to a booth and get seat of quality qib .There is a cost
of standing is given by τ (yi ) ≡ τ 1 yi + τ 2 ≥ 0.Hence,
η
Uib = qib [B (yi ) − pb − τ (yi )]
The utlity from the outside alternative is:
ηo
Uio = ξ −1
i [B (yi ) − pc ]
The expected denad for tickets in category j is equal to
dF (y)dG(ξ)
Sj (.) = M
(y,ξ)∈Aj
where Aj are consumer types who prefer option j. That is,
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Aj = [(yi , ξ) : Uij ≥ Uik , ∀k ∈ Ω] ⊂ R+
where Ω = {l, m, h, c, b, o} , that is the quality of seats.
1.3
Econometric Model
The distribution of potential income is estimated based on the survey by League
of American Theatres and Producers: proportion of people attending Broadway
theatre with annual income within n intervals.
The distribution of individual tastes for the show is given by the exponential
distribution:
ξ it ∼ exp(Xt β)
where X = {cons tan t, adv., day, awards, time, other shows} .
For the probability density of receiving a coupon:
λit =
exp(αyi − Zt γ)
1 + exp(αyi − Zt γ)
where Z contains day dummies, time and time squared. This density has
the appearance that for α > 0,probability of receiving a coupon decreases with
rise in yi .
From these distributions, we simulate consumers.
Capacity of the three seating arrangements is denoted by Cl , Cm , Ch . Once
the capacity of any region is reached within a a sequence of simulated consumers,
the option is no longer available for subsequent individuals in the sequence.
Let kijt be the number the tickets purchased by consumers ahead of individual i for region j in performance t. The tickets are only available if kijt < Cj .
To compute the seat quality in the high quality region that is offered to
individual i, start with best seat in high quality region Qmax . Assume that the
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subsequent quality rankings differ by 1. Medium quality seats are uniformly
different from the worst high quality region. Low quality seats have quality ql
(lower than qm )
Conditional on an individual receiving a coupon, the utlity specification that
is the basis for estimation is:
Uijt
qijt (δ 1 yiδ2 − pjt )η
qiht (δ 1 yiδ2 − pjt )η
=
qiht (δ 1 yiδ2 − pjt − τ 1 yi − τ 2 )η
ηo
ξ −1
i [B (yi ) − pc ]
for kijt < Cj and j ∈ {l, m, h}
f or kijt < Ch and j = c
f or kijt < Ch and j = c
f or j = o
If the individual does not receive a coupon, then the choice j = c is not
relevant.
The set of parameters to be estimated:
Θ = {ql , qm , Qmax , δ 1 , δ 2 , τ 1 , τ 2 , η, ηo , po , α, β, γ}
The predicted market share of product j is the expected demand for j subject
to quantity constraints. Hence
sjt (pt , Xt, Zt , Θ) =
dF (y)dG(ξ|Xt β)
(y,ξ)
Denote the actual number of individuals who chooise option j in time t as
Njt . where
Not = Mt −
Njt
j∈Ω/{o}
The market size, Mt , is the total number of people attending Broadway
theatre in the same week divided by eight (weekly number of performances for
all Broadway shows).
The log likelihood can be written as
l (., Θ) =
T Njt log sjt (., Θ)
t=1 j∈Ω
There are more normalizations for things that are not identified.
Once all the parameters are estimated, other counter factuals can be run.
The authors finds that price discrimination may improve the firm’s revenues
by 5 % relative to uniform pricing while the difference for aggregate consumer
welfare is negligible.
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