1 Price Discrimination
Price discrimination is always good for firms as they can increase revenue. How- ever 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 ofSeven 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
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. There- fore 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
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 cap- ture 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 consumeri. Letξ≥0denote consumer’si’s relative taste. The pair(yi, ξi)are known to individuals but not to the firm. Letyi ∼F(y)and ξi ∼G(ξ). Both distributions are known to the firm. F andGare independent.
There areMpotential consumers who come in a random sequence:{(y1, ξ1),(y2, ξ2), ...,(yM, ξM)}. Letqihdenote the quality of seat option in the high quality region. Letqim=qm
andqil=ql.Subject to availability, the net utlity to individualifrom choosing a full price ticket for seat qualityj∈ {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) =δ1yδi2
where δ1 > 0 and δ2 ∈ (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)con- sumerireceives a coupon that can be used to purchase a ticket for a high quality seat at pricepc< ph and obtain a utility:
Uic=qih[B(yi)−pc]η
The densityλ(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 qualityqib.There is a cost of standing is given byτ(yi)≡τ1yi+τ2≥0.Hence,
Uib=qib[B(yi)−pb−τ(yi)]η
The utlity from the outside alternative is:
Uio=ξ−1i [B(yi)−pc]ηo
The expected denad for tickets in categoryjis equal to
Sj(.) =M
(y,ξ)∈Aj
dF(y)dG(ξ)
whereAj are consumer types who prefer optionj. That is, Aj= [(yi, ξ) :Uij≥Uik,∀k∈Ω]⊂R+2 whereΩ ={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 withinnintervals.
The distribution of individual tastes for the show is given by the exponential distribution:
ξit∼exp(Xtβ)
whereX ={constant, adv., day, awards, time, other shows}.
For the probability density of receiving a coupon:
λit= exp(αyi−Ztγ) 1 + exp(αyi−Ztγ)
where Z contains day dummies, time and time squared. This density has the appearance that forα >0,probability of receiving a coupon decreases with rise inyi.
From these distributions, we simulate consumers.
Capacity of the three seating arrangements is denoted byCl, 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.
Letkijtbe the number the tickets purchased by consumers ahead of individ- ualifor regionj in performancet.The tickets are only available ifkijt< Cj.
To compute the seat quality in the high quality region that is offered to individuali,start with best seat in high quality regionQmax. Assume that the
subsequent quality rankings differ by 1. Medium quality seats are uniformly different from the worst high quality region. Low quality seats have qualityql (lower thanqm)
Conditional on an individual receiving a coupon, the utlity specification that is the basis for estimation is:
Uijt=
qijt(δ1yiδ2−pjt)η f or kijt< Cj andj∈ {l, m, h}
qiht(δ1yδi2−pjt)η f or kijt < Ch andj=c qiht(δ1yδi2−pjt−τ1yi−τ2)η f or kijt < Ch andj=c
ξ−1i [B(yi)−pc]ηo 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, δ1, δ2, τ1, τ2, η, ηo, po, α, β, γ}
The predicted market share of productjis the expected demand forjsubject to quantity constraints. Hence
sjt(pt, Xt,Zt,Θ) =
(y,ξ)
dF(y)dG(ξ|Xtβ)
Denote the actual number of individuals who chooise option j in time t as Njt.where
Not=Mt−
j∈Ω/{o}
Njt
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
t=1
j∈Ω
Njtlogsjt(.,Θ)
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.
