Lecture: Differentiated products I
So far, we have discussed the case of homogenous goods. But goods may differ, for
example, brands within product categories or in spatial location relative to consumers.
This brings us to the study of differentiated products.
Shepard (JPE 1991): Price Discrimination and Retail Configuration
It is well known that a monopolist can always price discriminate when the resale of its
product is difficult and consumers with different tastes can be separated. But the
possibility of price discrimination in multi-market firms is less established.
Demonstrating that price discrimination explains any of the observed price
differentials is tough for the reasons very similar to those discussed above: it is
difficult to separate out cost based differentials from discriminatory differentially,
especially in face of inadequate cost data.
This paper exploits a natural experminent: firms can differ in the ability to price
discriminate but not in the cost of production. Differences in the price structures
across firms are then argued to be evidence of price discrimination.
The comparison is in terms of prices for different quality service at petrol (gasoline)
stations. There are three kinds of gasoline stations: those that are full service (High
Quality), or are self service (Low Quality) or those that offer that offer both.
Consumer have preferences over quality and self select themselves.
Quality differentiation in products is often called vertical differentiation. The paper
compares the price differential between full service and self service petrol at stations
offering both service types (multi-product stations) with the price differential across
stations offering only full service and stations offering only self service (single
product stations).
Model:
1. Each consumer buys no more than one unit of gasoline per period
2. Consumers have identical preferences but differ in income
3. Because higher income individuals have a lower marginal utility of income,
they have a higher willingness to pay.
Consumers are of type t drawn from a uniform distribution U(0,1).
Let the utility to a consumer of type t from the various qualities be:
High Quality: Vh (t − ph )
Low Quality: Vl (t − Pl )
Don’t Buy: Vot
Single Product Firm (SP)
High Quality Product:
Marginal Consumer: Vh (t − ph ) = Vot
" V %
This implies t = $ h ' Ph
# Vh −Vo &
" V %
Demand: Dh (Ph ) = 1− t = 1− $ h ' Ph
# Vh −Vo &
Low Quality Product
" V %
Demand: Dl (Pl ) = 1− t = 1− $ l ' Pl
# Vl −Vo &
Multi Product Firm: (MP)
Find the consumer indifferent between high quality and low quality.
tl
Consumer of
this type
doesn’t buy
th
Consumer of
this type buys
Self Service
Consumer of
this type buys
Full Service
th is such that Vh (t h − ph ) = Vl (t h − pl )
" V %
" V %
t h = $ h ' ph − $ l ' pl
# Vh −Vl &
# Vh −Vl &
Therefore Demand for High Quality is
" V %
" V %
DhMP ( ph , pl ) = 1− t h = 1− $ h ' ph + $ l ' pl
# Vh −Vl &
# Vh −Vl &
Similarly tl is such that:
Vl (t l − pl ) = Vot l
Hence
" V %
t l = $ l ' pl
# Vl −Vo &
Therefore Demand for Low Quality is:
𝐷!!" 𝑝! , 𝑝! = 𝑡 ! − 𝑡 !
𝑉!
𝑉! 𝑉! − 𝑉!
=
𝑝! −
𝑉! − 𝑉!
𝑉! − 𝑉! 𝑉! − 𝑉!
Costs:
• Marginal costs (MC) depend on the quality of service. However they are
invariant to station type: that is full service costs the same in a single product
firm selling full service and in the multi product firm (critical identifying
assumption)
MC (self service)= w (which is cost of gas)
MC (Full service)= w + 𝛼 (cost of gas + cost of labour)
(MC is assumed to be constant)
Profit Maximizing:
1. SP firm: High Quality:
solve for
𝑝!!"
𝑚𝑎𝑥!! 𝑝! − 𝑤 − 𝛼 𝐷!
2. SP: Low Quality:
solve for
𝑝!!"
3. MP:
solve for 𝑝!!" 𝑝!!"
𝑚𝑎𝑥!! 𝑝! − 𝑤 𝐷!
𝑚𝑎𝑥!! ,!! 𝑝! − 𝑤 − 𝛼 𝐷!!" + 𝑝! − 𝑤 𝐷!!"
The solutions will give reduced form for prices under each scenario.
Define:
Δ!" = 𝑝!!" − 𝑝!!"
Δ!" = 𝑝!!" − 𝑝!!"
Δ = Δ!" − Δ!"
These differences are functions of 𝑤 and 𝛼. It is shown that for values of 𝑤 and 𝛼 that
give non negative quantity, the differences are positive (except a single point).
It is not surprising that the higher quality good costs more than the lower quality in
both Multi product and single product firm. But what about Δ?
To see this rewrite
Δ = 𝑝!!" − 𝑝!!" − 𝑝!!" − 𝑝!!"
as
Δ = 𝑝!!" − 𝑝!!" − 𝑝!!" − 𝑝!!"
It can be shown that
∆! = 𝑝!!" − 𝑝!!" ≥ 0
and
with at least one of them strict.
Hence 𝚫 > 𝟎
∆! = 𝑝!!" − 𝑝!!" ≤ 0
Intuition: 𝑝!!" − 𝑝!!" ≥ 0:
Suppose 𝑤 and 𝛼 are set to zero (simplification)
1
ph
pl
Dh
Using the
seen that
MRh
High
Quality
Dl
formulat for t, it can
𝑝! 𝑎𝑛𝑑 𝑝! will be
MRl
be
Low
Quality
!
charged such that 𝑡 ≥ !
!
In the MP, if prices stayed the same as SP, then for 𝑡 ≥ !, people will buy high
!
quality, and for 𝑡 < !, nothing will be bought. Lowering price of high quality good is
not optimal, because if it were, SP would already do that. Suppose there is a slight
increase in price of high quality. Then some consumer will not buy high quality good
but will buy lower quality good because it is available in a multi product form. Hence
proved!
Intuition: 𝑝!!" − 𝑝!!" ≤ 0.
A higher price has no effect in Multi product market.
A price decrease will have two effects in multiproduct market: Some consumers
switch from no purchase to purchase low quality. SAME AS IN SP
Some consumer may also switch from high quality to low quality: MAY NOT BE
GOOD
So net potential gain in MP from price decrease is lower than in SP.
However, in SP, this trade off has already been made. In MP, the SP price is not an
optimal price and at that SP price, no low quality product was being sold before price
decrease. So it is possible that MP firm will gain by reducing prices.
In a perfectly competitive industry all the 𝜟𝒔 are zero.
This theory is taken to a cross sectional data of 1500 gas stations (denoted by i ) in
MA, USA.
𝑝!"# = 𝛽! + 𝛽! 𝐷! + 𝛽! 𝐷! 𝐷! +. 𝑑𝑑 𝑠ℎ𝑖𝑓𝑡𝑒𝑟𝑠 + 𝑐𝑜𝑠𝑡 𝑠ℎ𝑖𝑓𝑡𝑒𝑟𝑠
Here 𝐷! = 1 𝑖𝑓 ℎ𝑖𝑔ℎ 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 and 0 otherwise. 𝐷! = 1 𝑖𝑓 𝑀𝑢𝑙𝑡𝑖𝑝𝑟𝑜𝑑𝑢𝑐𝑡 and 0
otherwise.
Reference: SP/ Low Quality: 𝛽!
SP/ High Quality: 𝛽! + 𝛽! (also contains any cost differences)
MP/Low Quality: 𝛽! + 𝛽!
MP/High Quality: 𝛽! + 𝛽! + 𝛽! + 𝛽!
Hence this is difference in difference specification and Δ = 𝛽!
Results: The paper finds evidence for price discrimination.