1. science
2. physics
3. thermodynamics

Employability and Careers Centre
International Careers Week
27 January – 31 January
Postgraduate Study in the US with the Fulbright
Commission
 Employability Skills for International Careers
 Paid Work Experience in the US with Parenthese
 International Talent Search with Universum Webinar
 International Volunteering Skills Talks
 Working in the UK after study
 Careers in EU Institutions and the Foreign and
Open to all International, EU
Commonwealth Office
and home students
 Chinese CV Masterclass
 International
Teaching
book
online atOpportunities
www.essex.ac.uk/careers

Get an exciting career using applied econometrics in
advertising!!
Wondering what career to pursue after your graduation? Ever considered Marketing Econometric
Consulting?
By 2015 marketers are forecast to spend $600bn globally. The multitude of channels available to
advertise on generates tons of data, meaning big brands are looking for econometricians to give them
the confidence to cut through the data and deliver invaluable insights for better decision making.
If you have a degree in Economics, this is one of the most exciting times to work in advertising.
To learn how you can use applied econometrics in the advertising world, join two University of Essex
alumni on 5th February from 5pm in room 4.722.
The two Essex Graduate presenters will be:
Vaska Atta Darkua, Senior Analyst at Ohal
Afo Babatunde, Project Manager at Ohal
EC930 Theory of
Industrial Organisation
Week 3:
Game Theory Review – Nash Equilibrium
Product Differentiation I – Representative
Consumer Models
2013-14, spring term
Outline
Review of Game Theory
Nash Equilibrium – Cournot/Bertrand
Subgame Perfect Nash Equilibrium -- Capacity Game/
Stackelberg
Product Differentiation I: Representative Consumer Models
Derivation of Demand/Basic Assumptions
Cournot/Bertrand Equilibria
Comparison
Example 1: Exam Question
Example 2: Unilever/Sara Lee
Reading: Lecture notes 2
Problem Set 3: Q1, 2
Discussion and Support: Cabral ch 12, Tirole ch 7.
(Game Theory Appendix)
4
Taxonomy of solution concepts
Complete
information
Static game
(normal form)
Dynamic game
(extensive form)
Nash equilibrium
Subgame perfect equilibrium
E.g. Cournot competition;
Public good provision
E.g. Entry game;
Capacity-building followed by
pricing
5
Example of Nash equilibrium (1)
“Prisoners’ dilemma” (payoffs: Andy, Bob)
Bob
Collude
Defect
Collude
2, 2
-1, 3
Defect
3, -1
1, 1
Andy
Unique Nash equilibrium: (defect, defect)
The Pareto dominant outcome (collude, collude) does
not occur in equilibrium
6
Example of Nash equilibrium (1)
Bertrand example (a=10, c=0, b=1 in earlier case)
Firm 2
Firm 1
Collude
Defect
Collude
100/8, 100/8
0, 100/4
Defect
100/4, 0
0, 0
Nash equilibrium: (defect, defect)
The Pareto dominant outcome (collude, collude) does
not occur in equilibrium
7
Example of Nash equilibrium (1)
Cournot example (a=10, c=0, b=1 in earlier case)
Firm 2
Firm 1
Collude
Defect
Collude
100/8, 100/8
300/32, 900/64
Defect
900/64, 300/32
100/9,100/9
Nash equilibrium: (defect, defect)
The Pareto dominant outcome (collude, collude) does
not occur in equilibrium
8
Example of Nash equilibrium (2)
“Battle of the sexes”
Pat
Chris
Football
Opera
Football
2, 1
0, 0
Opera
0, 0
1, 2
2 Nash equilibria in pure strategies

(football, football), (opera, opera)
9
Example of SPE
Entry game
Entrant
enter
stay out
Incumbent
fight
fight
accom
(-1, -1)
(1, 1)
accom
(2, 0)
(2, 0)
10
Example of SPE
Entry game
Fight
incumbent
Accom
Entrant
Enter
Stay Out
-1, -1
2, 0
1, 1
2, 0
Payoffs: (incumbent, entrant)
2 NE: (enter, accom), (stay out, fight)
One of these seems more reasonable than other,
though.
11
Example of SPE
Entry game
Entrant
enter
stay out
Incumbent
fight
fight
accom
(-1, -1)
(1, 1)
accom
(2, 0)
(2, 0)
2 NE: (enter, accom), (stay out, fight)
Latter involves incredible threat and is not a SPE
Only (enter, accom) is SPE
12
Example of SPE
Stackelberg game:
Incumbent chooses output…then entrant reacts to maximise profit.
We solve this backwards: first, we obtain the best response to any output chosen by
the incumbent…then we choose the best output for the incumbent, knowing what
will follow.
Q4 (2012 exam): P = 1-Q, MC = 0; 2 firms, homogeneous products.
a. Firms choose quantities simultaneously. Obtain Nash equilibrium
quantities/profits
b. Firm A commits to quantity before firm B. Obtain subgame perfect Nash
equilibrium. Are firms better/worse off than in (a)?
c. Firm A chooses, but does not commit before B. What is the SPNE now?
13
Example of SPE
Capacity/price game:
Firms first choose capacities simultaneously…then
firms then choose price to maximise profit simultaneously.
We solve this backwards: first, we obtain the best
response to any price chosen by the rival for any
capacity level…
then we choose the best capacity, knowing what will follow.
14
Product Differentiation (horizontal)
Recall: At same price, some of both/all products purchased.
Representative Consumer Models: Each consumer buys a single variety, all
consumers have the same preferences (we model the “typical” consumer).
Address Models: Each consumer buys a single variety, consumers
differ (we model brand positioning in characteristics space).
Eg: consumer products (shampoo, deodorant, toothpaste…)
breakfast cereals
petrol stations
pubs
15
Product Differentiation – Representative Consumer Models
(Singh/Vives Rand, 84)
U = V(𝑞1 , 𝑞2 ) + 𝑦
𝑉 𝑞1 , 𝑞2
𝑏
𝑏
2
= 𝑎𝑞1 + 𝑎𝑞2 − (𝑞1 ) − (𝑞2 )2 − 𝑑𝑞1 𝑞2
2
2
y is “numeraire” good, sold in competitive sector.
Consumer problem: Max U w.r.t. 𝑞1 , 𝑞2 ,y s.t. budget constraint, I = pq+y.
Substituting the constraint:
Max 𝑉 𝑞 + 𝐼 − 𝑝𝑞 , 𝑏𝑢𝑡 𝑎𝑠 𝐼 𝑖𝑠 𝑗𝑢𝑠𝑡 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, ↔ 𝑀𝑎𝑥(𝑉 𝑞 − 𝑝𝑞)
 So V’ = p is the FOC (so that the problem is separable into q and p maximisation)
 All income effects concentrated into y.
16
Product Differentiation – Representative Consumer Models
Consumer problem: Max V - [𝑝1 𝑞1 + 𝑝2 𝑞2 ] = CS
𝑏
2
𝑏
2
Max w.r.t 𝑞1 , 𝑞2 : 𝑎𝑞1 + 𝑎𝑞2 − (𝑞1 )2 − (𝑞2 )2 − 𝑑𝑞1 𝑞2 − 𝑝1 𝑞1 − 𝑝2 𝑞2
 𝑃1 = 𝑎 − 𝑏𝑞1 − 𝑑𝑞2
 𝑃2 = 𝑎 − 𝑏𝑞2 − 𝑑𝑞1
 b>d usually assumed, all parameters positive. (b=d, d=0 special cases)
 will need some other parameter restrictions to ensure all prices,
quantities remain positive.
 d < 0 means complementary. Note that demand slopes down in own price
but shifts in/out/not at all depending on whether goods are substitutes,
complements or independent.
 usually,
𝑑2
𝑏2
(ranging from 0 to 1) is used to express degree of homogeneity
17
Product Differentiation – Representative Consumer Models
Firm Behaviour:
Cournot, with constant marginal cost:
firm 1: Max w. r. t. 𝑞1 :
𝑞1 (𝑝1 -c)  𝑞1 (𝑎 − 𝑐 − 𝑏𝑞1 − 𝑑𝑞2 )
 FOC: 𝑎 − 𝑐 − 2𝑏𝑞1 − 𝑑𝑞2 = 0
𝑎−𝑐
𝑑
− 𝑞2
2𝑏
2𝑏
𝑎−𝑐
𝑑
− 𝑞1
2𝑏
2𝑏

𝑞1 =

𝑞2 =
 These are the reaction/best response functions.
 Reaction functions slope down as in homogeneous goods case.
18
Product Differentiation – Representative Consumer Models
Cournot Equilibrium:

𝑞1 =
𝑎−𝑐
2𝑏+𝑑
= 𝑞2
 Notice that if d = b, we have solution for homogeneous products,
if d=0, we have solution for monopoly (independent goods
in two separate markets)
as d moves from 0 to b, we gradually increase each output
as the markets meld.
 Little has changed, then, as diagrams look the same and a parameter
moves us smoothly from two separate markets into a single integrated
market.
 Notice that we can model (imperfect) complements with this method as
well.
19
Product Differentiation – Representative Consumer Models
Bertrand Equilibrium:
 Use same utility function and hence, same demands.
 Solve for quantity as a function of price:
𝑎(𝑏−𝑑)
𝑏
𝑑
−
𝑝
+
𝑝
𝑏2 −𝑑 2
𝑏2 −𝑑 2 1
𝑏2 −𝑑 2 2
𝑎(𝑏−𝑑)
𝑏
𝑑
𝑞2 = 2 2 − 2 2 𝑝2 + 2 2 𝑝1
𝑏 −𝑑
𝑏 −𝑑
𝑏 −𝑑
 𝑞1 =

 Or…𝑞1 =∝ −𝛽𝑝1 + 𝛾𝑝2
𝑞2 =∝ −𝛽𝑝2 + 𝛾𝑝1
So that as the price of good 2 rises, the quantity of good 1 consumed increases
…since they are substitutes.
Notice that b>d still holds and ensures that these are positive parameters.
20
Product Differentiation – Representative Consumer Models
Bertrand Equilibrium:
 Max (𝑝1 − 𝑐)𝑞1 𝑤. 𝑟. 𝑡. 𝑝1
 max (𝑝1 − 𝑐)(𝛼 − 𝛽𝑝1 + 𝛾𝑝2 )
𝛼+𝑐𝛽
𝛾
+ 𝑝2
2𝛽
2𝛽
𝛼+𝑐𝛽
𝛾
𝑝2 =
+ 𝑝1
2𝛽
2𝛽
 𝑝1 =

 These are upwards sloping reaction functions for d>0! In other words,
 as the price of good 2 rises, the price of good 1 rises as well.
 why?
21
Product Differentiation – Representative Consumer Models
Marginal revenue (for linear demand case):
𝑀𝑅1 = 𝑝1 + 𝑞1 𝑝1 , 𝑝2
𝑑𝑝1
𝑑𝑞1
When 𝑝2 falls, this decreases 𝑞1 , so that for same 𝑝1 the second term
of marginal revenue falls, (ie, if you make fewer sales, then the effect
of an decrease in price on those “inframarginal” sales falls).
This raises marginal revenue in total, increasing the incentive to increase
sales (so that 𝑝1 should fall).
22
𝑝2
Reaction function
of firm 1
•
Reaction function
of firm 2
equilibrium
𝛼+𝑐𝛽
2𝛽
𝛾
2𝛽
𝛼 + 𝑐𝛽
2𝛽
𝑝1
23
𝑝2
Reaction function
of firm 1
Reaction function
of firm 2
Profit rises
p
Profit increases along reaction function
𝑝1
24
Solving for equilibrium:
𝑝1 = 𝑝2 =
𝛼 + 𝑐𝛽
2𝛽 − 𝛾
And this exceeds marginal cost for 𝛼 > 𝛽 − 𝛾, which is
equivalent to a>1 in the original model.
Differentiation increases market power. In fact, as d
increases, equilibrium prices fall towards marginal cost.
Note that when d=0, we have monopoly pricing. When
d approaches b, the solution approaches the homogeneous case.
As in Cournot, we can also model imperfect complements with this method.
25
Comparing the two, for substitutes, Bertrand prices lower and
quantities higher in equilibrium than Cournot, and consumer surplus
higher in Bertrand (except where markets independent, where all
these are equal)
Profits lower in Bertrand than Cournot.
Opposite holds for complements case.
So what if firms can choose which they compete in (p or q)? Think of
a two stage game:
Firms non-cooperatively choose whether they compete in p or q
Firms compete non-cooperatively, as above.
If the goods are substitutes, it is a dominant strategy to select Cournot
(compete on quantities), whereas if the goods are complements, it
is a dominant strategy to compete in prices.
26
Firm 2
Price
Price
Quantity
Bertrand
Pr PQ
Firm 1
Quantity
Pr QP
Cournot
And PR QP > Bertrand Profit; Cournot profit > Pr PQ
Why? A Bertrand firm raises price against a rival assumed to keep
(lower) price fixed  raising price cuts sales a lot. A Cournot firm
assumes, implicitly, that rival will raise price when I do in order to
keep sales constant  raising price cuts sales much less.
27
How to interpret this result and this model?
Not clear that firms “choose” which game they are in or strategic
variable: this may be determined by technology of industry.
Not clear that representative consumer model is a good representation
of reality, as it assumes all products compete with each other equally
In most markets, some substitutes are closer than others. For this, we
would need a different type of model…
28
Example: Exam question (2012)
Let demands for two firms, w and p, be:
Dw(Pw, Pp) = ½ + (Pp-Pw)
Dp(Pw, Pp) = ½ + (Pw-Pp)
And marginal cost is constant at c
1.
Derive the Bertrand equilibrium prices and profits
Using our earlier notation:  = 1/2  =   =  (substitutes -- only price difference
“counts”)
Substituting into our expression, we obtain symmetric reaction functions:
1
Pp = 4𝜇 +
𝑃𝑤 +𝑐
2
1
𝑐
(𝑎𝑛𝑑 𝑎𝑛𝑎𝑙𝑜𝑔𝑜𝑢𝑠 𝑓𝑜𝑟 𝑃𝑤 ) so slope upwards with intercept 4𝜇 + 2
1
And equilibrium prices are obtained by solving simultaneously: P*=2𝜇 + 𝑐
1
1
1
So that profits are: (P*-c)q* = (2𝜇)(2) = 4𝜇.
29
2. Let  = 1= c. Now, suppose that at a cost of K = .3, W could obtain  = 1/2 . Would
it pay to make this investment?
1
Profit before = 4𝜇 = ¼.
1
1
Profit after = 4𝜇 = 2 . Net profit after = ½ - K = .2
 Net profit after less than profit before, so don’t invest.
 If the two firms share the cost equally, does the answer change?
Yes, since a cost of .15 means that net profit after = .35, which exceeds profit before
(for both firms).
Exercise: re-calculate your answers if  = 1/(2t). (question 5a, 2012 exam)
30
Example: Unilever acquisition of Sara Lee
Sara Lee
Unilever
EC concerned
about effect in
market for
deodorants
31
Example: Unilever acquisition of Sara Lee
Main relevant brands: Sara Lee – Sanex Deodorant
Unilever – Rexona, Dove Deodorants
Various formats (stick, roll on, spray)
gender appeal (male, female, unisex)
health appeal (skin-friendly, “no comment”)
National markets
 Differentiated product within national markets.
Market share of combined Sara Lee/Unilever brands = 35-70%,
depending on national market.
 4-5 other competitors in general.
32
Nested logit model using customer panels, switching analysis:
deodorants
Single nested
model
male
Sub-nest
non-male
Skin-friendly
outside good
No comment
Individual products
33
Nested logit used to estimate demands for the goods,
including own and cross price elasticities.
Merger simulation then conducted, removing Sara Lee as a
separate firm in a Bertrand, differentiated product model.
Find approximately 5% price increase, with higher effect in
non-male segment.
Compensating efficiencies in wide range depending on country
(5-25%)
Remedy: require divestment of Sanex brand
34
Critique:
Much of the model is reasonable: Nested logit assumes same elasticities (cross and
own) within categories. While nesting is somewhat arbitrary, good checks used
beforehand.
Statistical testing of nesting is possible, but only eliminates models: cannot identify a
single “best” nest (and did not in this case). Clearly, parties choose “best for them”.
The acquisition was based on expected development (growth) of market, not on static
effects – but modelling purely static. In other words, not only is competition static
in Bertrand model, but no changes in demand parameters occur over time in this
simulation – only ownership of brands changes (number of firms decreases by 1).
Results indicated a 40% price rise for Sara Lee brand in static model: but acquisition
was specifically for the purpose of growing the brand!
Final decision was the Unilever abandoned the acquisition, as the remedy completely
eliminated the original reason for the acquisition.
35
Summary:
Nash equilibrium and Subgame Perfect Nash Equilibrium are
flexible concepts to be used to characterise games of the type
we have discussed until now.
Cournot, Bertrand – examples of Prisoner’s Dilemma class
Stackelberg, Capacity Choice – examples of SPNE
(Horizontal) Differentiation: Representative Consumer Model
Works well for goods where a “typical” consumer choosing
among a set of goods that are equally substitutable is a good
description of reality.
These are static models so miss any dynamics
36