Industrial organization and strategy (7)

HOTELLING MODEL The Hotelling model is the

simplest Address model and it is

ased on the assumption that

products can be classified by 1

attribute. For example we have a

“Linear City” which is a city

made just of one street, so It can

be represented by a straight line,

and if I’m located at θ*, I’ll

choose the shop G4 because it’s

the closest one. Although this model simplifies a lot, its results can be generalized.

Hypothesis of the model:

-price is exogenus which means that

firms can’t decide the price but they

take it from the market

-the marginal cost is c and it’s fixed

for everybody

-product competition is played only

in 1 attribute

-We have M consumers in the

market and they are equally distributed with distribution M

-the competition is only based on localization, so LOCALISATION=DIFFERENTIATION

the profit of the firm I is pi=(p-c)*M*li

-p price

-c marginal cost

-M number of consumers

-li market segment captured by the firm

So, if we assume p-c=1 and M=1, the

profit is pi=li which means that in these

conditions the profit is equal to the

segment captured by the firm, so the

market share. now we need to distinguish firms in:

-interior firms: firms that are

located between other firms

-boundary firms: firms that are on

the boundary so they have a

competitor on the left/right but no

competitor on the other side. Interior firms are located in the middle θi

so they capture all the consumers that are

on the right and left until x and y because

these are indifferent consumers which

means that x and y are exactly in the

middle between 2 firms (x is in the middle

of θi-1 and θi and y is in the middle of θi

and θi+1) so they are perfectly indifferent

of going to one or to the other one. SO the profit is pi=li and li=y-x (the segment caputered by

θi). in the slide there is the

mathematical equation of what we

said:

-for x, the distance x- θi-1 is equal to

the distance θi – x, so we find the

location of x

-same for y

-so since li=y-x we can calculate li as

the formula, and since profit=li we

found the profit

for boundary firms

li-1= (θi-1 – 0) + (θi- θi-1/2)

-the first term is – 0 because in one

direction we don’t have other firms

-the second term is divided by 2

because it’s half of the distance

between θi an θi-1 where we have

the indifferent customer x

Same for li+1

Now the question is: how can we find

a position that is a Nash equilibrium?

Remember that here differentiation is

localization.

We need to satisfy the 2 points in the

slide 

Example: we have 4 products:

-θ1 and θ2 are at 0.25, θ1 as a

boundary firm and θ2 as a inner firm

-θ3 and θ4 are at 0.8, θ4 as a

boundary firm and θ3 as a inner firm

l1=0.25

l2=0.275

l3=0.275

l4=0.2 (the right boundary from 0.2 to 1→0.8)

this is not convenient for θ4 so he moves internally but also θ3 will move accordingly and they

move to 0.75. This is an equilibirum where all the firms have l1=l2=l3=l4=0.25

In this example the 2 firms start

at θ1 and θ2 and they both

have l1=l2=0.5. But this

situation is not an equilibrium

because point (ii) is not

satisfied so they start to

expand internally to maximize

their prfoti and end up both at

θ=0.5 which is a Nash

equilibrium. We can say that in a hoteling model,

with exogenus prices and no sunk or

fixed costs, firms try to group the

products in close locations and

since here

localisaiton=differentiation, firm

tend to DIFFERENTIATE AS LESS AS POSSIBLE→THIS IS THE PRINCIPLE OF MINIMUM

DIFFERENTIATION.

With more than 2 firms, companies tend to group in some specific positions= BUNCHING.

Examples of hoteling

model:

-Broadcast Tv:

prices are fixed (no one

pays for tv in chiaro)

and, although

producing programs

has sunk costs, the act

of programming has

not sunk cost, so the

hotleling model holds.

In this case channels

tend to differentiate as less as possible, in fact for example popular programs such as

telegiornali are more or less at the same hours in all the channels

LEZIONE 07: Product differentiation strategies - Part B We’ll remove some hypothesis in

order to introduce the concept of

free entry equilibrium in the linear

city model, the maximum

differentiation (the opposite of what

we have seen in part A) and then

we’ll remove the hypothesis of exogenus prices and consider endogenus prices.

First off we assume that we

have fixed cost (f) and they are

sunk, which means that they

cannot be recovered.

We also assume that (p-c)=1

fixed costs are:

-not sunk before entering

-sunk after entering

this means that firms before

entering need to do some

investments so they want to be

sure that if they enter they will

capture a segment of the market

li and since pi=li, it must be that

pi=li>f so the profit must be greater than the fixed cost they need to sustain, otherwise

companies won’t enter the market. firms can enter the market as

boundary firms or internal firms:

-v is the width of the boundary

segment

-w is the width of the internal

segment

As internal firms, companies get

w/2 so half of the internal

segment as we have seen in the

last chapter so the profit is pi=w*M/2 where M is the uniform distribution of customers.

Therefore firms will enter the market if w>sf/M.

As boundary firms, companies get the entire segment which is v, therefore firms will enter the

market if profit pi=v*M>f (slide a penna del prof sbagliata, è Maggiore), so v>f/M.

So, the 2 conditions are and since we don’t know if the firm enters as

a boundary or internal firm, we need to consider the condition that equipes all the

possibilities, so we need to maximize it and the maximum value is

so a firm that wants to enter the market must capture a segment lmax=2f/M.

In case lmax=2f/M the profit is

pi=f and the firm can consider to

enter in the market.

lmax is the minimum segment

the firm needs to achieve.

If all the firms achieve this

minimum segment, the number

of firms in the market will be

Nmin=1/lmax=M/2f. We can see

in the graph that

-the higher the costs (f), the less the firms in equilibrium in the market (Nmin)

-the more consumers in the market (M) the more the firms in equilibrium in the market (Nmin)

Now we need to understand where

the firms will position in the

market. If we have 1 firm it will be

localized in the middle at θ=1/2

because in this way he will take 0.5

on the left and 0.5 on the right.

If we have 2 firms they will be at

θ1=1/4 and θ2=3/4.

If we have 3 firms they will be at

θ1=1/6, θ2=3/6, θ3=5/6.

if we go further we will get the formula

in the slide […] 1/2*Nmin

So in the case of this chapter (with

sunk cost), the Minimum

differentiation principle doesn’t hold

anymore and the equilibrium is

characterized by the MAXIMUM

DIFFERENTIATION, so firms will be

localized at the maxmimum distance

possible, so they will try to

differentiate AS MORE AS POSSIBLE. What we learn is that if we move from

a model without sunk cost to a model

with sunk cost, we move from a

MINIMUM to a MAXIMUM

DIFFERENTIATION PRINCIPLE

because firms want to get a high

market length to recover fixed costs as

much as possible.

this is called localized competition.

LTT

NOW WE REMOVE THE LAST ASSUMPTION WE

DID BEFORE SO WE CONSIDER ENDOGENEOUS

PRICES, so firms can decide their prices.

assmptions:

-all firms have same marginal cost c=0

-the mismatching cost is T(D)=kD^2

-M=1

-duopoly situation

-PRINCIPLE MAXIMUM DIFFERENTIATION→ we

assume that the 2 firms are maximum

differentiated so firm A puts the product in zero and firm B in 1, so they are very differentiated.

(later we’ll remove also this hypothesis) Since prices are exogeneous we need to

play with the Utility function.

Considerig that firm A is at 0 and firm B

is at 1, their market segments θA and

θB depends on the positioning of x,

which is the customer for which buying

from A and buying from B is indifferent

(θB=1- θA).To find the position of x we

need to equalize the 2 utility functions

and we find the formula in 8.13. By deleting the common terms (we have θA and θB but θB=1-

θA) we get the position of the indifferent customer x which is θ’ in the next slide.

θ’ is in the formula.

(we’re always considering c=0)

By representing the 2 utility curves, the

matching point is θ’.

-If the 2 firms set the same prices pB=pA

the 2 firms get the same θ’=1/2

-if pB>pA, the market length of firm A is

greater than the market length of firm B,

because products are substitute so customers will choose the other one.

The formula 8.14 θ’ can be considered the demand function of firm A, because by increasing

pA, the demand will decrease. The cross-elasticity for the 2

products is formula 8.14.

We can see that of course cross

elasticity is directly proportional

to pB and inversely proportional

to qA, but the most important

thing is 1/2k. The parameter k is

the one we have in the formula of

mismatching cost T(D)=k*D^2 and it represents the perception of customers towards our

differentiation, in fact even if the distance D is very high, which means that differentiation is

very high, if k is zero, the mismatching cost is zero, which means that customers don’t

perceive the difference so it’s like it doesn’t matter; if k increases, even a small distance D is

important. Now let’s calculate the best

response of firm A.

profit of firm A is pi=pA*l where l is

the formula 8.14 we already found.

If we calculate the first derivate we

get 8.17 where the firs term is the

direct effect and second term is the

indirect effect.

by putting the 8.17 equal to zero we

get that price of firm A is 8.18 and if

we did the same thing for firm B we

would get the same for firm B.

By putting the 2 pA and pB in a

system we get that pA=pB=p=k so

the NAS EQUILIBRIUM IS EQUAL TO

k. what does this means?

This means that EQUILIBRIUM

PRICES ARE EQUAL TO THE DEGREE

OF PRODUCT DIFFERENTIATION,

THEREFORE THE MORE PRODUCTS

ARE DIFFERENTIATED THE HIGHER

ARE EQUILIBIRUM PRICES. Firms

want to differentiate products as

much as possible in order to set

higher prices.

back to the graph, since the