Externalities and the choice of strategic variable Consider the net utility function (1.) above. We can define

( ) (

i

) ( )

i i

i p p T

V =

αω

+ 1−

α ω

ˆ − as the surplus of joining network i. Then net utility can be written in two equivalent ways:

( ) ( ) ( )

i j i

i i j i i

x x t V

T x x t p p

−

−

≡

−

−

−

− + 1

α ω

ˆ

αω

On the one hand it seems reasonable to assume that firms compete by setting prices, on the other hand, it may be more convenient to consider net utility as the choice variable.¹⁸ Since (indirect) utility is measured in money there seems to be a one-to-one relationship between price and utility. In Laffont Rey and Tirole (1998b) it is argued (p. 52): Again we are back to a single-dimensional competition (competition in net surpluses or equivalently in fixed fees).” Armstrong (2002) however argues that this claim is not necessarily valid. When solving a model similar to the LRT model he states, in a footnote on page 359, that: “A subtle point is that one has to take care about the choice of strategic variables when network effects are present”.

Armstrong does not elaborate on this point however, and it may be worthwhile to take a closer look at this issue.¹⁹

Armstrong’s point can be illustrated by looking at a simplified²⁰ version of the stage 2 game in Foros and Hansen (2001). The simplifications are done to save notation and focus on the main aspect. Let the utility of being connected to network i be:

(

_i

)

_{i i}

i

i v k tx x p

U = +

α

+ 1−

α

− − −

18 See e.g. Armstrong and Vickers (2001)

19 The issue came to my attention when writing the paper on asymmetric costs and network competition. Since there were two equivalent ways of solving the same problem my idea was to check my calculations by doing both. I was not able to obtain the same result.

20 The simplification being that we do not take vertical differentiation into account, i.e. the parameter θ is set equal to zero. Furthermore the parameter β is set to unity (the degree of network externalities).

19

where v is the stand alone value of the network service. In this model network effects are linear. Total network size is normalized to unity, αi is the market share of firm i, thus

α

_i is the value of being able to communicate with others on the same network. The parameter k∈

[ ]

0,1 measures the quality when consumers communicate with subscribers on the other network, thus k

(

1−^α

)

is the utility from off-net communication.

Then net utility is defined as the utility of consuming the product, minus the price:

(

i

)

i i

i v k p

V = +

α

+ 1−

α

−

The market share functions are derived by identifying the location of the indifferent consumer:

(

Vi Vj

)

t − +

= 2

1 2

α

1 ,

or alternatively, if price is the strategic variable as:

( )

( ) (

_{i j}

)

i p p

k

t −

−

− −

= 2 1

1 2

α

1 .

We assume that production costs are normalised to zero. When price is the strategic variable, firms maximise:

α

_ip_i. When net utility is the strategic variable we must substitute from the definition of net utility in order to eliminate price from the profit expression:

( )

(

_{i i i}

)

i v+

α

+k −

α

−V

α

1

.

4.3.1. Price as strategic variable The first order condition for maximised profits is:

( )

( ) ( ( ) ) ( )

⁰

1 2

1 2

1 1

2

0 1 − =

−

− −

− +

− −

⇔

=

∂ +

∂ p

j p i p

i i

p p i i

i p p

k p t

k p t

p

α α

where superscript p denotes that price is the strategic variable. In a symmetric equilibrium (α =0.5, p^p = p_i^p = p^p_j ) prices and profits become:

20

( )

( )

2 1 2

1 k t

k t

p

p p

− −

=

−

−

=

π

4.3.2. Net utility as strategic variable Maximising profits:

( )

( )

[ ] ( ) ( )

0

2 1 2

1 1 2 1

max _{i i}+ − _i − _i ⇒ _i − + − _i − _i =

V k V t

k t V

k

i

α α

α α

α

In a symmetric equilibrium, net utility becomes: V_i =1−t. By inserting this expression back into the definition of net utility we find the equilibrium price and equilibrium profits:

( )

4 1 2

2 1

1

k t

t k p

V k

p

V V

i i i

i

− −

=

− −

=

⇔

−

− +

=

π

α α

where superscript V denotes that net utility is the strategic variable.

4.3.3. The two solutions compared

First of all, it is apparent from the calculations above that equilibrium prices and profits depend upon whether price or net utility is the strategic variable.

In the model above this is the case if k < 1, i.e. if the model exhibits network effects. Prices and profits are higher when net utility is the strategic variable.

In order to investigate whether the difference is significant I have illustrated the best response functions numerically for the two solutions below:

21

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35

price firm j

price firmi

Figure 9, The best response functions compared

In this example the parameter values: t = 0.4, k = 0.8, have been used. The solid lines are best response functions when price is the strategic variable and the dotted lines are best response functions (in the pricing dimension) when net utility is the strategic variable. In the games we consider here the difference in outcome is significant, equilibrium prices are 0.2 and 0.3 respectively.

In the simple model we consider here, the parameter restrictions that ensure a shared equilibrium are: t > (1 – k). Numerical simulations have revealed that parameter combinations close to this boundary²¹ yields a large difference between the two equilibria relative to parameter restrictions well inside the boundary.

When firms have net utility as a strategic variable they commit to a less aggressive behaviour. This becomes apparent if we look at the locus of the

21 One interpretation of the parameter restriction is that it assures that the network effect is not dominating the Hotelling differentiation. Thus the parameter restriction assures that the market share functions decrease in price.

22

best response functions at the previous page. For any price determined by the opponent, the best response price is higher in the game where net utility is the strategic variable as compared to the game where price is the strategic variable.

Consider a game where firms offer a contract where the level of the fixed fee is a function of market share, i.e. p_i = A_i +b_i

α

_i where, in the notation of the model above parameters are set such that: A_i =v+k−V_i and

k

b_i =1− . Then it becomes apparent that there is not a one-to-one relationship between pi and Vi since the price paid also depends on market share. Instead we can see that using net utility as a strategic variable is equivalent to using the parameter Ai as a strategic variable. Recall that network effects and horizontal differentiation have opposite effects on the willingness to pay for the marginal consumer as illustrated in figure 8 above.

When net utility is used as a strategic variable the competing firms commit to neutralise the effect of market share on the willingness to pay by the marginal consumer. By doing so, profits increase.

In the present thesis, the choice of strategic variable has impact on the solution in both the paper on cost asymmetries (paper 2) and the paper on internet competition (paper 3). In both cases price is used as the strategic variable. According to Armstrong (2002), this is (perhaps) the most plausible assumption.

In document Facility based competition in telecommunications : three essays on two-way access and one essay on three-way access (sider 26-30)