Chemistry Reference
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models. A typical model of this type specifies that the market share of a competi-
tor is equal to the attraction of its product, divided by the total attraction of all the
competitors' products in the market. Each competitor's attraction is given in terms
of its competitive effort allocations. To provide an example, let us consider the case
of two competitors, who compete against each other in the market on the basis of
marketing efforts expended. If x
1
>0denotes the marketing effort of competitor 1
and x
2
>0 the marketing effort of competitor 2, then ˛
1
x
ˇ
1
1
and ˛
2
x
ˇ
2
2
represent
the attractions of customers to the products of competitors 1 and 2, respectively.
The positive parameters ˛
1
and ˛
2
in this context denote the relative effectiveness
of efforts and the parameters ˇ
1
>0and ˇ
2
>0are the elasticities of the products'
attractions with respect to the marketing efforts. The competitors' market shares are
then given by
˛
1
x
ˇ
1
1
˛
2
x
ˇ
2
2
s
1
D
;
s
2
D
:
(4.1)
˛
1
x
ˇ
1
C
˛
2
x
ˇ
2
˛
1
x
ˇ
1
C
˛
2
x
ˇ
2
1
2
1
2
Such a specification has the theoretically appealing property that it is logically con-
sistent in the sense that it yields market shares that are between zero and one, and the
market shares sum to one across all the competitors in the market. If A>0denotes
the sales potential of the market (in monetary terms) and c
i
the marginal cost of
effort of firm i, then the one-period profits of firm 1 and 2 are
'
1
D
As
1
c
1
x
1
;
2
D
As
2
c
2
x
2
:
(4.2)
The reader should notice that by introducing the new decision variables
z
1
D
and cost functions C
1
.
z
1
/
D
c
1
z
˛
1
1=ˇ
1
˛
1
x
ˇ
1
,
z
2
D
˛
2
x
ˇ
2
2
and C
2
.
z
2
/
D
c
2
z
˛
2
1=ˇ
2
, the market share attraction game is identical to an oligopoly game
with isoelastic market demand function which we have discussed in the previous
chapter. Therefore, the results obtained there are valid for market share attraction
games as well.
Recall that ˇ
1
and ˇ
2
are the elasticities of the products' attractions with respect
to the marketing efforts. Hence, we typically have ˇ
i
2
.0;1/,or1=ˇ
i
>1,sothe
functions C
1
and C
2
are strictly convex. Consequently, in applications a unique
Nash equilibrium is obtained. In the general case, a closed-form solution for the
Nash equilibrium cannot be given. However, for the symmetric case, that is for
identical elasticities ˇ
1
D
ˇ
2
D
ˇ; identical marginal costs of effort c
1
D
c
2
D
c;
and identical effectiveness parameters ˛
1
D
˛
2
; the Nash equilibrium can be easily
calculated. It is characterized by identical efforts of the two competitors,
Aˇ
:
4c
;
Aˇ
E
D
(4.3)
4c
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