where αk≡xk/x is the market share of firm k and 1+αkr must be positive if goods are strategic
substitutes. Combining (44) and (45):
_ -[n÷l÷(l-¾)r]⅜÷(l÷<⅛r)<⅜.t (46)
t i>(n+2+r)
The cross effect is clearly positive provided goods are strategic substitutes. The own effect
is negative if demands are concave (r>0). With convex demands the effect could be positive.
However, this requires that demand be highly convex: r must be less than -(n+1 )/( 1-αk).
In addition, if goods are strategic substitutes (which implies that r>-1/αk), a positive own
effect requires that firm k be relatively small in the market: αk must be less than 1/(n+2).
A.3 Cournot Competition with Linear Demands and Differentiated Products
Return to the case of Cournot competition with linear demands, but now allow goods
to be symmetrically differentiated. Write the demand function facing firm k as:
(447 = 1 - [(l-e)xfe + ex] (47)
An identical series of steps to those in Section 3.1 now leads to:
_ 2-e-[2÷(n-l)φt÷<⅝-t (48)
‘ (2-e)(2+ne)
Clearly, this exhibits the desired properties, being decreasing in own access cost and
increasing in rivals’ access costs. In addition, as goods become more differentiated, so e falls
from one towards zero, the competition effect becomes less important, and the model
approaches the monopoly case of Section 2.
26
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