Reputations, Market Structure, and the Choice of Quality Assurance Systems in the Food Industry

16

The system of equations (6) implicitly defines the processor’s best response functions,
b_i(s_-i), i = 1,2 . A vector of investments in safety (s₁^*,s₂^*) is a Nash equilibrium for this model
if and only if s_i^*= b(s_-^*_i) for i = 1,2 . If an equilibrium for this stage exists, we can then
compute the output quantities and prices using the equilibrium rules for the Cournot game that
will be played at the second stage.

We now investigate the nature of the competition by studying a key property, the slope
of the best-response functions. To that end, we substitute the best-response function of firm i
into its first-order condition and differentiate it (at an interior solution) with respect to the
choice of its rival. After rearranging and omitting arguments for brevity, we get

∂²π^,    ∂m.      β ⁽ ∂∏        _i ∂m _i 1      '

+                      ^m-, + ∏ ^-i-

∂ b ( s—i ) = ^dsi^ds -, ^ds, (^{1 - βm},^m J₄/s-,________ds-iJl-emm-i^J

^ds -,                                  ^ai,i(^{1 - βm},^m -_i)'^λ

The denominator of this expression is negative by the sufficient conditions for a
maximum. Thus, the sign of the slope of the best-response function is the same as the sign of
the numerator (see Dixit). The first term is negative, since as one firm increases investments in
safety this reduces the benefits derived from increases in its rivals’ expenditures on QASs.⁸ By
a similar argument, and the assumption that more stringent systems yield better products, the
last two terms are positive. Overall, the sign is ambiguous without further structure, but close
inspection of equation (7) reveals some insights into its sign.

First, if the probability of success for both processors is high, and the future is valued,
the best-response function will likely slope upwards. Second, if the cross-partial terms are
close to zero, for example, if the products are not close substitutes, the slope will be positive.
But neither condition is necessary to obtain upward-sloping reaction functions. Downward-

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