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

all	a1,2	1	d^s1
_ a2,1	a2,2 _	(1-βm₁(s₁)m₂(s₂))²	_ds2_

~∂² E ∏¹	∂² E ∏¹ ^			~∂² E ∏¹ ' -----dω ∂s₁∂ω
^ds²	∂s₁∂s ₂	^ds1		~∂² E ∏¹ ' -----dω ∂s₁∂ω
∂² E ∏²	∂² E ∏²	_ds2_	_	∂² E ∏² ——z—dω [_ ∂s₂∂ω J
∂s ₂∂s₁	∂s ₂²			∂² E ∏² ——z—dω [_ ∂s₂∂ω J

where a_i_,_i and a_i_,_-_i, for i = 1,2 are as previously defined. The system can in principle be

solved to obtain

(8)	^ ds₁* ^ dω	1	- a 2,2	a1,2	~∂² E ∏¹
	^ ds₁* ^ dω	1	- a 2,2	a1,2	∂s₁∂ω
		=	^,	^,	∂² E ∏²
	ds *	Ω	_ a2,1	^-a1,1 _	∂² E ∏²
	_ dω _				∂s ₂∂ω

where again the stability conditions are Ω =(a_1,1a_2,2 - a_1,2a_2,1) >0 and a_i_,_i < 0 for i = 1,2 .

The decisions of both processors are equal, so it suffices to analyze the responses of
processor 1. From equation (8), the change in the optimal stringency of assurance for firm 1 is
given by

(9)

ds 1 ( ∂² E ∏¹ ∂² E ∏²

^^= = _ I ^- a 2 2 Z ⁺⁺ al 2 Z Z

dω Ω₄ ^, ∂s₁∂ω ^, ∂s ₂∂ω

The sign of expression (9) is ambiguous in general. However, we can further analyze
some cases. From the stability conditions, we know that Ω is positive and a_2,2 is negative.

Nothing more can be ascertained without imposing further structure. If symmetry is assumed
(as it is here) the cross-partial terms are equal and hence we need to sign only one of them.

We study the case in which QASs are strategic complements.⁹ From the previous

analysis we know that this case arises when a_1,2 > 0 . Using the stability conditions, we

• ʃɪ, ɪ, ɪ (ds* I (∂²E∏¹

immediately see that sgn ∣ —- I = sgn ∣------

У dω ) ∂s₁∂ω

Cross-partial differentiation of the objective

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