Non Linear Contracting and Endogenous Buyer Power between Manufacturers and Retailers: Empirical Evidence on Food Retailing in France

taste shocks Uij_t are independently and identically distributed according to a Gumbel (extreme
value type 1) distribution, so that the probability L_ij_t of buying j for consumer i at period t
conditional on ai and δ_i can be written :

exp(¼jt)

1 + ∑fc=ι exp(‰)

where V_ijt = β_b^ + β_r(_j) + δiXj - aip_jt + τη_jt∙

For simplicity, we assume that (y^δ ,vβ^ are independent and normalalize their variance to
one and mean to zero. Denoting f the standard normal probability distribution function, the
unconditional probability of the observed sequence of T choices for consumer i is then
where β is the vector of all β_b and β_r parameters in (18), j(i,t) is the chosen alternative by
consumer i at period t and f (α⅛∣α, σ) and f (δi ∣δ, σ^δ) are the p.d.f. of the random coefficients ai
and δi respectively.

Then, the log likelihood of the sample of choices over N individuals is :

∑^₁ In [Fi(α,σ≈,β, δ,σ^δ)] .

The probability of the observed sequence of choice for consumer i is approximated with simulation
for any given value of (a, σ^α, β, δ, σ^δ) and can be written :

SPi(a,σ^a,β,δ,σ^δ) = R ∑*_χ (∏^ Д_жф(а^г,δ^rɔ

where R is the number of simulations, a^r and δ^r are the r^th Halton draws of the distributions
f (αi∣α, σ) and f (δi∣δ, σ^δ) respectively.

Then, the model parameters are estimated by maximizing the simulated likelihood (Train, 2009)
which is

SLL(a, σ^a, β, δ, σ^δ) = V ' ɪ In [SP_i(a, σ^a, β, δ, σ^δ)^]

with respect to a, σ^a, β, δ, σ^δ.

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