If respondent j chooses the non-GM food, it implies that the utility of choosing the non-GM food
is greater than that of choosing GM food:
U1j > U0j (6)
By assuming the marginal utilities of money (income) for non-GM food and GM food are
identical, i.e. β1 = β0 = β, the probability of choosing non-GM food is:
Prob (Non-GM) = Prob [α1Zj + β1 (Yj - Pngmj) - α0 Zj - β0 (Yj - Pgmj) > 0]
= Prob [(α1 - αo) Zj - β(l,ιιgιrn - Pgmj) + (ε1j - εoj) > 0]
This can be written more compactly as:
Prob (Non-GM) = Prob [α Zj - β(∆P) + εj > 0] (7)
where,
α = (α1 - α0)
∆P = (Pngm - Pgm)
εj = (ε1j - ε0j)
Assume further that the error term has a logistic distribution and it is symmetrical. Therefore, we
can derive the probability of choosing non-GM food as:
Prob (non-GM) = Prob [α Zj - β∆P + ε > 0]
= Prob [- (α Zj - β∆P) < ε] = 1 - Prob [-(α Zj - β∆P) > ε]
= Prob [ε < (α Zj - β∆P)] (8)
Furthermore, with a logistic distribution, ε has a mean of zero and variance π2σ2/3. Normalizing
by σ creates a logistic variable with mean zero and variance n2/3. Equation (8) becomes:
Prob (non-GM) = Prob [θ < (α Zj - β∆P)/ σ] = Ψ [α Zj / σ - (β∆P /σ)] (9)
where θ = ɛ/ɑ, σ is the standard error, Ψ is the cumulative distribution function.
Therefore, by using a logistic distribution, the probability of choosing the non-GM product is:
Prob (non-GM) = [1 + exp (- (α Zj / σ - β∆P /σ))]-1 (10)