Consider a Nested Logit demand model with two nests, $N_1, N_2$: $N_1$ contains the outside option only (labelled "0"), $N_2$ contains all the remaining alternatives (labelled "$j=1,...,J$").

Suppose that the utility for consumer $i$ from picking alternative $j\in N_2$ is $$ U_{ij}\equiv \delta_j+v_i+\lambda \epsilon_{ij} $$ where $\lambda\in (0,1)$ and $(v_i, \epsilon_{ij})$ have a distribution which obeys the Nested Logit parametrisation.

Instead, the utility for consumer $i$ from picking alternative $0\in N_1$ is $$ U_{i0}\equiv \epsilon_{i0} $$

Question: could you help me to derive the market shares of a product $j\in N_1$ and of product $0\in N_2$?

I found the formulas for a generic nest structure (for example, this question is helpful), but when I try to apply them to my very simple case they do not seem to simplify a lot.


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