# Market shares of Nested Logit demand model

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.