# Consumer Surplus (Williams 1977 and Rosen 1981)

$$U_{nj} = V_{nj} + \epsilon_{nj}$$ for all $$j$$ and $$\epsilon_{nj}$$ captures the factors that affect utility but are not included in $$V_{nj}$$.

The decision maker chooses the alternative that provides the greatest utility.

Consumer surplus is, $$CS_{n} = (\frac{1}{\alpha_{n}})\thinspace max \thinspace (U_{nj}) \thinspace wrt \thinspace j$$.

Then, $$E(CS_{n}) = (\frac{1}{\alpha_{n}}) E[max \thinspace wrt \thinspace j \thinspace (V_{nj} + \epsilon_{nj} )]$$ where the expectation is over all possible values of $$\epsilon_{nj}$$.

My question :

If each $$\epsilon_{nj}$$ is iid extreme value, then we have

$$E(CS_{n}) = (\frac{1}{\alpha_{n}}) ln ( \sum_j e^{V_{nj}} ) + C$$ where C is unknown constant.

I could not understand where this $$C$$ is coming from ?

I was trying to calculate $$\int_{\infty}^{-\infty}(V_{ni} + \epsilon_{ni} ) P_{ni} f(\epsilon_{ni}) d\epsilon_{ni}$$

where $$P_{ni} = \frac{e^{V_{ni}}}{ \sum_j e^{V_{nj}} }$$ from logistic distribution.