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$U_{nj} = V_{nj} + \epsilon_{nj}$ 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.

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C is simply an unknown constant that represents the fact that the absolute value of utility cannot be measured. As welfare analysis is usually performed by comparing different situations ("states of the world") - This C is removed.

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