Optimal residential location and multivariate Frechet distribution

Consider a vector of stochastic variables $Z = (Z_1,...,Z_J)$. We assume each $Z_j$ is Frechet distributed

$$Z_j sim F(z_j) = exp(-z_j^{-theta}),$$

and that they are mutually independent such that $F_Z(z) = prod_{j=1}^J exp(- z_j^{-theta}) = exp(-sum_j z_j^{-theta})$. Furthermore, it is known that

$$ mathbb E[Z_j] = k(theta) = Gammaleft( 1-frac{1}{theta} right).$$

Given this set up we have the following properties (left unproven here):

1. When you scale $Z_j$ with a constant $A_j$ the distribution becomes $A_jZ_j sim F(z) = exp(-A_j^{theta}Z_j^{-theta}).$

2. The probability that $i = arg max_j {A_jZ_j}$ is given as

$$pi_i = frac{A_i^theta}{sum_j A_j^theta }.$$

We will use these properties for deriving equation (10) in the text you are reading. First we define the utility function og agent $h$ for alternative $j$ as

$$U_{jh}=Z_{jh}frac{W_{j}Q_{j}}{P_{j}^beta},$$

and we notice that the scaling constant $A_j = {W_{j}Q_{j}/P_{j}^beta}$. This then implies that the probability $pi_i$ of an agent choosing city $i$ is given as

$$pi_i = frac{left( {W_{i}Q_{i}/P_{i}^beta}right)^theta}{sum_j left({W_{j}Q_{j}/P_{j}^beta} right)^theta },$$

using that the labour force size is exogenously given the size of the labour force in city $i$ must be

$$ L_i = pi_i L = frac{left( {W_{i}Q_{i}/P_{i}^beta}right)^theta}{sum_j left({W_{j}Q_{j}/P_{j}^beta} right)^theta } L Leftrightarrow [8pt] (star) frac{P_i^beta L_i^{1/theta}}{Q_i} underbrace{left( frac{sum_j left({W_{j}Q_{j}/P_{j}^beta} right)^theta }{L} right)^{1/theta} }_{text{utility per worker}} = W_i,$$

where the utility per worker is simply a scaling constant independent of $i$ the index of the city under consideration.

How can we see it is utility per worker? The agent choose the alternative that provides max utility. Define the max utility

$$ hat U_i = max_j {U_{j}},$$

then the $Pr(hat U_i leq r) = Pr(text{all} U_j leq r) = prod_j Pr(U_j leq r)$ using independence. However

$$prod_j Pr(U_j leq r) = prod_j exp(-A_j^theta r^{-theta}) = expleft(-r^{-theta} cdot sum_j A_j^thetaright),$$ which is seen to be a Frechet distribution and define $s:=left(sum_j A_j^thetaright)^{1/theta}$ we can rewrite it to get

$$prod_j Pr(U_j leq r) = expleft(-r^{-theta} cdot s^thetaright) = expleft(-(r/s)^{-theta}right),$$

for which the expectation according to wiki is

$$mathbb E[hat U] = sk(theta) = left(sum_j A_j^thetaright)^{1/theta}k(theta) = left(sum_j left( W_{j}Q_{j}/P_{j}^betaright)^thetaright)^{1/theta}k(theta),$$

if you insert this in equation $(star)$ you have utility per worker except that workers are $L^{1/theta}$ - since $L$ is exogenous variable this is without consequence.