Optimal residential location and multivariate Frechet distribution

Question

I was reading the paper "Housing Constraints and Spatial Misallocation" by Chang-Tai Hsieh and Enrico Moretti.

First they define the utility function

$$ \ \ V_{ji}=\varepsilon_{ji}\frac{W_{i}Z_{i}}{P_{i}^\beta},$$

where $i$ denotes city and $j$ denotes individuals.

And where it is assumed that $\varepsilon_{ji}$ follows joint multivariate distribution such that $F_{g}(\varepsilon_{1},\cdots,\varepsilon_{N})=\exp(-\sum_{i=1}^{N}\varepsilon_{i}^{-\theta})$

Then suddenly the paper shows that $$(10) \ \ W_{i}=V\frac{P_{i}^{\beta}L_{i}^{\frac{1}{\theta}}}{Z_{i}},$$

where $V$ is the average worker utility in all cities.

I looked up basic properties of Frechet distribution. If distribution function is $F(x)=\exp(-x^{-\theta})$, then the mean is $\Gamma(1-\frac{1}{\theta})=\int_{0}^{\infty}x^{-\frac{1}{\theta}}\exp(-x)dx$. But I don't clearly see how the equation (10) relates to this property. Any hints or suggestions for the next step will be appreciated.

Answer 1

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

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

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

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

$$\prod_j Pr(U_j \leq r) = \exp\left(-r^{-\theta} \cdot s^\theta\right) = \exp\left(-(r/s)^{-\theta}\right),$$

for which the expectation according to wiki is

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

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.