Optimal residential location and multivariate Frechet distribution

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.

• Have you ever learned any discrete choice model or self-selection model? May 30 at 22:38
• @Alalalalaki Not that much. I will take a look at notes related to Gumbel, Frechet, and Weibull distribution
– hbkn
May 31 at 2:20

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.

• Amazing, thanks so much! Where did you pick up these math skills? You may think it's not high-level math, but this was out of my capacity
– hbkn
May 31 at 2:08
• Glad you like it .... May 31 at 2:35