<p>Consider a vector of stochastic variables <span class="math-container">$Z = (Z_1,...,Z_J)$</span>. We assume each <span class="math-container">$Z_j$</span> is Frechet distributed</p>
<p><span class="math-container">$$Z_j \sim F(z_j) = \exp(-z_j^{-\theta}),$$</span></p>
<p>and that they are mutually independent such that <span class="math-container">$F_Z(z) = \prod_{j=1}^J \exp(- z_j^{-\theta}) = \exp(-\sum_j z_j^{-\theta})$</span>. Furthermore, it is known that</p>
<p><span class="math-container">$$ \mathbb E[Z_j] = k(\theta) = \Gamma\left( 1-\frac{1}{\theta} \right).$$</span></p>
<p>Given this set up we have the following properties (left unproven here):</p>
<ol>
<li><p>When you scale <span class="math-container">$Z_j$</span> with a constant <span class="math-container">$A_j$</span> the distribution becomes <span class="math-container">$A_jZ_j \sim F(z) = \exp(-A_j^{\theta}Z_j^{-\theta}).$</span></p>
</li>
<li><p>The probability that <span class="math-container">$i = \arg \max_j \{A_jZ_j\}$</span> is given as</p>
</li>
</ol>
<p><span class="math-container">$$\pi_i = \frac{A_i^\theta}{\sum_j A_j^\theta }.$$</span></p>
<p>We will use these properties for deriving equation (10) in the text you are reading. First we define the utility function og agent <span class="math-container">$h$</span> for alternative <span class="math-container">$j$</span> as</p>
<p><span class="math-container">$$U_{jh}=Z_{jh}\frac{W_{j}Q_{j}}{P_{j}^\beta},$$</span></p>
<p>and we notice that the scaling constant <span class="math-container">$A_j = {W_{j}Q_{j}/P_{j}^\beta}$</span>. This then implies that the probability <span class="math-container">$\pi_i$</span> of an agent choosing city <span class="math-container">$i$</span> is given as</p>
<p><span class="math-container">$$\pi_i = \frac{\left( {W_{i}Q_{i}/P_{i}^\beta}\right)^\theta}{\sum_j \left({W_{j}Q_{j}/P_{j}^\beta} \right)^\theta },$$</span></p>
<p>using that the labour force size is exogenously given the size of the labour force in city <span class="math-container">$i$</span> must be</p>
<p><span class="math-container">$$ L_i = \pi_i L =  \frac{\left( {W_{i}Q_{i}/P_{i}^\beta}\right)^\theta}{\sum_j \left({W_{j}Q_{j}/P_{j}^\beta} \right)^\theta } L  \Leftrightarrow  \\[8pt]
(\star) \ \ \frac{P_i^\beta L_i^{1/\theta}}{Q_i} \underbrace{\left( \frac{\sum_j \left({W_{j}Q_{j}/P_{j}^\beta} \right)^\theta }{L} \right)^{1/\theta} }_{\text{utility per worker}} =  W_i,$$</span></p>
<p>where the utility per worker is simply a scaling constant independent of <span class="math-container">$i$</span> the index of the city under consideration.</p>
<p>How can we see it is utility per worker? The agent choose the alternative that provides max utility. Define the max utility</p>
<p><span class="math-container">$$ \hat U_i = \max_j \{U_{j}\},$$</span></p>
<p>then the <span class="math-container">$Pr(\hat U_i \leq r) = Pr(\text{all} \ U_j \leq r) = \prod_j Pr(U_j \leq r)$</span> using independence. However</p>
<p><span class="math-container">$$\prod_j Pr(U_j \leq r) = \prod_j \exp(-A_j^\theta r^{-\theta}) = \exp\left(-r^{-\theta} \cdot \sum_j A_j^\theta\right),$$</span>
which is seen to be a Frechet distribution and define <span class="math-container">$s:=\left(\sum_j A_j^\theta\right)^{1/\theta}$</span> we can rewrite it to get</p>
<p><span class="math-container">$$\prod_j Pr(U_j \leq r)  = \exp\left(-r^{-\theta} \cdot s^\theta\right) = \exp\left(-(r/s)^{-\theta}\right),$$</span></p>
<p>for which the expectation according to wiki is</p>
<p><span class="math-container">$$\mathbb E[\hat U] = sk(\theta) = \left(\sum_j A_j^\theta\right)^{1/\theta}k(\theta) = \left(\sum_j \left( W_{j}Q_{j}/P_{j}^\beta\right)^\theta\right)^{1/\theta}k(\theta),$$</span></p>
<p>if you insert this in equation <span class="math-container">$(\star)$</span> you have utility per worker except that workers are <span class="math-container">$L^{1/\theta}$</span> - since <span class="math-container">$L$</span> is exogenous variable this is without consequence.</p>

CC BY-SA 4.0
2021-05-31T02:08:38.880Z
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
CC BY-SA 4.0
2021-05-31T02:35:48.333Z
Glad you like it ....
Jesper Hybel
CC BY-SA 4.0
2021-05-31T00:27:48.453Z
2021-05-31T09:56:57.983Z
2021-05-31T09:56:57.983Z
Jesper Hybel
<p>I was reading the paper <a href="http://eml.berkeley.edu/%7Emoretti/growth.pdf" rel="nofollow noreferrer">&quot;Housing Constraints and Spatial Misallocation&quot;</a>
by Chang-Tai Hsieh and Enrico Moretti.</p>
<p>First they define the utility function</p>
<p><span class="math-container">$$ \ \ V_{ji}=\varepsilon_{ji}\frac{W_{i}Z_{i}}{P_{i}^\beta},$$</span></p>
<p>where <span class="math-container">$i$</span> denotes city and <span class="math-container">$j$</span> denotes individuals.</p>
<p>And where it is assumed that <span class="math-container">$\varepsilon_{ji}$</span> follows joint multivariate distribution such that <span class="math-container">$F_{g}(\varepsilon_{1},\cdots,\varepsilon_{N})=\exp(-\sum_{i=1}^{N}\varepsilon_{i}^{-\theta})$</span></p>
<p>Then suddenly the paper shows that  <span class="math-container">$$(10) \ \ W_{i}=V\frac{P_{i}^{\beta}L_{i}^{\frac{1}{\theta}}}{Z_{i}},$$</span></p>
<p>where <span class="math-container">$V$</span> is the average worker utility in all cities.</p>
<p>I looked up basic properties of Frechet distribution. If distribution function is <span class="math-container">$F(x)=\exp(-x^{-\theta})$</span>, then the mean is <span class="math-container">$\Gamma(1-\frac{1}{\theta})=\int_{0}^{\infty}x^{-\frac{1}{\theta}}\exp(-x)dx$</span>. 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.</p>

CC BY-SA 4.0
2021-05-30T22:38:45.660Z
Have you ever learned any discrete choice model or self-selection model?
Alalalalaki
CC BY-SA 4.0
2021-05-31T02:20:06.897Z
@Alalalalaki Not that much. I will take a look at notes related to Gumbel, Frechet, and Weibull distribution
hbkn
<macroeconomics><urban-economics><comparative-statics>
Optimal residential location and multivariate Frechet distribution
hbkn
# Optimal residential location and multivariate Frechet distribution

(Asked by: username_0 on 2021-05-30)

I was reading the paper ["Housing Constraints and Spatial Misallocation"](http://eml.berkeley.edu/%7Emoretti/growth.pdf)
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.

**Tags:** macroeconomics, urban-economics, comparative-statics
> **Comments:**
> **username_1**: Have you ever learned any discrete choice model or self-selection model?  
> **username_0**: @Alalalalaki Not that much. I will take a look at notes related to Gumbel, Frechet, and Weibull distribution  

---

## 1 answers

### Answer by username_2 on 2021-05-31. Score: 7 | Accepted answer

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.

> **Comments:**
> **username_0**: 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  
> **username_2**: Glad you like it ....  

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