<p>Use <span class="math-container">$W_{i}=V \cdot \frac{P_{i}^{\beta}}{Z_{i}}=VQ_i$</span>, then <span class="math-container">$$L_{i}=\left(\frac{\alpha^{1-\eta} \eta^{\eta}}{R^{\eta}} \cdot \frac{A_{i}}{(V_{i} Q_{i})^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}} \cdot T_{i}$$</span> and <span class="math-container">$$\sum L_i = {\left({\frac{\alpha}{V}}\right)}^{\frac{1-\eta}{1-\alpha-\eta}} {\left({\frac{\eta}{R}}\right)}^{\frac{\eta}{1-\alpha-\eta}} \sum\left(\frac{A_i}{Q_i^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}}T_i = 1$$</span> <span class="math-container">$$\frac{V}{\alpha} = {\left({\frac{\eta}{R}}\right)}^{\frac{\eta}{1-\eta}} \left(\sum\left(\frac{A_i}{Q_i^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}}T_i\right)^{\frac{1-\alpha-\eta}{1-\eta}}  $$</span>.</p>
<p>Use the FOC on labor, <span class="math-container">$W_i=\alpha \frac{Y_i}{L_i}$</span>, then <span class="math-container">$$\sum Y_i = \frac{V}{\alpha}\sum L_iQ_i = \frac{V}{\alpha} \bar{Q}$$</span>, and replace <span class="math-container">$\frac{V}{\alpha}$</span> with above equation then you get equation (7).</p>

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2021-03-08T04:05:09.843Z
Awesome. Thank you very much!!
hbkn
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2021-03-08T09:33:37.227Z
@baek, if you like the answer, consider accepting it!
VARulle
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2021-03-08T15:37:48.793Z
@VARulle I did it. Thanks for the tip!
hbkn
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2021-03-07T23:08:37.313Z
2021-03-07T23:15:30.533Z
2021-03-07T23:15:30.533Z
Alalalalaki
<p>I was reading the following paper:</p>
<p><a href="http://eml.berkeley.edu//%7Emoretti/growth.pdf" rel="noreferrer">http://eml.berkeley.edu//~moretti/growth.pdf</a></p>
<p>I got stuck at equation (7)</p>
<p>The firm's production function is <span class="math-container">$Y_{i}=A_{i}L_{i}^{\alpha}K_{i}^{\eta}T_{i}^{1-\alpha-\eta}$</span></p>
<p>Labor supply is <span class="math-container">$W_{i}=V\frac{P_{i}^{\beta}}{Z_{i}}=V\frac{\bar{P_{i}}^{\beta}L_{i}^{\beta \gamma_{i}}}{Z_{i}}$</span></p>
<p>Labor demand is <span class="math-container">$L_{i}=(\frac{\alpha^{1-\eta}\eta^{\eta}}{R^{\eta}}\frac{A_{i}}{W_{i}^{1-\eta}})^{\frac{1}{1-\alpha-\eta}}T_{i}$</span></p>
<p>The paper says that if we impose aggregate labor demand is equal to aggregate labor supply (normalized to one), then the aggregate output <span class="math-container">$Y=\sum_{i}Y_{i}$</span> is</p>
<p><span class="math-container">$Y=(\frac{\eta}{R})^{\frac{\eta}{1-\eta}}[\sum_{i}(A_{i}[\frac{\bar{Q}}{Q_{i}}]^{1-\eta})^{\frac{1}{1-\alpha-\eta}}T_{i}]^{\frac{1-\alpha-\eta}{1-\eta}}$</span></p>
<p>This step looks drastic to me. How can the aggregate output be derived from the above conditions?</p>

<supply-and-demand><labor-economics><aggregate-supply>
Deriving aggregate output from labor demand and supply
hbkn
# Deriving aggregate output from labor demand and supply

(Asked by: username_0 on 2021-03-07)

I was reading the following paper:

[http://eml.berkeley.edu//~moretti/growth.pdf](http://eml.berkeley.edu//%7Emoretti/growth.pdf)

I got stuck at equation (7)

The firm's production function is $Y\_{i}=A\_{i}L\_{i}^{\alpha}K\_{i}^{\eta}T\_{i}^{1-\alpha-\eta}$

Labor supply is $W\_{i}=V\frac{P\_{i}^{\beta}}{Z\_{i}}=V\frac{\bar{P\_{i}}^{\beta}L\_{i}^{\beta \gamma\_{i}}}{Z\_{i}}$

Labor demand is $L\_{i}=(\frac{\alpha^{1-\eta}\eta^{\eta}}{R^{\eta}}\frac{A\_{i}}{W\_{i}^{1-\eta}})^{\frac{1}{1-\alpha-\eta}}T\_{i}$

The paper says that if we impose aggregate labor demand is equal to aggregate labor supply (normalized to one), then the aggregate output $Y=\sum\_{i}Y\_{i}$ is

$Y=(\frac{\eta}{R})^{\frac{\eta}{1-\eta}}[\sum\_{i}(A\_{i}[\frac{\bar{Q}}{Q\_{i}}]^{1-\eta})^{\frac{1}{1-\alpha-\eta}}T\_{i}]^{\frac{1-\alpha-\eta}{1-\eta}}$

This step looks drastic to me. How can the aggregate output be derived from the above conditions?

**Tags:** supply-and-demand, labor-economics, aggregate-supply

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## 1 answers

### Answer by username_1 on 2021-03-07. Score: 7 | Accepted answer

Use $W\_{i}=V \cdot \frac{P\_{i}^{\beta}}{Z\_{i}}=VQ\_i$, then $$L\_{i}=\left(\frac{\alpha^{1-\eta} \eta^{\eta}}{R^{\eta}} \cdot \frac{A\_{i}}{(V\_{i} Q\_{i})^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}} \cdot T\_{i}$$ and $$\sum L\_i = {\left({\frac{\alpha}{V}}\right)}^{\frac{1-\eta}{1-\alpha-\eta}} {\left({\frac{\eta}{R}}\right)}^{\frac{\eta}{1-\alpha-\eta}} \sum\left(\frac{A\_i}{Q\_i^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}}T\_i = 1$$ $$\frac{V}{\alpha} = {\left({\frac{\eta}{R}}\right)}^{\frac{\eta}{1-\eta}} \left(\sum\left(\frac{A\_i}{Q\_i^{1-\eta}}\right)^{\frac{1}{1-\alpha-\eta}}T\_i\right)^{\frac{1-\alpha-\eta}{1-\eta}} $$.

Use the FOC on labor, $W\_i=\alpha \frac{Y\_i}{L\_i}$, then $$\sum Y\_i = \frac{V}{\alpha}\sum L\_iQ\_i = \frac{V}{\alpha} \bar{Q}$$, and replace $\frac{V}{\alpha}$ with above equation then you get equation (7).

> **Comments:**
> **username_0**: Awesome. Thank you very much!!  
> **username_2**: @baek, if you like the answer, consider accepting it!  
> **username_0**: @VARulle I did it. Thanks for the tip!  

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