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I was reading the following paper:

http://eml.berkeley.edu//~moretti/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?

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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).

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  • $\begingroup$ Awesome. Thank you very much!! $\endgroup$ – hbkn Mar 8 at 4:05
  • $\begingroup$ @baek, if you like the answer, consider accepting it! $\endgroup$ – VARulle Mar 8 at 9:33
  • $\begingroup$ @VARulle I did it. Thanks for the tip! $\endgroup$ – hbkn Mar 8 at 15:37

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