<p>I dont want to be rude but the only equation you copied correctly is the
productivity augmented Cobb-Douglas production function.</p>
<p>Equation 2 is equation 2 from Ottaviano, Peri (2008) (on page 8) it says:</p>
<p><span class="math-container">$$L=\left(\sum_{e=1}^{n} \theta_{e}L_{e}^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}$$</span>.</p>
<p>In the remaining equations <span class="math-container">$\theta$</span> is not under the exponent as well:</p>
<p><span class="math-container">$$L_{e}=\left(\sum_{\substack{1 \leq e \leq n\\1 \leq x \leq m}} \theta_{ex} L_{ex} ^{\frac{\eta - 1}{\eta}}\right)^{\frac{\eta}{\eta - 1}}$$</span></p>
<p>Is the equivalent of Ottaviano, Peri (2008) equation 3 on page 9 and</p>
<p><span class="math-container">$$\left(\theta_{exb} L_{exb}^{\frac{\sigma - 1}{\sigma}} + \theta_{exw} L_{exw} ^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}
$$</span></p>
<p>Is the equivalent of Ottaviano, Peri (2008) equation 4 on page 9 and</p>
<p>the expression for the logarithm of the wage is equation 2 on page 11 of Boustan(2008):</p>
<p><span class="math-container">$$ln w_{exr}=ln(A^{\frac{1}{\alpha}}k^{\frac{(1-\alpha)}{\alpha}})+\frac{1}{\delta}\cdot ln(L)+ln\theta_{e}-(\frac{1}{\delta}-\frac{1}{\eta})\cdot ln(L_{e})+ln\theta_{ex}-(\frac{1}{\eta}-\frac{1}{\sigma})\cdot ln(L_{ex})+ln\theta_{e_{exr}}-\frac{1}{\sigma}\cdot ln(L_{exr})$$</span></p>
<p>Whith the correct expressions its all boring algebra:</p>
<p>Fn. 29 on page 11 says:</p>
<blockquote>
<p>Following Ottaviano and Peri (2006), I first express output as a function of the capital-output ratio (κ = K/Y)</p>
</blockquote>
<p>Thus we have:</p>
<p><span class="math-container">$Y=AK^{1-\alpha}L^{\alpha} \Leftrightarrow Y=A\big(\frac{K}{Y}\big)^{1-\alpha}L^{\alpha}Y^{1-\alpha} \Leftrightarrow \\Y^{\alpha}=Ak^{1-\alpha}L^{\alpha}\Leftrightarrow Y=A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}L$</span></p>
<p>Observe that:</p>
<p><span class="math-container">$$W = x^{\frac{\zeta}{\zeta-1}} \Leftrightarrow \frac{\partial W}{\partial L_{exr}} = \frac{\zeta}{\zeta-1}x^{\frac{1}{\zeta-1}}\frac{\partial x}{\partial L_{exr}} =\frac{\zeta}{\zeta-1}W^{\frac{1}{\zeta}}\frac{\partial x}{\partial L_{exr}} (*)$$</span></p>
<p><span class="math-container">$\omega_{exr} = \frac{\partial Y}{\partial L_{exr}} = A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}\big(\sum_{e=1}^n\theta_eL_e^{\frac{\delta-1}{\delta}}\big)^{\frac{1}{\delta-1}}\theta_eL_e^{\frac{-1}{\delta}}\frac{\partial L_e}{\partial L_{exr}} \overset{(*)}= A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}\theta_e\frac{\partial L_e}{\partial L_{exr}}L^{\frac{1}{\delta}}L_e^{\frac{-1}{\delta}}(1)$</span></p>
<p><span class="math-container">$\frac{\partial L_e}{\partial L_{exr}} \overset{(*)}= \theta_{ex}\frac{\partial L_{ex}}{\partial L_{exr}}L_e^{\frac{1}{\eta}}L_{ex}^{\frac{-1}{\eta}} (2)$</span></p>
<p><span class="math-container">$\frac{\partial L_{ex}}{\partial L_{exr}} \overset{(*)}= \theta_{exr}L_{ex}^{\frac{1}{\sigma}}L_{exr}^{\frac{-1}{\sigma}} (3)$</span></p>
<p>Plug back (2) and (3) into (1), take logs and you will obtain the result.</p>
<p><em><strong>NOTE</strong></em> I dont quite understand why the derivative of <span class="math-container">$k = \frac{K}{Y}$</span> wrt L is not taken here.</p>

CC BY-SA 4.0
2021-02-28T16:45:38.770Z
The first equation in the solution shouldn't be $$\frac{\partial Y}{\partial L_{exr}} = A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}\big(\sum_{e=1}^n\theta_eL_e^{\frac{\delta-1}{\delta}}\big)^{\frac{1}{\delta-1}}\theta_eL_e^{\frac{-1}{\delta}}\frac{\partial L_e}{\partial L_{exr}} \overset{(*)}=A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}} \theta_e\frac{\partial L_e}{\partial L_{exr}}L^{\frac{1}{\delta}}L_e^{\frac{-1}{\delta}}$$?
Lucas
CC BY-SA 4.0
2021-02-28T16:48:47.253Z
@Lucas ah yes, copy-paste
Grada Gukovic
CC BY-SA 4.0
2021-02-28T12:56:01.877Z
2021-02-28T16:50:31.850Z
2021-02-28T16:50:31.850Z
Grada Gukovic
<p>I'm reading &quot;Competition in the Promised Land – Black Migrants in Northern Cities and Labor Markets&quot;, by Leah Boustan, and I'm trying to understand her computation of black and white elasticity of substitution in labor market</p>
<p>She uses a Cobb-Douglas function:</p>
<p><span class="math-container">$$Y=AL^{\alpha}K^{1-\alpha}$$</span></p>
<p>and then defines aggregate labor as:</p>
<p><span class="math-container">$$L=\left(\sum_{e=1}^{n} \left(L_{e} \theta_{e}\right)^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}$$</span></p>
<p>aggregate labor by educational group as:</p>
<p><span class="math-container">$$L_{e}=\left(\sum_{\substack{1 \leq e \leq n\\1 \leq x \leq m}} \left(L_{ex} \theta_{ex}\right)^{\frac{\eta - 1}{\eta}}\right)^{\frac{\eta}{\eta - 1}}$$</span></p>
<p>and aggregate labor by experience and educational group by:</p>
<p><span class="math-container">$$\left(\left(L_{exb} \theta_{exb}\right)^{\frac{\sigma - 1}{\sigma}} + \left(L_{exw} \theta_{exw}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}
$$</span></p>
<p>Then she derivates the production function relative to <span class="math-container">$L_{exr}$</span> (where <span class="math-container">$r=w,b$</span>) and obtains:</p>
<p><span class="math-container">$$ln w_{exr}=ln(A^{\frac{1}{\alpha}}k^{\frac{(1-\alpha)}{\alpha}})+\frac{1}{\delta}\cdot ln(L)+ln\theta_{e}-(\frac{1}{\delta}-\frac{1}{\eta})\cdot ln(L_{e})+ln\theta_{ex}-(\frac{1}{\eta}-\frac{1}{\delta})\cdot ln(L_{ex})+ln\theta_{e_{exr}}-\frac{1}{\delta}\cdot ln(L_{exr})$$</span></p>
<p>I cannot figure out how to reach this expression. I thought what I need is just 1) to substitute the L terms backwardly, 2) take the derivative, and 3) apply log. The substitution gives:</p>
<p><span class="math-container">$$A K^{1 - \alpha} \left(\left(\sum_{e=1}^{n} \left(\theta_{e} \left(\sum_{\substack{1 \leq e \leq n\\1 \leq x \leq m}} \left(\theta_{ex} \left(\left(L_{exb} \theta_{exb}\right)^{\frac{\sigma - 1}{\sigma}} + \left(L_{exw} \theta_{exw}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}\right)^{\frac{\eta - 1}{\eta}}\right)^{\frac{\eta}{\eta - 1}}\right)^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}\right)^{\alpha}
$$</span></p>
<p>But the derivative of this expression relative, for example, to <span class="math-container">$L_{exb}$</span> is a equation very different from the expression showed by the author (even after the log).</p>
<p>I must be making a very silly mistake here, but how can I get the desired expression from the equations of L? Is there a textbook where I can find a discusson about elasticity of substitution between demographic groups?</p>
<p><strong>EDIT:</strong>  paper version of this part of the book can be found <a href="https://www.nber.org/system/files/working_papers/w13813/w13813.pdf" rel="nofollow noreferrer">here</a>. In my research I noted that Boustan's work is based on <a href="https://www.nber.org/papers/w12497" rel="nofollow noreferrer">Ottaviano</a> (2006), and <a href="https://www.nber.org/papers/w7655" rel="nofollow noreferrer">Card and Lemieux</a> (2001).</p>
<p><a href="https://github.com/lucascr91/playground/blob/master/labor_elasticity/es_groups.ipynb" rel="nofollow noreferrer">Here</a> is a jupyter notebook with the equations</p>

CC BY-SA 4.0
2021-02-27T17:30:53.263Z
Are you sure that the two first inner powers are $(\sigma-1)/\sigma$ and $1/\sigma$? In most cases aggregate labour inputs are homogeneous of degree one in the elementary labour inputs.
Bertrand
CC BY-SA 4.0
2021-02-27T17:41:46.143Z
You are right. I will edit the question to correct theses expoents.
Lucas
<microeconomics><labor-economics><elasticity-of-substitution>
How to compute elasticity of substitution in labor market of blacks and whites using experience-education groups?
Lucas
# How to compute elasticity of substitution in labor market of blacks and whites using experience-education groups?

(Asked by: username_0 on 2021-02-25)

I'm reading "Competition in the Promised Land – Black Migrants in Northern Cities and Labor Markets", by Leah Boustan, and I'm trying to understand her computation of black and white elasticity of substitution in labor market

She uses a Cobb-Douglas function:

$$Y=AL^{\alpha}K^{1-\alpha}$$

and then defines aggregate labor as:

$$L=\left(\sum\_{e=1}^{n} \left(L\_{e} \theta\_{e}\right)^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}$$

aggregate labor by educational group as:

$$L\_{e}=\left(\sum\_{\substack{1 \leq e \leq n\\1 \leq x \leq m}} \left(L\_{ex} \theta\_{ex}\right)^{\frac{\eta - 1}{\eta}}\right)^{\frac{\eta}{\eta - 1}}$$

and aggregate labor by experience and educational group by:

$$\left(\left(L\_{exb} \theta\_{exb}\right)^{\frac{\sigma - 1}{\sigma}} + \left(L\_{exw} \theta\_{exw}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}
$$

Then she derivates the production function relative to $L\_{exr}$ (where $r=w,b$) and obtains:

$$ln w\_{exr}=ln(A^{\frac{1}{\alpha}}k^{\frac{(1-\alpha)}{\alpha}})+\frac{1}{\delta}\cdot ln(L)+ln\theta\_{e}-(\frac{1}{\delta}-\frac{1}{\eta})\cdot ln(L\_{e})+ln\theta\_{ex}-(\frac{1}{\eta}-\frac{1}{\delta})\cdot ln(L\_{ex})+ln\theta\_{e\_{exr}}-\frac{1}{\delta}\cdot ln(L\_{exr})$$

I cannot figure out how to reach this expression. I thought what I need is just 1) to substitute the L terms backwardly, 2) take the derivative, and 3) apply log. The substitution gives:

$$A K^{1 - \alpha} \left(\left(\sum\_{e=1}^{n} \left(\theta\_{e} \left(\sum\_{\substack{1 \leq e \leq n\\1 \leq x \leq m}} \left(\theta\_{ex} \left(\left(L\_{exb} \theta\_{exb}\right)^{\frac{\sigma - 1}{\sigma}} + \left(L\_{exw} \theta\_{exw}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}\right)^{\frac{\eta - 1}{\eta}}\right)^{\frac{\eta}{\eta - 1}}\right)^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}\right)^{\alpha}
$$

But the derivative of this expression relative, for example, to $L\_{exb}$ is a equation very different from the expression showed by the author (even after the log).

I must be making a very silly mistake here, but how can I get the desired expression from the equations of L? Is there a textbook where I can find a discusson about elasticity of substitution between demographic groups?

**EDIT:** paper version of this part of the book can be found [here](https://www.nber.org/system/files/working_papers/w13813/w13813.pdf). In my research I noted that Boustan's work is based on [Ottaviano](https://www.nber.org/papers/w12497) (2006), and [Card and Lemieux](https://www.nber.org/papers/w7655) (2001).

[Here](https://github.com/lucascr91/playground/blob/master/labor_elasticity/es_groups.ipynb) is a jupyter notebook with the equations

**Tags:** microeconomics, labor-economics, elasticity-of-substitution

> **Comments:**
> **username_1**: Are you sure that the two first inner powers are $(\sigma-1)/\sigma$ and $1/\sigma$? In most cases aggregate labour inputs are homogeneous of degree one in the elementary labour inputs.  
> **username_0**: You are right. I will edit the question to correct theses expoents.  

---

## 1 answers

### Answer by username_2 on 2021-02-28. Score: 3 | Accepted answer

I dont want to be rude but the only equation you copied correctly is the
productivity augmented Cobb-Douglas production function.

Equation 2 is equation 2 from Ottaviano, Peri (2008) (on page 8) it says:

$$L=\left(\sum\_{e=1}^{n} \theta\_{e}L\_{e}^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}$$.

In the remaining equations $\theta$ is not under the exponent as well:

$$L\_{e}=\left(\sum\_{\substack{1 \leq e \leq n\\1 \leq x \leq m}} \theta\_{ex} L\_{ex} ^{\frac{\eta - 1}{\eta}}\right)^{\frac{\eta}{\eta - 1}}$$

Is the equivalent of Ottaviano, Peri (2008) equation 3 on page 9 and

$$\left(\theta\_{exb} L\_{exb}^{\frac{\sigma - 1}{\sigma}} + \theta\_{exw} L\_{exw} ^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}
$$

Is the equivalent of Ottaviano, Peri (2008) equation 4 on page 9 and

the expression for the logarithm of the wage is equation 2 on page 11 of Boustan(2008):

$$ln w\_{exr}=ln(A^{\frac{1}{\alpha}}k^{\frac{(1-\alpha)}{\alpha}})+\frac{1}{\delta}\cdot ln(L)+ln\theta\_{e}-(\frac{1}{\delta}-\frac{1}{\eta})\cdot ln(L\_{e})+ln\theta\_{ex}-(\frac{1}{\eta}-\frac{1}{\sigma})\cdot ln(L\_{ex})+ln\theta\_{e\_{exr}}-\frac{1}{\sigma}\cdot ln(L\_{exr})$$

Whith the correct expressions its all boring algebra:

Fn. 29 on page 11 says:

> Following Ottaviano and Peri (2006), I first express output as a function of the capital-output ratio (κ = K/Y)

Thus we have:

$Y=AK^{1-\alpha}L^{\alpha} \Leftrightarrow Y=A\big(\frac{K}{Y}\big)^{1-\alpha}L^{\alpha}Y^{1-\alpha} \Leftrightarrow \\Y^{\alpha}=Ak^{1-\alpha}L^{\alpha}\Leftrightarrow Y=A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}L$

Observe that:

$$W = x^{\frac{\zeta}{\zeta-1}} \Leftrightarrow \frac{\partial W}{\partial L\_{exr}} = \frac{\zeta}{\zeta-1}x^{\frac{1}{\zeta-1}}\frac{\partial x}{\partial L\_{exr}} =\frac{\zeta}{\zeta-1}W^{\frac{1}{\zeta}}\frac{\partial x}{\partial L\_{exr}} (\*)$$

$\omega\_{exr} = \frac{\partial Y}{\partial L\_{exr}} = A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}\big(\sum\_{e=1}^n\theta\_eL\_e^{\frac{\delta-1}{\delta}}\big)^{\frac{1}{\delta-1}}\theta\_eL\_e^{\frac{-1}{\delta}}\frac{\partial L\_e}{\partial L\_{exr}} \overset{(\*)}= A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}\theta\_e\frac{\partial L\_e}{\partial L\_{exr}}L^{\frac{1}{\delta}}L\_e^{\frac{-1}{\delta}}(1)$

$\frac{\partial L\_e}{\partial L\_{exr}} \overset{(\*)}= \theta\_{ex}\frac{\partial L\_{ex}}{\partial L\_{exr}}L\_e^{\frac{1}{\eta}}L\_{ex}^{\frac{-1}{\eta}} (2)$

$\frac{\partial L\_{ex}}{\partial L\_{exr}} \overset{(\*)}= \theta\_{exr}L\_{ex}^{\frac{1}{\sigma}}L\_{exr}^{\frac{-1}{\sigma}} (3)$

Plug back (2) and (3) into (1), take logs and you will obtain the result.

***NOTE*** I dont quite understand why the derivative of $k = \frac{K}{Y}$ wrt L is not taken here.

> **Comments:**
> **username_0**: The first equation in the solution shouldn't be $$\frac{\partial Y}{\partial L_{exr}} = A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}}\big(\sum_{e=1}^n\theta_eL_e^{\frac{\delta-1}{\delta}}\big)^{\frac{1}{\delta-1}}\theta_eL_e^{\frac{-1}{\delta}}\frac{\partial L_e}{\partial L_{exr}} \overset{(*)}=A^{\frac{1}{\alpha}}k^{\frac{1-\alpha}{\alpha}} \theta_e\frac{\partial L_e}{\partial L_{exr}}L^{\frac{1}{\delta}}L_e^{\frac{-1}{\delta}}$$?  
> **username_2**: @username_0 ah yes, copy-paste  

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