<p>I believe the conceptual problem is that you are considering the share of labour associated with high and low-skilled workers separately. What <span class="math-container">$s_j^L$</span> actually captures is the share of sector <span class="math-container">$j$</span>'s output that is produced by labour, both from high and low-skilled workers. You can see that this is what <span class="math-container">$s_j^L$</span> means from equations (A32)-(A35) in the appendix.</p>
<p>Something that makes the algebraic derivations more difficult is that the stated labour shares correspond to the case when <span class="math-container">$\eta=0$</span>, i.e., when no output is paid to intermediaries. I don't think this is clearly stated in the Appendix. I shall proceed under the general case when <span class="math-container">$\eta \in (0,1)$</span> to make it clear why the stated labour shares only hold for the limiting case of <span class="math-container">$\eta=0$</span>. This <span class="math-container">$\eta=0$</span> condition is also imposed in Acemoglu and Restrepo (2019) from which the result is cited.</p>
<p>By Proposition 1 of the paper, sector <span class="math-container">$j$</span>'s output is produced according to <span class="math-container">$$Y_j=\frac{P_j^{\frac{\eta}{1-\eta}}}{1-\eta}\left[[I_j-N_j+1]^{\frac{1}{\hat{\sigma}}}K_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}+\left(\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right)^{\frac{1}{\hat{\sigma}}}\mathcal{L}_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right]^{\frac{\hat{\sigma}}{\hat{\sigma}-1}}$$</span></p>
<p>From equations (A30) and (A31) of the paper, <span class="math-container">$$\begin{align*}W_j &amp;=\left[\frac{[1-\eta]Y_jP_j^{\sigma}}{\mathcal{L}_j}\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}} \\
R_j &amp;=\left[\frac{[1-\eta]Y_jP_j^{\sigma}}{K_j}[I_j-N_j+1]\right]^{\frac{1}{\hat{\sigma}}} \end{align*}$$</span></p>
<p>Therefore, <span class="math-container">$$\begin{align*}\mathcal{L}_jW_j &amp;=\mathcal{L}_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y_jP_j^{\sigma}\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}} \\
K_jR_j &amp;=K_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y_jP_j^{\sigma}[I_j-N_j+1]\right]^{\frac{1}{\hat{\sigma}}} \end{align*}$$</span></p>
<p>Summing these and using the <span class="math-container">$Y_j$</span> equation, <span class="math-container">$$\begin{align*}\mathcal{L}_j W_j +K_j R_j &amp;=\mathcal{L}_j^{\frac{\sigma-1}{\sigma}}\left[[1-\eta]Y_jP_j^{\sigma}\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}} + K_j^{\frac{\sigma-1}{\sigma}}\left[[1-\eta]Y_jP_j^{\sigma}[I_j-N_j+1]\right]^{\frac{1}{\hat{\sigma}}} \\
&amp;=\left([1-\eta]P_j^{\sigma}\right)^{\frac{1}{\hat{\sigma}}}Y_j^{\frac{1}{\hat{\sigma}}}\left([I_j-N_j+1]^{\frac{1}{\hat{\sigma}}}K_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}+\left(\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right)^{\frac{1}{\hat{\sigma}}}\mathcal{L}_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right) \\
&amp;=\left([1-\eta]P_j^{\sigma}\right)^{\frac{1}{\hat{\sigma}}}Y_j^{\frac{1}{\hat{\sigma}}} \left[\frac{[1-\eta]Y_j}{P_j^{\frac{\eta}{1-\eta}}}\right]^{\frac{\hat{\sigma}-1}{\hat{\sigma}}} \\
&amp;=[1-\eta]P_jY_j \end{align*}$$</span> where the final inequality follows since <span class="math-container">$\frac{\eta}{1-\eta}=\frac{\sigma-\hat{\sigma}}{\hat{\sigma}-1}$</span>.</p>
<p>The labour share of sector <span class="math-container">$j$</span> is, therefore, <span class="math-container">$$s_j^L=\frac{\mathcal{L}_jW_j}{P_jY_j}=[1-\eta]\frac{\mathcal{L}_jW_j}{\mathcal{L}_jW_j+K_jR_j}$$</span>  Plugging in the above expressions: <span class="math-container">$$\begin{align*}s_j^L&amp;=[1-\eta]\frac{\mathcal{L}_jW_j}{\mathcal{L}_jW_j+K_jR_j} \\
&amp;=[1-\eta]\frac{\mathcal{L}_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y_jP_j^{\sigma}\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}}}{\mathcal{L}_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y_jP_j^{\sigma}\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}}+K_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y_jP_j^{\sigma}[I_j-N_j+1]\right]^{\frac{1}{\hat{\sigma}}}}\\
&amp;= [1-\eta]\left[1+\left(\frac{[I_j-N_j+1]}{\int_{I_j}^{N_j}\gamma_j(i)^{\hat{\sigma}-1}di}\right)^{\frac{1}{\hat{\sigma}}}\left(\frac{K_j}{\mathcal{L}_j}\right)^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right]^{-1} \\
&amp;=[1-\eta]\left[1+\left(\frac{1-\hat{\Gamma}_j}{\hat{\Gamma}_j}\right)^{\frac{1}{\hat{\sigma}}}\left(\frac{K_j}{\mathcal{L}_j}\right)^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right]^{-1} \end{align*}$$</span></p>
<p>Where <span class="math-container">$\hat{\Gamma}_j$</span> is the same as <span class="math-container">$\Gamma_j$</span> but with <span class="math-container">$\hat{\sigma}$</span> rather than <span class="math-container">$\sigma$</span>.</p>
<p>Clearly, unless <span class="math-container">$\eta=0$</span>, the stated result will not hold. Note that since <span class="math-container">$\hat{\sigma}=(1-\eta)\sigma+\eta$</span>, when <span class="math-container">$\eta=0$</span> we have <span class="math-container">$\sigma=\hat{\sigma}$</span>.</p>

CC BY-SA 4.0
2024-08-20T14:13:13.317Z
Thank you so much for the effort of going through the article. I appreciate it. If I consider equation (A32), $W_L L_U$, tells us the income of low skilled labor employed in sector $U$. Are you sure about the fact that $S^{L}_U$ denotes the labor share in sector $U$ of the composite labor input, $\mathcal{L}_u$? About the intermediates, I agree with you.
Maximilian
CC BY-SA 4.0
2024-08-20T15:09:02.003Z
@Maximilian To see $s_U^L$ is the labour share in sector $U$ of the composite labour input $\mathcal{L}_U$ you can consider the following. The total revenue of sector $U$ is $P_U Y_U$. From (A32) and (A34), the revenue going to labour is $$W_L L_U+W_H H_U=\alpha^U s_U^L P_U Y_U+(1-\alpha^U)s_U^L P_U Y_U=P_U Y_U s_U^L$$ Therefore, the share of sector $U$ revenue going to labour is $\frac{W_L L_U+W_H H_U}{P_U Y_U}=s_U^L$. Additionally, the stated expression for $s_j^L$ doesn't vary between $L$ and $H$ so cannot express the labour share of low or high-skill workers in sector $j$.
Joseph Basford
CC BY-SA 4.0
2024-08-20T15:24:53.767Z
You're right. Congrats on the well-articulated answer. However, the notation in the paper didn't help.
Maximilian
CC BY-SA 4.0
2024-08-21T07:46:47.000Z
@Maximilian agreed, the notation does make the result quite opaque, especially since it seems to not be used in the rest of the Appendix.
Joseph Basford
CC BY-SA 4.0
2024-08-20T11:54:13.080Z
2024-08-20T11:54:13.080Z
Joseph Basford
<p>I'm studying a recent article published by Autor et al. (QJE, 2024). The model is presented in Appendix, Section J. I'm working on solving the model, but I'm stuck at the final step, particularly with the aggregation. I hope you can help. Below is a brief summary of the model:</p>
<p>There are two sectors, producing skill-intensive and skill-non-intensive goods, <span class="math-container">$Y_s$</span> and <span class="math-container">$Y_u$</span>, respectively. A representative household consumes both commodities according to
<span class="math-container">\begin{equation}
    U(Y_u,Y_s) = Y_u^{\beta} Y_s^{1 - \beta}
\end{equation}</span>
where <span class="math-container">$\beta \in (0,1)$</span>, and <span class="math-container">$p_j$</span> denotes the price of good <span class="math-container">$j \in \{ u, s\}$</span>. Let <span class="math-container">$P$</span> denote the ideal price index; and <span class="math-container">$P_u Y_u + P_s Y_s = P Y$</span>. Let <span class="math-container">$Y$</span> be the numeraire so that <span class="math-container">$P \equiv 1$</span>. Suppose there is no leisure and therefore the labor supply is inelastic.</p>
<p>Each sector produces a unique final output by combining a unit measure of tasks <span class="math-container">$i \in [N_{j-1},N_j]$</span>:</p>
<p><span class="math-container">\begin{equation}
    Y_j = \left[\int_{N_{j-1}}^{N_j} y_j(i)^{\frac{\sigma-1}{\sigma}} \, di \right]^{\frac{\sigma}{\sigma-1}}
\end{equation}</span></p>
<p>where <span class="math-container">$y_j(i)$</span> is the output of task <span class="math-container">$i$</span> in sector <span class="math-container">$j$</span>; <span class="math-container">$\sigma$</span> is the elasticity of substitution between tasks (assumed identical across sectors <span class="math-container">$j \in \{U, S\}$</span>).</p>
<p>Each task is produced by combining a labor composite of high- and low-skill types, <span class="math-container">$n_j(i)$</span>, or capital, <span class="math-container">$k_j(i)$</span>, with a task-specific intermediate <span class="math-container">$q_j(i)$</span>. The production function for task <span class="math-container">$i$</span> is given by:</p>
<p><span class="math-container">\begin{equation}
    y_j(i) =
    \begin{cases} 
    B_j q_j(i)^\eta k_j(i)^{1-\eta} &amp; \text{if } i \in [N_j - 1, I_j] \\
    B_j q_j(i)^\eta [\gamma_j(i) n_j(i)]^{1-\eta} &amp; \text{if } i \in (I_j, N_j]
    \end{cases}
\end{equation}</span></p>
<p>where <span class="math-container">$B_j \equiv \psi_j^\eta [1 - \eta]^{\eta-1} \eta^{-\eta}$</span> for notational convenience; the parameter <span class="math-container">$\eta \in (0, 1)$</span> is the share of output paid to intermediates; <span class="math-container">$\gamma_j(i)$</span> is the productivity of the labor composite <span class="math-container">$n_j(i)$</span> (relative to capital); and <span class="math-container">$I_j$</span> and <span class="math-container">$N_j$</span> are the equilibrium thresholds for automation and new task creation, respectively, meaning tasks from <span class="math-container">$N_j - 1$</span> to <span class="math-container">$I_j$</span> are produced by machines and those from <span class="math-container">$I_j$</span> to <span class="math-container">$N_j$</span> are produced by labor.</p>
<p>The key assumption is: <span class="math-container">$\gamma_j(i)$</span> is strictly increasing in <span class="math-container">$i$</span>, meaning that labor has a comparative advantage in performing higher index (more complex) tasks.</p>
<p>The labor composite <span class="math-container">$ n_j(i) $</span> in each sector is a Cobb-Douglas combination of <span class="math-container">$H $</span> and <span class="math-container">$L$</span> - type labor:
<span class="math-container">\begin{equation}
    n_j(i) = l_j(i)^{\alpha_j} h_j(i)^{1-\alpha_j}.
\end{equation}</span></p>
<p>Let <span class="math-container">$L_U$</span>, <span class="math-container">$L_S$</span>, <span class="math-container">$H_U$</span>, and <span class="math-container">$H_S$</span> be the equilibrium labor allocations to each sector.
<span class="math-container">\begin{equation}
L_U + L_S = L \quad \text{and} \quad H_U + H_S = H.
\end{equation}</span>
The wage index <span class="math-container">$W_j$</span> is equal to
<span class="math-container">\begin{equation}
    W_j \equiv \alpha^{-\alpha_j} (1 - \alpha_j)^{\alpha_j - 1} \cdot W_L^{\alpha_j} \cdot W_H^{1 - \alpha_j}
\end{equation}</span></p>
<p>Finally, capital is sector-specific, with sectoral capital stocks <span class="math-container">$K_U$</span> and <span class="math-container">$K_S$</span> taken as given. The capital rental rate for sector-specific capital is denoted by <span class="math-container">$R_j$</span> for each sector <span class="math-container">$j \in \{U, S\}$</span>.
Since <span class="math-container">$\gamma_j(i)$</span> is strictly increasing in <span class="math-container">$i$</span>, there exists a unique threshold <span class="math-container">$\tilde{I}_j$</span> in each sector such that:
<span class="math-container">\begin{equation}
    \frac{W_j}{R_j} = \gamma_j (\tilde{I}_j)
    \label{threshold}
\end{equation}</span></p>
<p>Tasks are competitively supplied, and the price of task <span class="math-container">$i$</span>, <span class="math-container">$p_j(i)$</span>, is
<span class="math-container">\begin{equation}
p_j(i) =
\begin{cases} 
    R_j^{1-\eta} &amp; \text{if } i \in [N_j - 1, I_j] \\
    \left( \frac{W_j}{\gamma_j(i)} \right)^{1-\eta} &amp; \text{if } i \in (I_j, N_j]
\end{cases}
\label{price_intermediate}
\end{equation}</span></p>
<p>The demand for sector task output, <span class="math-container">$y_j(i) $</span> reads
<span class="math-container">\begin{equation}
    y_j(i) = \left( \frac{P_j}{p_j(i)} \right)^{\sigma} Y_j = \beta_j Y P_j^{\sigma-1} p_j(i)^{-\sigma}
    \label{demand_intermediate1}
\end{equation}</span></p>
<p>The supply of <span class="math-container">$y_j(i)$</span> is a Cobb-Douglas aggregate of labor, capital, and intermediates. The sectoral demands for capital and labor for each task <span class="math-container">$i$</span> are as follows:</p>
<p><span class="math-container">\begin{equation}
k_j(i) =
\begin{cases} 
    [1 - \eta]\beta_j Y P_j^{\sigma-1} R_j^{-\hat{\sigma}} &amp; \text{if } i \in [N_j - 1, I_j] \\
    0 &amp; \text{if } i \in (I_j, N_j]
\end{cases}
\end{equation}</span></p>
<p><span class="math-container">\begin{equation}
n_j(i)=
\begin{cases}
    0 &amp; \text{if } i \in [N_j - 1, I_j] \\
    [1 - \eta]\beta_j Y_j P_j^{\sigma-1} \frac{1}{\gamma_j(i)} \left[ \frac{W_j}{\gamma_j(i)} \right]^{-\hat{\sigma}} &amp; \text{if } i \in (I_j, N_j]
\end{cases}
\end{equation}</span>
with <span class="math-container">$\hat{\sigma} = 1 - (1 - \sigma)(1 - \eta)$</span>.</p>
<p>The capital and labor markets clear in each sector, so that</p>
<p><span class="math-container">\begin{equation}
\int_{N_j - 1}^{I_j} [1 - \eta]\beta_j Y_j P_j^{\sigma - 1} R_j^{-\hat{\sigma}} \, di = K_j 
\label{clear_k}
\end{equation}</span></p>
<p><span class="math-container">\begin{equation}
\int_{I_j}^{N_j} [1 - \eta]\beta_j Y_j P_j^{\sigma - 1} \frac{1}{\gamma_j(i)} \left[\frac{W_j}{\gamma_j(i)}\right]^{-\hat{\sigma}} \, di = \mathcal{L}_j 
\label{clear_l}
\end{equation}</span></p>
<p>where <span class="math-container">$\mathcal{L}_U \equiv L_U^{\alpha_U} H_U^{1-\alpha_U}$</span> and <span class="math-container">$\mathcal{L}_S \equiv (L - L_U)^{\alpha_S} (H - H_U)^{1-\alpha_S}$</span>.</p>
<p>Factor prices satisfy the following ideal price index condition:
<span class="math-container">\begin{equation}
P_j^{1-\sigma} = [I_j - N_j + 1] R_j^{1-\hat{\sigma}} + W_j^{1-\hat{\sigma}} \int_{I_j}^{N_j} \gamma_j(i)^{\hat{\sigma}-1} \, di
\end{equation}</span>
Aggregate output reads:
<span class="math-container">\begin{equation}
(1 - \eta) Y_j = P_j^{\frac{\eta}{1- \eta }} \left[ \left( I_j - N_j + 1 \right)^{\frac{1}{\hat{\sigma}}} \left( K_j \right)^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}} 
+ \left( \mathcal{L}_j \right)^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}} \left( \int_{N_j}^{I_j} \gamma_j(i)^{\hat{\sigma} - 1} \, di \right)^{\frac{1}{\hat{\sigma}}} \right]^{\frac{\hat{\sigma}}{\hat{\sigma} - 1}}
\end{equation}</span>
where <span class="math-container">$\frac{\sigma - \hat{\sigma}}{\hat{\sigma} - 1} = \frac{\eta}{1- \eta }$</span>.</p>
<p>Wage-bills must satisfy:
<span class="math-container">\begin{align}
W_{L}L_U &amp;= \alpha_U s^{L}_{U} P_U Y_U = \alpha_U s^{L}_{U} \beta Y \\
W_{L}L_S &amp;= \alpha_S s^{L}_{S} P_S Y_S = \alpha_S s^{L}_{S} (1- \beta)  Y \\
W_{H}L_U &amp;= (1-\alpha_U) s^{L}_{U} P_U Y_U = (1-\alpha_U) s^{L}_{U} \beta Y \\
W_{H}L_S &amp;= (1-\alpha_S) s^{L}_{S} P_S Y_S = (1-\alpha_S) s^{L}_{S} (1- \beta)  Y
\end{align}</span></p>
<p>What I'm not able to derive are the labor shares. The article derives just one (4 in total) and I'm trying to derive it, i.e., the share of output in sector <span class="math-container">$j$</span> going to unskilled workers (<span class="math-container">$s^{L}_{j}$</span>). Can you show me how to derive it:
The labor share in sector ( j ) is given by:
<span class="math-container">\begin{equation}
s^{L}_{j} = \left[ 1 + \left( \frac{1 - \Gamma_j}{\Gamma_j} \right)^{\frac{1}{\sigma}} \left( \frac{K_j}{ \mathcal{L}_j} \right)^{\frac{\sigma - 1}{\sigma}} \right]^{-1}
\label{l_share}
\end{equation}</span>
with <span class="math-container">$\sigma$</span> the elasticity between tasks in goods production and
<span class="math-container">\begin{equation}
\Gamma_j \equiv \frac{\int_{N_j}^{I_j} \gamma_j(i)^{\sigma - 1} \, di}{\left[I_j - N_j + 1\right]^{\sigma - 1} + \int_{N_j}^{I_j} \gamma_j(i)^{\sigma - 1} \, di}.
\end{equation}</span></p>
<p>My attempt (which does not match the paper's result):
<span class="math-container">\begin{equation}
\frac{\partial Y_j}{\partial L_j} = Y^{\frac{1}{\hat{\sigma}}} \alpha_j L_j^{\alpha_j - 1} H_j^{1 - \alpha_j} \left( \int_{I_j}^{N_j} \gamma(i)^{\hat{\sigma} - 1} \, di \right)^{\frac{1}{\hat{\sigma}}} = W_L.
\end{equation}</span></p>
<p>Then, since <span class="math-container">$\mathcal{L} = L^{\alpha_j}_j H^{1-\alpha_j}_j$</span>,
<span class="math-container">\begin{equation}
s^{L}_j = \frac{W_L L_j}{\alpha_j P_j Y_j} = \frac{\mathcal{L}_j \left( \int_{I_j}^{N_j} \gamma(i)^{\hat{\sigma} - 1} \, di \right)^{\frac{1}{\hat{\sigma}}}}{Y^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}} P_j},
\end{equation}</span>
I eventually get
<span class="math-container">\begin{equation}
s^{L}_j = \frac{1}{\mathcal{L}^{-\frac{1}{\hat{\sigma}}}(1-\eta)^{- \frac{\hat{\sigma} - 1}{\hat{\sigma}}} P_j^{\frac{\hat{\sigma} - 1}{\hat{\sigma} (1-\eta)}} \left [\left( \frac{I_j - N_j + 1}{\int_{I_j}^{N_j} \gamma(i)^{\hat{\sigma} - 1} \, di} \right)^{\frac{1}{\hat{\sigma}}} \left( \frac{K_j}{\mathcal{L}_j} \right)^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}  }+1 \right]}.
\end{equation}</span>
which seems to be not the right one.</p>

<microeconomics><macroeconomics><mathematical-economics>
Aggregation and Labor Shares in a Static Model
Maximilian
# Aggregation and Labor Shares in a Static Model

(Asked by: username_0 on 2024-08-18)

I'm studying a recent article published by Autor et al. (QJE, 2024). The model is presented in Appendix, Section J. I'm working on solving the model, but I'm stuck at the final step, particularly with the aggregation. I hope you can help. Below is a brief summary of the model:

There are two sectors, producing skill-intensive and skill-non-intensive goods, $Y\_s$ and $Y\_u$, respectively. A representative household consumes both commodities according to
\begin{equation}
U(Y\_u,Y\_s) = Y\_u^{\beta} Y\_s^{1 - \beta}
\end{equation}
where $\beta \in (0,1)$, and $p\_j$ denotes the price of good $j \in \{ u, s\}$. Let $P$ denote the ideal price index; and $P\_u Y\_u + P\_s Y\_s = P Y$. Let $Y$ be the numeraire so that $P \equiv 1$. Suppose there is no leisure and therefore the labor supply is inelastic.

Each sector produces a unique final output by combining a unit measure of tasks $i \in [N\_{j-1},N\_j]$:

\begin{equation}
Y\_j = \left[\int\_{N\_{j-1}}^{N\_j} y\_j(i)^{\frac{\sigma-1}{\sigma}} \, di \right]^{\frac{\sigma}{\sigma-1}}
\end{equation}

where $y\_j(i)$ is the output of task $i$ in sector $j$; $\sigma$ is the elasticity of substitution between tasks (assumed identical across sectors $j \in \{U, S\}$).

Each task is produced by combining a labor composite of high- and low-skill types, $n\_j(i)$, or capital, $k\_j(i)$, with a task-specific intermediate $q\_j(i)$. The production function for task $i$ is given by:

\begin{equation}
y\_j(i) =
\begin{cases}
B\_j q\_j(i)^\eta k\_j(i)^{1-\eta} & \text{if } i \in [N\_j - 1, I\_j] \\
B\_j q\_j(i)^\eta [\gamma\_j(i) n\_j(i)]^{1-\eta} & \text{if } i \in (I\_j, N\_j]
\end{cases}
\end{equation}

where $B\_j \equiv \psi\_j^\eta [1 - \eta]^{\eta-1} \eta^{-\eta}$ for notational convenience; the parameter $\eta \in (0, 1)$ is the share of output paid to intermediates; $\gamma\_j(i)$ is the productivity of the labor composite $n\_j(i)$ (relative to capital); and $I\_j$ and $N\_j$ are the equilibrium thresholds for automation and new task creation, respectively, meaning tasks from $N\_j - 1$ to $I\_j$ are produced by machines and those from $I\_j$ to $N\_j$ are produced by labor.

The key assumption is: $\gamma\_j(i)$ is strictly increasing in $i$, meaning that labor has a comparative advantage in performing higher index (more complex) tasks.

The labor composite $ n\_j(i) $ in each sector is a Cobb-Douglas combination of $H $ and $L$ - type labor:
\begin{equation}
n\_j(i) = l\_j(i)^{\alpha\_j} h\_j(i)^{1-\alpha\_j}.
\end{equation}

Let $L\_U$, $L\_S$, $H\_U$, and $H\_S$ be the equilibrium labor allocations to each sector.
\begin{equation}
L\_U + L\_S = L \quad \text{and} \quad H\_U + H\_S = H.
\end{equation}
The wage index $W\_j$ is equal to
\begin{equation}
W\_j \equiv \alpha^{-\alpha\_j} (1 - \alpha\_j)^{\alpha\_j - 1} \cdot W\_L^{\alpha\_j} \cdot W\_H^{1 - \alpha\_j}
\end{equation}

Finally, capital is sector-specific, with sectoral capital stocks $K\_U$ and $K\_S$ taken as given. The capital rental rate for sector-specific capital is denoted by $R\_j$ for each sector $j \in \{U, S\}$.
Since $\gamma\_j(i)$ is strictly increasing in $i$, there exists a unique threshold $\tilde{I}\_j$ in each sector such that:
\begin{equation}
\frac{W\_j}{R\_j} = \gamma\_j (\tilde{I}\_j)
\label{threshold}
\end{equation}

Tasks are competitively supplied, and the price of task $i$, $p\_j(i)$, is
\begin{equation}
p\_j(i) =
\begin{cases}
R\_j^{1-\eta} & \text{if } i \in [N\_j - 1, I\_j] \\
\left( \frac{W\_j}{\gamma\_j(i)} \right)^{1-\eta} & \text{if } i \in (I\_j, N\_j]
\end{cases}
\label{price\_intermediate}
\end{equation}

The demand for sector task output, $y\_j(i) $ reads
\begin{equation}
y\_j(i) = \left( \frac{P\_j}{p\_j(i)} \right)^{\sigma} Y\_j = \beta\_j Y P\_j^{\sigma-1} p\_j(i)^{-\sigma}
\label{demand\_intermediate1}
\end{equation}

The supply of $y\_j(i)$ is a Cobb-Douglas aggregate of labor, capital, and intermediates. The sectoral demands for capital and labor for each task $i$ are as follows:

\begin{equation}
k\_j(i) =
\begin{cases}
[1 - \eta]\beta\_j Y P\_j^{\sigma-1} R\_j^{-\hat{\sigma}} & \text{if } i \in [N\_j - 1, I\_j] \\
0 & \text{if } i \in (I\_j, N\_j]
\end{cases}
\end{equation}

\begin{equation}
n\_j(i)=
\begin{cases}
0 & \text{if } i \in [N\_j - 1, I\_j] \\
[1 - \eta]\beta\_j Y\_j P\_j^{\sigma-1} \frac{1}{\gamma\_j(i)} \left[ \frac{W\_j}{\gamma\_j(i)} \right]^{-\hat{\sigma}} & \text{if } i \in (I\_j, N\_j]
\end{cases}
\end{equation}
with $\hat{\sigma} = 1 - (1 - \sigma)(1 - \eta)$.

The capital and labor markets clear in each sector, so that

\begin{equation}
\int\_{N\_j - 1}^{I\_j} [1 - \eta]\beta\_j Y\_j P\_j^{\sigma - 1} R\_j^{-\hat{\sigma}} \, di = K\_j
\label{clear\_k}
\end{equation}

\begin{equation}
\int\_{I\_j}^{N\_j} [1 - \eta]\beta\_j Y\_j P\_j^{\sigma - 1} \frac{1}{\gamma\_j(i)} \left[\frac{W\_j}{\gamma\_j(i)}\right]^{-\hat{\sigma}} \, di = \mathcal{L}\_j
\label{clear\_l}
\end{equation}

where $\mathcal{L}\_U \equiv L\_U^{\alpha\_U} H\_U^{1-\alpha\_U}$ and $\mathcal{L}\_S \equiv (L - L\_U)^{\alpha\_S} (H - H\_U)^{1-\alpha\_S}$.

Factor prices satisfy the following ideal price index condition:
\begin{equation}
P\_j^{1-\sigma} = [I\_j - N\_j + 1] R\_j^{1-\hat{\sigma}} + W\_j^{1-\hat{\sigma}} \int\_{I\_j}^{N\_j} \gamma\_j(i)^{\hat{\sigma}-1} \, di
\end{equation}
Aggregate output reads:
\begin{equation}
(1 - \eta) Y\_j = P\_j^{\frac{\eta}{1- \eta }} \left[ \left( I\_j - N\_j + 1 \right)^{\frac{1}{\hat{\sigma}}} \left( K\_j \right)^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}}
+ \left( \mathcal{L}\_j \right)^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}} \left( \int\_{N\_j}^{I\_j} \gamma\_j(i)^{\hat{\sigma} - 1} \, di \right)^{\frac{1}{\hat{\sigma}}} \right]^{\frac{\hat{\sigma}}{\hat{\sigma} - 1}}
\end{equation}
where $\frac{\sigma - \hat{\sigma}}{\hat{\sigma} - 1} = \frac{\eta}{1- \eta }$.

Wage-bills must satisfy:
\begin{align}
W\_{L}L\_U &= \alpha\_U s^{L}\_{U} P\_U Y\_U = \alpha\_U s^{L}\_{U} \beta Y \\
W\_{L}L\_S &= \alpha\_S s^{L}\_{S} P\_S Y\_S = \alpha\_S s^{L}\_{S} (1- \beta) Y \\
W\_{H}L\_U &= (1-\alpha\_U) s^{L}\_{U} P\_U Y\_U = (1-\alpha\_U) s^{L}\_{U} \beta Y \\
W\_{H}L\_S &= (1-\alpha\_S) s^{L}\_{S} P\_S Y\_S = (1-\alpha\_S) s^{L}\_{S} (1- \beta) Y
\end{align}

What I'm not able to derive are the labor shares. The article derives just one (4 in total) and I'm trying to derive it, i.e., the share of output in sector $j$ going to unskilled workers ($s^{L}\_{j}$). Can you show me how to derive it:
The labor share in sector ( j ) is given by:
\begin{equation}
s^{L}\_{j} = \left[ 1 + \left( \frac{1 - \Gamma\_j}{\Gamma\_j} \right)^{\frac{1}{\sigma}} \left( \frac{K\_j}{ \mathcal{L}\_j} \right)^{\frac{\sigma - 1}{\sigma}} \right]^{-1}
\label{l\_share}
\end{equation}
with $\sigma$ the elasticity between tasks in goods production and
\begin{equation}
\Gamma\_j \equiv \frac{\int\_{N\_j}^{I\_j} \gamma\_j(i)^{\sigma - 1} \, di}{\left[I\_j - N\_j + 1\right]^{\sigma - 1} + \int\_{N\_j}^{I\_j} \gamma\_j(i)^{\sigma - 1} \, di}.
\end{equation}

My attempt (which does not match the paper's result):
\begin{equation}
\frac{\partial Y\_j}{\partial L\_j} = Y^{\frac{1}{\hat{\sigma}}} \alpha\_j L\_j^{\alpha\_j - 1} H\_j^{1 - \alpha\_j} \left( \int\_{I\_j}^{N\_j} \gamma(i)^{\hat{\sigma} - 1} \, di \right)^{\frac{1}{\hat{\sigma}}} = W\_L.
\end{equation}

Then, since $\mathcal{L} = L^{\alpha\_j}\_j H^{1-\alpha\_j}\_j$,
\begin{equation}
s^{L}\_j = \frac{W\_L L\_j}{\alpha\_j P\_j Y\_j} = \frac{\mathcal{L}\_j \left( \int\_{I\_j}^{N\_j} \gamma(i)^{\hat{\sigma} - 1} \, di \right)^{\frac{1}{\hat{\sigma}}}}{Y^{\frac{\hat{\sigma} - 1}{\hat{\sigma}}} P\_j},
\end{equation}
I eventually get
\begin{equation}
s^{L}\_j = \frac{1}{\mathcal{L}^{-\frac{1}{\hat{\sigma}}}(1-\eta)^{- \frac{\hat{\sigma} - 1}{\hat{\sigma}}} P\_j^{\frac{\hat{\sigma} - 1}{\hat{\sigma} (1-\eta)}} \left [\left( \frac{I\_j - N\_j + 1}{\int\_{I\_j}^{N\_j} \gamma(i)^{\hat{\sigma} - 1} \, di} \right)^{\frac{1}{\hat{\sigma}}} \left( \frac{K\_j}{\mathcal{L}\_j} \right)^{\frac{\hat{\sigma} - 1}{\hat{\sigma}} }+1 \right]}.
\end{equation}
which seems to be not the right one.

**Tags:** microeconomics, macroeconomics, mathematical-economics

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## 1 answers

### Answer by username_1 on 2024-08-20. Score: 1 | Accepted answer

I believe the conceptual problem is that you are considering the share of labour associated with high and low-skilled workers separately. What $s\_j^L$ actually captures is the share of sector $j$'s output that is produced by labour, both from high and low-skilled workers. You can see that this is what $s\_j^L$ means from equations (A32)-(A35) in the appendix.

Something that makes the algebraic derivations more difficult is that the stated labour shares correspond to the case when $\eta=0$, i.e., when no output is paid to intermediaries. I don't think this is clearly stated in the Appendix. I shall proceed under the general case when $\eta \in (0,1)$ to make it clear why the stated labour shares only hold for the limiting case of $\eta=0$. This $\eta=0$ condition is also imposed in Acemoglu and Restrepo (2019) from which the result is cited.

By Proposition 1 of the paper, sector $j$'s output is produced according to $$Y\_j=\frac{P\_j^{\frac{\eta}{1-\eta}}}{1-\eta}\left[[I\_j-N\_j+1]^{\frac{1}{\hat{\sigma}}}K\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}+\left(\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right)^{\frac{1}{\hat{\sigma}}}\mathcal{L}\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right]^{\frac{\hat{\sigma}}{\hat{\sigma}-1}}$$

From equations (A30) and (A31) of the paper, $$\begin{align\*}W\_j &=\left[\frac{[1-\eta]Y\_jP\_j^{\sigma}}{\mathcal{L}\_j}\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}} \\
R\_j &=\left[\frac{[1-\eta]Y\_jP\_j^{\sigma}}{K\_j}[I\_j-N\_j+1]\right]^{\frac{1}{\hat{\sigma}}} \end{align\*}$$

Therefore, $$\begin{align\*}\mathcal{L}\_jW\_j &=\mathcal{L}\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y\_jP\_j^{\sigma}\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}} \\
K\_jR\_j &=K\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y\_jP\_j^{\sigma}[I\_j-N\_j+1]\right]^{\frac{1}{\hat{\sigma}}} \end{align\*}$$

Summing these and using the $Y\_j$ equation, $$\begin{align\*}\mathcal{L}\_j W\_j +K\_j R\_j &=\mathcal{L}\_j^{\frac{\sigma-1}{\sigma}}\left[[1-\eta]Y\_jP\_j^{\sigma}\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}} + K\_j^{\frac{\sigma-1}{\sigma}}\left[[1-\eta]Y\_jP\_j^{\sigma}[I\_j-N\_j+1]\right]^{\frac{1}{\hat{\sigma}}} \\
&=\left([1-\eta]P\_j^{\sigma}\right)^{\frac{1}{\hat{\sigma}}}Y\_j^{\frac{1}{\hat{\sigma}}}\left([I\_j-N\_j+1]^{\frac{1}{\hat{\sigma}}}K\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}+\left(\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right)^{\frac{1}{\hat{\sigma}}}\mathcal{L}\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right) \\
&=\left([1-\eta]P\_j^{\sigma}\right)^{\frac{1}{\hat{\sigma}}}Y\_j^{\frac{1}{\hat{\sigma}}} \left[\frac{[1-\eta]Y\_j}{P\_j^{\frac{\eta}{1-\eta}}}\right]^{\frac{\hat{\sigma}-1}{\hat{\sigma}}} \\
&=[1-\eta]P\_jY\_j \end{align\*}$$ where the final inequality follows since $\frac{\eta}{1-\eta}=\frac{\sigma-\hat{\sigma}}{\hat{\sigma}-1}$.

The labour share of sector $j$ is, therefore, $$s\_j^L=\frac{\mathcal{L}\_jW\_j}{P\_jY\_j}=[1-\eta]\frac{\mathcal{L}\_jW\_j}{\mathcal{L}\_jW\_j+K\_jR\_j}$$ Plugging in the above expressions: $$\begin{align\*}s\_j^L&=[1-\eta]\frac{\mathcal{L}\_jW\_j}{\mathcal{L}\_jW\_j+K\_jR\_j} \\
&=[1-\eta]\frac{\mathcal{L}\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y\_jP\_j^{\sigma}\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}}}{\mathcal{L}\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y\_jP\_j^{\sigma}\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di\right]^{\frac{1}{\hat{\sigma}}}+K\_j^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\left[[1-\eta]Y\_jP\_j^{\sigma}[I\_j-N\_j+1]\right]^{\frac{1}{\hat{\sigma}}}}\\
&= [1-\eta]\left[1+\left(\frac{[I\_j-N\_j+1]}{\int\_{I\_j}^{N\_j}\gamma\_j(i)^{\hat{\sigma}-1}di}\right)^{\frac{1}{\hat{\sigma}}}\left(\frac{K\_j}{\mathcal{L}\_j}\right)^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right]^{-1} \\
&=[1-\eta]\left[1+\left(\frac{1-\hat{\Gamma}\_j}{\hat{\Gamma}\_j}\right)^{\frac{1}{\hat{\sigma}}}\left(\frac{K\_j}{\mathcal{L}\_j}\right)^{\frac{\hat{\sigma}-1}{\hat{\sigma}}}\right]^{-1} \end{align\*}$$

Where $\hat{\Gamma}\_j$ is the same as $\Gamma\_j$ but with $\hat{\sigma}$ rather than $\sigma$.

Clearly, unless $\eta=0$, the stated result will not hold. Note that since $\hat{\sigma}=(1-\eta)\sigma+\eta$, when $\eta=0$ we have $\sigma=\hat{\sigma}$.

> **Comments:**
> **username_0**: Thank you so much for the effort of going through the article. I appreciate it. If I consider equation (A32), $W_L L_U$, tells us the income of low skilled labor employed in sector $U$. Are you sure about the fact that $S^{L}_U$ denotes the labor share in sector $U$ of the composite labor input, $\mathcal{L}_u$? About the intermediates, I agree with you.  
> **username_1**: @Maximilian To see $s_U^L$ is the labour share in sector $U$ of the composite labour input $\mathcal{L}_U$ you can consider the following. The total revenue of sector $U$ is $P_U Y_U$. From (A32) and (A34), the revenue going to labour is $$W_L L_U+W_H H_U=\alpha^U s_U^L P_U Y_U+(1-\alpha^U)s_U^L P_U Y_U=P_U Y_U s_U^L$$ Therefore, the share of sector $U$ revenue going to labour is $\frac{W_L L_U+W_H H_U}{P_U Y_U}=s_U^L$. Additionally, the stated expression for $s_j^L$ doesn't vary between $L$ and $H$ so cannot express the labour share of low or high-skill workers in sector $j$.  
> **username_0**: You're right. Congrats on the well-articulated answer. However, the notation in the paper didn't help.  
> **username_1**: @Maximilian agreed, the notation does make the result quite opaque, especially since it seems to not be used in the rest of the Appendix.  

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