<p>I have made some partial progress towards the result, but there is still some discrepancy between what I am getting and the stated result. I've written out my thinking in case it might be helpful to you or others in deriving the correct result.</p>
<p>I will omit time subscripts for ease of reading.</p>
<p>The production function of the economy is <span class="math-container">$F(K,N^e)=h_a^{\gamma}AK^{\beta}(N^e)^{1-\beta}$</span>. So, the real wage  is <span class="math-container">$$w=F_N(K,N^e)=(1-\beta)h_A^{\gamma}AK^{\beta}(N^e)^{-\beta}$$</span> Lucas makes the assumption that all agents are of some fixed skill level <span class="math-container">$h$</span>, so <span class="math-container">$N^e=uhN$</span> and <span class="math-container">$h_a=h$</span>. Therefore, <span class="math-container">$$w=(1-\beta)AK^{\beta}(uN)^{-\beta}h^{-\beta+\gamma}$$</span></p>
<p>We are interested in how wages in the BGP are affected by an increase in capital. That is, we want to know the elasticity of wages as we move along the line of balanced growth paths in <span class="math-container">$(z_1,z_2)$</span> space. To answer this question, we can convert our wage equation into the <span class="math-container">$(z_1,z_2)$</span> space by back-substitution. <span class="math-container">$$w=(1-\beta)A(uN_0)^{-\beta}z_1^{\beta}z_2^{-\beta+\gamma}$$</span> The locus of BGPs is implicitly defined by equation (28), <span class="math-container">$$(\beta A N_0^{1-\beta}u^{1-\beta})z_1^{\beta-1}z_2^{1-\beta+\gamma}=\rho+\sigma \kappa$$</span> Taking logarithms, <span class="math-container">$$\ln z_2 = \frac{1}{1-\beta+\gamma}[\ln (\rho+\sigma \kappa )-\ln (\beta AN_0^{1-\beta}u^{1-\beta})]+\frac{1-\beta}{1-\beta+\gamma}\ln z_1$$</span> Therefore, along this path, the elasticity of <span class="math-container">$z_2$</span> with respect to <span class="math-container">$z_1$</span> is, <span class="math-container">$$\frac{d \ln z_2}{d\ln z_1}=\frac{1-\beta}{1-\beta+\gamma}\in (0,1)$$</span> this being less than one means that when <span class="math-container">$z_1$</span> is plotted on the <span class="math-container">$y$</span>-axis and <span class="math-container">$z_2$</span> is plotted on the <span class="math-container">$x$</span>-axis, as in Fig 1, the BGP locus will be convex.</p>
<p>The logarithm of wages is <span class="math-container">$$\ln w=\ln[(1-\beta)A(uN_0)^{-\beta}] +\beta \ln z_1 + (-\beta+\gamma)\ln z_2$$</span> Therefore, the elasticity of wages with respect to capital along the curve in Fig 1 is, <span class="math-container">$$\frac{d\ln w}{d\ln K}=\beta \frac{d\ln z_1}{d\ln K}+(-\beta+\gamma)\frac{d\ln z_2}{d\ln z_1}\frac{d\ln z_1}{d\ln K}$$</span> simple calculation gives <span class="math-container">$\frac{d\ln z_1}{d\ln K}=1$</span>, so <span class="math-container">\begin{align*}\frac{d\ln w}{d\ln K}&amp;=\beta+(-\beta+\gamma)\frac{1-\beta}{1-\beta+\gamma}=\frac{\gamma}{1-\beta+\gamma}  \end{align*}</span></p>

CC BY-SA 4.0
2024-09-11T06:55:29.753Z
2024-09-11T06:55:29.753Z
Joseph Basford
<p>I'm currently reading one of the most cited papers in Economics, <em>On the Mechanics of Economic Development</em> by Robert Lucas Jr. (1988), published in the <em>Journal of Monetary Economics</em>.</p>
<p>The technology for aggregate output is given by:</p>
<p><span class="math-container">\begin{equation}
N(t) c(t) + \dot{K}(t) = A K(t)^{\beta} \left[u(t)h(t)N(t) \right]^{1-\beta} h(t)^{\gamma},
\end{equation}</span></p>
<p>where <span class="math-container">$N(t)$</span> denotes the population size, which grows at the exogenous rate <span class="math-container">$\lambda$</span>. Here, <span class="math-container">$K(t)$</span> represents the total stock of capital, and <span class="math-container">$\dot{K}(t)$</span> is its rate of change. The term <span class="math-container">$u(t)h(t)N(t)$</span> represents the effective workforce, where <span class="math-container">$h(t)$</span> denotes human capital and <span class="math-container">$u(t)$</span> denotes the time devoted to production. Finally, <span class="math-container">$h(t)^{\gamma}$</span> captures the external effects of human capital.</p>
<p>We have two state variables: <span class="math-container">$K(t)$</span> and <span class="math-container">$h(t)$</span>.</p>
<p>It is assumed that:</p>
<p><span class="math-container">\begin{equation}
\dot{h}(t) = h(t)[1-u(t)]\delta,
\end{equation}</span></p>
<p>where <span class="math-container">$1-u(t)$</span> denotes the effort devoted to the accumulation of human capital.</p>
<p>Let <span class="math-container">$\kappa$</span> denote the growth rate of consumption, i.e., <span class="math-container">$\kappa = \dot{c}(t)/c(t)$</span>. The following condition holds for the marginal product of capital:</p>
<p><span class="math-container">\begin{equation}
\beta A K(t)^{\beta -1} \left[u(t)h(t)N(t) \right]^{1-\beta} h(t)^{\gamma} = \rho + \sigma \kappa,
\end{equation}</span></p>
<p>where <span class="math-container">$\rho$</span> is the discount rate and <span class="math-container">$\sigma$</span> is the inverse of the intertemporal elasticity of substitution.</p>
<p>Now, let <span class="math-container">$\nu = \dot{h}(t)/h(t)$</span>. On a balanced growth path, it is clear that:</p>
<p><span class="math-container">\begin{equation}
\nu = \delta(1-u).
\end{equation}</span></p>
<p>The common growth rate of consumption and per capita capital is then given by:</p>
<p><span class="math-container">\begin{equation}
\kappa = \left(\frac{1-\beta + \gamma}{1-\beta}\right)\nu.
\end{equation}</span></p>
<p>The article states that it is possible to define a curve linking the two normalized variables <span class="math-container">$z_1(t) = e^{-(\kappa + \lambda)t} K(t)$</span> and <span class="math-container">$z_2(t) = e^{-\nu t} h(t)$</span>. Using these two variables in the marginal product of capital equation, we have:</p>
<p><span class="math-container">\begin{equation}
(\beta A N^{1 - \beta}_0 u^{1-\beta}) z^{\beta - 1}_1 z^{1-\beta + \gamma}_2 = \rho + \sigma \kappa.
\end{equation}</span></p>
<p>Finally, for <span class="math-container">$\gamma \geq 0$</span>, the real wage increases as one moves up the curve just defined above. Along this curve, we have the elasticity formula:</p>
<p><span class="math-container">\begin{equation}
\frac{K}{w} \frac{\partial w }{\partial K} = \frac{(1+\beta)\gamma}{1-\beta+\gamma}.
\end{equation}</span></p>
<p>Could you show me how to derive this elasticity?</p>
<p>I attempted the following, but it didn't work:</p>
<p><span class="math-container">\begin{equation}
\partial{Y}/\partial{(uhN)} = w,
\end{equation}</span></p>
<p>and <span class="math-container">$\frac{K}{w} \frac{\partial w }{\partial K}$</span> simply gives me <span class="math-container">$\beta$</span>.</p>

CC BY-SA 4.0
2024-09-03T09:28:25.810Z
Since you are moving along the locus of BGP points in $(z_1,z_2)$ space, you have to consider the total derivative of $w$ with respect to $K$. That is, you have to consider not only the direct effect of changing $K$ (which is $\beta$) but also the effect changing $K$ has on human capital $h$ as you move along the locus and how this alters wages. You should be able to get this indirect effect from the equation $(\beta AN_0^{1-\beta}u^{1-\beta})z_1^{\beta-1}z_2^{1-\beta+\gamma}=\rho +\sigma \kappa$.
Joseph Basford
<macroeconomics><mathematical-economics><economic-growth>
Lucas Model 1988, deriving the elasticity
Maximilian
# Lucas Model 1988, deriving the elasticity

(Asked by: username_0 on 2024-08-28)

I'm currently reading one of the most cited papers in Economics, *On the Mechanics of Economic Development* by Robert Lucas Jr. (1988), published in the *Journal of Monetary Economics*.

The technology for aggregate output is given by:

\begin{equation}
N(t) c(t) + \dot{K}(t) = A K(t)^{\beta} \left[u(t)h(t)N(t) \right]^{1-\beta} h(t)^{\gamma},
\end{equation}

where $N(t)$ denotes the population size, which grows at the exogenous rate $\lambda$. Here, $K(t)$ represents the total stock of capital, and $\dot{K}(t)$ is its rate of change. The term $u(t)h(t)N(t)$ represents the effective workforce, where $h(t)$ denotes human capital and $u(t)$ denotes the time devoted to production. Finally, $h(t)^{\gamma}$ captures the external effects of human capital.

We have two state variables: $K(t)$ and $h(t)$.

It is assumed that:

\begin{equation}
\dot{h}(t) = h(t)[1-u(t)]\delta,
\end{equation}

where $1-u(t)$ denotes the effort devoted to the accumulation of human capital.

Let $\kappa$ denote the growth rate of consumption, i.e., $\kappa = \dot{c}(t)/c(t)$. The following condition holds for the marginal product of capital:

\begin{equation}
\beta A K(t)^{\beta -1} \left[u(t)h(t)N(t) \right]^{1-\beta} h(t)^{\gamma} = \rho + \sigma \kappa,
\end{equation}

where $\rho$ is the discount rate and $\sigma$ is the inverse of the intertemporal elasticity of substitution.

Now, let $\nu = \dot{h}(t)/h(t)$. On a balanced growth path, it is clear that:

\begin{equation}
\nu = \delta(1-u).
\end{equation}

The common growth rate of consumption and per capita capital is then given by:

\begin{equation}
\kappa = \left(\frac{1-\beta + \gamma}{1-\beta}\right)\nu.
\end{equation}

The article states that it is possible to define a curve linking the two normalized variables $z\_1(t) = e^{-(\kappa + \lambda)t} K(t)$ and $z\_2(t) = e^{-\nu t} h(t)$. Using these two variables in the marginal product of capital equation, we have:

\begin{equation}
(\beta A N^{1 - \beta}\_0 u^{1-\beta}) z^{\beta - 1}\_1 z^{1-\beta + \gamma}\_2 = \rho + \sigma \kappa.
\end{equation}

Finally, for $\gamma \geq 0$, the real wage increases as one moves up the curve just defined above. Along this curve, we have the elasticity formula:

\begin{equation}
\frac{K}{w} \frac{\partial w }{\partial K} = \frac{(1+\beta)\gamma}{1-\beta+\gamma}.
\end{equation}

Could you show me how to derive this elasticity?

I attempted the following, but it didn't work:

\begin{equation}
\partial{Y}/\partial{(uhN)} = w,
\end{equation}

and $\frac{K}{w} \frac{\partial w }{\partial K}$ simply gives me $\beta$.

**Tags:** macroeconomics, mathematical-economics, economic-growth

> **Comments:**
> **username_1**: Since you are moving along the locus of BGP points in $(z_1,z_2)$ space, you have to consider the total derivative of $w$ with respect to $K$. That is, you have to consider not only the direct effect of changing $K$ (which is $\beta$) but also the effect changing $K$ has on human capital $h$ as you move along the locus and how this alters wages. You should be able to get this indirect effect from the equation $(\beta AN_0^{1-\beta}u^{1-\beta})z_1^{\beta-1}z_2^{1-\beta+\gamma}=\rho +\sigma \kappa$.  

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## 1 answers

### Answer by username_1 on 2024-09-11. Score: 2 | Accepted answer

I have made some partial progress towards the result, but there is still some discrepancy between what I am getting and the stated result. I've written out my thinking in case it might be helpful to you or others in deriving the correct result.

I will omit time subscripts for ease of reading.

The production function of the economy is $F(K,N^e)=h\_a^{\gamma}AK^{\beta}(N^e)^{1-\beta}$. So, the real wage is $$w=F\_N(K,N^e)=(1-\beta)h\_A^{\gamma}AK^{\beta}(N^e)^{-\beta}$$ Lucas makes the assumption that all agents are of some fixed skill level $h$, so $N^e=uhN$ and $h\_a=h$. Therefore, $$w=(1-\beta)AK^{\beta}(uN)^{-\beta}h^{-\beta+\gamma}$$

We are interested in how wages in the BGP are affected by an increase in capital. That is, we want to know the elasticity of wages as we move along the line of balanced growth paths in $(z\_1,z\_2)$ space. To answer this question, we can convert our wage equation into the $(z\_1,z\_2)$ space by back-substitution. $$w=(1-\beta)A(uN\_0)^{-\beta}z\_1^{\beta}z\_2^{-\beta+\gamma}$$ The locus of BGPs is implicitly defined by equation (28), $$(\beta A N\_0^{1-\beta}u^{1-\beta})z\_1^{\beta-1}z\_2^{1-\beta+\gamma}=\rho+\sigma \kappa$$ Taking logarithms, $$\ln z\_2 = \frac{1}{1-\beta+\gamma}[\ln (\rho+\sigma \kappa )-\ln (\beta AN\_0^{1-\beta}u^{1-\beta})]+\frac{1-\beta}{1-\beta+\gamma}\ln z\_1$$ Therefore, along this path, the elasticity of $z\_2$ with respect to $z\_1$ is, $$\frac{d \ln z\_2}{d\ln z\_1}=\frac{1-\beta}{1-\beta+\gamma}\in (0,1)$$ this being less than one means that when $z\_1$ is plotted on the $y$-axis and $z\_2$ is plotted on the $x$-axis, as in Fig 1, the BGP locus will be convex.

The logarithm of wages is $$\ln w=\ln[(1-\beta)A(uN\_0)^{-\beta}] +\beta \ln z\_1 + (-\beta+\gamma)\ln z\_2$$ Therefore, the elasticity of wages with respect to capital along the curve in Fig 1 is, $$\frac{d\ln w}{d\ln K}=\beta \frac{d\ln z\_1}{d\ln K}+(-\beta+\gamma)\frac{d\ln z\_2}{d\ln z\_1}\frac{d\ln z\_1}{d\ln K}$$ simple calculation gives $\frac{d\ln z\_1}{d\ln K}=1$, so \begin{align\*}\frac{d\ln w}{d\ln K}&=\beta+(-\beta+\gamma)\frac{1-\beta}{1-\beta+\gamma}=\frac{\gamma}{1-\beta+\gamma} \end{align\*}


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