<p>As indicated in my comment above, for <span class="math-container">$\psi=1$</span>, your model is nested in the one of Mankiw/Romer/Weil (1992). You can rewrite production as</p>
<p><span class="math-container">$$
Y = A{K^\alpha }{H^\beta } = {K^\alpha }{H^\beta }{\left( {\tilde AL} \right)^{1 - \alpha  - \beta }}
$$</span></p>
<p>with <span class="math-container">$\tilde A= A^{\frac{1}{1-\alpha-\beta}}$</span> denoting labour-augmenting technological progress and <span class="math-container">$L=1$</span>. This allows you to divide by <span class="math-container">$\tilde AL$</span> to get your system in intensive form:</p>
<p><span class="math-container">$$
\begin{align}
  y &amp;= \left( k \right)^\alpha \left( h \right)^\beta \\
  \dot k &amp;= {s_K}y - \left( {\delta  + g} \right)k \\ 
  \dot h &amp;= {s_H}y - \left( {\delta  + g} \right)h \\ 
\end{align}
$$</span></p>
<p>In steady state with <span class="math-container">$\dot k = \dot h =0$</span>, this gives you immediately that the ratios of the capital stocks to output are constant.</p>
<p>But with <span class="math-container">$\psi&lt;1$</span>, there will be a term depending on technology left in the steady state equation for human capital</p>
<p><span class="math-container">$$
0 = {s_h}y^\psi{\left( {\tilde AL} \right)^{\psi -1}} - \left( {\delta  + g} \right)h
$$</span>
which is a contradiction. So from what I can see, there is no BGP in this case.</p>
<p><strong>Addendum</strong>:
To clarify the definition, Acemoglu (2008)'s textbook definition is</p>
<blockquote>
<p>By balanced growth, we mean a path of the economy consistent with the Kaldor facts (Kaldor, 1963), that is, a path where, while output per capita
increases, the capital-output ratio, the interest rate, and the
distribution of income between capital and labor remain roughly
constant.</p>
</blockquote>
<p>That would not be the case here with respect to human capital.</p>
<p>If you don't care about a balance between factor to output ratios, i.e. a steady state in some appropriately normalized form, then there is at least a steady growth path in this model with
<span class="math-container">$$
{g_Y} = g + \alpha {g_K} + \beta {g_H} = \left( {1 + \alpha } \right)g + \beta {g_H} = \left( {1 + \alpha } \right)g + \beta \psi {g_Y} = \left( {1 + \alpha  + \beta \psi } \right)g
$$</span></p>
<p>Income shares with competitive markets where factors are paid their marginal product, in contrast, will be constant due to the Cobb-Douglas assumption. E.g.:
<span class="math-container">$$
\begin{align}
  {F_K} &amp;= \alpha A{K^{\alpha  - 1}}{H^\beta } = \alpha \frac{Y}{K} \\
  \frac{{{F_K}K}}{Y} &amp;= \alpha  \\ 
\end{align} 
$$</span>
But due to non-CRS you need to assume that the rest of output <span class="math-container">$1-\alpha-\beta$</span> goes to the fixed factor labor <span class="math-container">$L=1$</span></p>

CC BY-SA 4.0
2022-03-11T13:14:46.900Z
What do you mean "a contradiction"? The transform variables of $k$, $h$ could be different when $\psi<1$.
Alalalalaki
CC BY-SA 4.0
2022-03-11T13:18:05.053Z
Following the derivation in the question, we have $g_Y=g_K$, $(1-\psi)g_Y = g_H$, and thus $g_Y = g + \alpha g_Y + \beta (1-\psi)g_Y$. Why isn't this a BGP?
Alalalalaki
CC BY-SA 4.0
2022-03-11T13:31:27.407Z
1. Commonly, a BGP also involves the ratio of capital stocks to output to remain constant. Of course you may have a different definition, but there is no mentioning in the OP. Here, you would have human capital growing slower than physical capital in each period, so the ratio of K/H would go to infinity. 2. If you impose that $\dot h=0$, but something on the right still grows, this is a contradiction.
jpfeifer
CC BY-SA 4.0
2022-03-11T18:02:42.050Z
Thanks! I will recheck about definition of BGP buy the approach still give me ideas about approaching these problems.
Dayne
CC BY-SA 4.0
2022-03-13T12:21:24.137Z
@jpfeifer: I have added the clarification regarding the BGP definition. I have also added my approach but I still have a doubt about how to check whether BGP exists.
Dayne
CC BY-SA 4.0
2022-03-13T12:28:26.660Z
@Dayne What you now define is arguably not a BGP, but a steady state equilibrium where adjustment dynamics have ceased and growth rates do not change anymore.  The "Balanced" in BGP typically indicates that growth rates of relevant variables have a particular relation so that great ratios like capital to output ratios or investment to output ratios stay constant. That will not be the case in your model. As mentioned above, K/H will not be stable over time, but rather go to infinity. That is not "balanced".
jpfeifer
CC BY-SA 4.0
2022-03-13T12:42:20.413Z
@jpfeifer: if capital to output ratios are ever going to be constant then it would mean same growth rates. If K/H has to stable over time then their growth rates necessary have to be equal isn't it? It appears that in this definition of BGP, growth rates have to be same. Isn't it? I did read somewhere that in BGP the share of contribution to output of factors is stable but that would be different from what you are suggesting.
Dayne
CC BY-SA 4.0
2022-03-13T13:04:51.883Z
I added a textbook definition to clarify the issue. In the end, you need to decide what you want. Steady growth or additionally some kind of balance.
jpfeifer
CC BY-SA 4.0
2022-03-13T14:07:07.687Z
@jpfeifer: Actually I also read Acemoglu' lecture notes. He doesn't seem to be suggesting that the ratios such as K/H are constant. Rather just that their shares in income are stable. That would require for example $F_K K/Y$ is constant.
Dayne
CC BY-SA 4.0
2022-03-13T14:08:55.650Z
In any case I think I understand why your approach is important. In my approach I can probably find the steady state growth rates but not the steady state k* and h*, which is also part of the BGP. So need to employ your method also to complete the answer I guess
Dayne
CC BY-SA 4.0
2022-03-13T15:13:11.690Z
@Dayne In his textbook definition, he explicitly talks about capital to output ratios. Income shares will indeed be constant.
jpfeifer
CC BY-SA 4.0
2022-03-13T17:03:58.000Z
@jpfeifer: okay. So you mean H/Y must also be constant as per the standard BGP definition. But then the equivalent definition is that all the endogenous variables grow at same rate. That's quite a strong condition!
Dayne
CC BY-SA 4.0
2022-03-14T09:34:30.430Z
Yes, it is quite strong. That's the reason why there are generalizations like https://www.jstor.org/stable/2695912
jpfeifer
CC BY-SA 4.0
2022-03-15T08:57:32.553Z
@jpfeifer: thanks will look into the link...looks like something I was looking for. But to solve the immediate question, is my solution correct as per the mentioned definition? Also I have posted another question on similar lines. Interestingly in this new question the definition was not given and perhaps what you mentioned is required. Requesting to please have a look: https://economics.stackexchange.com/q/50749/24967
Dayne
CC BY-SA 4.0
2022-03-15T10:10:35.170Z
@jpfeifer: also see these two links: (1) https://economics.stackexchange.com/questions/10115/solow-model-steady-state-v-balanced-growth-path?rq=1; (2) https://economics.stackexchange.com/questions/12645/growth-rate-of-variables-on-a-balanced-growth-path-bgp?rq=1. In the second answer the definition is as per you mentioned in the point 3. So that should make your answer correct. It would be great if others confirm so I can award bounty to you and accept the answer.
Dayne
CC BY-SA 4.0
2022-03-15T12:07:08.880Z
@Dayne I don't think there is a precise accepted definition of a BGP in the literature with respect to your model. Your model satisfies the Kaldor facts in that physical capital to GDP is indeed constant. It is human capital to GDP that is not constant. Is that required in the definition of a BGP? People can reasonably disagree.
jpfeifer
CC BY-SA 4.0
2022-03-11T12:35:12.927Z
2022-03-13T15:18:29.613Z
2022-03-13T15:18:29.613Z
jpfeifer
<p>This is a self study question. I am novice at this and have only basic knowledge of solving such problems.</p>
<p><span class="math-container">\begin{align}
Y &amp;= AK^\alpha H^\beta \\
\dot{K} &amp;= s_KY - \delta K \\
\dot{H} &amp;= s_HY^\psi - \delta H \\
\end{align}</span></p>
<p>Given, <span class="math-container">$\alpha + \beta &lt;1$</span> (DRS) and <span class="math-container">$\psi \in (0,1]$</span>. Also <span class="math-container">$A(t) = A(0)e^{gt}$</span></p>
<p>Objective is to find Balanced Growth path (BGP).</p>
<p>Now I read that from <a href="https://en.wikipedia.org/wiki/Uzawa%27s_theorem" rel="nofollow noreferrer">Uzawa's theorem</a>, growth rates of all will be equal in BGP. But the theorem is applicable for only CRS. Since Uzawa is not applicable I am unable to design a good effective capital per labor variable to put <span class="math-container">$\dot{k}=0$</span>.</p>
<p>By using that growth rates are constant I could only reach till:</p>
<p><span class="math-container">\begin{align}
(s_k Y/K)/(s_HY^\psi/H) &amp;= constant \\
Y^{1-\psi}H/K &amp;= constant \\
\implies (1-\psi)g_Y + g_H - g_K &amp;= 0
\end{align}</span></p>
<p>Second equation comes from production function: <span class="math-container">$g_Y = g + \alpha g_K + \beta g_H$</span>.</p>
<p>Still leaves me with two equations and three variables. Not sure how to go ahead or if this is useful at all.</p>
<hr />
<p><strong>EDIT</strong>:
A small clarification: BGP is defined in question as the state when all factors grow at constant rate.</p>
<p>Now based on comments, the second and third equation can be used to get: <span class="math-container">$g_Y = g_K$</span> and <span class="math-container">$g_H = \psi g_Y$</span>. Using these two I can quickly get the BGP path. Now the doubt is whether this approach is right in general? Because I have directly assumed here that BGP exists, i.e., a stable steady path exists. Is this correct??</p>

CC BY-SA 4.0
2022-03-11T07:55:56.407Z
Your are missing some optimality condition for the factor inputs, i.e. the relative use of the two capital stocks.
jpfeifer
CC BY-SA 4.0
2022-03-11T08:49:46.653Z
Do you mean that as a hint or that the question is incomplete?
Dayne
CC BY-SA 4.0
2022-03-11T09:09:29.590Z
As a hint. Usually, in equilibrium you balance the factor inputs for each unit to have the same net return. That may give you the missing equations/restrictions. Alternatively, try following the logic of Mankiw/Romer/Weil (1992) at https://eml.berkeley.edu/~dromer/papers/MRW_QJE1992.pdf, Section II.A: Rewrite production as $Y=K^\alpha H^\beta (\tilde AL)^{1-\alpha-\beta}$ with $\tilde A=A^{\frac{1}{1-\alpha-\beta}}$ and $L=1$. At least for $\psi=1$ your model is nested in theirs
jpfeifer
CC BY-SA 4.0
2022-03-11T12:18:19.673Z
Why do you have $\psi$? The human capital investment is not the same good as consumption? Then what is the level of consumption in your model?
Alalalalaki
CC BY-SA 4.0
2022-03-11T12:20:46.773Z
Why write $\left(s_{k} Y / K\right) /\left(s_{H} Y^{\psi} / H\right)$ in a combination? If you do it separately, you will find $g_Y = g_K$.
Alalalalaki
CC BY-SA 4.0
2022-03-11T17:57:57.830Z
@Alalalalaki: i really feel stupid that I didn't see this. I think it ks straight forward to see that $g_y = g_K$ and $g_H = \psi g_Y$. Thanks for this. I didn't fully understand your second last comment though. Also post your comment as answer so I can accept and others can also confirm that it is correct.
Dayne
CC BY-SA 4.0
2022-03-11T17:59:15.257Z
@jpfeifer: thanks for your comments. The reference seems like a good source for novice like me. Although I prefer Alalalalaki's solution as it seems rather simple.
Dayne
<macroeconomics><economic-growth><self-study><solow><endogenous-growth>
Solow Model with DRS and Human Capital
Dayne
# Solow Model with DRS and Human Capital

(Asked by: username_0 on 2022-03-09)

This is a self study question. I am novice at this and have only basic knowledge of solving such problems.

\begin{align}
Y &= AK^\alpha H^\beta \\
\dot{K} &= s\_KY - \delta K \\
\dot{H} &= s\_HY^\psi - \delta H \\
\end{align}

Given, $\alpha + \beta <1$ (DRS) and $\psi \in (0,1]$. Also $A(t) = A(0)e^{gt}$

Objective is to find Balanced Growth path (BGP).

Now I read that from [Uzawa's theorem](https://en.wikipedia.org/wiki/Uzawa%27s_theorem), growth rates of all will be equal in BGP. But the theorem is applicable for only CRS. Since Uzawa is not applicable I am unable to design a good effective capital per labor variable to put $\dot{k}=0$.

By using that growth rates are constant I could only reach till:

\begin{align}
(s\_k Y/K)/(s\_HY^\psi/H) &= constant \\
Y^{1-\psi}H/K &= constant \\
\implies (1-\psi)g\_Y + g\_H - g\_K &= 0
\end{align}

Second equation comes from production function: $g\_Y = g + \alpha g\_K + \beta g\_H$.

Still leaves me with two equations and three variables. Not sure how to go ahead or if this is useful at all.

---

**EDIT**:
A small clarification: BGP is defined in question as the state when all factors grow at constant rate.

Now based on comments, the second and third equation can be used to get: $g\_Y = g\_K$ and $g\_H = \psi g\_Y$. Using these two I can quickly get the BGP path. Now the doubt is whether this approach is right in general? Because I have directly assumed here that BGP exists, i.e., a stable steady path exists. Is this correct??

**Tags:** macroeconomics, economic-growth, self-study, solow, endogenous-growth
> **Comments:**
> **username_1**: Your are missing some optimality condition for the factor inputs, i.e. the relative use of the two capital stocks.  
> **username_0**: Do you mean that as a hint or that the question is incomplete?  
> **username_1**: As a hint. Usually, in equilibrium you balance the factor inputs for each unit to have the same net return. That may give you the missing equations/restrictions. Alternatively, try following the logic of Mankiw/Romer/Weil (1992) at https://eml.berkeley.edu/~dromer/papers/MRW_QJE1992.pdf, Section II.A: Rewrite production as $Y=K^\alpha H^\beta (\tilde AL)^{1-\alpha-\beta}$ with $\tilde A=A^{\frac{1}{1-\alpha-\beta}}$ and $L=1$. At least for $\psi=1$ your model is nested in theirs  
> **username_2**: Why do you have $\psi$? The human capital investment is not the same good as consumption? Then what is the level of consumption in your model?  
> **username_2**: Why write $\left(s_{k} Y / K\right) /\left(s_{H} Y^{\psi} / H\right)$ in a combination? If you do it separately, you will find $g_Y = g_K$.  
> **username_0**: @Alalalalaki: i really feel stupid that I didn't see this. I think it ks straight forward to see that $g_y = g_K$ and $g_H = \psi g_Y$. Thanks for this. I didn't fully understand your second last comment though. Also post your comment as answer so I can accept and others can also confirm that it is correct.  
> **username_0**: @jpfeifer: thanks for your comments. The reference seems like a good source for novice like me. Although I prefer Alalalalaki's solution as it seems rather simple.  

---

## 1 answers

### Answer by username_1 on 2022-03-11. Score: 1

As indicated in my comment above, for $\psi=1$, your model is nested in the one of Mankiw/Romer/Weil (1992). You can rewrite production as

$$
Y = A{K^\alpha }{H^\beta } = {K^\alpha }{H^\beta }{\left( {\tilde AL} \right)^{1 - \alpha - \beta }}
$$

with $\tilde A= A^{\frac{1}{1-\alpha-\beta}}$ denoting labour-augmenting technological progress and $L=1$. This allows you to divide by $\tilde AL$ to get your system in intensive form:

$$
\begin{align}
y &= \left( k \right)^\alpha \left( h \right)^\beta \\
\dot k &= {s\_K}y - \left( {\delta + g} \right)k \\
\dot h &= {s\_H}y - \left( {\delta + g} \right)h \\
\end{align}
$$

In steady state with $\dot k = \dot h =0$, this gives you immediately that the ratios of the capital stocks to output are constant.

But with $\psi<1$, there will be a term depending on technology left in the steady state equation for human capital

$$
0 = {s\_h}y^\psi{\left( {\tilde AL} \right)^{\psi -1}} - \left( {\delta + g} \right)h
$$
which is a contradiction. So from what I can see, there is no BGP in this case.

**Addendum**:
To clarify the definition, Acemoglu (2008)'s textbook definition is

> By balanced growth, we mean a path of the economy consistent with the Kaldor facts (Kaldor, 1963), that is, a path where, while output per capita
> increases, the capital-output ratio, the interest rate, and the
> distribution of income between capital and labor remain roughly
> constant.

That would not be the case here with respect to human capital.

If you don't care about a balance between factor to output ratios, i.e. a steady state in some appropriately normalized form, then there is at least a steady growth path in this model with
$$
{g\_Y} = g + \alpha {g\_K} + \beta {g\_H} = \left( {1 + \alpha } \right)g + \beta {g\_H} = \left( {1 + \alpha } \right)g + \beta \psi {g\_Y} = \left( {1 + \alpha + \beta \psi } \right)g
$$

Income shares with competitive markets where factors are paid their marginal product, in contrast, will be constant due to the Cobb-Douglas assumption. E.g.:
$$
\begin{align}
{F\_K} &= \alpha A{K^{\alpha - 1}}{H^\beta } = \alpha \frac{Y}{K} \\
\frac{{{F\_K}K}}{Y} &= \alpha \\
\end{align}
$$
But due to non-CRS you need to assume that the rest of output $1-\alpha-\beta$ goes to the fixed factor labor $L=1$

> **Comments:**
> **username_2**: What do you mean "a contradiction"? The transform variables of $k$, $h$ could be different when $\psi<1$.  
> **username_2**: Following the derivation in the question, we have $g_Y=g_K$, $(1-\psi)g_Y = g_H$, and thus $g_Y = g + \alpha g_Y + \beta (1-\psi)g_Y$. Why isn't this a BGP?  
> **username_1**: 1. Commonly, a BGP also involves the ratio of capital stocks to output to remain constant. Of course you may have a different definition, but there is no mentioning in the OP. Here, you would have human capital growing slower than physical capital in each period, so the ratio of K/H would go to infinity. 2. If you impose that $\dot h=0$, but something on the right still grows, this is a contradiction.  
> **username_0**: Thanks! I will recheck about definition of BGP buy the approach still give me ideas about approaching these problems.  
> **username_0**: @jpfeifer: I have added the clarification regarding the BGP definition. I have also added my approach but I still have a doubt about how to check whether BGP exists.  
> **username_1**: @Dayne What you now define is arguably not a BGP, but a steady state equilibrium where adjustment dynamics have ceased and growth rates do not change anymore.  The "Balanced" in BGP typically indicates that growth rates of relevant variables have a particular relation so that great ratios like capital to output ratios or investment to output ratios stay constant. That will not be the case in your model. As mentioned above, K/H will not be stable over time, but rather go to infinity. That is not "balanced".  
> **username_0**: @jpfeifer: if capital to output ratios are ever going to be constant then it would mean same growth rates. If K/H has to stable over time then their growth rates necessary have to be equal isn't it? It appears that in this definition of BGP, growth rates have to be same. Isn't it? I did read somewhere that in BGP the share of contribution to output of factors is stable but that would be different from what you are suggesting.  
> **username_1**: I added a textbook definition to clarify the issue. In the end, you need to decide what you want. Steady growth or additionally some kind of balance.  
> **username_0**: @jpfeifer: Actually I also read Acemoglu' lecture notes. He doesn't seem to be suggesting that the ratios such as K/H are constant. Rather just that their shares in income are stable. That would require for example $F_K K/Y$ is constant.  
> **username_0**: In any case I think I understand why your approach is important. In my approach I can probably find the steady state growth rates but not the steady state k* and h*, which is also part of the BGP. So need to employ your method also to complete the answer I guess  
> **username_1**: @Dayne In his textbook definition, he explicitly talks about capital to output ratios. Income shares will indeed be constant.  
> **username_0**: @jpfeifer: okay. So you mean H/Y must also be constant as per the standard BGP definition. But then the equivalent definition is that all the endogenous variables grow at same rate. That's quite a strong condition!  
> **username_1**: Yes, it is quite strong. That's the reason why there are generalizations like https://www.jstor.org/stable/2695912  
> **username_0**: @jpfeifer: thanks will look into the link...looks like something I was looking for. But to solve the immediate question, is my solution correct as per the mentioned definition? Also I have posted another question on similar lines. Interestingly in this new question the definition was not given and perhaps what you mentioned is required. Requesting to please have a look: https://economics.stackexchange.com/q/50749/24967  
> **username_0**: @jpfeifer: also see these two links: (1) https://economics.stackexchange.com/questions/10115/solow-model-steady-state-v-balanced-growth-path?rq=1; (2) https://economics.stackexchange.com/questions/12645/growth-rate-of-variables-on-a-balanced-growth-path-bgp?rq=1. In the second answer the definition is as per you mentioned in the point 3. So that should make your answer correct. It would be great if others confirm so I can award bounty to you and accept the answer.  
> **username_1**: @Dayne I don't think there is a precise accepted definition of a BGP in the literature with respect to your model. Your model satisfies the Kaldor facts in that physical capital to GDP is indeed constant. It is human capital to GDP that is not constant. Is that required in the definition of a BGP? People can reasonably disagree.  

---

