<p>Given</p>
<p><span class="math-container">$$Y = Af(K,L,Z)$$</span></p>
<p>it follows that</p>
<p><span class="math-container">$$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$</span></p>
<p>where dotted expression are time derivatives and dividing with <span class="math-container">$Y$</span> it follows</p>
<p><span class="math-container">$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{1}{Y}\right] \dot L + \left[A \frac{\partial f}{\partial Z} \frac{1}{Y}\right]\dot Z ,$$</span></p>
<p>and dividing and multiplying with <span class="math-container">$L$</span> and <span class="math-container">$Z$</span> in relevant summands it follows that</p>
<p><span class="math-container">$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{L}{Y}\right] \frac{\dot L}{L} + \left[A \frac{\partial f}{\partial Z} \frac{Z}{Y}\right]\frac{\dot Z}{Z} ,$$</span></p>
<p>and then terms in square brackets are defined as <span class="math-container">$\alpha_1,\alpha_2$</span> and <span class="math-container">$\alpha_3$</span> being respectively</p>
<ol>
<li>The marginal productivity of capital <span class="math-container">$\alpha_1 = A \frac{\partial f}{\partial K}$</span></li>
<li>The elasticity of output with respect to labor <span class="math-container">$\alpha_2 = A \frac{\partial f}{\partial L} \frac{L}{Y}$</span> and</li>
<li>The elasticity of output with respect to other factors  <span class="math-container">$\alpha_3 = A \frac{\partial f}{\partial Z} \frac{Z}{Y}$</span></li>
</ol>
<p>In the text the authors write <span class="math-container">$\partial Y/\partial X$</span> with <span class="math-container">$X\in \{K,L,Z\}$</span> instead of
<span class="math-container">$\partial f/\partial X$</span> which must be a typo. For example it is hard to convince yourself that with <span class="math-container">$Y = A f(K,L,Z)$</span> the marginal productivity of capital is <span class="math-container">$A \frac{\partial Y}{\partial K}$</span> and not instead <span class="math-container">$\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$</span>.</p>
<p>Obviously the typo is of no consequence since the terms are parameterized for the purpose of estimation.</p>

CC BY-SA 4.0
2021-03-16T16:53:50.067Z
Thank you, along with @callculus you both help me a lot.
John M. Riveros
CC BY-SA 4.0
2021-03-16T06:33:07.030Z
2021-03-16T06:33:07.030Z
Jesper Hybel
<p>I'm struggling to understand how Khan &amp; Reinhart (1990) go from the next production function.</p>
<p><span class="math-container">$$y=A f(K,L,Z)$$</span></p>
<p>Where <span class="math-container">$y$</span> is the production of the economy, <span class="math-container">$A$</span> is a variable which contains the technological change, <span class="math-container">$K$</span> is the capital stock of the economy, <span class="math-container">$L$</span> is the labor (work) of the economy, <span class="math-container">$Z$</span> is a vector of variables containing usual covariates.</p>
<p>Into the following result as an expression of growth-rates.</p>
<p><span class="math-container">$$\frac{dy}{y} = [A \frac{\partial y}{\partial K}]\frac{dK}{y} + [A \frac{\partial y}{\partial L}  \frac{L}{y}]\frac{dL}{L} + [A \frac{\partial y}{\partial Z}\frac{Z}{y}]\frac{dZ}{Z} + \frac{dA}{A}$$</span></p>
<p>The information is that <span class="math-container">$y,K,L,Z$</span> are variables (redundant I know but I want to make it clear), however, in the article, they mention <span class="math-container">$A$</span> as assumed to grow an exogenous rate (I assume that <span class="math-container">$A$</span> is also taken as a variable in the first equation). They don't make clear anymore transformations, simply &quot;growth terms&quot;.</p>
<p>The last expression is mentioned by the authors to be &quot;growth terms&quot; of <span class="math-container">$y=Af(K,L,Z)$</span>. but I lost it that way. Usually for expressing growth rates I simply use logarithms and differentiate over time. but this approach is not clear to me.</p>
<p>My intuition is that <span class="math-container">$y$</span> was calculated with the total differential of the form of
<span class="math-container">$$dy=\frac{\partial y}{\partial K}dK + \frac{\partial y}{\partial L}dL + \frac{\partial y}{\partial A}dA$$</span></p>
<p>and then it was divided by <span class="math-container">$y$</span>. However, I never saw the total differentials while I was studying, so I don't know if <span class="math-container">$A$</span> is multiplying <span class="math-container">$f$</span> and we may need to proceed with the product rule of the derivatives? or what is actually the process to get the result of <span class="math-container">$dy/y$</span> as they get.</p>
<p>Reference:
Khan, M. S. &amp; Reinhart, C. M. (1990) Private investment and economic growth in developing countries World Development, Volume 18, Issue 1, January 1990, Pages 19-27, DOI: <a href="https://doi.org/10.1016/0305-750X(90)90100-C" rel="noreferrer">https://doi.org/10.1016/0305-750X(90)90100-C</a></p>

CC BY-SA 4.0
2021-03-16T06:11:52.857Z
I think it is just a typo. All $\partial y/\partial x$ with $x \in \{K,L,Z\}$ should be $\partial f/\partial x$ otherwise the definitions given in the text of $\alpha_1,\alpha_2$ and $\alpha_3$ are also wrong. For example they say $\alpha_1$ is the marginal productivity of capital and then state $\alpha_1 = A \frac{\partial y}{\partial K}$ but clearly when $Y = A f(K,L,Z)$ the marginal productivity of capital is $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$.
Jesper Hybel
<macroeconomics><mathematical-economics><economic-growth><production-function><empirical-methods>
How to get this Production Function in Growth Rates
John M. Riveros
# How to get this Production Function in Growth Rates

(Asked by: username_0 on 2021-03-15)

I'm struggling to understand how Khan & Reinhart (1990) go from the next production function.

$$y=A f(K,L,Z)$$

Where $y$ is the production of the economy, $A$ is a variable which contains the technological change, $K$ is the capital stock of the economy, $L$ is the labor (work) of the economy, $Z$ is a vector of variables containing usual covariates.

Into the following result as an expression of growth-rates.

$$\frac{dy}{y} = [A \frac{\partial y}{\partial K}]\frac{dK}{y} + [A \frac{\partial y}{\partial L} \frac{L}{y}]\frac{dL}{L} + [A \frac{\partial y}{\partial Z}\frac{Z}{y}]\frac{dZ}{Z} + \frac{dA}{A}$$

The information is that $y,K,L,Z$ are variables (redundant I know but I want to make it clear), however, in the article, they mention $A$ as assumed to grow an exogenous rate (I assume that $A$ is also taken as a variable in the first equation). They don't make clear anymore transformations, simply "growth terms".

The last expression is mentioned by the authors to be "growth terms" of $y=Af(K,L,Z)$. but I lost it that way. Usually for expressing growth rates I simply use logarithms and differentiate over time. but this approach is not clear to me.

My intuition is that $y$ was calculated with the total differential of the form of
$$dy=\frac{\partial y}{\partial K}dK + \frac{\partial y}{\partial L}dL + \frac{\partial y}{\partial A}dA$$

and then it was divided by $y$. However, I never saw the total differentials while I was studying, so I don't know if $A$ is multiplying $f$ and we may need to proceed with the product rule of the derivatives? or what is actually the process to get the result of $dy/y$ as they get.

Reference:
Khan, M. S. & Reinhart, C. M. (1990) Private investment and economic growth in developing countries World Development, Volume 18, Issue 1, January 1990, Pages 19-27, DOI: <https://doi.org/10.1016/0305-750X(90)90100-C>

**Tags:** macroeconomics, mathematical-economics, economic-growth, production-function, empirical-methods

> **Comments:**
> **username_1**: I think it is just a typo. All $\partial y/\partial x$ with $x \in \{K,L,Z\}$ should be $\partial f/\partial x$ otherwise the definitions given in the text of $\alpha_1,\alpha_2$ and $\alpha_3$ are also wrong. For example they say $\alpha_1$ is the marginal productivity of capital and then state $\alpha_1 = A \frac{\partial y}{\partial K}$ but clearly when $Y = A f(K,L,Z)$ the marginal productivity of capital is $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$.  

---

## 1 answers

### Answer by username_1 on 2021-03-16. Score: 5 | Accepted answer

Given

$$Y = Af(K,L,Z)$$

it follows that

$$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$

where dotted expression are time derivatives and dividing with $Y$ it follows

$$\frac{\dot Y}{Y} = \frac{\dot A}{A} + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{1}{Y}\right] \dot L + \left[A \frac{\partial f}{\partial Z} \frac{1}{Y}\right]\dot Z ,$$

and dividing and multiplying with $L$ and $Z$ in relevant summands it follows that

$$\frac{\dot Y}{Y} = \frac{\dot A}{A} + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{L}{Y}\right] \frac{\dot L}{L} + \left[A \frac{\partial f}{\partial Z} \frac{Z}{Y}\right]\frac{\dot Z}{Z} ,$$

and then terms in square brackets are defined as $\alpha\_1,\alpha\_2$ and $\alpha\_3$ being respectively

1. The marginal productivity of capital $\alpha\_1 = A \frac{\partial f}{\partial K}$
2. The elasticity of output with respect to labor $\alpha\_2 = A \frac{\partial f}{\partial L} \frac{L}{Y}$ and
3. The elasticity of output with respect to other factors $\alpha\_3 = A \frac{\partial f}{\partial Z} \frac{Z}{Y}$

In the text the authors write $\partial Y/\partial X$ with $X\in \{K,L,Z\}$ instead of
$\partial f/\partial X$ which must be a typo. For example it is hard to convince yourself that with $Y = A f(K,L,Z)$ the marginal productivity of capital is $A \frac{\partial Y}{\partial K}$ and not instead $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$.

Obviously the typo is of no consequence since the terms are parameterized for the purpose of estimation.

> **Comments:**
> **username_0**: Thank you, along with @callculus you both help me a lot.  

---

