# How to get this Production Function in Growth Rates

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

• 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 $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$. Mar 16 at 6:11

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.

• Thank you, along with @callculus you both help me a lot. Mar 16 at 16:53