# Growth Accounting Relations

Growth accounting relations is a quantitative model Robert Solow developed in 1957. It is used to measure the effect of different factors of economic growth. In addition, it indirectly estimates the technological progress in an economy. In other words, it is a production function regarding growth rates.

## The Growth Accounting Equation

This equation is based on the Cobb-Douglas production function, broken down into percentage changes in output factors associated with labor, capital, and technology. It is given by:

$$\frac{\Delta Y}{Y}=\frac{\Delta A}{A}+\alpha\frac{\Delta K}{K}+\left(1-\alpha\right)\frac{\Delta L}{L} \ldots\ldots\ldots (i)$$

From the equation above, the growth accounting equation mentions that:

\begin{align*} \text{The growth rate of output} &=\text{Technological change} \\ & + \alpha(\text{growth rate of capital}) \\ & +(1- \alpha)(\text{Growth rate of labor}) \end{align*}

$$\alpha$$ is the elasticity of our output relative to the capital since a 1% rise in capital causes an $$\alpha$$% rise in output. Similarly, $$(1-\alpha)$$ is the elasticity of output concerning labor. Recall also that $$\alpha$$ and $$(1-\alpha)$$ are the proportion of income paid to each factor. Any other unspecified factor is taken care of by the TFP factor.

### Example: Interpreting Elasticities of Capital and Labor in Growth Accounting Equation

Economic data of a developed country reveals that shares of capital ($$\alpha$$) and labor $$(1-\alpha)$$ are roughly 0.2 and 0.8, respectively. What can be deduced from these results?

This implies that a rise in the labor growth rate will significantly impact potential GDP growth more than the capital growth rate while keeping other factors constant. A 1% increase in capital for each worker raises the output by 0.2%, and an equivalent rise in labor increases production by 0.8%.

## Uses of Growth Rate Equation

1. Estimation of the Contribution of Technological Progress to Economic Growth

Solow approximated TFP from equation (1) by making $$\frac{ {\Delta A}}{A}$$ the subject of the formula, then substituting in the values of $$\frac{{\Delta K}}{K},\frac{{\Delta L}}{L}$$ and $$\alpha$$. TFP represents the quantity of output that growth in capital or labor does not explain.

2. Measurement of Sources of Growth in the Economy

The growth accounting equation is used to measure the impact of each production factor on the economy’s long-term growth.

3. Measurement of Potential GDP

The potential GDP is approximated using equation (1) as a function of growth rates of capital, labor, the TFP (this is residual in the growth accounting equation), and the factor $$\alpha$$.

## The Labor Productivity Growth Accounting Equation

The labor productivity growth accounting equation is a substitute for measuring potential GDP. It possesses the same characteristics as Solow’s approach. However, it is easy and represented as the function of labor input and labor input productivity. Therefore, there is no need to approximate the capital input and total factor productivity (TFP).

However, the major shortcoming involves both capital deepening and TFP progress in the productivity period, making it hard to analyze and anticipate over a long period.

The labor productivity growth accounting equation is given by:

\begin{align*} & \text{The growth rate in Potential GDP} \\ & = \text{Growth rate of the labor force over a long-term period} \\ & + \text{Growth rate of labor productivity over a long-term period.} \end{align*}

## Question

The labor force of country X grows by 2% per year and labor productivity by 4%; the growth rate of the potential GDP per year is closest to:

1. 2%.
2. 4%.
3. 6%.

#### Solution

According to the labor productivity growth accounting equation,

\begin{align*} & \text{The growth rate in Potential GDP} \\ & = \text{Growth rate of the labor force over a long-term period} \\ & + \text{Growth rate of labor productivity over a long-term period.} \\ & = 2\% + 4\% = 6\% \text{ per year} \end{align*}

LOS 9 (e) Demonstrate forecasting potential GDP based on growth accounting relations.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success

Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
Nyka Smith
2021-02-18
Every concept is very well explained by Nilay Arun. kudos to you man!