# 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 Solowâ€™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.

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