Understanding Growth Theories with Application in Nepal's Case

Growth theory is central to macroeconomics that aims to explain the long-term growth path of economies. It delineates the factors that contribute to the sustained growth of an economy and delves into the mechanisms that drive increases in output, productivity, and living standards over extended periods. Over the years, growth theory has evolved significantly with several key developments and notable contributions from eminent economists.

Economists are keen on exploring the growth path of the economy since antiquity. The roots of growth theory can be traced back to classical economists such as Adam Smith, David Ricardo, and Thomas Malthus, who discussed the role of factors like labor, capital, and population growth in economic progress. However, the formalization of growth theory began in the mid-20th century after two eminent economists, Harrod and Domar, independently ushered the pioneering theory to elucidate the dynamics of economic growth.

In this article, we will be exclusively expounding two prominent growth models, namely, Harrod and Domar model and the Solow-Swam model or simply the Solow model. These growth theories are rudimentary in their superficial analysis; however, turn up involved involving intricate economic phenomena. Harrod's and Domar's growth model, aforementioned, are the pioneering growth model to delineate the long-run path of an economy. Many textbooks prefer to explain the growth theory combinedly as the Harrod-Domar growth model due to their similarity in multiple aspects. However, this article will deal with them separately or we will first deal with Harrod's growth model and tersely allude the Domar's growth model. Subsequently, we explore Solow-Swan or Solow growth model. Eventually, the article concludes with a comparison between the Harrod-Domar growth model and the Solow growth model. 

Harrod's growth model has been named after an eminent economist Sir Roy Harrod. Harrod's growth model presents the delicate relationship between economic growth, the efficiency of capital (or capital-output ratio), and savings (or investment). Harrod's growth model relies on some of the promising assumptions such as constant capital-output ratio, constant savings ratio, fixity in a ratio of inputs, and so on. Adhering to these assumptions, the fundamental equation of the Harrod growth model is obtained as:

`\dotY = \frac{\Delta Y}{Y} = \frac{\text{Saving ratio (s)}}{\text{Capital Output Ratio (v)}}---(i)`

Using the fundamental equation in equation (i), the economic growth for an economy can be estimated.

Let's use the data of the National Statistics Office (NSO) to estimate Nepal's economic growth. For this purpose, we will use Gross National Saving (GNS) to GDP ratio and Capital-output ratio. The GNS to GDP ratio is about 31 percent in FY2022/23. The capital-output ratio of 4.9 has been borrowed from the 15th plan of Nepal. Plugging these values in equation (i) yields the estimated growth.

`\dotY=\frac{s}{v}=\frac{31}{4.9}=6.32`

Given the savings ratio of 31 percent and capital-output ratio of 4.9, the estimated economic growth rate is 6.32 percent. The estimated economic growth and actual growth rate may have significant departure due to limitations in data while estimating economic growth. For example, Harrod growth model is oversimplified with some fragile assumptions, such as constant capital-output ratio, constant savings ratio, Leontief-type production function and so on. These assumptions are stepping stones, yet are stumbling blocks for convergence. Let's do a few mathematical exercises to precisely derive the long-run growth path of the economy following the Harrod growth model. For this, we will differentiate equation (i) with respect to time.

`\frac{\DeltaY}{Y} \times \frac{1}{\Delta t}=\frac{s}{v}`

`\frac{\DeltaY}{Y}=(\frac{s}{v})\Delta t---(ii)`

Integrating equation (ii), we get

`\int_{}^{} \frac{\Delta Y}{Y}=\int_{}^{} \frac{s}{v} \Delta t`

`lnY_t = \frac{s}{v}t+c` 

`Y_t = e^{\frac{s}{v}t+c}`

`Y_t = e^{\frac{s}{v}t+ e^c`

`Y_t = Y_0 e^{\frac{s}{v}t ---(iii)`

Now, let's ponder on the growth path of an economic following equation (iii). Saving ratio (s) and capital output ratio (v) are positive and constant, and time (t) grows consecutively starting from 0. Then, `Y_t` must indubitably grow exponentially exhibiting an explosive path.

Figure 1: Long-run growth path of economy based on Harrod model

The Harrod growth model explicates the long-run growth model but with a pitfall of non-convergence. Why Harrod's model does not yield convergence? The answer to this question relies on its assumptions. It assumes a constant capital-output ratio which ensures that an economy achieves a constant economic growth infinitely with savings in 't' time period. It ignores diminishing return of capital, which is the fundamental limitation of the Harrod model and precludes convergence. Harrod's equilibrium is also called Razer edge equilibrium and Prof. Robinson has named it as a happy incident. Thus, attending equilibrium is an ordeal and to stabilize the equilibrium is indeed a harrowing ordeal in Harrod's growth model.

Domar model, named after Evsey Domar, is also a Keynesian growth model. Domar model is analogous to Harrod's model in terms of long-run path of an economy, but they differ significantly in terms of their perspective towards economic growth. Domar takes a slightly different course compared to Harrod. Domar's model focuses on investment and considers investment as the sole factor responsible for growth. Investment has two effects; income generating effect and capacity generating effect. Domar refer income generating effect as multiplier side of investment, while capacity generating effect as sigma (σ) side of investment. Moreover, the theory ingeniously refers multiplier side as aggregate demand and sigma side as aggregate supply.

`\DeltaY=\sigmaI---(iv)`

`\DeltaY = \frac{1}{1-b}\DeltaI`

`\DeltaY=\frac{1}{\alpha}\DeltaI---(v)`

Equating equation (iv) and equation (v), we get

`\sigmaI = \frac{1}{\alpha}\DeltaI`

`\frac{\DeltaI}{I}=\sigma\alpha---(vi)`

Equation (vi) is the fundamental equation of Domar model. Nevertheless, equation (vi) is in terms of investment. Income and investment grow at a same rate in Domar model, so

`\frac{\DeltaY}{Y}=\sigma\alpha---(vii)`

If we do some mathematical stuffs as in equation (ii), we get

`Y_t = Y_o e^{\sigma \alpha t}---(viii)`

In equation (viii), α and σ are constant and positive, and time 't' increases consecutively starting from 0, `Y_t` must growth exponentially and take an explosive path as in Harrod model. Though Harrod and Domar initially chose a different course, but ultimate converge to a same long-run equilibrium path of an economy. Moreover, both growth models are based on Keynesian school of thought. Thus, the majority of textbooks prefer to use them in unison rather than explaining them separately.

Now, it's time to explore the Solow growth model. The Solow growth model, developed by Robert Solow in the 1950s, introduces a more comprehensive framework that accounts for factors such as capital accumulation, technological progress, population growth, and capital depreciation. By emphasizing the long-term equilibrium and steady-state growth paths, the Solow model offers a deeper understanding of how sustained economic growth is achieved over time. The model's primary focus is on steady-state equilibrium, where key economic variables stabilize and maintain a consistent growth rate. The Solow growth model is an improvement over the Harrod and Domar growth model. Solow relaxes some of the unrealistic assumptions of Harrod and Domar growth model such as constant capital-output ratio, fixity of ratio of inputs, and so on. Solow growth model is also called the neoclassical growth model as it is fundamentally based on the Cobb-Douglas production function which is one of the empirical production functions developed by neoclassical economists Paul Douglas and Charles Cobb in 1928. 

Solow growth model has two perceptible departures from Harrod and Domar model; (i) Harrod and Domar model are based on Leontief production function while Solow model is based on Cobb-Douglas production function and (ii) Solow assumes that capital becomes obsolete thus yields diminishing returns suggesting that capital-output ratio is not constant. Other major assumptions of Solow model are (iii) homogeneous production function, (iv) constant returns to scale, (v) marginal products of inputs are declining but positive, (vi) labour force grows at 'n' exogenous rate, (vii) economies save a constant fraction of their output and invest that savings in new capital, and (viii) the existing capital stock depreciates over time at a rate δ, meaning some capital becomes obsolete.

`Y=f(L,K)---(ix)`

Equation (ix) is a neoclassical production function with a usual isoquant. Y is yielded if labour and capital both used simultaneously.

`\frac{dK}{dt} = I = sY - dK --- (x)`

For simplicity, Solow expressed equation (ix) in per-capita terms.

`y=f(k)---(xi)`

To derive the fundamental equation, we shall do simple mathematical exercise.

Per-capita capital (k) = `\frac{K}{L}---(\text{xii})`

Differentiating equation (xii) with respect to time,

`\frac{dk}{dt}=\frac{d(\frac{K}{L})}{dt}---(\text{xii.1})`

Solving equation `\text{xii.1}`, we get (see PDF for proof) 

`\dot k = sy - (\delta + n)k---(\text{xiii})`

Equation xiii is the fundamental equation of Solow model. `\dot k` represents the rate of change of the capital stock with respect to time. It indicates how the amount of capital in the economy is changing over a specific time period. `sy` signifies investment in the economy. s represents the savings rate, indicating the proportion of per capita output (y) that is saved and invested for future capital formation. Here, `n` and `\delta` represents population growth rate and depreciation rate respectively. `(n+\delta)k` represents the amount of capital that is being used up due to population growth and depreciation. If sy is greater than `(δ+n)k`, then the capital stock will increase over time and vice-versa. This equation is a fundamental element of the Solow Growth Model as it illustrates the dynamic balance between investment and capital depletion factors that shape the evolution of an economy's capital stock over time.

At steady state, `\dot k` does not change, that is, `sy=(n+δ)k`. Solow model solves the non-convergence problem encountered by Harrod and Domar model. When investment `(sy)` is greater than disinvestment or depletion `((n+δ)k)`, then capital stock expands, else capital stock contracts. Expansion of capital stock increases aggregate income, while depletion of capital stock shrinks aggregate income. Moreover, Solow model incorporated diminishing returns of capital, which suggests that economies with low capital stock grow faster compared to the economies with high capital stock. Hence, developed countries have lower growth rates compared to developing countries. Evidently, China and India have higher growth rates compared to the USA, Japan and other developed economies. However, in the long-run China and India cannot maintain high growth rates due to limited scope in increasing capital stock. Consequently, they too converge.

Solow model can also be applied in case of Nepal. All seven provinces of Nepal are not equally developed. Currently, Bagmati province may have the highest per-capita capital while Karnali province may have the lowest per-capita capital. Capital yields diminishing returns suggesting that with every increase in capital Karnali province will grow faster compared to same increase of capital in Bagmati province. This is true for all the provinces. Thus, in the long run, all the provinces will have similar growth rates and economic landscapes.

Let's explain Solow model using a hypothetical example.

Saving rate (s) = 0.25

Population growth rate (n) = 0.02

Depreciation rate (`\delta`) = 0.05

Solow residual (A) = 300

Output elasticity of capital (`\alpha`) = 0.4

Per capita capital (k) = Rs. 8000

Now, 

`y=Ak^\alpha=300×8000^0.4=10923.38`

Now, 

`\dot k=0.25×10923.38-(0.05+0.02)×8000`

`\dot k=2170.85`

Now, 

New per-capita capital stock `(k_(t+1))=8000+2170.85=10170.85`

New per-capita income `(y_(t+1))=300×10170.85^0.4=12024.42`

In subsequent iterations, we observe the increase in capital stock at a decreasing rate.

Figure 2: Path of  `\frac{dk}{dt}`

The capital stock increases in the diminishing rate due to application of diminishing returns. Thus, every economy converges in the long-run. Moreover, Solow model is an improvement over the Harrod and Domar model. The breaking down of Solow's fundamental equation in equation xiii yields same long-run growth proposed by Harrod and Domar model, that is, in long-run economy grows at 'n' exogenous rate.

In a nutshell, the Harrod-Domar growth model and the Solow growth model represent distinct approaches to understanding economic growth. The Harrod-Domar model emphasizes short-term investment-driven dynamics, offering a straightforward view of how changes in investment influence economic output and employment. In contrast, the Solow model provides a more comprehensive framework by incorporating long-term considerations such as technological progress, capital accumulation, and population growth. While the Harrod-Domar model is suited for addressing immediate economic challenges, the Solow model's depth offers insights into sustained growth patterns and convergence. The choice between these models depends on the analytical scope and policy context. Moreover, growth theories are not limited to mere Harrod and Domar and Solow growth models. There are multitude of growth theories such as Kaldor growth model, AK model, Schumpeterian growth theory, Romer model, Lucas-Uzawa model, and many more.