# Solow Swan Model of Growth

__Introduction__

Economist Solow constructs his model of fiscal development as an alternative to the Harrod Domar line of thought devoid of its critical postulations of fixed rations in manufacturing. Solow guesses an incessant manufacturing function connecting productivity to the raw materials of capital and labour which can be surrogated.

__Postulations__

Solow constructs his model based on the following postulations.

- One compound product is manufactured

- Productivity is considered as net productivity after making provision for the depreciation of capital

- There are invariable proceeds to level. Or otherwise the manufacturing function is standardised of the first level

- The two aspects of manufacturing labour and capital are paid according to their marginal physical outputs

- Prices and remuneration are amendable

- There is continuous full employment of the available stock of capital

- Labour and capital can be surrogated for one and other

- There is unbiased technical development

- The saving proportion is invariable

__The Solow Model__

Based on these postulations Solow depicts in his model that with variable technical co-efficient there would be an inclination for capital labour proportion to regulate itself in the course of symmetry proportion.

Is the nominal proportion of capital is large, capital and productivity would grow more slowly than labour force and vice versa. Solow's scrutiny is convergent to symmetry path to begin with any capital labour proportion.

Solow considers productivity as a whole the only product in the financial system. Its annual rate of production is elected as Y(t) which shows the actual earnings of the society, part of it is consumed and the rest is saved and invested.

That which is saved is an invariable s and the rate of saving is sY(t). K(t) is the stock of capital. Therefore net investment is the rate of enhancement of this stock of capital i.e. dk / dt or ḱ. So the basic identity is

ḱ = sY …..Equation (1)

As the productivity is produced with capital and labour technological possibilities are represented by the production function

Y = F (K, L) …..Equation (2)

That depicts invariable returns to scale.

Merging Equations (1) and (2), we get the following

ḱ = sF(K,L) …..Equation (3)

In Equation (3), L represents aggregate employment.

As the population is growing exogenously, the labour force hikes at an invariable relative rate n.

Therefore,

nt

L(t) = L …..Equation
(4)

oe

Solow considers n as Harrod’s natural rate of growth in the non-presence of technological variation and L(t) as the availability supply of labour at time (t).

The tight hand side of equation (4) represents the compound rate of the growth of labour force from period o to period t. Conversely, equation (4) can be considered as a supply curve of labour.

It says that the exponentially growing labour force is offered for employment absolutely inelastic. The labour supply curve is a vertical line which moves to the right in time as the labour force improves according to equation (4).

Then the actual wages ascertains the rate regulates so that all accessible labour is employed and the marginal output equation ascertains the wage rate which will actually administer.

By merging equation (4) and (3), Solow gives basic equation

nt

ḱ = sF(K,
L ) …..Equation
(5)

oe

He considers this basic equation as determining the time path of capital accumulation, ḱ that should be followed if all accessible labour is to be fully employed. It provides the time profile of the society’s capital stock which will fully employ the available labour.

Once the time paths of the capital stock and of the force are known the according time path of actual productivity can be calculated from the production function.

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__Increase in productive Capacity__

Domar describes the supply side as the annual rate of investment is I and the annual productive capacity per dollar of newly created capital is equal on the average to s and this represents the ratio of increase in actual earnings or productivity to an enhancement in capital or the reciprocal of the accelerator or the marginal capital productivity ratio. Therefore the productive capacity of I dollar invested will be I.s dollar a year.

But some new investment will be at the expense of the old. It will thus, compete with the latter for labour markets and other aspects of production. Consequently the productivity of old plants will be curtailed and the enhancement in the annual productivity of the fiscal system will be rather less than I.s.

This can be depicted as I σ, where σ is the sigma which represents the net latent social average output of investment (= Δ Y / I). Hence I σ is less than I.s. I σ is the aggregate net potential enhancement in productivity which the fiscal system and is termed as the sigma effect.

__Required Enhancement in Total Demand__

The demand side is described by the Keynesian multiplier. Let us assume the annual enhancement is denoted by Δ Y and the enhancement in investment by Δ I and the propensity to save by α – alpha (= Δ S / Δ Y). Then the enhancement in earnings will be equal to the multiplier (1 / α) times the enhancement in investment.

It is given by: Δ Y = Δ I
* __1__

α

__Symmetry__

To maintain full employment symmetry level of earnings, total demand must be equal to total supply. Thus we accomplish at the fundamental equation of the model:

ΔI
*__1 __ = I α

Α

Solving this equation by dividing both sides by I and multiplying by α we obtain the following:

__ΔI__ = α σ

I

This equation depicts that to uphold full employment the growth rate of net independent investment ΔI / I should be equal to α σ (the MPS times the output of capital). This is the rate at which investment must grow to assure the use of potential capacity in order to uphold a steady growth rate of the fiscal system at full employment.

A numerical illustration could explain us better the Domar Model.

__Illustration 60__

Assume σ be 20% per annum, α be 10% and Earnings Y be $ 300 billions per annum.

- Compute to achieve full employment what would be amount of investment to be made.
- Also ascertain the productive capacity of investment and
- The comparative hike in earnings with the given values

__Solution__

- The amount to invested to accomplish full employment is

Y * α / 100

**300 * 10 / 100 = $30
billions**

**Hence, in order to achieve full employment, the investment to be made is $ 30
billion dollars.**

- To know the productivity of capital investment,

Y * (α / 100) * (σ / 100)

300 * (10 / 100) * (20 / 100)

**$ 6 billion dollars**

**Therefore, the capital investment productivity would increase by 6 billion dollars.**

- The comparative hike in income is,

Y * __(α / 100) * (σ / 100)__

Y

300 * __(10 / 100) * (20 / 100)__

300

**6 / 300 % = 2%**

**Hence the comparative increase in income would be 2%.**

__The Harrod Model__

The Harrod model depends on three discrete rates of growth. Primarily, there is the actual growth rate symbolized by G which is ascertained by the saving ratio and the capital-productivity ratio. It depicts short run cyclical changes in the rate of growth. Next to it, there is the warranted growth rate represented by Gw which is the full capacity growth rate of earnings of a fiscal system. Lastly, there is a original growth rate depicted by Gn which is considered as “the welfare optimum” by Harrod. It is termed as capable or full employment rate of growth.

__The Actual Growth Rate__

In the Harrodian model the first fundamental equation is:

GC = s

Where, G is the rate of growth of productivity in a given period of time and can be expressed as Δ Y / Y; C is the net adding up to capital and is denoted as the ratio of investment to the enhancement in earnings, i.e. I / Δ Y and s is the average inclination to save, i.e. SlY. By substituting these ratios in the above equation we obtain:

(Δ Y / Y) * (I / Δ Y) = S / Y or I / Y = S / Y or I = S

The equation is merely a summary of the axiom that original savings parities original investment.

__The Warranted Rate of Growth__

The warranted rate of growth is as per to Harrod, the rate at which producers will be content with what they are performing. It is the industrial symmetry, it is the line of sophistication which if accomplished will content profit takers that they have done the correct obsession. Therefore this growth rate is primarily associated to the behaviour of businessmen. At the warranted rate of growth, demand is huge enough for business men to sell what they have manufactured and they will carry on manufacturing the same percentage of growth rate.

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- Double Entry Book Keeping
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