# The Solow Growth Model with one Endogenous Growth Model

In order to compare two models of economic growth, I will look at the

primary model of exogenous growth, the Solow model, and ArrowÂ?s

endogenous growth theory, based on research and development generated

within the system. I will define the models and identify their

similarities and differences.

The Solow model, or Neoclassical growth model as it is sometimes

known, is an example of exogenous growth models. This is to say that

the level of economic growth depends on externally determined rates of

growth in certain variables. The Solow model was devised to show the

relationship between the inputs of labour (L), capital (K) and

knowledge (A) on the output level (Y). these are modelled as a

function of time, which does not directly feature in the model:[image].

Therefore an example of this would be the Cobb Douglas function

F(K,AL) = KÎ?(AL)1-Î?, 0<Î? 0 and fÂ?(k) <0.

this means that the marginal product of capital is positive, but it

declines as the level of capital rises , i.e. there is diminishing

marginal product of capital.

Using the intensive form, our production function now becomes [image].

We now make assumptions concerning how the stocks of the inputs vary

over time. Given initial levels of K, L and A, we assume that labour

and knowledge grow at constant, exogenously determined rates, n and g

respectively. That is to say

.

The level of change in the stock of capital depends on two things,

firstly the amount of output invested, the proportion denoted by s,

and the level of depreciation of the existing capital stock, denoted

by Î´. This is illustrated in the equation [image]. This is

endogenously set, in comparison to the previous equations. There

therefore needs to be an amount of investment that counters the

depreciation. This will be determined by considering the model in

terms of k.

Because the level of effective labour, and its growth, is exogenously

set. It can be seen that the economic growth will be determined by the

behaviour of capital. Throughout time, there will be pressures on the

economy as capital depreciates and the amount of effective labour

increases. Now, an equation can be derived that illustrates the rate

of change of capital stock of effective labour: [image]. This can be

worked out using the chain rule, and this equation is very important,

indeed the central equation of the model. The rate of change of k is

the difference between the injections (actual investment per unit of

effective labour) and withdrawals (break-even investment that is the

amount necessary to maintain the level of k). Looking at the

composition of the withdrawals, it can be seen that a number of things

affect the capital stock. Firstly, capital depreciates at a rate Î´

therefore the difference needs replacing. As well as this, the level

of effective labour is growing at a rate of n+g. because of this; more

investment is needed in order to keep the ratio fixed.

[image]

Figure 1 Investment, Actual and Break Even

For small values of k, actual investment exceeds breakeven investment.

As the capital stock grows, the economy moves towards the optimal

level of capital stock, k*. For large levels of k, depreciation and

other factors will cause the level of capital stock per unit of

effective labour to decrease. As a result, the level of investment

will settle at a point where it covers the leakages. At this level of

k, growth is steady; raising the total level of capital stock, K, by

n+g. constant returns therefore implies that the level of income, Y,

is also increasing at this rate. The implication of the model is that

no matter where you start, the economy converges every time to a

balanced growth path, growing at the exogenously determined rate n+g.

the importance of this model shows that it doesnÂ?t matter the amount

of capital within the economy, only the rate of growth of labour and

technological change.

This has an implication for government policy. Any movement to force a

change in the savings rate, s, will have an impact on the growth rare

in the short run, as the rate of change of k is positive. Because of k

rising, it will eventually lead to convergence with a new, higher

level of k*. from this it can be seen that although short run effects

on the level of economic growth will be positive, it will converge

back to the level exogenously determined and therefore policies have

no impact in the long term.

Although the model demonstrating the effect of technology on economic

growth is relevant, it is not necessarily suitable to consider

technological growth as an exogenously determined variable. Instead a

concept of accumulating knowledge through experience, of Learning by

Doing (Arrow 1962), can be introduced.

The determinant of the source of technological progress is the amount

of new knowledge generated through every day economic activities. This

is different from the previous model, which takes the level as

exogenously determined. The Cobb Douglas function can be used again,

where all inputs are concentrated into the production of goods:[image].

The variable A has been replaced by B, which will be discussed later.

The production of capital has the by-product of producing knowledge.

This endogenises the variable, as the stock of knowledge is now a

function of the stock of capital. The variable B represents the level

of skill generated: [image] and when introduced into the production

function yields[image], or, in terms of income per unit of labour:

[image]. Although this is a different value to the one cited in the

Solow model, with this new value[image], the workings concerning

inputs and leakages to the capital stock is still used. Because there

is no exogenously determined growth rate g, the equation for the rate

of change in capital stock per unit of labour is now [image] and the

growth rate of [image] is[image]. The model now is dependent on the

value of[image]. This opens up the process of economic growth to a

number of alternatives to SolowÂ?s model. If [image]=1, [image]. As a

result, growth is constant and any policies implemented by government,

that could affect the savings rate will have no impact in the long

run, as before.

The difference between this and the Solow model becomes more apparent

when [image]<1. From this[image]. There is no growth in the long run;

however, learning by doing will have a positive effect on the steady

state level. If[image], any government policy will have an effect on

the growth of the economy.

Although The Solow model identifies the level of capital per worker

and the effectiveness of labour both as the ability to create

permanent growth in the output per worker of the economy. ArrowÂ?s

model does not dispute that, it simply identifies the determinant of

the effectiveness of labour, i.e. the level of knowledge available and

utilised, is determined within the system, rather than exogenously

determined.

The Solow Growth Model with one Endogenous Growth Model
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