# 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 9.9 of 10 on the basis of 3513 Review.