1. Chapter 9
Alternative
Theories
Of Endogenous
Growth
Charles I. Jones

2. The AK Model This model is an alternative model of endogenous growth.
The name “AK” derives from the assumed production function:
This is related to the Cobb-Douglas with
We assume that A is some positive constant.
Capital is accumulated in the usual, Solow-type manner:
where s is the savings rate, and is the rate of depreciation.
We assume that there is no population growth, so all capital letters can be interpreted as per-capita values.

3. Solow Diagram for the AK Model
Since the production function in the AK model is just a straight line, the Solow diagram looks differently (assuming investments exceed depreciation)):
The savings/investment line sY does not intersect the depreciation line
As a result, at any value of capital growth continues indefinitely: no steady state!

4. Constant and Diminishing Returns to Capital
Because in the AK model the production function is constant returns to scale, the marginal product of capital is always equal to A.
In contrast, in the basic Solow model, the marginal product of capital is diminishing with the level of capital.

5. Growth and Investment Rates in AK Model
To see what affects the growth rate in the AK model, let us log and differentiate the production function:
Dividing both sides of the capital accumulation equation by K, we obtain:
because Y=AK
The economy’s growth rate follows immediately to be equal to:

6. Policy Implications in AK Model
The growth rate in AK model is proportional to the savings rate s:
As a result, if the government conducts a policy that favors savings, it permanently increases the growth rate of the economy!
Policy actions did not have a permanent growth effect in the original Solow model.
In fact, the AK model is a subcase of the original Solow model with the parameter
Mathematically, in this case the steady state is so far away from K=0 that realistically, it never gets reached, the economy grows forever.

7. Linearity in Differential Equations
It appears that permanent growth is possible whenever a variable’s growth rate is a linear function of this variable. Indeed, in the standard Solow model we know that
In case , the growth rate of capital stock declines as the economy accumulates more capital. If , the growth rate of capital is constant:
Similarly, if technology growth rate is linear in A, we get permanent growth rate increases if g grows:

8. Lucas’s Human Capital Model
Suppose each individual has 1 unit of time (e.g. 24 hours) to divide between working and studying. Working takes fraction u of this time unit, so studying takes 1-u.
Human capital per worker then evolves according to
Given the human capital-augmented production function
we can show that the growth rate of the economy will be proportional to the growth rate of human capital (show that).
In this model, permanent increases in growth rate are possible if individuals decide to dedicate a larger fraction of their time to studying, i.e. if they decide to reduce u.

9. Knowledge as Externality
Consider a standard production function:
Like in Chapter 4, this function is constant returns to scale in capital K and labor L, but if B is accumulated endogenously, this function is increasing returns to scale (IRS).
IRS requires imperfect competition since otherwise no output is left to compensate innovators for their efforts (show this).
Suppose that accumulation of capital generates innovation:

10. Knowledge as Externality
Knowledge accumulation according to is an externality since firms do not realize that more capital stock results in more knowledge.
Firms act as if the stock of knowledge is unaffected by the economy-wide stock of capital.
As a result, firms are paying capital its marginal product
Combining the production function and the knowledge accumulation equation , we come up with
In case L=1, we are back to the AK model! (show why assuming L=1 is not essential for this analysis).

11. Endogenous Accumulation of Knowledge
In general, there are two ways to deal with the increasing returns to scale that are required if one wishes to endogenize the accumulation of knowledge:
Imperfect competition
Knowledge externalities
Realistically, most knowledge appears to be produced according to directed research efforts rather than by accident.

12. Externalities AND Research Effort
Consider the knowledge accumulation function from Chapter 5, assuming :
For this function exhibits increasing returns to scale: the return to labor is one, and the return to A is , so that total return to knowledge-producing factors is
Knowledge created in the past makes research today more effective: this is a positive externality of knowledge creation we referred to as standing on shoulders effect.
While today’s researchers are producing positive externalities for the future generations of researchers, they do not get compensated for it.

13. Do Policies Matter to Growth?
We know have models where government policies do have a permanent effect on growth… and we have models where they do not.
The property rights instituted and protected by the government do increase growth rates since otherwise nobody will innovate.
However, we also know that policy actions result in temporary increases in growth rates. How long does “temporary” last? If it lasts for 100 years, we might as well say that in our model where policies technically only have a level effect actually produce a long-lasting permanent growth effect.
A very long transitory effect approaches very close to a permanent effect!

14. Linear Differential Equations
Lack of Evidence for
Linear Differential Equations
We have seen how the introduction of linear differential equations can result in permanent growth effects.
However, the empirical evidence appears to reject this sort of linearity.
For example, in the AK model we assume the capital share is 1, while the majority of empirical studies found
Another example: with the knowledge accumulation equation in Chapter 5 becomes
However, the number of researchers has grown enormously, while growth rates remained at roughly 1.8%, meaning

15. The Role of Varieties
To reconcile the empirical observation of growing researchers numbers with the observed low growth rates, Young (1998) and Howitt (1999) suggest the following modification of the production function for ideas:
where M is the number of varieties: larger numbers of researchers do not affect growth rates because more researchers have to work on more varieties. At the same time, we retain the linear nature of the differential equation describing the evolution of A.