3.1. The AK growth model
The models described so far all have the implication that changes in government policies, such as subsidies to research or capital investment, have level effects but no long-run growth effects. That is, these policies raise the growth rate temporarily as the economy grows to a higher level of the balanced growth path. But in the long run, the growth rate returns to its initial level.
There are two meanings of the phrase endogenous growth:
 Long-run growth is not driven by some exogenous process, like exogenous technological progress. Instead the long-run growth rate depends on the economic decisions of economic agents (households and firms).  Public policy is potentially capable of affecting the long-run growth rate.
3.1.1. A simple model
The simplest endogenous growth model is a straightforward extension of the Solow model. In the Solow model the production function reads Y = A K a L1-a . If we set a = 1 and assume A = const. we get Y = AK
A = const. > 0
For simplicity, we assume that the savings rate s is exogenously given. The capital accumulation equation hence is °
K =sAK -dK
and the growth rate of capital turns out to read K = s A - d . In addition, the growth rate of output is equal to the growth rate of capital
Y = K =sA-d
Since we assume that there is no population growth, overall output is equal to per capita output. Assuming that s A > d `
(i.e. the growth condition holds), we have Y > 0. The explanation for this perpetual growth is immediatley seen by comparing the two diagrams below. The neoclassical convergence mechanism (i.e. the average product of capital falls as the capital stock increases) is absent in the AK model since the production function is linear in capital 4
The AK model
The Solow model
 Growth is endogenous in the sense that we did not revert to an exogenous engine of growth, like exogenous technological progress.
 The growth rate of the economy depends positively on the savings / investment rate. Hence any public policy measure that increases the savings rate accelerates economic growth permanently.
 The model implies divergence in international income. If two economies start out with different intial stocks of capital, then the absolut gap gets larger as time proceeds. If two economies have different savings rates and hence different growth rate, the ratio of international income level explodes (collapses). How plausible is the assumption "output is linear in capital" in economic terms? Is there any plausible economic reasoning for this crucial assumption? There is one sloppy answer and two more elaborated answers to this question:  capital is considered to comprise physical as well as human capital;  there are positive knowledge spill-over effects;
 the AK model can be considered as a reduced form (or a short cut formulation) of more elaborated models .
3.1.2. A model with knowledge spill-overs (Arrow, 1962)
The production function for final output is given by
Y = B K a L1-a
which is CRS in private inputs K and L. Population, equal to labor input L, can be normalized to one. The individual firm takes total factor productivity B as given. However, we suppose that B is in fact endogenously determined. Specifically, the accumulation of capital generates new knowledge about production in the economy as a whole. In particular, we assume that
B = A K 1-a with A = const. > 0
That is, an incidental by-product of capital accumulation by firms in the economy is the improvement of the technology that firms use to produce. Technological progress, modelled as a by-product of capital accumulation, is external to the firm. Combining the two preceding equations gives
Y = A K L1-a
This is exactly the AK...
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