Applying the maximum principle, we obtain the result where the ratio of the two capital stocks K/H is constant and equal to α/(1 - α). One can substitute this result for the production function to get
Y = AK[(1 - α)/ α]1 - α (1.8)
which is a simple version of the AK model. Following Jones ( 1995), if one takes logs and totally differentiates the above equation, one can get
where A[(1 - α)/ α]1 - α is a constant. This equation is useful for empirical tests because it indicates that the growth rate of output, gt, is an affined transformation of the investment rate for physical capital. When gt is expressed in terms of its time series properties, the dynamics of the growth rates of income should be similar to the dynamics of the investment rates. Hence one of the following three cases is expected to appear in the empirical results:
Case 1 If both growth rate and investment rates are I(1), it might be interesting to examine whether a cointegration relationship exists between growth rates and investment rates.
Case 2 If the growth rate is I(0) and the investment rate is I(1), this implies that permanent changes in investment rates have no permanent effect on the growth rate.
Case 3 If both the growth rate and the investment rate are I(0), then the result is inconclusive in terms of the AK-type endogenous growth model discussed above.
These properties of the endogenous growth model will be subjected to a time series test in the next section.
Annual data on the growth rate of GDP per capita, investment ratio and openness from 1960 to 1990 were used in this study. All data are extracted from the Penn World Tables ( Summers and Heston, 1991). The Summers and Heston data are annual data produced in conjunction with the United Nations Income Comparison Project and is contained in a data set called the Penn World Tables- Mark 5 (or PWT5). Graphs of this data are provided in Figures 1.1-1.5