The hicksian model with investment floor and income ceiling

Abstract

As we saw in Chapter 3, Hicks (1950) modified the Samuelson (1939) linear multiplier-accelerator model through introducing two constraints. The linear multiplier-accelerator model itself only has two options: Exponentially explosive or damped motion. According to Hicks, only the explosive case is interesting, as only this produces persistent motion endogenously, but it had to be limited through two (linear) constraints for which Hicks gave factual explanations. When the cycle is in its depression phase it may happen that income decreases so fast that more capital can be dispensed with than what disappears through depreciation, i.e., natural wear and aging. As a result, the linear accelerator would predict not only negative investments (disinvestments), but to an extent that implies active destruction of capital. To avoid this, Hicks introduced his floor to disinvestment at the natural depreciation level. When the cycle is in its prosperity phase, then it may happen that income would grow at a pace which does not fit available resources. Hicks has a discussion about what then happens, in terms of inflation, but he contended himself with stating that (real) income could not grow faster than available resources which put a ceiling on the income variable. Hicks never formulated his final model with floor and ceiling mathematically, it seems that this was eventually done by Rau (1974), where the accelerator-generated investments were limited downwards through the natural depreciation floor, and where the income is limited upwards through the ceiling, determined by available resources. Formally: (Equation presented) Eliminating Ct and It, one has the single recurrence equation: (Equation presented) It remains to say that Hicks's original discussion included an exponential growth in autonomous expenditures, combined with the bounds If and Yc growing at the same pace, but taking the bounds as constant and deleting the autonomous expenditures, gives a more clear-cut version. It was this that was originally analyzed in detail by Hommes (1991), and the notation above comes from Hommes. However, there were some pieces missing in his discussion, such as two-dimensional bifurcation diagrams, which makes it motivated to make a new attack on this model. © Springer-Verlag BerHn Heidelberg 2006.