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Let's see how the investment game changes the simplest and most basic "Keynesian" economic model. As you recall, this model is based on a relationship between income and expenditure. On the one hand, consumption expenditure depends on income, and consumption increases less than proportionately with income. This is shown in Figure 2 by the line labelled C. On the other hand, total expenditure has to equal total income in equilibrium. Accordingly, we add together the various components of expenditure to obtain total expenditure for each respective level of income. In this simplest case there are just two components of expenditure: consumption and investment. It is here that we see the difference: there are two possible levels of investment, corresponding to two equilibria in the investment game. Accordingly, in Figure 1 we see two total expenditure lines. One labelled C+I1, results when we add the low level investment to the consumption function. Therefore, it corresponds to a low-investment equilibrium in the investment game. The other line, C+I2, corresponds to a high-investment equilibrium.
But that's not the whole story, of course. Which investment level will occur? That will depend on the profitability of investment. Remember, the profitability of investment rises (faster for the high-level investment strategy) because investment stimulates expenditure, which in turn raises the profitability of investment. That means there is a total expenditure just large enough to make high investment the profitable strategy, but no larger. In Figure 2, that expenditure is Y0. The idea is that Y0 in Figure 2 corresponds to y in Figure 1. When C+I<Y0, the low-level equilibrium of the investment game prevails; when C+I>Y0, the high-level equilibrium prevails.
Now suppose the economy starts out at a in Figure 2. For whatever reason, income declines steadily. Then expenditure falls along C+I2 to point b. At that point, however, the low-level investment strategy becomes the more profitable one, and expenditure drops discontinuously to c. Continued declines in income would proceed along C+I1. But suppose that the process is reversed and income begins to rise. Expenditure rises along C+I1 past c. Expenditure is still below Y0, so the low-investment strategy is still the more profitable one. Instead, expenditure rises along C+I1 to d, where it jumps discontinuously up to C+I2 and resumes its path toward a.
Accordingly, we have two different expenditure functions, depending on which direction expenditure is changing. Figure 3 shows the expenditure function for declining income. Here we have shown the traditional 45deg. line to help spot income-expenditure points, that is, points at which income (on the horizontal axis) is equal to expenditure (on the vertical axis). We see that there are two equilibria, one at f and one at g. This may suggest that if some disaster were to move the equilibrium far below the equilibrium at f, it would settle into the other one at g instead of returning to f -- the beginning of a long-term slump. Figure 4 shows the expenditure function for a rising income. We see that with income rising from f, the higher-level equilibrium no longer exists! Thus the slump would not only be long-term, but it would be permanent, unless something were to change that would modify the behavior that underlies the consumption function and the profitability of investment.
Thus, it seems that the investment game can be the basis of a simple macroeconomic model that shows many of the key characteristics of the macroeconomic system as post-Walrasians understand it. There are multiple equilibria, and they are path-dependent. Recoveries cannot simply reverse the pattern of the slumps; but the economy must find a new path out of a slump. Because of these difficulties, slumps may be irreversible without change in the structure of the economy. Most important, all of this requires only a minor complication of our most simple macroeconomic model.
It remains to see if we can "tell stories" based on this model that plausibly explain key events of economic history, such as the great depression, the post-war recovery, and the European slump of the 1980's. But that will have to be left for another document.
Roger A. McCain 