The Great Depression led to a profound renewal of how problems of governmental fiscal policy were tackled. With reference to the notion of the multiplier introduced by John Bates Clark, Richard Kahn and John Maynard Keynes, the idea that public expenditures could be used to pull the economy out of its slump gained more acceptance in the 1930s. In a series of articles published in 1939 and 1940, Paul Samuelson, then aged 25, urged his contemporaries to examine that idea on the basis of a macro-dynamic model.
Inspired by Hansen’s works, in particular his interpretation of the 1937 crisis (collected in his 1938 Full Recovery or Stagnation?) which was discussed during his seminar at Harvard, Samuelson crafted a model to highlight how a temporary change in public expenditures could leave the final position of the economy unchanged but influenced its trajectory.
Samuelson viewed his model as an introduction to Tinbergen’s macro-dynamic works (see our other posts on this subject here and there); in this perspective, Samuelson was the direct heir of the macro-dynamic approach of cycles developed during the 1930s by Frisch, Tinbergen and Kalecki. Like them, he started from economic assumptions which he found in the work of “literary” economists, in his case Hansen, Harrod, Keynes, Kahn, or Clark, and he developed from those ideas a closed model made up of three simple equations, whose combinations was nevertheless able to give rise to various dynamic trajectories.
By analyzing these different trajectories, and what determined them, Samuelson was able to show how different economic policies would impact the final position of the economy and its trajectory toward it. The recession experienced by the US in 1937 was the case that he wanted to illustrate, to show when and how pump priming could lift the economy, but the determinants of the models clearly showed that expansions resulting from a temporary rise in public expenditures would always come at the price of severe recessions. Placed in the context of the 1937 crisis, when a contraction in public expenditures coincided with a sharp recession at a time when the economy had not completely recovered from the 1929-1933 depression, Samuelson’s model turned out to be a clear and sharp warning on the necessity to consider the dynamics of the economy.
Assuming that a macroeconomic equilibrium is reached at every point in time so that whatever is produced is entirely sold, Samuelson introduced the following equation:
where 
is global output, 
is the part of output sold to consumers (consumption), It is the part sold to firms (investment) and 
is the part sold to the government (public expenditures).
To determine what drives Samuelson introduced two behavioral equations accounting for the changes in and 
, which are two of the components in global demand or global spending. In accordance with Hansen’s reading of Keynes, consumption is assumed to result from income generated one period earlier:
where 
is the marginal propensity to consume, that is, a coefficient telling how much any additional dollar received in 
is spent in 
.
On the other hand, investment itself depends on changes in consumption. The decisions of firms to invest, or spend, depend on their ability to sell goods to consumers, and they base their decisions on the changes in consumption rather than its current level: they decide to spend more on investment when consumption is increasing and less when it is decreasing:
where 
is the value of the “relation,” that is, a coefficient representing how much investment changes when consumption changes between and .
In the end, the combination of these two hypotheses showed that income in time t depends on income in and in 
:
This shows that income in the present period depends positively on the previous period and negatively on what happened two periods before. Now, asked Samuelson, what would happen if the government decided to implement a temporary increase in public expenditures? What would happen if the pump was turned on and then stopped? The following application reproduces the trajectory of income in Samuelson’s model for temporary and permanent increases in public expenditures. The model starts in the equilibrium determined by 
, and at time 
a shock can be added, either permanently or only temporarily; different trajectories will result from different combinations of the parameters and .
On the basis of his model, Samuelson distinguished four possibilities, and examined in each case the effect of different expenditure policies. These regions depended on the parameters and , and for specific values of those parameters, one can see that the economy will get back to its initial position and that the stationary state will be reached either cyclically or monotonically. In those “stable cases,” if the economy is very sensitive to investment, the restoration of equilibrium will happen with cyclical fluctuations: the cycle is caused by the fact that changes in consumption have an effect on investment which is increased by higher values of beta. Once increases in consumption start to diminish in magnitude, the fall in investment has a more important consequence on global demand, which is transmitted back to consumption via a smaller output, which was assumed to be always equal to global demand. In the end, when the model is stable, the economy is back to its initial stationary state but not without, as we can see, having experienced a severe recession: thus, a policy of pump priming based merely on a temporary impulse will not be able to drive the economy forward, and is doomed to failure.
It is only for larger values of the coefficient , when the model becomes unstable, that a temporary rise in public expenditures may cause a never ending expansion of the economy while a temporary decrease in public expenditures may cause a never ending recession. Thus “pure” pump-priming relies on an assumption of instability in the economy.
At the end of the day, if one rules out that case of instability, which Samuelson did on the basis of a nonlinear extension of his simple model, there remains no reason to bet on the possibility that a temporary change in public expenditures may generate a never ending rise in output as the supporters of the pump priming believed. Worse, one must be prepared to experience recession as a consequence of economic policy. The conclusion is that it is not enough to prime the pump, one needs to pump it permanently!