In the 72 years since its publication, Michał Kalecki's Political Aspects of Full Employment has received little in the way of faithful formalization. The currency it has found has been mostly in grave departures from its premises and conclusions. For the most part, the paper has been treated as if it were a theory of the business cycle. Perhaps the most notable of these endeavors was William Nordhaus' 1975 paper, although a nod to this direction was recognized by Joan Robinson in her 1972 speech at Cambridge. This view generally sees Kalecki's theory as saying that political power determines macroeconomic outcomes.
To be sure, the theory does derive from the realization of a consequence of part of what may be considered the business cycle – widespread unemployment. However, the meat of Kalecki's argument is not (necessarily) about what causes economic downturns per se so much as the curious way in which large business interests react to them.
In my reading, the curiosity motivating Kalecki is not one of what causes widespread unemployment, but rather why businesses would have an interest in seeing it persist. In this model, Kalecki takes rampant unemployment as a prior condition to his model, and is interested in why business interests would oppose a full employment program even when such a program is to their own benefit.
What Kalecki argues instead is that the very fact of full employment itself gives workers the leverage to bargain for higher wages. For Kalecki, lack of job security acts as a discipline device. With a larger pool of unemployed, workers are easily replaced, and they know it. Thus, any attempt to bargain for higher wages, better working conditions, etc. can be easily remedied by businesses with the sack. Thus, any attempt to take this discipline tool from the capitalist class is viciously opposed in spite of the fact that they stand to make higher profits.