The law of Falling Rate of Profit is a fundamental conjecture of Marxian economics and a pillar of Marxist theory with significant political and historical effects.

To understand its scope, we consider a simple economy with many capital goods and a single wage good, as well labour times across these sectors. Technology is represented by a Leontief matrix.

While finding that the profit rate is strictly less than the exploitation rate, a classic result of Marx, we prove also a version of the Fundamental Marxian Theorem: these rates are equal if and only if they are zero. Interestingly, we provide necessary and sufficient conditions for a specific Transformation Problem of values into prices, that are exogenous, involving only technologies and labor times; and we show that the profit rate is inversely proportional to the total unit cost of production.

Under a constant exploitation rate, the law of FRP is a mere tautology. Importantly, without this empirically questionable assumption, we prove in a new way that the law can be violated: the key of our reasoning is the transformation of values into prices in the spirit of Morishima [9].

Keywords: Values, labor times, Leontief technology, exploitation rate, profit rate, transformation of values into production prices, the law of Falling Rate of Profit.