Can cost-reducing technical change lead to a fall in the long run rate of profit if class struggle manages to keep the rate of exploitation constant? In this paper, we derive three results that, taken together, answer this question in the affirmative. First, we identify three properties that new real wage bundles must satisfy to keep the rate of exploitation constant and lead to a falling rate of profit. Second, we derive sufficient conditions for existence of an infinite number of such real wage bundles. Third, we show that, if the initial real wage bundle is such that the maximum price-labor value ratio is larger than 1 plus the rate of exploitation, then starting from any configuration of technology, there always exists a viable, capital-using labour-saving technical change that satisfies the sufficient conditions of the previous result. These results vindicate Marx's claim that if the rate of exploitation remains unchanged then technical change in capitalist economies can lead to a fall in the long run rate of profit.