Oh wait, the Soviets did not actually reciprocate Reagan's increases in defense spending:
If American defense spending had bankrupted the Soviet economy, forcing an end to the Cold War, Soviet defense spending should have declined as East-West relations improved. CIA estimates show that it remained relatively constant as a proportion of the Soviet gross national product during the 1980s, including Gorbachev's first four years in office. Soviet defense spending was not reduced until 1989 and did not decline nearly as rapidly as the overall economy.
What really happened:
The Soviet economy was not the only economy burdened by very high levels of defense spending. Israel, Taiwan, and North and South Korea have allocated a disproportionate share of resources to defense without bankrupting their economies. Indeed, some of these economies have grown dramatically. A far more persuasive reason for the Soviet economic decline is the rigid "command economy" imposed by Stalin in the early 1930s. It did not reward individual or collective effort; it absolved Soviet producers from the discipline of the market; and it gave power to officials who could not be held accountable by consumers. Consequently much of the investment that went into the civilian sector of the economy was wasted. The command economy pre-dated the Cold War and was not a response to American military spending. The Soviet Union lost the Cold War, but it was not defeated by American defense spending.
http://www.theatlantic.com/past/politics/foreign/reagrus.htm
Nice source!
I like 'The Soviet Collapse' that describes another aspect of the downfall.
Mises Wiki | Economic Resources and Books (search engine)
Penned by a very infamous guy.
You are right, he does seem like a rather controversial fellow! The analysis by itself, seems to be sound, though.