Keynes argued with many while writing the General Theory. Why not say Keynes versus Dennis Robertson is "the clash that defined modern economics"? I think one could argue that Hayek influenced mainstream macroeconomics through J. R. Hick's exposition of temporary equilibrium in Value and Capital, but this seems a quite attenuated influence.
Keynesians can accept that Hayek had the better of the early 1930s debate with Keynes over Keynes's Treatise on Money. The Treatise is not the General Theory. But Sraffa and Kaldor embarrassed Hayek in his attempt to develop the ABCT. I think the two most important errors in the ABCT are:
- The natural rate of interest is undefined in intertemporal General Equilibria (also known as, more or less, "plan coordination").
- No simple relation exists between the optimal allocation of resources among orders of goods and interest rates.
Qualitative stories about the embodiment in capital goods of misaligned plans are, perhaps, the best one can get from Hayek's capital theory. Ludwig Lachmann and Peter Lewin are my favorite Austrian-school economists to read on this point. (Did Lachmann have any enduring effect on mainstream economics?) But it is hard to quantitatively estimate the effects of such misalignments, and I don't find convincing that recovering from these misalignments are the source of the largest worldwide business cycles that we have seen. Anyways, I prefer Joan Robinson on capital theory in disequilibrium.
The importance one should assign the clash between Hayek and Keynes in the history of macroeconomics seems to me to depend crucially on the impact of Hayek on mainstream views of Keynes' General Theory. But Hayek had no such impact. Hayek tried to address the General Theory in his Pure Theory of Capital. But Hayek's book was ignored at the time and is generally considered a failure. About the only thing else Hayek had to say about Keynes's General Theory occurred in interviews and other transient popular pieces. So, whatever you may think about the worth of Hayek's writings, where can you locate their impact?