Misses and Hayek were battling against the socialists such as H. Dickinson, O. Lange, and F. Taylor on the “economic calculation debate.” Keynes was not on either side. He did not subscribe to the Austrian view, nor was he a socialist.

Although Keynes postulate a role for the government in crises situation, he did not accept the central planning or management view of the socialists. He will go so far as to speak of the “socialization of investment…But beyond this no obvious case is made out for a system of State Socialism”. (General Theory (GT) p. 378).

I think Keynes could have chimed into the debate at the point where the debate focused on information. Government must have information in order to help steer the economy towards full employment. Keynes was hesitant to use probability theory in the GT, although he wrote the famous “Treatise on Probability”. In a fuzzy way, this idea can be put into the same set of thought of Hayek, namely that information cannot be organized as it resides in the brain cells where it is inaccessible to the hands of the planners or policy makers. So both of them have a Fuzzy set of information, one that economic ideas is non-ergodic, and the other that information is only subjectively accessible. Now, the Fuzzy set has picked up another uncertainty element regarding how agents partition information differently.

Keynes was arguing with the Austrians on other matters. His debate with Hayek on monetary matters is well known, starting with Hayek’s review of Keynes’s “Treatise on Money”. Hayek however, did not pay as much attention to Keynes” “General Theory”, dismissing as an inadequate piece on employment.

Well  a few brief observations:

First, if you look at the Economic Journal during the period, Keynes pieces were overwhelmingly about exchange rates, public finance and 'macroeconomic topics' rather than the welfare economic issues argued in the calculation debate.

Second, In the second volume of Skidelsky's biography of Keynes (pp,232-33), Skidelsky notes that Keynes view of the USSR led him to see socialism as 'ideological, obsolete, irrelevant... likely to involve gross interferences with individual liberty' (p.233).

Third, Hayek recognised that Keynes was considering planning from a very different point of view than the market socialists. Note Hayek's review of How to Pay for the War doesn't discuss calculation arguments.

So overall, Keynes wasn't that interested in the debate.

The "calculation debate" mainly focused on the (static) microeconomics of price formation and resource allocation, while Keynes looked at the shortcomings of traditional (laissez faire) capitalism from a quite different perspective.

An intersting contribution on Keynes's view on capitalism is R.E. Backouse and B.W. Bateman, "Keynes and Capitalism", History of Political Economy, Winter 2009 41(4): 645-671; 

Note that the same could be said, in a different theoretical context, of Schumpeter as well.

This is our answer to this question I made in 2014: Article What would have been Keynes' position in the socialist econo...

Some hypotheses:

(1) Much of the debate was in German and Keynes did not speak this language very fluently.

(2) Keynes had a very low opinion of Marxist economics and socialism in general. Perhaps more importantly, Keynes knew that most of the policymakers whom he was trying to influence in the UK and US shared his opinion of Marxism and socialism. Therefore, he tended to focus on debating economists like Hayek and Hawtrey, who were listen to by the people who were also listening to Keynes.

(3) The Socialist Calculation Debates made a variety of charitable assumptions towards central planning that Keynes would think implausible, e.g. that the incentive problem for socialism could be solved and that an Actonian corruption of the planners (of the kind that Keynes noted when talking about The Road to Serfdom) would not be a problem. Given his focus on disequilibria and his experience as a speculative investor, I think that Keynes would have been sympathetic to both the Misesian argument (the necessity of entrepreneurial capital markets for choosing among multiple technologically feasible plans of production) and the Hayekian argument (the necessity of capital markets for aggregating and using dispersed information).

Now some evidence: the only place I know where Keynes talks (indirectly) about central planning is his letter to Hayek regarding the Road to Serfdom. Let's look at what he says:

"I come finally to what is really my ONLY serious criticism of the book. You admit here and there that it is a question of knowing where to draw the line. You agree that the line has to be drawn somewhere [between a laissez-faire market economy and planning], and that the logical extreme is not possible...

I should say that what we want is not no planning, or even less planning, indeed I should say that we almost certainly want more. But the planning should take place in a community in which as many people as possible, both leaders and followers, wholly share YOUR moral position... What we need is the restoration of right moral thinking - a return to proper moral values in our social philosophy."

(emphasis added)

I interpret that to mean that (a) Keynes agreed with Hayek's critique of social planning and socialism in general in the Road to Serfdom, but (b) he wanted there to be more social direction of investment under liberal governments who generally favour individualism and mostly laissez-faire economics. An activist fiscal policy, but with a Hayekian government and population.