However, your main criticism of the Keynesian cross is that is a simplification – which is true.
If you want to add the complexity of unemployed people’s consumption, then you have to explicitly model that in a two consumer model and look at their incomes. But it makes no difference to the model – the idea is that people decide to spend less, and there is unemployment and income falls. Who really cares to what decimal this income is distributed?

The problem with the Keynesian cross is that the interest rate is not in the consumption function – and hence does not change savings. Thus people are blinding saving regardless of how much others are trying to invest. Thus, in order to equalize savings and investment, income adjusts.
It makes no difference if unemployed people are consuming a lot, a little or something or nothing.
The Modigliani IS-LM curve drops this assumption and equalizes savings and investment. Income is tied instead to the labor market, so you always have full employment if real wages are flexible.

To me, this is a ridiculous assumption, and instead the Keynesian Cross is actually a better model than the more complicated IS-LM. And it is likely it is accurate in the short run – people usually do not save depending on the interest rate much, nor do investors do real investment so much because of it. Therefore, there is likely a lot of income adjustment in the real world.

Y = a + bY

‘a’ is actually a store of wealth that people consume when they are unemployed. Your criticism is irrelevant.

However, there are a million things ‘wrong’ with the Keynesian cross – it’s a simplification, obviously. All it does is describe the market for savings and investment. The problem is that Income adjusts the market and not the interest rate because consumption and investment have little to do with the interest rate.

But in reality, the only possible consumption prior to production is “the consumption of capital”, if an economy has any.

For example: unemployment could occur by people working less, with no individuals actually unemployed, therefore no extra ‘unemployment’ consumption is required to maintain them, just a reduction in living standards for everyone.

If the Keynesian cross was intended to explain the situation you describe, I might have looked at that too, but this is not the case.

It seems to me that this article is easily refuted. For example: unemployment could occur by people working less, with no individuals actually unemployed, therefore no extra ‘unemployment’ consumption is required to maintain them, just a reduction in living standards for everyone.

Huh? How can there be unemployment if “no individuals actually unemployed?”

(Also, if the author is correct, there cannot be a reduction in production output without also having to produce some quantity to give away for free!)

The point of the article is to begin with the Keynesian assumptions and show that they’re self-contradictory.

(Also, if the author is correct, there cannot be a reduction in production output without also having to produce some quantity to give away for free!)

I think the author is mistaken in the belief that an unemployed person suddenly no longer becomes part of the aggregate totals. Everyone is still in the aggregate totals reguardless of employment.

Keynes based his 1936 treatise The General Theory of Employment, Interest and Money on one key assumption, that involuntary unemployment is a possible market-equilibrium outcome.

Wrong. That’s not an assumption, it’s an observation (using Keynes’s definition of involuntary unemployment). The supporting assumptions of his theory are other than that. However, those assumptions and that theory support the conclusion that that can happen, so the observation is a “confirming instance”. That doesn’t make it necessarily true, since other things could do that too.

the true logic of the Keynesian model implies that the equilibrium is restored by the cessation of the physical existence of the unemployed (i.e., death). Any other outcome contains unresolved internal contradictions.

That is exactly how Stalin, Hitler and Mao attempted to restore equilibrium during their respective heydays. Keynes famously wrote in the forward to the German-language edition of his General Theory:

The theory of aggregate production, which is the point of the following book, nevertheless can be much easier adapted to the conditions of a totalitarian state than the theory of production and distribution of a given production put forth under conditions of free competition and a large degree of laissez-faire.

Keynesianism = Totalitarianism = Death

With respect to D2, Keynes spent some time arguing why he thinks there should be no particular reason for the current investment and current consumption to be related.

-pc and φ(N) should be interdependent, right? It doesn’t seem correct that pc is exogenous. As N fluctuates up and down, I think that pc would fluctuate also.

-Should D2 and pc be inversely releated? I think that when consumption decreases, investment would increase, and vice versa. Figure 2 suggests that D2 is constant, and that does not seem right to me either.

Regarding Krugman, how can he possibly know whether the Austrian School is “worthy of serious study” without having studied it? Sounds like proof positive to me that he, like all other “critics” of the Austrian School, not only haven’t seriously studied it, but are completely unfamiliar with its basic concepts. But I guess that’s modern science, eh?