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It seems hard for me to understand the underlying logic of this sentence in Chapter 2 of this book:

They do not seem to have realised that, unless the supply of labour is a function of real wages alone, their supply curve for labor will shift bodily with every movement of prices.

Keynes is talking about the demands of labor is more likely to be a minimum of money wage rather than real wage. Why the supply curve for labor will shift bodily with different prices(prices of wage-goods?) if the supply of labor is a function with other variables(what else variables in this function?)?

I imagine the supply & demand curve intersects on a sector with the x axis being the amount of employment y axis wages, but I'm not sure if this wage should be money wage or real wage or something else.

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This sentence says something very simple: If labor supply is a function of many arguments, not just the wage, $L^s = f(w,X,Y,...)$, then the supply curve, which is a line in a two-dimensional space of $(L^s,w)$ keeping everything else constant, will shift as a whole("bodily") if these $X's$ and $Y's$ change.

What could these "$X$'s and $Y$'s" be? First, if the author defined labor supply as a function of nominal wages, then certainly, a change in prices of consumption goods would shift the labor supply curve.

But even if the labor supply is defined as a function of real wages already, then there are other "$X$'s and $Y$'s" that may shift the whole curve, even if the real wage has not changed: for example, direct transfers from the Government (not as a windfall, but as a systematic policy). Since we assume that working is a "bad", people will want to work less for every real wage level, if they get an increased supplementary income through the Government.

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Breaking down your question into parts:

They do not seem to have realised that, unless the supply of labour is a function of real wages alone, their supply curve for labor will shift bodily with every movement of prices.

The quote implies that firms should calculate their costs using real wages instead of NOMINAL wages. Nominal wages are only important in the short run, nominal values are neutral in the long run. Repricing constantly leads to the well-known menu costs or shoe leather costs problems.

Why the supply curve for labor will shift bodily with different prices (prices of wage-goods)?

I don't think different prices make the supply curve shift according to Keynes. Rather the price-change will cause unemployment! Read more about this here: In classical macroeconomics, why Labor supply responds to real wages and not money wages?

Which other variables are included in the labor supply function?

A good estimation of the labor supply is the marginal rate of substitution (MRS) of labor and consumption. A typical form of labor supply then: $$L_t ^\eta = w_t C_t ^{\sigma}$$ where $L_t$: labor, $w_t$: real wage, $C_t$: consumption, and $\eta, \sigma$: elasticities of labor and consumption.

According to utilitarian school, working is considered a negative, and consumption is considered a desirable good. The goal is to optimize the path of lifetime consumption.

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    $\begingroup$ I think your answer does not really address the question, which is to understand what Keynes meant with that phrase. Instead, you judged the phrase from the viewpoint of (your interpretation of) other theories. $\endgroup$ – luchonacho Jul 15 '17 at 7:51
  • $\begingroup$ Well, we need to ask Keynes to be 100% sure. $\endgroup$ – Übel Yildmar Jul 15 '17 at 8:14
  • $\begingroup$ We do not need certainty on this. You can give your interpretation, which the OP can judge. Your answer does not even attempt to provide such interpretation. $\endgroup$ – luchonacho Jul 15 '17 at 8:59
  • $\begingroup$ Then I think I don't understand the question properly. Could you please shed light on it? $\endgroup$ – Übel Yildmar Jul 24 '17 at 15:49
  • $\begingroup$ To be fair, the question is not very clear. But the quotation of Keynes is "... their supply curve for labor will shift bodily with every movement of prices." And then you say: "I don't think different prices make the supply curve shift according to Keynes.". So to me you are stating that Keynes was wrong rather than interpreting what Keynes meant, and what I think the OP is asking ("what is the logic behind this sentence"). $\endgroup$ – luchonacho Jul 24 '17 at 16:23

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