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I'm struggling to understand the intuition behind why the profit function is homogenous of degree one jointly in all prices (i.e. input prices and output prices). the Intuition feels like it should be degree 0. You double, costs, and you double revenue. Profit is unchanged?

What's also confusing me is P.384 in Wlater Nicholson Microeconomic Theory 11th says

A doubling of all the prices in the profit function will precisely double profits—that is, the profit function is homogeneous of degree 1 in all prices. We have already shown that marginal costs are homogeneous of degree 1 in input prices; hence a doubling of input prices and a doubling of the market price of a firm’s output will not change the profit-maximizing quantity it decides to produce. However, because both revenues and costs have doubled, profits will double. This shows that with pure inflation (where all prices rise together) firms will not change their production plans, and the levels of their profits will just keep up with that inflation.

Surely if profits have doubled, profits haven't just kept up with inflation, they've beat inflation? Because profit takes into account Costs?

I would understand if they said costs had doubled, but Revenue's had also doubled, and so profit stays the same, i.e. homogeneous of degree 0, in all pries. That sounds to me like the definition of "Just keeping up with inflation."

What's the glaringly obvious piece of the puzzle I'm missing? Thanks!

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Profits in nominal terms double when the price of both inputs and output double i.e. Nominal profits are homogeneous of degree 1 in all the prices. However, if we look at profits in real terms i.e. profits in terms of units of that good, so Real profits $:= \frac{\text{Nominal Profits}}{\text{Price}}$, and that will not change when all the prices double. So one can say that real profits are homogenous of degree $0$ in all the prices.

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