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What is the interpretation of the expenditure function's being concave?

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The expenditure function is always concave.

Suppose you consume two goods. Suppose the price of one of those goods, say $p1$, increases. Suppose also that, rather than rebalance your demand, you increase consumption of good one enough to maintain some given level of utility (let's say whatever level of utility you had whenever consuming goods 1,2 before the price of good 1 increased). Then your expenditure rises linearly. Suppose instead that you rebalance your demand. Then your expenditures rise but not as much as whenever you do not rebalance your demand. Thus, concavity.

So, I suppose an interpretation here is that a consumer always has the ability to rebalance demands s.t. he/she maintains some desired level of utility in the cheapest way possible. Worst case, expenditures increase linearly. But linear functions are both concave and convex.

This is a very loose way to discuss this. However, I think it gives intuition, which is what I think you are looking for here.

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