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Let us assume all these functions exist and are differentiable. Furthermore let us denote the inverse of the function $\overline u^1(x_1)$ by $x_1\left(\overline u^1\right)$. Note that for any invertible function $f$ we have $$\left.\frac{\text{d} f(x)}{\text{d} x}\right|_{x=x_0} = \frac{1}{\left.\frac{\text{d} f^{-1}(y)}{\text{d} y}\right|_{y=f(x_0)}}.$$...
Example. Each month, God gives Adam $60$ apples and Eve $40$ (for a total of $100$ apples). Let's write this allocation as $X=(60,40)$. The Devil now comes along and offers to increase their total monthly allotment of apples to $101$, but on the condition that the allocation must be $Y=(59,42)$. Observe: $X$ is Pareto efficient but not Kaldor-Hicks ...