= Utility conjugate
{title2=$\widehat U(y)=\sup_{x>0}(U(x)-xy)$}
For an increasing concave utility on positive wealth, its <utility conjugate> uses the indicated supremum and is a <convex function> of the positive price variable. If $F(x)=-U(x)$ on positive $x$, extended by infinity elsewhere, then $\widehat U(y)=F^*(-y)$. Under the <Inada conditions> its optimizer is the <inverse marginal utility>, and its <derivative> is the negative of that optimizer. The sign convention is part of the definition.
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