= Hölder bound for discounted CRRA consumption
{c}
{title2=$\mathbb E\int e^{-bt}U(c_t)dt\leq U(x)\left(\mathbb E\int e^{-bt/R}Z_t^{1-1/R}dt\right)^R$}
For $0<R<1$, nonnegative <consumption> with state-price budget $\mathbb E\int Z_tc_tdt\leq x$ satisfies this bound under <CRRA utility>. Apply <Holder inequality> on probability-times-time measure to $(Zc)^{1-R}$ and $e^{-bt}Z^{R-1}$, with conjugate exponents $1/(1-R)$ and $1/R$. If the price integral $D$ is finite, equality forces full budget use and $c_t=(x/D)e^{-bt/R}Z_t^{-1/R}$. Financing this process is a separate feasibility issue, so the equality pattern alone does not assert attainability in an incomplete market.
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