Maximum price a consumer is willing to offer?
There is indeed and it is called a bid function. Consider a standard set up where preferences of the agent are given by the utility function $$u(c,h) = \bar u$$ with $c$ being a composite good and where the budget constraint is given as $$c + ph = I.$$ In this case the price $p$ must satisfy $$p = \frac{I-c}{h}$$ and the maximal price $\phi(\bar u,I)$ that the consumer can offer for a unit of $h$ conditional on attaining a utility level of at least $\bar u$ given the income $I$ can be defined as
$$\phi(\bar u,I):=\max_{c,h} \left\{\ \frac{I-c}{h} \ \Bigg\lvert \ u(c,h) \geq \bar u \right\}, $$
under standard assumptions on $u(c,h)$ the constraint is binding and therefore using that $h = u^{-1}(c,\bar u)$ the optimization problem can be reformulated to
$$\phi(\bar u,I):=\max_{c,h} \ \frac{I-u^{-1}(c,\bar u)}{h}.$$
The bid function complements duality by satisfying
$$V(\phi(\bar u,I),I) = \bar u \phantom{xxx} \wedge \phantom{xxx}\ \phi(V(p,I),I) = p,$$
in relation to the indirect utility function $V(p,I) :=\max_{c,h}\{u(c,h)\lvert c+ph=I\}$ as well as the relations
$$E(\phi(\bar u,I),\bar u) = I \phantom{xxx} \wedge \phantom{xxx}\ \phi(\bar u,E(p,\bar u)) = p,$$
in relation to the expenditure function $E(p,\bar u) :=\min_{c,h}\{c + ph\lvert u(c,h) = \bar u\}$.
This implies that the bid-function can be found by inverting the indirect utility function or the expenditure function. So with Cobb-Douglas preferences
$$u(c,h) = \left( \frac{c}{\alpha}\right)^\alpha\left(\frac{h}{1-\alpha} \right)^{1-\alpha},$$
the indirect utility function is given as $V(p,I) = I/p^\alpha$ implying that the bid function is
$$\phi(\bar u,I) = \left(\frac{I}{\bar u} \right)^{1/\alpha}.$$
For an overview of the more well-known duality relations see this post duality and for the perhaps most important application of the bid function see this post monocentric city dealing with the monocentric city model.