In microeconomics it is standard to maximize utility subject to a constraint. For this problem income and prices are normally and the solution is the Marshall which plugged into direct utility function gives the indirect utility function.

Similarly, taking instead the utility to be achieved and prices as given, the expenditure minimization problem finds the expenditure (income) necessary to achieve the given level of utility at given prices and the solution is the Hicksian demand and the expenditure function.

Is there in a similar fashion a problem where utility and income are taken for granted but where prices are solved for?

The idea would be to solve for the maximal price that a consumer is willing to pay for a good if the consumer is to achieve a certain level of utility given a certain income.


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.


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