# Dual Problems in Economics - Conceptual Question

Assume that I am solving a typical consumer problem in Economics in which I want to maximize the utility of an agent that consumes good $$x$$ and good $$y$$ and has a utility function $$U=xy$$ and a budget equal to $$W$$. Then, the problem is

$$$$\max_{x,y} xy \; \;\text{s.t} \; \;p_{x}x + p_{y}y \leq W.$$$$

It is often said in Economics classes that the dual problem corresponds to minimizing the cost of achieving a particular utility level, i.e

$$$$\min_{x,y} p_{x}x + p_{y}y \; \;\text{s.t} \; \; U^{0} \leq xy.$$$$

However, using duality optimization theory I compute the dual for the primal problem and I get

$$$$\min_{\mu} \mu W - \mu^{2}p_{x}p_{y}.$$$$

Of course, all of the three achieve the same solutions, however, I wanted to understand two things:

1. Is there a way to arrive from duality theory to what people in economics call the dual problem?
2. Can anybody provide an intuitive interpretation for the third problem?

I guess that the consumer wants on one hand to minimize how costly the constraint is, but at the same time wants to make $$\mu$$ as large as possible in order to maximize the utility.

• Someone please? Oct 4, 2022 at 10:53