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
\begin{equation} \max_{x,y} xy \; \;\text{s.t} \; \;p_{x}x + p_{y}y \leq W. \end{equation}
It is often said in Economics classes that the dual problem corresponds to minimizing the cost of achieving a particular utility level, i.e
\begin{equation} \min_{x,y} p_{x}x + p_{y}y \; \;\text{s.t} \; \; U^{0} \leq xy. \end{equation}
However, using duality optimization theory I compute the dual for the primal problem and I get
\begin{equation} \min_{\mu} \mu W - \mu^{2}p_{x}p_{y}. \end{equation}
Of course, all of the three achieve the same solutions, however, I wanted to understand two things:
- Is there a way to arrive from duality theory to what people in economics call the dual problem?
- 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.
