Here, the equivalence of the maximization- and minimization- problem is
explained. In the previous pages we have solved the problem of the household
optimum in the form of a maximum, i.e. we have looked for the combination of goods
$x$ and
$y$ which maximizes the
utility $U\left(x,y\right)$ for a given
budget $B$ when the
prices of goods are ${p}_{x}$
and ${p}_{y}$.
An alternative formulation would be to minimize the costs
$x{p}_{x}+y{p}_{y}$ in order to achieve a
certain level of utility ${U}_{0}$.
Hence,
The duality principle states that the solutions to both problems are identical if the budget
$B$ corresponds to the
utility level ${U}_{0}$. If the
maximum utility level ${U}_{0}$ is
reached with the budget $B$
(maximum problem), then the minimum costs to reach the utility level
${U}_{0}$ (minimum problem) are
exactly $B$ and the respective
optimal combinations of $x$
and $y$
are the same. This can be shown easily by means of the Lagrange equation
systems.
Maximum problem | Minimum problem |
Lagrange function | Lagrange function |
$\mathbb{L}\left(x,y,\lambda \right)=U\left(x,y\right)+\lambda \left(x{p}_{x}+y{p}_{y}-B\right)$ | $\mathbb{L}\left(x,y,\lambda \right)=x{p}_{x}+y{p}_{y}+\stackrel{\u0303}{\lambda}\left(U\left(x,y\right)-{U}_{0}\right)$ |
First order conditions: | First order conditions: |
$\frac{d}{\mathit{dx}}U\left(x,y\right)+\lambda {p}_{x}=0$ | ${p}_{x}+\stackrel{\u0303}{\lambda}\frac{d}{\mathit{dx}}U\left(x,y\right)=0$ |
$\frac{\frac{d}{\mathit{dx}}U\left(x,y\right)}{\frac{d}{\mathit{dy}}U\left(x,y\right)}=\frac{{p}_{x}}{{p}_{y}}$
| $\frac{\frac{d}{\mathit{dx}}U\left(x,y\right)}{\frac{d}{\mathit{dy}}U\left(x,y\right)}=\frac{{p}_{x}}{{p}_{y}}$ |