\( \newcommand{\ll}{\mathcal L} \newcommand{\ff}{\mathcal F} \definecolor{cornellred}{RGB}{179,27,27} \newcommand{\cred}[1]{\textcolor{cornellred}{#1}} \newcommand{\tcred}[1]{\textcolor{cornellred}{\textrm{#1}}} \newcommand{\scr}[1]{\scriptscriptstyle #1} \newcommand{\tn}[1]{\tiny #1} \)

Example 3

$\cred\ff$
$\cred 1$ $\cred 2$ $\cred 3$
$\cred 1$ $1$ $\cred 8$ $3$
$\cred\ll$ $\cred 2$ $3$ $1$ $\cred 8$
$\cred 3$ $\cred 8$ $3$ $1$
$\cred 4$ $7$ $7$ $7$

Beyond non-negativity there are 12 dual constraints.

  • The primal constraint on worker 4 is not binding, so $\scr w_{\tn 4}=0$.
  • $\scr w_{\tn 4}+\pi_{\tn i}\geq\scr 7$, so all $\scr \pi_{\tn i}\geq 7$.
  • $\scr \pi_{\tn i+1}=8-w_{\tn i}$ ($\scr i\mod 4$)
  • so $\scr 0\leq w_{\tn i}\leq 1$ for $\scr i\leq 3$.

The unemployed worker constrains every worker's wages.