Two typical firm optimization problems are considered: primal on the left plot and dual on the right plot. Orange surfaces on each plot correspond to long-run costs (long-run because all cost components

and

are variable), and transparent gray surfaces correspond to production volume. The production function

has fixed

on the left plot and variable

on the right; the long-run cost function

has fixed

on the right plot and variable

on the left. Also, the vertical axis of the left plot is in units of

, and that of the right plot is in units of

. When you drag the sliders, you may see that parameters

,

,

,

affect both plots, but the sliders for the constraints

and

affect only their corresponding plot. Red and blue lines are the trajectories of optimal solutions, given that the parameters do not change. The optimal solution on each plot is the point located on the intersection between the two surfaces and the line.

Since

and

change independently, each model gives its own optimal results. But if you use the buttons, the models become connected; that is, each model will take an optimal solution from the counterpart problem as its constraint, and so both primal and dual models give the same optimal combination of resources

. The numerical panels under the plots show the effect of the buttons (be warned that optimal solutions might be out of the plot's range).

As a side note, the same logic and modeling is fully applicable to the consumer choice problem, with appropriate interpretations of parameters and variables of the model. The main difference is that the primal problem is utility maximization given budget constraint, and the dual problem is budget minimization given required utility level.