Firm Costs Optimization Problem in Primal and Dual Form

This Demonstration shows how primal and dual optimization problems relate to each other. The primal problem considered is the minimization of a long-run cost function of the form , where is a vector of factors and is a vector of factor prices, given the planned production . The production function is of the Cobb–Douglas form , where and are factor shares. In such a setting, the dual problem is to maximize given the costs limit . Those problems can be considered separate if they use their own constraints. But if we take the optimal value of the primal object function as the constraint for the dual problem, or vice versa, we get the same optimal solution .

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DETAILS

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).
The solution of each model gives different forms of the demand function for resources.
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.
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