As in consumer’s theory (where consumption duality is analysed), the firm´s input decision has a dual nature. Finding the optimum levels of inputs, can not only be seen as a question of choosing the lowest isocost line tangent to the production isoquant (as seen when minimising cost), but also as a question of choosing the highest production isoquant tangent to a given isocost line (maximising production). In other words, having a cost function that sets a budget constraint, solving for the inputs allocation that gives the highest output.
The way to solve either problem is very similar: we look for the Lagrangian function and obtain first order conditions, then solve the system.
When dealing with primal demand, that is, output maximisation, our Langrangian is as follows:
That is, our Lagrangian is our production function, which depends on K, L, minus the restriction- our budget.
When dealing with dual demand, that is, cost minimisation, our Lagrangian system is as follows:
Our Lagrangian is our cost function (depends on K, L), minus our production function, which must equal a constant.
Video – Production duality: