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: So that: 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: Subj. to: So that: Our Lagrangian is our cost function (depends on K, L), minus our production function, which must equal a constant. |

**Video – Production duality:**