#### Summary

Two things determine profits: income, or turnover (the price at which we sell something) and costs (how much we spent making what we sell). Therefore, knowing how much our costs are going to be is essential when planning the viability of a business.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: That is, our Lagrangian is our cost function, which depends on K, L, minus our production function, which must equal a constant. |

**Video – Production duality:**