SummaryIn this first LP on production, we examine the decisions that lead to optimal levels of production. This is crucial, as it mirrors the same decisions that we saw consumers making: assigning our limited (and expensive!) resources in the best way possible in order to maintain optimal levels of production.
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: