Question

If a firm is producing in the long run using capital and labor inputs, how the least cost 

combination of the inputs will be determined? Show graphically and mathematically.

Answer 1

$\frac{min}{(L,K)} ​ wL+rK ....(1.1)$

$Considering; q=f=(L,K)... (1.2)$$The$ $Lagrangian$ $Function$ $Can be$ $Define$ $as$

$∧(L,K,λ)=wL+rK−λ(f(L,K)−q) ....(1.3)$$where; λ => Lagrange multiplier$

The initial order conditions for an interior solution when L > 0 and K > 0 include:

$\frac{∂∧}{∂L} ​ =0⇒w=\frac{λ∂f(L,K)}{∂L} ​ ..... (1.4)$

$\frac{∂∧}{∂L} ​ =0⇒r= \frac{λ∂f(L,K)}{∂L} ​ ...... (1.5)$

$\frac{∂∧}{∂L}=0⇒Q= \frac{λ∂f(L,K)}{∂L} ... (1.6)$

Based on the 6th module

$MPL= \frac{∂f(L,K)}{∂L} ​ and MPK=\frac{∂f(L,K)}{∂K} ​$

Substituting 1.4 as well as 1.5 to eliminate Lagrange multiplier yields (expression 1.1):

$\frac{MPl}{MPk} ​ =\frac{w}{r} r w ​ ........ (1.7)$

While 1.6 tends to be the constraint:

$q=f(L,K) ........(1.8)$