Question

True or false? ‘If λ(w, y) is the Lagrangian multiplier associated with the firm’s cost-minimisation
problem, then mc(w, y) = λ(w, y).’

Answer 1

Cost function is TC(q)=wL+yK=wL(q)+yK(q)=f(q,y,w) and Marginal Cost is MC=dTC(q)/dq

We have fixed level of production such that q(L,K)=Z

Hence setting up lagrnagian function (M)

M=TC(q)-λ(w, y)*(q(L,K)-Z)

We have Z is constant therefore when we differentiate above equation then we get

dM/dq=dTC/dq-λ(w, y) as dZ/dq=0

Now for given level of q we have optimality for cost minimisation problem when

dM/dq=0 and d2M/dq2>0 for given level of "q"

therefore

MC(w,y)=λ(w, y)

Hence proved.