How do you derive a cost function for a production function?

Let say the production function is given by q = f(k,l)---(1) , where q is the maximum output that can be produced using labor(l) and capital(k).

Hence, the cost function is given by C(w,r,q) = wl + vk-----(2), where w is the wage rate, r is the per unit cost of capital, l and k are functions of w,v,and q.

Assuming that a production function is continuous and differentiable, we can set up Lagrangian for minimizing the cost for a given level of output, say q0q0.

The Lagrangian is given by:

L=wl+vk+λ(q0−f(l,k))L=wl+vk+λ(q0−f(l,k))

differentiating the above function wrt k, l and λλ, we get the first order conditions as follows:

∂L∂k=v−λfk(l.k)=0∂L∂k=v−λfk(l.k)=0

∂L∂l=w−λfl(l.k)=0∂L∂l=w−λfl(l.k)=0

∂L∂λ=q0−f(l.k)=0∂L∂λ=q0−f(l.k)=0

Using the three first order conditions, we get conditional input demand of k and l in terms of w,v and q.

Using (2) and the input demand derived via FOC's, we get the cost function.