Differetiating MC = P

[patter2809]

Q. Let C(w,q*) be the cost-minimising function for input prices w, and output level, q*, where q* is the profit-maximising output. q* = S(w,p), the firm's supply to the market given w and p.

We are given that C_q(w, S(w,p)) = p. Differentiate with respect to w_j.

Attempt RHS differentiates to 0 because it isn't a function of w_j.

LHS is a function of w_j, and a function of a function, which is a function of wj. I tried just applying the chain rule to give: C_qq(w,S(w,p))S_j(w,p), but that's just one term in the answer I've been given.

How do I correctly differentiate this? Thanking you for any services rendered.

 

[HallsofIvy]

Q. Let C(w,q*) be the cost-minimising function for input prices w, and output level, q*, where q* is the profit-maximising output. q* = S(w,p), the firm's supply to the market given w and p.

We are given that C_q(w, S(w,p)) = p. Differentiate with respect to w_j.

You say what w is but I don't see any statement of what \(\displaystyle w_j\) is. Is \(\displaystyle w_j\) a component of the vector w? If so then you are doing a partial derivative. The partial derivative of \(\displaystyle C_q(w, S(w, p))\) with respect to \(\displaystyle w_j\) is \(\displaystyle \dfrac{\partial C}{\partial w_j}+ \dfrac{\partial C_q}{\partial S}\dfrac{\partial S}{\partial w_j}\)

Attempt

RHS differentiates to 0 because it isn't a function of w_j.

LHS is a function of w_j, and a function of a function, which is a function of wj. I tried just applying the chain rule to give: C_qq(w,S(w,p))S_j(w,p), but that's just one term in the answer I've been given.

How do I correctly differentiate this? Thanking you for any services rendered.

 

[patter2809]

Thanks for the reply.

W_j is a component of w.

Why is it ∂C/∂w_j + ∂C/∂S * ∂S/∂w_j? Are we not trying to find ∂C/∂w_j in the first place?

 

[HallsofIvy]

"\(\displaystyle \dfrac{\partial C}{\partial w_j}\)" means just the derivative of C with respect to its "immediately apparent" dependence on \(\displaystyle w_j\). For example if \(\displaystyle C= w_j^2+ P^2\) then \(\displaystyle \dfrac{\partial C}{\partial w_j}= 2w_j\). But if P is also a function of \(\displaystyle w_j\) we have a "hidden" dependence on \(\displaystyle w_j\) and we need to consider that also:\(\displaystyle \dfrac{dC}{dw_j}= \dfrac{\partial C}{\partial w_j}+ \dfrac{\partial C}{\partial P}\dfrac{\partial P}{\partial w_j}= 2w_j+ 2P\dfrac{\partial P}{\partial w_j}\).

 

[patter2809]

Thank you so much! This has been driving me up the wall.