The setup is simple but a bit lengthy. Please bear with me. Suppose that I have a production function $F(K,L)$ that is:
- constant return to scale;
- increasing in each factor: $F_K>0$, $F_L>0$ (these are partials);
- satisfying diminishing returns: $F_{KK}<0$ and $F_{LL}<0$.
Define an auxiliary function $f(k)=F(k,1)$. Consider the following 2 problems
(A) Profit maximization: with $W,R>0$ given, $$ \max_{K,L}(F(K,L)-WL-RK) $$ Solving this yields an increasing relationship between $w=\frac{W}{R}$ and $k=\frac{K}{L}$ described by $$ w=\frac{f(k)}{f'(k)}-k. $$ We can invert this relationship to obtain an increasing function $k(w)$.
(B) Unit-cost minimization: $$ \min_{a_K,a_L}(Ra_K+Wa_L)\quad\text{s.t.}\quad F(a_K,a_L)=1. $$ Solving this yields the optimal $a_K(w)$ and $a_L(w)$ where as earlier $w=\frac{W}{R}$.
My question: my instructor claimed in class that $\frac{a_K(w)}{a_L(w)}=k(w)$. How can I show this?
Thank you for your help. I have been struggling with this for the past 2 hours.
Due to constant return to scale, the cost minimization problem subject to $F(K,L)=q$ yields the optimal choices $K=qa_K(w)$ and $L=qa_L(w)$. Thus, the profit optimization problem can be written as $$ \max_q[q-R(qa_K(w))-W(qa_L(w))].\tag{C} $$ Regardless of which $q$ solves (C), the optimal $K$ and $L$ for (C) (which are the same optimal $K$ and $L$ for (A)) are proportional to $a_K(w)$ and $a_L(w)$ by a common factor (i.e. the optimal $q$). It follows that $$ \frac{a_K(w)}{a_L(w)}=\frac{K}{L}=k=k(w). $$