Short run cost function derivation

Question

What would be the short-run cost function if labor is fixed at

• $l>y/2$
• $l<y/2$

given the production function $f(k,l)=\min(3k,2l)$?

Answer 1

With the given production function, to produce an output of $y$ requires that $3k\geq y$ and $2l\geq y$.

If $l$ is fixed with $2l<y$, then it is not possible to produce $y$. So in your second case ($l<y/2$) there is no solution to the short-run cost minimization problem.

If $l$ is fixed with $2l\geq y$ (as in your first case) then is it is possible to produce $y$. Assuming the cost of capital ($r$) is positive, then to minimize cost you should choose the smallest amount of $k$ such that $3k\geq y$. Thus you should choose $3k=y$ or $k=y/3$. The short-run cost function is thus $r(y/3)+wl$, where $w$ is the wage rate.