Concave production function implies convex cost function

Question

Let's assume we have an increasing production function $f:\mathbb{R^+} \to \mathbb{R^+}$

Now, assume this production function is concave and that the price of input z is fixed (this is a single-input and single output case). I want to show this implies the corresponding cost function $C^f(w,q)$ is convex.

My thoughts:

let $z,z' \in \mathbb{R^+}$ where W.L.O.G $z'>z$ and let $f(z)=q$ and $f(z')=q$

Since $f$ is concave, we take $\alpha \in [0,1]$ $s.t$:

$$f(\alpha z +(1- \alpha)z') \geq \alpha f(z) +(1- \alpha)f(z')$$

let $\alpha z +(1- \alpha)z'=z''\in \mathbb{R^+}$ where by necessity $z\leq z'' \leq z'$

Since cost of $z=w$ is fixed at some $w \in \mathbb{R^+}$

we can rewrite our cost function as $C^f(q)$ which is convex if for $\alpha \in [0,1]$:

$$C(\alpha q +(1- \alpha)q') \leq \alpha c(q) +(1- \alpha)c(q')$$

Now, since $f$ is concave, it cannot be the case that $f$ experiences increasing returns to scale. Then $f$ has constant returns to scale or diminishing returns to scale.

Then:

Since $z''$ is clearly feasible then we have that:

$$C(w,q'') \leq w*z''$$ $$=\alpha w*z + (1- \alpha) w*z'$$ $$= \alpha c(w,q) + (1- \alpha) c(w,q')$$

Does this look correct?

Answer 1

Given the fixed input price $w$, the cost function can be written as $$ C(q)=f^{-1}(q)\times w $$ where $f^{-1}$ is the inverse of the production function $f$. From the discussion here, one can conclude that the inverse of a concave strictly increasing function is convex. Thus, $C(q)$ is convex as well.

Going back to your approach, you might like to have this clearly stated. Let $q''=\alpha q + (1-\alpha)q'$, $f(z)=q$, $f(z')=q'$, and $f(z'')=q''$ Then $$ \begin{align} f(z'')&=&q''\\ &=&(\alpha q + (1-\alpha)q')\\ &=&\alpha f(z)+(1-\alpha) f(z')\\ &\leq&f(\alpha z +(1-\alpha z')) \end{align} $$ Then $f(z'')\leq f(\alpha z +(1-\alpha z'))$ implying that $\alpha z +(1-\alpha z')\geq z''$ since $f$ is (strictly) increasing. Hence, $$ z''w=C(q'')\leq \alpha C(q)+ (1-\alpha) C(q') $$

Answer 2

The economic intuition here is clear: if the single-input production function is concave, the marginal product is diminishing. So for equal increases of the single input (and given fixed unit cost), say unit per unit, production increases less and less, while cost increases equally. Flip it, and then for sequential unit increases in output, cost increases more and more per incremental output unit. This is the primitive description of a strictly positive second derivative, which is the condition for a strictly convex function.