Marginal and Average costs for constant returns to scale production function being constant

Suppose that we are dealing with a production function $$q = f(k,l)$$, of inputs capital and labor. If this function exhibits constant returns to scale then I know that both the marginal cost and average cost functions $$C(q)$$ are equal and constant for each of the inputs and the total. i.e: $$\frac{\partial C(q)}{\partial q} = \frac{C(q)}{q}$$This means that the contingent demand for labor and capital must be linear in $$q$$. Why is this is case mathematically and what is the economic interpretation?

• Hi! What do you mean by "what is the economic interpretation"? Do you understand what "linear in $q$" means? Nov 29, 2023 at 7:19
• I get the economic interpretation that the marginal and average costs increase at a constant rate with respect to q, which makes sense for constant returns to scale. So I guess I just wanted to have a mathematical proof.
– Nick
Nov 30, 2023 at 6:38

Constant returns to scale (CRS) means that for any positive real number $$a$$ $$f(a \cdot k,a \cdot l) = a \cdot f(k,l).$$ Let us assume that industry wants to produce one unit of output. Then they face the unit cost minimization problem $$\min_{k,l} \ r \cdot k + w \cdot l$$ s.t. $$1 = f(k,l).$$ Suppose this problem is solved by a pair $$(k^*,l^*)$$. If we consider the general cost minimization problem $$\min_{k,l} \ r \cdot k + w \cdot l$$ s.t. $$q = f(k,l),$$ the pair $$(q \cdot k^*,q \cdot l^*)$$ has to be an optimal solution because of CRS.
The pair $$(k^*,l^*)$$ was optimal for the unit cost minimization problem, thus for all pairs $$(k',l')$$ where $$1 = f(k', l')$$, we have $$r \cdot k^* + w \cdot l^* \leq r \cdot k' + w \cdot l'.$$ Now let us move on to the general cost minimization problem where the constraint is $$q = f(k,l).$$ For any feasible pair $$(\hat{k},\hat{l})$$, define $$k' = \frac{\hat{k}}{q} \\ l' = \frac{\hat{l}}{q}.$$ Because of CRS, this new pair $$(k',l')$$ produces exactly one unit of output, thus we know that $$r \cdot k^* + w \cdot l^* \leq r \cdot k' + w \cdot l'.$$ holds. Multiplying this by $$q$$ we have $$r \cdot q \cdot k^* + w \cdot q \cdot l^* \leq r \cdot q \cdot k' + w \cdot q \cdot l' = r \cdot \hat{k} + w \cdot \hat{l}.$$ Again because of CRS, the pair $$(q \cdot k^*,q \cdot l^*)$$ results in $$q$$ units of output. Thus we have seen that $$(q \cdot k^*,q \cdot l^*)$$ is a feasible solution that costs no more than any other pair $$(\hat{k},\hat{l})$$, making $$(q \cdot k^*,q \cdot l^*)$$ an optimal solution to the general cost minimization problem. QED
The solution to the cost minimization problem shows the demanded/used amount of capital and labor, thus these are indeed linear in $$q$$.