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?

  • $\begingroup$ Hi! What do you mean by "what is the economic interpretation"? Do you understand what "linear in $q$" means? $\endgroup$
    – Giskard
    Nov 29, 2023 at 7:19
  • $\begingroup$ 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. $\endgroup$
    – Nick
    Nov 30, 2023 at 6:38

1 Answer 1


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 $q$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.