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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?

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  • $\begingroup$ 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

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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.



Proof.
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 $q$.

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