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

CC BY-SA 4.0
2023-11-30T10:21:31.470Z
2023-11-30T10:21:31.470Z
Giskard
<p>Suppose that we are dealing with a production function <span class="math-container">$q = f(k,l)$</span>, 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 <span class="math-container">$C(q)$</span> are equal and constant for each of the inputs and the total. i.e:
<span class="math-container">$$\frac{\partial C(q)}{\partial q} = \frac{C(q)}{q}$$</span>This means that the contingent demand for labor and capital must be  linear in <span class="math-container">$q$</span>. Why is this is case mathematically and what is the economic interpretation?</p>

CC BY-SA 4.0
2023-11-29T07:19:25.767Z
Hi! What do you mean by "*what is the economic interpretation*"? Do you understand what "linear in $q$" means?
Giskard
CC BY-SA 4.0
2023-11-30T06:38:26.400Z
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
<microeconomics><production-function>
Marginal and Average costs for constant returns to scale production function being constant
Nick
# Marginal and Average costs for constant returns to scale production function being constant

(Asked by: username_0 on 2023-11-28)

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?

**Tags:** microeconomics, production-function
> **Comments:**
> **username_1**: Hi! What do you mean by "*what is the economic interpretation*"? Do you understand what "linear in $q$" means?  
> **username_0**: 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.  

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## 1 answers

### Answer by username_1 on 2023-11-30. Score: 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.

---

**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

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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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