<p>How can I prove that for a production function <span class="math-container">$F:\mathbb X \rightarrow \mathbb R$</span> with constant returns to scale</p>
<p><span class="math-container">$$\forall x\in \mathbb X, \forall t &gt; 0: \ \ F(tx) = t F(x)$$</span></p>
<p>and with the cost function</p>
<p><span class="math-container">$$C(p,y):=\min_{x \in \mathbb X} \{p^\top x\lvert F(x) \geq y\}$$</span></p>
<p>it must be the case that <span class="math-container">$$C(p,ty) = tC(p,y).$$</span></p>
<p>My first strategy has been to make a constructive proof along the following lines</p>
<p><span class="math-container">$$C(p,ty) = \min_{x \in \mathbb X} \{p^\top x\lvert F(x) \geq ty\}
=\min_{x \in \mathbb X} \{p^\top x\lvert F(x/t) \geq y\},$$</span>
with the second identity using the constant returns to scale assumption. I then define <span class="math-container">$z=x/t$</span> and rewrite the variable over which to minimize
<span class="math-container">$$= \min_{x/t} \{t p^\top(x/t) \lvert F(x/t) \geq y\} = t \min_z\{p^\top z \lvert F(z) \geq y\} = t C(p,y).$$</span></p>
<p>However I am not entirely sure that this proof is solid so I guess my question boils down to whether this proof really is solid or if there is another way to proove it?</p>

CC BY-SA 4.0
2020-12-27T21:29:15.113Z
Yes, your proof is fine. There are several other ways to show this, but this one is among the smartest.
Bertrand
CC BY-SA 4.0
2020-12-27T23:58:02.137Z
Thx. Feel free to supplement though if you find the time.
user270396
<cost-functions><economies-of-scale>
Constant returns to scale and cost function: $C(p,ty) = tC(p,y)$
user270396
# Constant returns to scale and cost function: $C(p,ty) = tC(p,y)$

(Asked by: username_0 on 2020-12-27)

How can I prove that for a production function $F:\mathbb X \rightarrow \mathbb R$ with constant returns to scale

$$\forall x\in \mathbb X, \forall t > 0: \ \ F(tx) = t F(x)$$

and with the cost function

$$C(p,y):=\min\_{x \in \mathbb X} \{p^\top x\lvert F(x) \geq y\}$$

it must be the case that $$C(p,ty) = tC(p,y).$$

My first strategy has been to make a constructive proof along the following lines

$$C(p,ty) = \min\_{x \in \mathbb X} \{p^\top x\lvert F(x) \geq ty\}
=\min\_{x \in \mathbb X} \{p^\top x\lvert F(x/t) \geq y\},$$
with the second identity using the constant returns to scale assumption. I then define $z=x/t$ and rewrite the variable over which to minimize
$$= \min\_{x/t} \{t p^\top(x/t) \lvert F(x/t) \geq y\} = t \min\_z\{p^\top z \lvert F(z) \geq y\} = t C(p,y).$$

However I am not entirely sure that this proof is solid so I guess my question boils down to whether this proof really is solid or if there is another way to proove it?

**Tags:** cost-functions, economies-of-scale
> **Comments:**
> **username_1**: Yes, your proof is fine. There are several other ways to show this, but this one is among the smartest.  
> **username_0**: Thx. Feel free to supplement though if you find the time.  

---

## 0 answers

