<p>Assume that <span class="math-container">$Y$</span> has constant returns to scale, which means that if for all <span class="math-container">$t &gt; 0$</span>
<span class="math-container">$$
(-z_1, -z_2, q) \in Y \to (-tz_1, -tz_2, tq) \in Y
$$</span>
We want to show that <span class="math-container">$\overline{Y}$</span> has non-increasing returns to scale, which means that for all <span class="math-container">$t \in (0,1]$</span>
<span class="math-container">$$
(-z_2,  q) \in \overline{Y} \to (-tz_2, tq) \in \overline{Y},
$$</span></p>
<p>Fix <span class="math-container">$z_1$</span> at <span class="math-container">$\bar z_1$</span>.
Then for <span class="math-container">$t \in (0,1]$</span> as <span class="math-container">$\bar z_1 \ge t z_1$</span>, we have:
<span class="math-container">$$
(-t\bar z_1, -t z_2, ty) \ge (-\bar z_1, -tz_2, ty).
$$</span>
So by free disposal:
<span class="math-container">$$
(-t \bar z_1, -t z_2, ty) \in Y \to (-\bar z_1 - t z_2, t y) \in Y \tag{1}
$$</span>
So for all <span class="math-container">$t \in (0,1]$</span>
<span class="math-container">$$
\begin{align*}
&amp;(-z_2, y) \in \overline{Y},\\
\leftrightarrow &amp;(-\bar z_1, -z_2, y) \in Y,\\
\to &amp;(-t\bar z_1, -tz_2, ty) \in Y,\\
\to &amp;(-\bar z_1, -tz_2, t y) \in Y,\qquad \tag{ from (1)}\\
\leftrightarrow &amp;(-tz_2, ty) \in \overline{Y}.
\end{align*}
$$</span></p>

CC BY-SA 4.0
2023-12-09T09:10:43.007Z
2023-12-09T09:10:43.007Z
tdm
<p>Suppose that <span class="math-container">$Y\subset R^3$</span> is a production set satisfying the free disposal condition: if <span class="math-container">$y\in Y$</span> and <span class="math-container">$y'\leq y$</span> then <span class="math-container">$y'\in Y.$</span> Suppose the technology of production uses good 1 and good 2 as inputs to produce good 3.</p>
<p>Define the set <span class="math-container">$\bar{Y}=\{(-z_2,q):(-\bar{z}_1,-z_2,q)\in Y\}\subset R^2$</span> as the production possibilities available when the use of the first good is fixed at level <span class="math-container">$\bar{z}_1$</span>. How do i show that if <span class="math-container">$Y$</span> exhibits constant returns to scale, the <span class="math-container">$\bar{Y}$</span> exhibits nonincreasing returns to scale?</p>
<p>In addition, is it possible for <span class="math-container">$Y$</span> to be such that <span class="math-container">$\bar{Y}$</span> also exhibits CRTS no matter at what level <span class="math-container">$z_1$</span> is fixed?</p>
<p>Here is my attempt. If <span class="math-container">$Y$</span> is CRTS, then <span class="math-container">$(-z_1,-z_2,q)\in Y$</span> and <span class="math-container">$(-tz_1,-tz_2,tq)\in Y$</span>. Let <span class="math-container">$t\in (0,1)$</span>. But in this case, <span class="math-container">$z_1$</span> is fixed, so how do I proceed? I am unsure of how to use the &quot;free disposal&quot; condition.</p>

CC BY-SA 4.0
2023-12-08T17:58:23.750Z
Have you tried directly applying the definitions to either of your questions? Can you edit in your attempt so that we can see where you get stuck?
Giskard
CC BY-SA 4.0
2023-12-08T18:02:28.793Z
Also have you considered accepting [these](https://economics.stackexchange.com/questions/56909/how-can-i-show-that-the-optimal-savings-are-0-for-all-time-periods) [answers](https://economics.stackexchange.com/questions/56951/why-is-the-budget-correspondence-lower-hemicontinuous) to your older questions?
Giskard
CC BY-SA 4.0
2023-12-08T23:46:43.170Z
$(-\bar{z}_1,-tz_2,tq)\leq(-t\bar{z}_1,-tz_2,tq)$ for $t\in [0,1]$.
Michael Greinecker
<production-function>
A problem with "Returns to Scale"
Ludwig Gershwin
# A problem with "Returns to Scale"

(Asked by: username_0 on 2023-12-08)

Suppose that $Y\subset R^3$ is a production set satisfying the free disposal condition: if $y\in Y$ and $y'\leq y$ then $y'\in Y.$ Suppose the technology of production uses good 1 and good 2 as inputs to produce good 3.

Define the set $\bar{Y}=\{(-z\_2,q):(-\bar{z}\_1,-z\_2,q)\in Y\}\subset R^2$ as the production possibilities available when the use of the first good is fixed at level $\bar{z}\_1$. How do i show that if $Y$ exhibits constant returns to scale, the $\bar{Y}$ exhibits nonincreasing returns to scale?

In addition, is it possible for $Y$ to be such that $\bar{Y}$ also exhibits CRTS no matter at what level $z\_1$ is fixed?

Here is my attempt. If $Y$ is CRTS, then $(-z\_1,-z\_2,q)\in Y$ and $(-tz\_1,-tz\_2,tq)\in Y$. Let $t\in (0,1)$. But in this case, $z\_1$ is fixed, so how do I proceed? I am unsure of how to use the "free disposal" condition.

**Tags:** production-function
> **Comments:**
> **username_1**: Have you tried directly applying the definitions to either of your questions? Can you edit in your attempt so that we can see where you get stuck?  
> **username_1**: Also have you considered accepting [these](https://economics.stackexchange.com/questions/56909/how-can-i-show-that-the-optimal-savings-are-0-for-all-time-periods) [answers](https://economics.stackexchange.com/questions/56951/why-is-the-budget-correspondence-lower-hemicontinuous) to your older questions?  
> **username_2**: $(-\bar{z}_1,-tz_2,tq)\leq(-t\bar{z}_1,-tz_2,tq)$ for $t\in [0,1]$.  

---

## 1 answers

### Answer by username_3 on 2023-12-09. Score: 3

Assume that $Y$ has constant returns to scale, which means that if for all $t > 0$
$$
(-z\_1, -z\_2, q) \in Y \to (-tz\_1, -tz\_2, tq) \in Y
$$
We want to show that $\overline{Y}$ has non-increasing returns to scale, which means that for all $t \in (0,1]$
$$
(-z\_2, q) \in \overline{Y} \to (-tz\_2, tq) \in \overline{Y},
$$

Fix $z\_1$ at $\bar z\_1$.
Then for $t \in (0,1]$ as $\bar z\_1 \ge t z\_1$, we have:
$$
(-t\bar z\_1, -t z\_2, ty) \ge (-\bar z\_1, -tz\_2, ty).
$$
So by free disposal:
$$
(-t \bar z\_1, -t z\_2, ty) \in Y \to (-\bar z\_1 - t z\_2, t y) \in Y \tag{1}
$$
So for all $t \in (0,1]$
$$
\begin{align\*}
&(-z\_2, y) \in \overline{Y},\\
\leftrightarrow &(-\bar z\_1, -z\_2, y) \in Y,\\
\to &(-t\bar z\_1, -tz\_2, ty) \in Y,\\
\to &(-\bar z\_1, -tz\_2, t y) \in Y,\qquad \tag{ from (1)}\\
\leftrightarrow &(-tz\_2, ty) \in \overline{Y}.
\end{align\*}
$$


---

