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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.

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  • $\begingroup$ 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? $\endgroup$
    – Giskard
    Dec 8, 2023 at 17:58
  • $\begingroup$ Also have you considered accepting these answers to your older questions? $\endgroup$
    – Giskard
    Dec 8, 2023 at 18:02
  • $\begingroup$ @Giskard Hi - I have accepted answers and added my attempt. Thank you! $\endgroup$ Dec 8, 2023 at 18:50
  • $\begingroup$ $(-\bar{z}_1,-tz_2,tq)\leq(-t\bar{z}_1,-tz_2,tq)$ for $t\in [0,1]$. $\endgroup$ Dec 8, 2023 at 23:46

1 Answer 1

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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 $t \in (0,1]$ $$ \begin{align*} &(-z_2, y) \in \overline{Y},\\ \leftrightarrow &(-\bar z_1, -z_2, y) \in Y,\\ \to &(-t\bar z_1, -tz_2, ty) \in Y,\\ \to &(-\bar z_1, -tz_2, t y) \in Y,\qquad \tag{ from (1)}\\ \leftrightarrow &(-tz_2, ty) \in \overline{Y}. \end{align*} $$

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