A problem with "Returns to Scale"

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.

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

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*}