# In a box diagram, why does efficiency locus lie on one side of the diagonal, if both sectors haves constant returns to scale function?

The following is what I understand, so far. If we measure labour in the $$x$$-axis and capital in the $$y$$-axis, the slope of diagonal of the box is the capital-labour ratio $$K/L$$ in the economy. Let $$A$$ be the upright good and $$B$$ the upside-down good.

If the efficiency locus lies entirely on the right of the diagonal, that means if we consider any non-corner point on the locus, the slope of capital-labour ratio line in sector $$A,$$ $$K_A/L_A < K/L$$. Also, in sector $$B,$$ $$K_B/L_B > K/L$$. This implies $$K_A/L_A < K_B/L_B$$ i.e., $$A$$ is always relatively labour intensive. Therefore, using this logic, efficiency locus lying on one side of the diagonal just means that there is no factor intensity reversal taking place.

But according to Giancarlo Gandolfo International Trade Theory and Policy, efficiency locus will always lie on one side of the diagonal if the production functions in both sectors exhibit constant returns to scale. This implies that constant returns to scale function never exhibit factor intensity reversals, but this is not true; the same book mathematically shows CES production functions can exhibit them once.

Where am I going wrong? And why does constant returns to scale imply efficiency locus lying on one side of the diagonal.

• Hi! What do you mean by "Let $A$ be the upright good and $B$ the upside-down good." in this context? It seems that you defined the $y$-axis as capital? Nov 20, 2021 at 19:55
• I mean an Edgeworth box diagram. A is the good whose isoquants are drawn upright (origin is at lower left), B is the good whose isoquants are drawn upside down (origin is at upper right) Nov 21, 2021 at 6:58
• Are we still talking about a situation where the $y$-axis represents the capital? If not, can you please reread your question and edit where necessary? Nov 21, 2021 at 7:35
• Yes, I am talking about a standard Edgeworth box diagram with two goods (A,B) and two inputs (labour, capital) that you would use in something like HO model. Nov 21, 2021 at 8:43