# Concave preferences have negative SE [Proof]

Question from Intermediate Microeconomics by Hal Varian: Suppose that preferences are concave. Is it still the case that the substitution effect is negative? This is my point: If preferences are concave, then the consumer prefers extreme bundles to some combination. This is, no matter how much it cost he prefers to specialize his consumption in some good. Let's suppose that the consumer optimal bundle is $$(x_1,0)$$ and prices are $$(p_1,p_2)$$. Also, suppose that price of good one increases to $$p_1'$$. To achieve the same bundle, the consumer only needs to adjust their income ($$\Delta m = \Delta p \hspace{0.05cm} x_1$$) in such a way that the new bundle is achievable at the new price (if $$\Delta p \geq 0 \rightarrow \Delta m \geq 0$$). So, in order to achieve the corner solution $$(x_1,0)$$ (adjust prices keeping purchasing power constant), the only way to do is by rotating the budget line around the same horizontal point $$(x_1,0)$$ which means that the substitution effect becomes 0, and this should be true even if the price goes up or down. What part of my reasoning is missing? How does this differ from perfect substitutes case?