# Optimal point and MRS

I read that the tangency condition is not sufficient for optimality, and that one other condition is that the MRS must equal the slope of the budget line at an interior optimum. My confusion is that since MRS is the slope of an indifference curve, then wouldn't it automatically be equal to the slope of the budget line of the budget line is tangent to the indifference curve?

• 1. Where did you read this? 2. What exactly do you mean by "tangency condition"? – Giskard Jul 23 '19 at 6:23
• In this context, the "tangency condition" is that the MRS is equal to to slope of the budget line. That is, your two conditions are equivalent. However, there can still be a corner solution. – Bayesian Jul 26 '19 at 15:15

Below is an example where the tangency (point $$A$$) is not the optimum (assuming both $$x$$ and $$y$$ are "goods" so that more of either good is preferred to less). In fact, the optimum occurs at the corner (point $$B$$) where the indifference curve is not tangent to the budget line. This happens because the preference over $$x$$ and $$y$$ is not convex, which is manifested by a set of indifference curves that is concave to the origin.
A non-convex preference represents the opposite taste for variety. Suppose $$x$$ is pop and $$y$$ is coffee, and I, as most people, would prefer to have them separately than combined. Then my preference over pop and coffee would be non-convex.