# How do I figure out whether the ICs are convex or concave?

Question: Randy Ratpack hates studying both economics and history. The more time he spends studying either subject, the less happy he is. But Randy has strictly convex preferences.

(a) Sketch an indifference curve for Randy where the two commodities are hours per week spent studying economics and hours per week spent studying history. Will the slope of an indifference curve be positive or negative?

(b) Do Randy’s indifference curves get steeper or flatter as you move from left to right along one of them?

My attempt: Since he hates studying both economics and history and the more time he spends studying either subject, the less happy he is, his marginal utilities for economics and history are negative each. Therefore the slope of the ICs will be negative ($$\frac{dy}{dx}=-\frac{U_x}{U_y}$$). I cannot figure out whether the indifference curve is convex or concave. $$\frac{d^2y}{dx^2}=\frac{d(-\frac{U_x}{U_y})}{dx}=-\frac{U_{xx}Uy-U_xU_{yx}}{U_y^2}$$. How do I determine the sign of this expression. I do not know what the signs of $$U_{xx}$$ and $$U_{yx}$$ are.

• Hint: 1) Both studying history and studying economics are "non goods". 2) Convexity and concavity are the exact opposite of one another. – the_rainbox Jul 20 at 6:33
• The question says "Randy has strictly convex preferences", which should mean "Randy thinks averages are better than the extremes". One example which is consistent with this could be that Randy sees 25 hours each of economics and history as being equivalently horrible as 10 hours of one and 30 hours of the other. That would suggest the indifference curves should get steeper as you move from left to right along one of them. This is the opposite of what would happen if Randy enjoyed both and had convex preferences, so it is possible that the question may be expecting a different shape – Henry Jul 20 at 9:52
• @Henry Please post answers as answers. I cannot fathom why you always post these frequently great insights in the comments. – Giskard Jul 20 at 10:20