# Is comparative advantage only beneficial with convex utility functions?

I've been playing around with the numbers associated with the traditional comparative advantage problem provided by Ricardo. (https://en.wikipedia.org/wiki/Comparative_advantage)

I looked at comparative advantage in my Econ 101 class, but I was surprised to find when reviewing it that it seems like comparative advantage is itself not enough to ensure trade. (I had thought this was sufficient to ensure trade for all optimizing agents.)

Am I correct in believing you'd need the utility function for at least one of the countries in the standard comparative advantage example to be convex, with a diminishing marginal rate of substitution (e.g., a Cobb-Douglass function of the form $U(x,y) = (x)^{0.5} (y)^{0.5}$)? Or am I missing something? It seems to me that without diminishing marginal rate of substitution (which isn't a huge assumption, of course), there's no benefit to trade.

• The utility function you give as an example is not convex. – Alecos Papadopoulos May 29 '17 at 20:43
• The preferences are convex, which lead to concavity of the utility function. – Alecos Papadopoulos May 29 '17 at 20:59

Though I believe your comment about the functional form of utility is overall correct. If, for example, the utility function were $$u(x_{1},x_{2})=\max\{x_{1},x_{2}\}$$ both countries would produce whatever they were more productive in, and neither country would be incentivized to trade.