# 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

What traditionally matters is a quasiconcave utility function (that is, the individual/country at least weakly prefers mixing such that all better sets are convex). I assume that's what you're referring to when you describe the Cobb-Douglass example.

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.

(But again, I think the key is quasiconcavity of the utility functions, not convexity.)

• Thanks for the answer -- this is what I was wondering. However, I thought generic Cobb-Douglas utility functions were described as being "convex" since graphing an isoquant curve (x-y product bundles with equal utility) yields some version of a U shape. But maybe I'm missing something, since I'm not really aware of the formal definitions of convex or "quasi-concave" functions of multiple variables. – Brandon May 28 '17 at 22:40
• Oh, I still mix up definitions all the time. And I'm not sure what your background in econ is, but personally, the differences and implications of functional forms wasn't stressed until graduate school. My experience might have been abnormal, but it was largely glossed over in undergrad. If interested, see this proof of why Cobb Douglas functions, where the sum of exponents is weakly less than one, are concave. And this explains some too – AndrewC May 28 '17 at 22:50
• Side note- my grammar is apparently terrible on days when I'm working alone.I apologize for butchering my answers. – AndrewC May 28 '17 at 22:55