When are there NO gains from trade?

OK you trade theorists. I've got a question for you. It's the exact opposite question to the one you normally ask. When are there no gains from trade?

Biologists need to know.

Let me set the ground rules.

2x2 model. Two people, and two goods. Each person has a Production Possibility Frontier and a utility function. The PPF and indifference curves are not restricted to be concave or convex, but we do assume that the autarky point for each individual is an interior solution (both people choose to produce and consume positive quantities of both goods if trade is forbidden). Zero transactions costs.

I can think of two cases where there are no gains from trade:

Case 1. Linear PPFs, where both have exactly the same slope. (So no comparative advantage.)

Case 2. PPFs that are concave to the origin, and which have exactly the same slope at the two autarky points. (Both people have the same MRS=MRT). This would be the case, for example, if the two individuals are identical in both PPFs and utility functions; but could also be the case if different PPFs and different utility functions exactly cancelled out.

(Case 1 is really just a subset of case 2, if I replace "concave" with "weakly concave".)

I'm trying to think of an example where the PPFs are convex to the origin (economies of scale) but there are no gains from trade, but I can't. Are there always gains from trade with economies of scale?

There are probably cases where the goods are indivisible, so that the PPF and indifference curves are kinky or discontinuous, right?

Is there some sense in which it can be proven that for two randomly chosen individuals the probability of there being no gains from trade is vanishingly small? Lebesgue measure zero, or something?

Here's why I ask. Because Jeremy Fox emailed me a link to this paper (pdf) about market models in biology. When should we expect to see "mutualism" between two species? And the paper is using basic trade theory. (Except it implicitly assumes that the two goods are perfect complements in consumption, so that indifference curves are L-shaped, and preferences can be represented by a ray from the origin.)

Have a look at Figure 4 in the paper. I disagree with the authors' statement that "resource trade is NOT beneficial" in box 2 top row, and box 5 bottom row of that table. (Unless I am misreading the table.)

My sense is that, absent transactions costs, "mutualism" (i.e. trade) is almost always beneficial for two species that both consume positive quantities of two goods. So, when we don't see mutualism in such cases, the reason must almost always be transactions costs. (Not that surprising really, since fish and birds don't have good contract laws and stuff, and I'm not sure whether Evolutionary Stable Strategies can make up for their absence).

Comments

[Nathan Tankus]

economics should keep it's dogma out of the natural sciences.

[K]

Good question. Do you put information asymmetry in the category of transaction cost? Cause that seems like a pretty good reason not to trade.

[Jeremy Fox]

Since I'm the one who asked for Nick's perspective as an economist on 'biological market models', I feel like I'd better comment (and not just to thank Nick for going from zero to substantive blog post in the space of a few hours!)

The implicit assumption of two perfectly complementary goods is, in the biological case considered in the paper Nick links to, entirely reasonable. All organisms need (say) carbon and phosphorus, and neither of those resources can substitute for the other (you can't stick C atoms in your DNA where P atoms are supposed to go, for instance). The assumption of fixed preferences is also empirically reasonable for many organisms (though actually not for plants). Many species have quite homeostatic chemical composition, and so require different, perfectly complementary resources in fixed ratios. So some aspects of biological market models that look unusual to economists are unusual for good biological reasons. But I'm suspicious that some other aspects of current biological market models may be economically unusual for less-good reasons. Hence my curiosity (which Nick satisfied insanely fast!) as to how an economist would react to these models.

In addition to the points Nick raised here, I'm also chewing on the issues raised in his previous post, to do with what happens (in a biological context) when future expectations of market players aren't satisfied, perhaps because it's impossible for them all to be satisfied...

Anyway, thanks again to Nick for taking the time to offer his thoughts and make an economically-ignorant ecologist a little less economically-ignorant.

[Nick Rowe]

Nathan: It's far too late for that, I'm afraid. If you add Thomas Malthus and Adam Smith together, and apply it to biology (OK and do a lot of thinking and observation) you get Charles Darwin!

K: yes. Asymmetric Information underlies a lot of transactions costs, in one way or another.

Jeremy: "The implicit assumption of two perfectly complementary goods is, in the biological case considered in the paper Nick links to, entirely reasonable."

Aha! I really did wonder about that. I assumed (uncharitably) that it was just because they weren't comfortable with more general preferences, and drawing indifference curves. If critters really do consume in fixed proportions, regardless of relative prices, then that is a much simpler way of drawing preferences.

[Greg Ransom]

Let me recommend Michael Ghiselin's _The Economy of Nature and the Evolution of Sex_ to anyone interest in the economics opand biology of the division of labor, among other economic / biological topics.

Wonderful stuff. Better then Richard Dawkins.

[Nick Rowe]

OK. Is this a genuine "proof" that there are always gains from trade under increasing returns to scale (PPFs convex to the origin)?

At autarky, either MRTi=MRTj or not. Lets take each in turn.

1. If MRTi is not = MRTj then it is clear that total output of both goods will increase if both agents make a small move in opposite directions towards specialising in that good in which they have a (local) comparative advantage.

2. Draw a straight line between the 2 agents' autarky points. The mid-point of that line represents average production per agent. Double it, and we get total production. If MRTi=MRTj, then let both agents make a medium-sized move in opposite directions away from autarky. The mid-point of the straight line drawn between the two production points must move North-East as they move away from the autarky points. So we get more of both goods produced.

Thus, there are always gains from trade with increasing returns to scale.

[Eric Pedersen]

Well, wasted a good chunk of morning thinking about this, but I think you can prove it more generally than that[1]:

Assume 2 goods, X and Y, and two individuals, A and B. The PPF's for A and B are, respectively[2], Y_A = F(X_A) and Y_B = G(X_B). Assume both F and G have negative first derivatives. [X_A*, F(X_A*)] is the autarky point for A, and [X_B*, G(X_B*)] is for B.

To get gains from trade (when goods are perfect complements), we have to find a new combo, X_A, X_B, where X_A + X_B >= X_A* + X_B* and F(X_A) + G(X_B)>= F(X_A*)+G(X_B*). To simplify, I'll hold the sum of X's constant, and look at small changes along the ppf's: X_A = X_A+n, X_B = X_b-n.

The change in the sum of Y's is:
delta_Y = Y_A + Y_B - Y_A* -Y_B* = F(X_A*+n) + G(X_B*-n) - F(X_A*) - G(X_B*)

After Taylor expanding and simplifying:
delta_Y = n*F'(X_A*) - n*G'(X_B*) + n^2/2*F''(X_A*) + n^2/2*G''(X_B*) + ...

If F' != G', we can ignore the higher order terms (since they vanish for small n). In that case, either F' > G', so n>0 leads to delta_y > 0, or F' < G', so n<0 leads to delta_y >0.

If F' == G', you still get gains from trade if F'' & G'' >0, (both convex) or if F'' or G'' > 0, and F'' + G'' > 0.

I think that covers all the bases. These are, though, only sufficient, not necessary conditions to get gains from trade.

[1]I'm absolutely positive this proof, or its equivalent, isn't new, but I don't have any trade textbooks handy to pull it from.

[2] God, I wish latex worked in Typepad comments.

[Nick Rowe]

Well done Eric! A morning well-wasted!

There was one case I missed: what if one agent has a convex and the other a concave PPF?

[Stephen Gordon]

God, I wish latex worked in Typepad comments.

Testify!

[Eric Pedersen]

If the first derivatives aren't equal at their respective autarky points, the convexity/concavity of the ppf's don't matter. If they are equal, their'll still be gains from trade if the second derivative of the ppf of the agent with a convex ppf is greater in magnitude than the second derivative of the agent with a concave function.

So:

If A'' (convex) > 0 > B'' (concave), then, to have gains:

A'' > 0 > B'' > -A''.

[Nick Rowe]

Eric: that makes sense, intuitively.

The bottom line to all this: given the assumptions (especially both agents consume both goods in autarky), there really are almost always gains from trade. If two critters don't exhibit mutualism, it must almost certainly be because of transactions costs (one critter would eat the other critter if both appeared in the marketplace).

[Nathan Tankus]

Nathan: It's far too late for that, I'm afraid. If you add Thomas Malthus and Adam Smith together, and apply it to biology (OK and do a lot of thinking and observation) you get Charles Darwin!

i should have been more clear. Modern Economics should keep it's dogma out of the social sciences. i have much more respect for classical economics then i do for neoclassical economics

[Adam]

If there is no gain to be had from the trade, why are the parties trading?

[Nick Rowe]

Adam: If there are no gains from trade, they won't trade. If there are gains from trade, they will trade, unless the transactions costs exceed the gains from trade.