Tankus Notes

Since I did a post on increasing returns to devoting resources to producing rice, I think it is time to focus on the opposite. That is, it’s time to focus on resource depletion and decreasing returns.

To do this, it may be useful to introduce some concepts from ecological economics. First there is throughput. Throughput is the amount of energy and matter involved in a production process. A big argument of ecological and biophysical economics is that many of what have appeared as “productivity increases” to economists has historically actually been increasing throughput of energy. Regeneration meanwhile is the idea that when natural resources are left unused they start to regain their original characteristics. Thus the essential distinction between non-renewable and renewable resources is their rate of regeneration. Technically fossil fuels, even considering the greenhouse gas effect, do have a rate of regeneration. However since we are thinking in terms of human resource use the time scale for regeneration, and the level of resource unemployment required to get there, these resources can be called non-renewable given how extremely slow their rate of regeneration is. That last point is worth emphasizing, renewability only matters relative to human time scales. No humans and the time it will take for oil to reform in the ground is irrelevant- at least to us.

This leads us to our final concept: the maximum sustainable scale. This refers to the idea that there is a level of employing a natural resource where the rate of throughput, ie the rate at which this material or energetic natural resource is used, is equal to the rate of regeneration. In other words there is a rate of use where the stock of this natural resource remains constant. A scale of production beyond that will deplete the natural resource and, for our purposes here especially important, lead to diminishing returns over time. This can potentially end with the resource being used up all together. Thus, to simplify here we’re only going to be dealing with homogenous nature. In reality of course, there are many different material forms of nature that are important to different production processes and techniques and have different rates of regeneration.

Thus, rather than dealing with increasing returns in rice, I’m going to deal with decreasing returns in grapes. In this example “Monocropping” grapes leads to an overuse of the soil and less returns each year. At maximum production, Friday produces one less grape meal each year. The less resources devoted to producing grapes, the slower the “decreasing returns” are. For simplicity’s sake I’ve not only assumed the midpoint of the production possibility frontier is the point that maximizes utility but that it is also the spot of maximum sustainable production for Grapes. For those not up on the jargon, I’m assuming that the combination of rice and grape Robinson Crusoe or Friday would choose to produce without specialization both leads to them getting the most enjoyment but also involves producing grapes at the maximum sustainable scale. In other words they would be led by their “short run” self interest to choose to produce the level of grapes and rice that leaves their stock of natural resources constant. With all that in mind, let’s go to the data. I’m starting with the same numerical values as i used in the first post and will try to stick to those numerical values for every Crusoe post I do.

Year One	Friday	Robinson Crusoe
Rice meals produced per year	10	30
Grape meals produced per year	40	90

Year One	Friday	Robinson Crusoe
Opportunity cost of producing one Rice Meal	4 grape meals	3 grape meals
Opportunity cost of producing one Grape meal	0.25 rice meals	0.3333 rice meals

Year Eleven	Friday	Robinson Crusoe
Rice meals produced per year	10	30
Grape meals produced per year	30	90

Year Eleven	Friday	Robinson Crusoe
Opportunity cost of producing one Rice Meal	3 grape meals	3 grape meals
Opportunity cost of producing one Grape meal	0.3333 rice meals	0.3333 rice meals

So what’s going on here? What happened is that there was a jump to the no specialization “special case”. That is, there is a point at which Crusoe’s absolute advantage is identical for both goods and thus there are no gains from trade. Because of decreasing returns in grape production, the gains from trade were eliminated by changing relative prices . This is a point that never gets discussed enough- the difference in the absolute advantage Robinson Crusoe has and thus the difference between the opportunity costs of production are changed under any situation that isn’t constant returns to scale. This is why the handwaving when discussing the situation where Crusoe’s absolute advantage of both goods in undergraduate economics classes are so misleading. Yes it is a “special case” but in a situation where some fundamental forces are driving changes in relative prices it becomes a point that all models will pass through. Thus as we’ve seen, if you assume “short run” maximization in a situation where relative prices aren’t constant (which is most situations) then eventually short run gains from trade will disappear or explode. As we’ve also seen however, in both cases “short run” optimization is inconsistent with intertemporal optimization.

The reason we get the result that output is actually lower in “equilibrium” even as the decreasing returns end is the nature of “maximum sustainable production”. The jump to no specialization led to production at the midpoint of both Crusoe’s and Friday’s production possibilities frontier. This led to no further decreases in natural resources but also not a return to the original “stock”.

What is truly galling about this result is decreasing returns to scale with regard to natural resources was an essential part of Ricardo’s rent theory. That following static comparative advantage when one good experienced diminishing returns not only eliminated gains from trade but lowers equilibrium output relative to the no specialization case makes discussions of comparative advantage infuriating to those with some knowledge of the history of economic thought and the inherent limitations of this perspective.

The point of this isn’t that professors should be using the Robinson Crusoe analogy to convince their students of protectionism and militate against specialization instead of convincing them of free trade and specialization. The point is, are you sure your students could follow either of these broadsides against the way the usual comparative advantage model is used? If not, and i suspect the answer is no, then this part of every undergraduate economics education is propaganda and not education as such. Students should be able to explain and use economic concepts and because the basic example isn’t intuitive and isn’t really connected to their lived experiences, they generally don’t. Instead they are rushed along to more baroque and mathematically complex versions of the same models that are also not very understandable. This leaves students in a mass of confusion, but they still remember the slogans long after which, to a cynical man, might appear to be the point.

The Comparative Advantage story is famous. It is an essential part of nearly every undergraduate economics education. It’s propagandistic benefit is also well known. To review, I’m going to run through a numerical example.

Year One	Friday	Robinson Crusoe
Rice meals produced per year	10	30
Grape meals produced per year	40	90

Here we see that Robinson Crusoe has an absolute advantage in everything. That is, he can produce more of either type of output than Friday can. Oh no! Poor Friday! But wait, David Ricardo is to the rescue! As long as Crusoe’s absolute advantage over Friday is different for producing Rice than producing Grapes, the opportunity cost (what else he could be doing with that time and resources) of Crusoe producing grapes is higher than Friday producing grapes! Boom, gains from specialization and trade.

Year One	Friday	Robinson Crusoe
Opportunity cost of producing one Rice Meal	4 grape meals	3 grape meals
Opportunity cost of producing one Grape meal	0.25 rice meals	0.3333 rice meals

You can see here that this is precisely what we find. The table is clearly telling us that Friday should produce grapes while Crusoe produces rice. However, what happens if there are increasing returns to scale? Below I run through such an exercise.

Below I’m assuming that for each year Crusoe spends producing rice with specialization he is able to produce 4 more rice meals the next year.Poor friday can only produce the same amount of rice year after year and he just continues producing grape meals. For each year Crusoe spends producing Rice without specialization, he can produce 3 more rice meals the next year. For each year Friday produces Rice without specialization, he can produce 2 more rice meals the next year. Of course you could do these assumptions in all different ways, that is sort of the overarching point with regard to how these models are used.

nonetheless, I chose these assumptions specifically. First, output isn’t growing by some percentage rate of the previous years output because recall that in this model all factors of production are fully employed and constant. There is no investment here. Thus the growing returns are relative to the underlying stock of capital and land and are thus exponential relative to them. There is no reason to think exponentially accelerating returns are possible under these conditions. However, there is reason to suppose that applying all factors of production will lead to returns growing more quickly than just applying some percentage of your factors of production. This is why rice output is growing faster for Crusoe under specialization than without specialization.

Second, Friday does not only start out less efficient but he is also less efficient at expanding output. I made this assumption to show that if you move to increasing returns there is a case against specialization even if Friday starts out less efficient at producing rice and is always less efficient at producing rice even if he focuses on it. I didn’t do the analysis here because it was obvious, but if you had perverse specialization- that is Friday specializes in rice- Friday produces 3 more rice meals every year, leading to the least total rice output of all the cases.

Year 25 With Specialization	Friday	Robinson Crusoe
Rice meals produced per year	10	122
Grape meals produced per year	40	90

Year 25 With Specialization	Friday	Robinson Crusoe
Opportunity cost of producing one Rice Meal	4 grape meals	0.7377 grape meals
Opportunity cost of producing one Grape meal	0.25 rice meals	1.3555 rice meals

Year 25 Without Specialization	Friday	Robinson Crusoe
Rice meals produced per year	51	84
Grape meals produced per year	40	90

Year 25 Without Specialization	Friday	Robinson Crusoe
Opportunity cost of producing one Rice Meal	0.7843 grape meals	1.071 grape meals
Opportunity cost of producing one Grape meal	1.275 rice meals	0.9333 rice meals

There are a number of interesting things here to say. This does a good job of making Cameron Murray’s point, which I’ve agreed with ever since I took a look at “Robinson Crusoe’s Economic Man”, that the Robinson Crusoe story is a great pedagogical tool for getting across a number of important concepts in economics. Here we have not only increasing returns and time but Baumol’s cost disease, inequality/distribution and intertemporal optimization. Relative prices are shifting in this economy because of labor productivity increases that exist in one good but don’t exist in another. Within an economy, those working in the low/no productivity growth sectors get wage increases to induce them to do these jobs. Hence the “cost disease” is the rising prices in these industries. This example clearly and easily expresses that idea. On the flip side, in other social contexts there may be no inherent reason for productivity increases to be shared.

If I felt particularly inclined to troll economists, I’d say that what they ignore in their arguments over comparative advantage is intertemporal optimization. That is a bit of a dig but it does get at an important truth. At each Individual point the traditional argument for comparative advantage applies. In fact, the gains to trade as conventionally measured are growing along with Crusoe's labor productivity! Yet taken together this is wildly suboptimal. The argument for protectionism in a very real sense has for a long time been about intertemporally increasing output by inducing production and investment decisions that seem perverse given current production possibilities and prices. The model also makes the important point that protectionism doesn’t have to ever give the protected country's protected industries absolute advantages in producing output over other countries for there to be an intertemporal maximization (or at least improvement) case for protectionism.

Then there is the issue of distribution which increasing returns under specialization brings to mind. If Robinson Crusoe and Friday were more conflictual, say because of property rights or because they represent different countries, in the specialization case Crusoe may not share any gains at all. After all Friday was gaining from specialization in the first year and should be willing to specialize for that benefit 25 years later. Of course the cost of not specializing to Crusoe is increasing all the time precisely because of his productivity increases. If the opportunity cost of not specializing for Friday was lowered by some probability of Crusoe giving in and sharing gains not specializing could be an optimal strategy for Friday for a sustained period of time. That is Friday may be well positioned to strike for higher wages and this example could be a case where striking improved overall welfare. After all economists typically say there is declining marginal utility to more of the same kind of output.

I hope you found this as interesting and entertaining as I did. I suspect that the different ways you can use the Robinson Crusoe story to express concepts in economics will become a running series on this blog.