# The Inverse Farm Size–Productivity Relationship: “Proof” that Smallholders Can Feed the World?

I spent the last week in Montreal for the McGill Global Food Security Conference, where I had been asked to present some of the work I have been doing on food prices.

The conference was very interesting, and it allowed me to finally meet colleagues whom I only knew through their work, meet new colleagues, and interact with a number of students interested in food policy.

At some point, however, one of the presenters mentioned the inverse relationship between farm size (i.e., the amount of land a farmer cultivates) and productivity (i.e., the amount of output per unit of land a farmer can get, or that farmer’s yield), an empirical regularity in the developing world which constitutes an old puzzle in development economics and on which I have done some work a few years ago. The following figure is an example of such a relationship for rice farms in Madagascar — the y-axis measures the logarithm of a plot’s yield in kilograms of rice for one hundredth of a hectare, and the x-axis measures a plot’s size in hundredths of a hectare.

What that presenter said was that the existence of an inverse relationship between farm size and productivity is proof that smallholders can feed the world. Nothing, however, could be further from the truth.

Indeed, the inverse relationship between farm size (or plot size, since the inverse relationship has also been observed within farms) and productivity can be the result of three different things, none of which indicates that smaller farms should be relied upon for food security.

### Three Explanations for the Inverse Relationship

The first explanation for the inverse relationship has to do with market failures. In one of the earliest studies on the inverse relationship, for example, Sen (1966) posited that everything else equal, households whose farms were smaller simply had more labor per hectare than larger farms. Though the labor market can usually absorb surplus labor, high unemployment might push households whose farms are smaller to use more labor than is optimal on their farms, with the end result being that households whose farms are smaller will have higher yields.

The second explanation for the inverse relationship has to do with omitted variables. Most empirical researchers working on the inverse relationship do not have precise measures of soil quality (e.g., carbon, nitrogen, and potassium in the soil as well as soil pH). But since most people choose to cultivate good lands first and will tend to choose relatively worse lands as they increase the size of their farms, it is very likely that larger farms have lower soil quality on average. Once again, the end result is that larger farms appear less productive than smaller farms.

The third explanation for the inverse relationship has to do with measurement error. This happens when the size of a farmer’s landholdings is measured with error and that error is negatively correlated with the true size of a farmer’s landholdings. For example, if farmers consistently over-report the size of their landholdings (say, because land is a measure of prestige and political power, as in many developing countries), one would find a spurious inverse relationship between farm size and productivity.

In a 2010 World Development article on the inverse relationship, my coauthors and I had access to precise soil quality measurements for a sample of Malagasy farms. We found that those usually omitted soil quality measurements explain only a very small fraction of the observed inverse relationship and that market failures explain most (though not all) of it.

### Policy (Non)Implications

None of the explanations above mean that smallholders are in a better position to feed the world, and to claim otherwise is dangerous, as it leaves people with gravely mistaken beliefs about food security.

To see this, you only need to ask yourself whether you really believe that the key to improving food security and feeding the world lies in breaking up large farms into smaller ones.

1. Jeremy Cherfas

Are there other possible explanations. For example, what happens when you differentiate farms that have hired labour from farms that do not? Is it a myth that the same farmers who performed so poorly on Soviet farms produced bumper crops on their own land? And could there possibly be a role for greater agricultural biodiversity on smaller plots, which are less intensive and so less genetically uniform?

2. Marc F. Bellemare

Thanks for your comment, Jeremy. There is certainly room for other explanations, but bear in mind that in my work on the topic:

1. Controlling for the types of labor (adult, children, hired, reciprocal help) hardly makes a dent in the inverse relationship,
2. Soviet farmers, as I see it, started experiencing abysmal yields once Lenin’s New Economic Policy was ditched and Stalin de-kulakized, and
3. In my work, everyone grows the same thing (i.e., makalioka rice, or the pink Malagasy rice), and controlling for the technology, seeds, etc. used hardly makes a dent in the inverse relationship.

As I mentioned on Twitter, empirically, the inverse relationship is both a correlation (i.e., an unconditional relationship between yield, or kg/ha, and plot size, or ha) and an empirical relationship that holds ceteris paribus.

Now, it is perfectly possible that smallholders can feed the world — short of running a randomized controlled trial on that question, we’ll never know for sure. My point here is just that the inverse relationship is not evidence that that’s the case.

Hi,

Could you explain me the ‘optimal’ amount of labor for a farm? This is an arbitrary figure set by market prices which are far from ‘optimal’ due to the non-inclusion of both social as well as ecological externalities. Your neoclassical reasoning only holds if your only goal is to maximize monetary value which does not equate with land productivity, welfare and especially food security.

4. Marc F. Bellemare

Yes, absolutely. Suppose the farmer produces some output q according using labor L following the process q = F(L), where F’>0 (the more labor he uses, the more output he gets) and F”<0 (the more labor he uses, the more output he gets, but at a decreasing rate). Suppose further that the price of output q is p and the price of labor (the prevailing wage) is w.

In that case, profit maximization implies that the farmer choose to allocate labor up to the point L*, where L* is such that pF'(L*) = w, or F'(L*) = w/p, or the point where the marginal product of labor pF'(L) is equal to its marginal cost w, since that's the point where profit is maximized.

This L* is not an arbitrary figure: it is driven by the interplay of (i) the price of output p, (ii) the price of labor w, and (iii) production technology and soil characteristics F'(L). And it is not purely driven by market prices, as you imply, because it is also driven by production technology and soil characteristics. My neoclassical reasoning holds for profit maximization. And of course profit maximization is not the same as land productivity, thought higher profits do translate into higher welfare.

Now, you will counter, not everyone is out to maximize profit. And you would be right for many cases in developing countries. In the presence of enough market failures, the farmer can no longer behave as a profit maximizer, because his production decision is no longer separable from his consumption decision (Singh et al., 1986). Such cases can lead to the inverse relationship between farm size and productivity, but do note that when that happens, the farmer almost surely attains a lower level of welfare relative to when there are no market failures.

If only reality would be as simplistic as the underlying assumptions of microeconomics.

If you are really following the microeconomics model, the farmer has a tougher choice than only maximizing his farm production function. In fact, he is faced with an even greater challenge, given his skills, talents, learning ability and given market prices for all goods and services in the market (which undoubtedly vary over time), how much of his labor and time should he invest in his farming activities at one time? After all, to maximize his life income (of which maximizing farm income is but a part), he has to make a continuous choice what goods or services to produce. So given all these constraints, he has to continuously monitor all his abilities and the market prices to see what mix of goods and services he needs to produce. Because what happens when the farmer who allocates the optimal amount of labor in his farming activity would actually make twice the income as a banker? You would have a violation of the rationality assumption underlying all microeconomics.

Now, you can argue that the farmer acts ‘as if’ he has made all these calculations in his head and is a farmer because that is what he is best at. Or I may argue that reality is much more influenced by social, technological, political and cultural phenomena than it is by solving for the derivative of a simplistic production function. Because even with such a production function and only looking at his farming activities, it is highly unlikely that the farmer has the ability or time to find the optimal allocation of his labor thanks to the white noise term. Farmers should solve their equation taking account of endogeneity, multicollinearity and other econometrical issues to really get a grip on their optimal amount of labor. And all that in a continuously changing world with technological, social and environmental shocks. I pity the farmer.

And we haven’t talked about externalities yet. As market prices do not take account of externalities, his optimal choice might actually not at all maximize his profits as his current production methods are eating away his natural capital.

In short, reality is so much more complex than what microeconomics is assuming. Of course there is a merit in simplifying the reality and drawing conclusion but in the case of microeconomics it is plainly oversimplified.

It is exactly the understanding of the shaky foundations of economics that led to me to not pursue a career in agricultural economics (while having a masters in economic research) and to pursue a career in agriculture instead. Don’t talk it, walk it.

In my worldview, the key to understanding the farmer’s decision is an interplay between cultural, social and technological factors on the economic condition that profits have to be sufficient (not maximized). Understanding those factors is much more the key to understanding and influencing behavior than it is by drawing conclusion from assumed production functions.

My two cents.

Because empirical research validates the strength of a model to describe the data chosen by the researcher.

Empirical research is not objective as even the research question and the choice of data are already infused with subjective choices. What empirical research does is to give the ability to compare one model with another, not to say whether or not a model describes reality in a sufficient matter.

If that was indeed the case, there would be no hayek vs keynes or there would be no unexpected financial crises as our financial models are (or rather were) empirically sound.

What bothers me the most is that you take your subjective results and draw policy (non)implications that bigger is better because your model shows that this maximizes profits. Furthermore, you link this with food security which is much more an issue of accessibility than availability. The world is producing enough food to feed everyone. However, bringing the control over food to a couple of gigantic actors who only have profits as incentive does indeed lead to food insecurity, even for demographic groups within the USA.

Your microeconomic model is correct if you limit yourself to answer the question: how do we maximize profits?

However, it is a woefully inadequate to answer any question about among others land productivity, food accessibility and hence food security.

7. Marc F. Bellemare

You write that I “take [my] subjective results and draw policy (non)implications that bigger is better because [my] model shows that this maximizes profits.”

Did you actually read our paper? Nowhere do we assume profit maximization. The empirical results in our paper are not based on any microeconomic model: We regress the logarithm of rice yield on a bunch of control variables, including some soil quality measurements and household fixed effects; full stop.

If you want to criticize our empirics — and God knows there is plenty for you to criticize — you can do so more accurately by questioning our functional form, the stable unit treatment value assumption across the plots owned by a given household, or the fact that we do not have a nationally representative sample.

Also, nowhere do I advocate that food be produced by a monopoly or an oligopoly, and I doubt that’ll actually ever happen (on that, read Ronald Coase’s 1937 “The Nature of the Firm).

I’m merely saying that (i) there is no empirical evidence that smallholders can feed the world, but (ii) there is plenty of evidence that shows that the inverse relationship between farm size and productivity is spurious or the results of market failures which, once they are addressed (as they inevitably will in the process of economic development) will mean smallholders are no longer more productive.

I realize you have an ax to grind with economists and that you dislike what I have to say. That, however, is not a license for you to post rubbish on my blog.