# **Introduction**

The existence of an inverse farm size–productivity relationship (IR), defined as a smooth tendency of productivity to decline with farm size, has been considered as a well-established finding of agricultural economics, and a stylized fact to guide rural development strategies and land reform to promote efficiency and overall equity. First observed in Russia (Chayanov, 1926), then in the dualistic agriculture labor market in India (Bardhan 1973, Sen 1975, Srinivasan 1972), IR was initially explained by labor market imperfections and supervision constraints, with small-scale farmers having lower opportunity and supervision costs of their labor than operators of larger farms.

From the economic perspective, IR implies that diseconomies of scale characterize agriculture systems for several possible reasons, including the failure of land and labor markets to equalize production efficiency across farm size distribution. From the policy perspective, reforms to redistribute land from large towards small farms appear justified not only on equity but also on efficiency grounds.

In most empirical studies, IR appears to be a smooth tendency of land productivity to decline with farm size and thus is not limited to a different pattern of resource uses between large and small farms. While different reservation wages could account for family versus non-family farms, this would not explain why land productivity appears to decline within small family farms as well.

The IR hypothesis has been tested in a variety of studies using households and plot level data, in cross sectional or country frameworks, over time using panel data to account for farmers’ and plots’ unobserved heterogeneity, and controlling for land quality. The availability of a more accurate measure of land area by means of GPS has seemed to add further proof for the existence of IR, even after eliminating farmers’ self-reported land bias. More recently, the use of geo-referenced measures of soil quality and infrastructure has helped in rejecting the hypothesis that IR might be the fruit of statistical artefacts.

In this paper, we analyze the land productivity–farm size relationship using a quantile regression approach to test whether farm size affects differently farmers with low productivity as compared with farmers of average or higher levels of productivity. In this framework, we aim to understand how farm size affects farmers’ productivity achievements across the whole distribution of land productivity.

The large majority of the IR studies have modeled farm productivity as a function of land area and other covariates using an OLS approach. The use of georeferenced variables that correct for measurement errors, objective measures of soil quality, and use of panel data to account for farmer and plot heterogeneity do not change this core framework, which is essentially based on the study of the effect of farm size on the conditional mean of land productivity. However, this approach clearly provides only a partial view of the relationship, as we might be interested in describing the relationship at different points in the conditional distribution of agriculture productivity. By supplementing the estimation of conditional mean functions with techniques for estimating an entire family of conditional quantile functions, we aim to provide a more complete statistical analysis of the stochastic relationships among random variables.

Quantile regression is not a novel approach. The econometric specification was introduced in 1968 by Koenker and Basset. More recent empirical applications concerned education (Koenker and Bilias, 2001) and peer effects (Levin, 2001), analysis of duration of employment spell (Horowitz and Neumann, 1987; Fitzenberger,1997), ecology (Cade and Noon, 2003; Chamaillé- Jammes and Blumstein, 2012; and Lemke K., 2012), health economics (Johar and Katayama, 2012), and in the context of political economy aspects of trade policy (Billger, 2009; Imai et al., 2012). To our knowledge, application to agriculture policies are found only in Evenson and Mwabu (E-M, 2001) and Scandizzo and Savastano (S-S, 2017). E-M studied the impact of Kenya’s extension service, and found that productivity response to acreage, measured through quantile regression, was not significantly different across quantiles, but displayed a concave shape, first rising and then falling with the size of the cultivated area. Testing the IR on national representative surveys from five Sub-Saharan African countries, S-S pointed out that IR may be an artifact of the central tendency indicators 2