# **3. Estimating the inverse productivity relationship**

Production is commonly shown to be negatively correlated with farm area. This inverse productivity relationship has been tried and tested by economists for decades, most recently by Carletto, Savastano and Zezza (2013). While some, such as Bhalla and Roy (1988), have argued that land quality differences fully or near-fully explain the inverse relationship, others, like Lamb (2003) and Barrett, Bellemare and Hou (2010) conclude that land quality does contribute to the relationship, but does not explain it away. However, Lamb (2003) notes that results from fixed and random effects models imply that the land size variable may be subject to measurement error suggesting that part of the inverse relationship may be explained by land mismeasurement. If measurement error is a contributing factor, the use of GPS could ameliorate some of the inverse relationship between productivity and land area. To analyze whether the measurement methodology influences this relationship, descriptive statistics are first examined.

Table 5 provides means for the value of output<sup>11</sup> per farm as well as the value of output per acre using both the GPS and self-reported areas as the basis. The means are disaggregated by land area designations to provide an initial assessment of the inverse productivity relationship. The existence of a basic inverse relationship is clear in the descriptive statistics. While the value of output per farm increases with size, as is expected, the value of output per acre falls. This is generally true for both the GPS-measured land area as well as with the self-reported areas although there are some exceptions within the countries. In Malawi, the mean values for both GPS and self-reported land areas do not explicitly exhibit an inverse productivity relationship with rising productivity for larger plot areas, which may be due to high returns in key crops such as tobacco. Niger also has one category that does not match the pattern although given the sample sizes for smaller areas this may be due to outliers.

While looking at the means, an inverse productivity relationship appears to emerge for both GPS and self-reported areas. With simple mean relationships, however, it is difficult to compare across the two measures and hard to know whether other factors besides land size, which are not controlled for, are driving the relationship. To assess whether the measurement methodology used impacts the strength of the relationship to any significant degree, we turn to regression analysis.

# _Analysis of value of output and productivity_

Following the standards for productivity (Carlson, Zilberman, & Miranowski, 1993), to assess the inverse productivity relationship one or two of the following reduced-form empirical relationships are estimated: where Yi is the value of output on farm “i”, Li is land area in acres, Pi are plot characteristics— including farmer-reported soil type—and control for land quality variation, Hi are household characteristics, Xi are farm inputs, and θr are locational (enumeration area) effects that control for local factors not controlled for otherwise (market prices, agroecology, etc.).

> 11 Output values are calculated using household-provided conversion factors to kilograms and imputing market prices based on enumeration area medians (with a minimum of three observations per crop and enumeration area to minimize the influence of seasonality and outliers). Unit prices are specific to crop and crop condition where applicable (i.e, fresh, dry, shelled, unshelled, etc). The value of output was separately constructed for some countries using unit-specific prices rather than kilogram conversions to validate the robustness of the output calculations.

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