adverse effects on land productivity (Blarel _et al._ 1992). Despite Rwanda is characterized by higher levels of fragmentation (6 parcels with a median Simpson index of 0.66 per household), there is no significant relationship between levels of fragmentation and yields, irrespectively of whether efforts are made to control for endogeneity. While pointing towards risk reduction as a potential reason for persistence of fragmentation, links between fragmentation and variability of output remain tentative at best (Blarel _et al._ 1992).

# **2.2 Measurement issues and analytical approach**

With the highest population density in Africa (384 inhabitants/km<sup>2</sup> overall and 526/km<sup>2</sup> of agricultural land) and some 85% of the population having agriculture as their main income source, effective land use and land-related investment are critical for poverty reduction in Rwanda (Republic of Rwanda 2009). In 2008, the average household had only 0.72 ha of land, below the threshold of 0.75 ha estimated to be required to meet a family’s subsistence needs with available technology. Negative impacts from population growthinduced fragmentation are widely viewed as having led to serious environmental challenges (Republic of Rwanda 2004) and even conflict (Andre and Platteau 1998). In fact, to stem fragmentation, the law prohibits fragmentation of land below a size of 1 ha,<sup>5</sup> though more than 90% of parcels fall below this size (Ali and Deininger 2014).

A measure of fragmentation that is widely used in the literature, in addition to the number of parcels, is the Simpson index, defined as _SI_ =1- Σ _ai_<sup>2</sup> / (Σ _ai_ )<sup>2</sup> where _a_ is the area of parcel _i_ =1,.., _n_ . This index varies between 0 and 1 with zero for a completely consolidated operation. To account for the ability to grow a larger number of crops as one of the potential benefits of fragmentation, we complement this with the Shannon crop diversity index, defined as _SH_ =- Σ ( _Pi_ * ln _Pi_ ) where _Pi_ is the share of area covered by a specific crop. This index which combines both richness of type (number of crops) and relative abundance (land allocation among crops), equals zero if there is only one crop (i.e., no diversity) and increases with the number of cultivated crops as well as with more even shares by different crops, reaching its maximum when crops are cultivated in equal shares (i.e., _Pi_ =1/ _J_ ) where _J_ is the maximum possible number of crops cultivated. Finally, parcel altitude and its variability provide a third indicator of fragmentation.

As they do not incorporate information on plot location, the above measures are at best imperfect proxies for the cost of having to travel between plots. That is, households with given number and size distribution of plots will have the same Simpson index irrespectively of whether these plots are located next to each other and the homestead or kilometers away and separated by rivers or mountain ridges. In fact, some surveys have included questions regarding travel time between homestead and plots to fill this gap. We go

> 5 The 2005 Organic Land Law (OLL) prohibited sub-division of agricultural land parcels less than one ha and required approval for any subdivision of parcels between 1 and 5 ha. The prohibition of sub-division is upheld in the 2013 OLL revision.

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