beyond this by recording parcel-level GPS coordinates for all of a household’s parcels in the cell of residence<sup>6</sup> so as to allow computation of the mean Euclidian distance between homestead and parcel, the length of the path that has to be traversed to visit all parcels, and the time required to walk these. Use of the latter is justified because, in Rwanda’s hilly terrain, a straight line will often not be the most effective way of getting from location A to location B. To implement this, we use the publicly available Aster Global DEM to derive a 30 by 30 meter slope grid and apply appropriate friction parameters to compute the least cost walking distance between any two points.<sup>7</sup> Figure 1, which displays the DEM together with the location of the homestead and parcels for one of our sample households with 4 parcels (at an altitude between about 1,500 and 2,100 m), one of which at the homestead, illustrates this procedure graphically. It is easily verified that the straight line or Euclidian distance (in red) may be different from the least cost path between any pair of parcels (in blue). Figure 2 displays the minimum path to visit all the household’s parcels starting and ending at the homestead graphically.

# **2.3 Econometric approach**

Before considering the impact of fragmentation on productivity or farm efficiency, it is helpful to first identify the extent to which fragmentation helps Rwanda’s smallholder farmers manage risks. To this end, we use responses to the question of whether a crop shock was experienced on a given parcel to compute the share of land used by a household that was affected by a crop shock.<sup>8</sup> We then regress this variable on indicators for fragmentation (number of parcels, Simpson index, Shannon index, number of crops), farm-level characteristics such as the coefficient of variation in parcels’ altitude, the Herfindahl index of diversity in soil quality and slope<sup>9</sup> , as well as standard attributes including size and composition of the household and age and education of the head to obtain a reduced-form estimate of the extent to which any of these factors may help to reduce the likelihood of being affected by a crop shock.

> 6 Expanding beyond the cell would have been too costly.

> 7 The Aster model is available at <u>http://asterweb.jpl.nasa.gov/. We use Tobler’s (1993) hiking function—estimated using Imhof’s (1950) empirical</u> data—that models for walking on footpath by taking into account the slope of the terrain. It is an exponential function of slope ( _S_ ), defined as _W_ = 6*exp{-3.5*abs( _S_ + 0.05)}, that takes an average speed of 5 km/h on a flat terrain.

> 8 The question was specifically phrased as: “While in the field or at the time of harvest, was there any crop damage/loss?” 9 The Herfindahl index (fractionalization index for soil type and topography) that captures the probability that two randomly selected parcels from a given household will not belong to the same soil type or topography is defined as H=∑ ) where _i_ indexes groups, _n_ is the total number of groups (soil type or topography) in each household and _pi_ is the share of group _i_ in the household (i.e., the area share of a given soil type or topography). nn ii=1 ii ii It ranges between zero and one, moving from a homogenous to a more diversified land type. pp (1 −pp

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