As a result, the relative size of the migrant network should be a function of both whether or not cards have been issued and the time since cards have been issued in the village. Given that the size of the potential network has an upper bound, we expect years-since-IDs-issued to have a non-linear relationship with the share of the village labor force working as migrants, and growth in the size of this potential migrant network should decline after initially increasing with distribution of IDs. In Figure 4, we show a lowess plot of the relationship between years since IDs were distributed and the change in the share of village residents working as migrants from year t − 1 to t. Immediately after IDs are distributed, the share of the village labor force working as migrants grows sharply, and then slows after seven years. This pattern suggests non-linearity in the relationship between ID distribution and new participants in the village migrant labor force. We thus specify our instrument as a dummy variable indicating that IDs had been issued interacted with years since issue, and then experiment with quadratic, cubic and quartic functions of years-since-IDs were issued. We settle on the quartic function for our instruments because we find it fits the pattern of expanding village migrant share better than the quadratic or the cubic functions.<sup>20</sup> In order to exploit additional heterogeneity across villages in how the timing of ID card distribution affects the growth of migrant networks, we interact the quartic with the variance of historic village rainfall during important periods of the crop calendar.<sup>21</sup> Whether or not it is appropriate to interact the years-since-IDs were issued with the rainfall variance merits careful consideration. We expect that in villages with low rainfall variance, households would be less likely to respond to ID cards with migration and IDs will have less impact on growth of the migrant network from these villages. The interaction terms are valid instruments under the assumption that a village fixed effect controls for fixed differences across villages in the riskiness of the local environment, and that the rainfall variance interactions pick up differences in the rate of growth in networks across villages subsequent to distribution of IDs.

Since the differenced interaction term in equation (9), ∆(cit−1 · Mjt) is comprised of two endogenous regressors, we also include instruments for this term. We identify it using interactions between consumption in periods t − 3 and t − 4 and the eight instruments for the size of the migrant

> 20 Results in the paper are robust to using the quadratic or cubic functions of years-since-IDs were issued.

> 21 Giles and Yoo (2007) analyze the crop calendar and different combinations of monthly rainfall shocks, and demonstrate that for the villages and households of Anhui, Jiangsu, Henan and Shanxi, negative shocks between July and November are the strongest predictor of negative shocks to agricultural production during the following year. We have similarly examined the relationship between rainfall and the crop calendar for Hunan, Sichuan and Zhejiang, and found the shock from April to November to be more important, which makes some sense due to the longer growing season in these areas. Jilin’s crop calendar is more similar to Henan and Shanxi, so we use the July-November period for Jilin.

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