“Immigration, Search and Redistribution: A Quantitative Assessment of Native Welfare” A Reply

Abstract Stark and Byra (2025) identify a mistake in how capital income is calculated in the supplementary files of Battisti et al. (2018) and point out that our wage bargaining equation does not correspond to the Nash bargaining solution in the presence of wage taxation. We acknowledge both points and thank the authors for pointing out these issues. The bargaining solution we use in Battisti et al. (2018) is the Kalai (1977) proportional bargaining solution with fixed surplus shares, not Nash bargaining. In our reply, we show that if we calibrate the model to wage tax rates, correcting the coding mistake has only a minor effect on our original results if maintaining the original Kalai bargaining. If we also switch to the Nash bargaining solution, the estimated gains from immigration become smaller. Welfare effects depend on the precise nature of bargaining and tend to be larger with Kalai proportional bargaining. Average total welfare gains from immigration are then 0.37% for both low-skilled and high-skilled natives, instead of 1.25% for high-skilled and 1.00% for low-skilled natives in Battisti et al. (2018).