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A useful addition to the model would be adding hand-to-mouth consumers. These would be represented by households where consumption is equal to labor income and savings is zero (and thus no need to solve a FOC for savings).
The distribution of these agents would need to be accounted for. One way to do this would be to make the lambdas parameter a matrix with two rows, where the first represents the fraction of hand-to-mouth consumers in each j group and the second row is the fraction of forward looking consumers in each j group.