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Your post is unclear and confusing. In some places you assume a marginally employable worker that is employed. Elsewhere you assume a not employed person. You seem to be asking if a not employed person receiving UBI creates value for society. I will answer to that in the hope that it is a succinct re-expression of your intended question. I will ignore non-...


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Notice that the value of $w_r(n_r)$ will be such that $V(e_r) = V_r(e_0)$ so we can just put this equal to $V_r$. Then the three conditions are $$ \begin{align*} &V_r = w_r - e_r + \beta (q V_u + (1 - q) V_r) \tag{1}\\ &V_r = w_r - e_0 + \beta (q V_u + (1 - q) V_r) - \beta \sigma(V_r - V_u) \tag{2}\\ &V_u = \bar w_0 + \beta (n_r V_r + (1 - n_r) ...


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