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$U(x^\star,y^\star) = a \ln( \frac{am}{p_1}) + b \ln(\frac{bm}{p_2})$ $= a \ln(a) + a \ln(m) - a \ln(p_1) + b \ln(b) + b \ln(m) - b \ln(p_2)$ Since $a + b = 1$, $= \ln(m) - a \ln(p_1) - b \ln(p_2) + ... • 2,006 1 vote Accepted ### Non-Negativity Constraints KKT The way I’d do it is to simply optimize the usual Lagrangian$\mathcal{L} = f(x,y) + \lambda (k - g(x,y))$. If all variables end up being non-negative, that is your solution. If any variable, let’s ... • 2,006 1 vote ### Utility maximization for a household consisting of a woman and a man, with gender discrimination I found BakerStreet's answer useful to intuitively make conclusions about the model, however I was then able to explicitly solve for the household's agents' optimal time allocations myself with a ... • 2,006 1 vote Accepted ### Missing Non-Negativity Constraint? The only thing that changes is the$k$first order condition. The one you got with the non-negativity constraint Lagrange multiplier is$-r + \lambda \alpha L^{1-\alpha} + \mu_k \leq 0$Substracting$\...
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