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This is a generic approach (find a transformation that make the utility problem easier to work with. Assume a household has utility $$U(x,y) = x^\alpha y^\beta$$. A utility function is a convenient way to represent preferences. However, we saw in the chapter that utility functions have many limitations. One limitation is that although utility ...


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Objective function of Lagrangian can be set up either with $+\lambda$ or $-\lambda$, depending on how you solve the budget constraint. Actually, for the solution it does not matter if $\lambda$ has negative or positive sign in the equation. You can clearly see it from the formula if you expand the second term: $$ \mathcal{L}(x,y, \lambda) = U(x,y) + \lambda(...


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I think what it’s referring to optimal $y$ (i.e. what would be normally in calculus denoted as $y^*$- but you already use that for the initial stock). It makes sense that to maximize the utility MRS should be equal to marginal productivity since MRS should be equal to ratio of prices and in case like this where the person produces goods for themselves the ...


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