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You want to prove that $U(\alpha) = \int_z u(\alpha\cdot z) dF(z)$ is continuous, given that $u(w)$ is continuous? By definition $|U(\alpha)-U(\alpha')| = |\int_z u(\alpha\cdot z) - u(\alpha' \cdot z) dF(z) | <\int_z |u(\alpha\cdot z) - u(\alpha' \cdot z)| dF(z)$. Take the supremum over $z$, to get $$ \int_z |u(\alpha\cdot z) - u(\alpha' \cdot z)| dF(z) &...


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If $p_1, p_2 >0$, you should consume no $x_2$, because it enters your objective negatively. So you should consume $x_1^* = (p_1 e_1 + p_2 e_2)/p_1$ and $x_2^* = 0$. If the prices are not necessarily positive, you might be able to get infinite utility. For example, if $p_2 = -2$ and $p_1 = 1$, adding 1 unit of $x_2$ relaxes your budget constraint by 2, so ...


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From your formula for $x$ when $y=0$, you should be able to find $U$ in terms of $M$ and $p_x$ when $y=0$. Similarly, $U$ in terms of $M$ and $p_y$ when $x=0$. The key then is to find the critical price ratio at which, to maximise $U$, the switch needs to occur from $y=0$ to $x=0$. Can you take it from there?


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If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over $n$ goods, $x_1,\dots,x_n$) takes the form \begin{equation} u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho}, \end{...


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At first the individual consumes 9 units of $x_1$ and 5 units of $x_2$. But then the consumption of $x_1$ drops to 4, and you get enough $x_2$ to keep the previous profit unchanged. Determine the units of $x_2$ that the individual consumes. Profit is an odd word here, in more mathematical terms what you are being asked to do is to evaluate $$u(9,4)=u(4,x_2)...


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The above answer by Regio covers most of the answer Just to emphasize further : Consider a 2 goods case. We define MRS as the opportunity cost of consuming one more unit of good 1. Opportunity cost means "What Am I Giving Up ?". Since you are consuming only two goods, you are giving up some amount of good 2 in order to consume this one more unit of good 1. ...


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It is implicit in the interpretation: Mas-Collel: the amount that must be given (+) to compensate for a reduction (-). Reny: The rate at which good j can be exchanged (+ & -) for good i. The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good $i$...


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