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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 $$\int_z |u(\alpha\cdot z) - u(\alpha' \cdot z)| dF(z) &... 0 If p_1, p_2 >0, you should consume no x_2, because it enters your objective negatively. So you should consume x_1^* = (p_1 e_1 + p_2 e_2)/p_1 and x_2^* = 0. If the prices are not necessarily positive, you might be able to get infinite utility. For example, if p_2 = -2 and p_1 = 1, adding 1 unit of x_2 relaxes your budget constraint by 2, so ... 1 From your formula for x when y=0, you should be able to find U in terms of M and p_x when y=0. Similarly, U in terms of M and p_y when x=0. The key then is to find the critical price ratio at which, to maximise U, the switch needs to occur from y=0 to x=0. Can you take it from there? 3 If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over n goods, x_1,\dots,x_n) takes the form u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho}, \end{... 0 At first the individual consumes 9 units of x_1 and 5 units of x_2. But then the consumption of x_1 drops to 4, and you get enough x_2 to keep the previous profit unchanged. Determine the units of x_2 that the individual consumes. Profit is an odd word here, in more mathematical terms what you are being asked to do is to evaluate$$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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