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Assume that preferences are given by a utility function is given

$$u(x_1,x_2) = (x_1^\rho + x_2^\rho)^{1/\rho}$$

what then are the Marshall demand given budget constraint

$$p_1x_1 + p_2x_2 \leq I$$

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    $\begingroup$ Could you please show us your attempt per our policy on self-thought/homework questions? $\endgroup$ – 1muflon1 Dec 15 '20 at 17:30
  • $\begingroup$ I will answer the question myself. $\endgroup$ – Jesper Hybel Dec 15 '20 at 17:31
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    $\begingroup$ 1) Currently I don't know the answer but can easily find it 2) It is a standard question the answer to which may be of benefit to other users and hence a contribution to the site. If the question and answer already exist you are welcome to close as a duplicate, in which case it would not contribute to the site. $\endgroup$ – Jesper Hybel Dec 15 '20 at 17:38
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    $\begingroup$ Yes, I am familiar with those rules. I do not see how I am breaking any of those rules but let me know if you think otherwise. Also I have seen other user ask and answer questions themselves, questions that I myself have often benefited from reading so I do not believe I am doing anything detrimental to site or not in line with site policy. $\endgroup$ – Jesper Hybel Dec 15 '20 at 17:56
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    $\begingroup$ You are allowed to answer your own questions, however according to our rules: When posting a homework question, it is therefore essential that you demonstrate some evidence of having attempted to answer the question independently. Regardless of whether it is for actual homework or not (which we cannot verify) any homework-esque question should follow those rules or be closed. Your question does not demonstrate any attempt at solving the question $\endgroup$ – 1muflon1 Dec 15 '20 at 18:00
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To answer this question I will first generalize slightly the question to deal with the utility function

$$u(x) = \left(\sum_j x_j^\alpha\right)^{1/\alpha}$$

The Marshall demand can be written as

$$x_k^\star(p,I) = \left(\frac{p_k}{\bar p}\right)^{\frac{1}{\alpha - 1}} \frac{I}{\bar p} = \frac{p_k^\frac{1}{\alpha - 1} I}{\sum_j p_j^\frac{\alpha}{\alpha-1}},$$

and the value function as

$$V(p,I) := u(x^\star) = \frac{I}{\bar p}$$

where $\bar p := \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}$ is the price index as derived here Dixit-Stiglitz Pricing Index.

To find the Marshal demand start from the standard condition that relative prices equal MRS

$$\frac{p_j}{p_k} = \frac{\partial u/\partial x_j}{\partial u/\partial x_k} = \frac{x_j^{\alpha - 1}}{x_k^{\alpha - 1}},$$

in order to get

$$p_k^{\frac{1}{\alpha - 1}} x_j = p_j^{\frac{1}{\alpha - 1}} x_k,$$

which implies that

$$p_k^{\frac{\alpha}{\alpha - 1}} x^\alpha_j = p_j^{\frac{\alpha}{\alpha - 1}} x^\alpha_k,$$

and by summing over $j$ this results in the equation

$$p_k^{\frac{\alpha}{\alpha - 1}} \sum_j x^\alpha_j = x^\alpha_k \sum_j p_j^{\frac{\alpha}{\alpha - 1}} ,$$

which implies that

$$(A)\ \ \ p_k^{\frac{1}{\alpha - 1}} \left(\sum_j x^\alpha_j \right)^{1/\alpha}= x_k \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} ,$$

where multiplying with $p_k$ and summing over $k$ results in the equation

$$\sum_k p_k^{\frac{\alpha}{\alpha - 1}} \left(\sum_j x^\alpha_j \right)^{1/\alpha} = I \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha},$$

from which I isolate the factor including $x_j$'s to get

$$\left(\sum_j x^\alpha_j \right)^{1/\alpha} = I \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{\frac{1-\alpha}{\alpha}} = \frac{I}{\bar p},$$

where the LHS is the utility function equal to an expression including only income $I$ and prices which therefore are the value function $V(p,I) = I/\bar p$. Inserting this expression in (A) and isolation $x_k$ gives the Marshall Demand

$$p_k^{\frac{1}{\alpha - 1}} \frac{I}{\bar p}= x_k \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} \Leftrightarrow x_k^\star(p,I) = \left(\frac{p_k}{\bar p} \right)^{\frac{1}{\alpha - 1}} \frac{I}{\bar p}.$$

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