Marshall demand for simple CES utility

Question

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

$$p_1x_1 + p_2x_2 \leq I$$

Answer 1

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

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