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I have recently encountered (in a few applications), the Dixit-Stiglitz Pricing index. Consider a Constant elasticity of substitution form of utility $$U=\left(\sum_{i=1}^{N}c_{i}^{1-\frac{1}{\sigma}}\right)$$

This index basically aggregates prices to give a general term as a function of prices that yields a unit cost of utility. How is the index derived?

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This index is derived solving the cost minimization problem

$$\min_x \{ p^\top x \lvert \bar u(x) \geq \bar u\},$$

where $p=(p_1,...,p_J)$ is price vector and $x=(x_1,...,x_J)$ is the consumption bundle. The problem can therefore also be written as

$$\min_{x_1,...,x_J} \sum_{j=1}^J p_j x_j \\[8pt] s.t. \ \ u(x_1,...,x_J)\geq \bar u.$$

One interpretation of this problem is that the underlying question asked and answered by solving the problem is:

How much money is the consumer required to spend at the price $p$ in order to achieve the utility $\bar u$?

Solving the problem you get the Hicks demand functions

$$x^\star(p,\bar u),$$

and inserting these in the budget equation $p^\top x$ you get the expenditure function

$$E(p,\bar u) = p^\top x^\star(p,u) = \sum_j p_j x_j^\star(p,u)$$

It is worth noting that the problem are analogous to the cost minimization problem for firms

Firms minimize cost of producing output $y$ given a production technology captured by production function $f(x)$, hence the firm solve the problem

$$\min_x \{p^\top x \lvert f(x) \geq y\},$$ here prices are now factor prices. The solution here is conditional demand functions $x^\star(p,y)$ and the cost function $C(p,y) = p^\top x^\star(p,y) = \sum_j p_j x^\star_j(p,y)$.

Deriving the unit cost of utility with CES preferences

I will now show how to derive the unit cost of utility given CES preferences. As noted about this is the same as deriving the cost function of firm using CES production technology. I will not show that this is a price index, which would require setting up the formal properties of what a price index is, which is not trivial.

So the problem to be solved is

$$\min_x \sum_j p_j x_j \lvert u(x) \geq u$$ where

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

To solve the problem I set up Lagrangian function and find FOC thereby arriving at the result that relative prices equal MRS. The Lagrangian is

$$\mathcal L(x,\lambda) = \sum_j p_j x_j - \lambda(\bar u - u(x)),$$

with FOC

$$\frac{\partial \mathcal L(x,\lambda)}{\partial x_k} = p_k + \lambda \frac{\partial u(x)}{\partial x_k} = p_k - \lambda \left(\sum_j x_j^\alpha\right)^{\frac{1}{\alpha}-1} x^{\alpha - 1} = 0,$$

this implies - using that constraint is binding so $\lambda > 0$ - that

$$\frac{p_k}{p_j} = \frac{x_k^{\alpha - 1}}{x_j^{\alpha-1}}\phantom{xxx} \Leftrightarrow \phantom{xxx} x_k = \left(\frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{1}{\alpha - 1}},$$

which is inserted in the constraint $u(x) = \bar u$ to get

$$\bar u = \left(\sum_k \left( \left(\frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{1}{\alpha - 1}}\right)^\alpha\right)^{1/\alpha} = \left(\sum_k \left( \frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha},$$

the sum index is running over $k$ so not binding $x_j$ and $p_j$ in the expression and these can therefore freely be moved outside the sum to get

$$\bar u = \left(\sum_k \left( \frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} = \frac{x_j}{p_j^{\frac{1}{\alpha-1}}}\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha},$$ which gives us the result that

$$x_j^\star(p,\bar u) = \frac{p_j^{\frac{1}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u,$$

which is Hicks demand. To get expenditure function use

$$E(p, u) = \sum_j p_j x_j^\star(p,u) =\sum_j p_j \frac{p_j^{\frac{1}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u = \frac{\sum_j p_j^{\frac{\alpha}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u = \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}\cdot \bar u,$$

where it is easy to see that $\bar p:= \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}$ is the cost of each utility point since the derivative of $E(p,u)$ with respect to $u$ does not depend on the level of utility.

It is perhaps worth adding a final property, namely that the Hicks demand can be reexpressed in terms of the price index:

$$x^\star_j(p,u) = \frac{p_j^{\frac{1}{\alpha-1}}}{\sum_j p_j^{\frac{\alpha}{\alpha - 1}}}u \bar p = s_j(p) u \bar p = s_j(p)E(p,u),$$

where $s_j(p):=\frac{p_j^{\frac{1}{\alpha-1}}}{\sum_j p_j^{\frac{\alpha}{\alpha - 1}}}$ is denoted with "s" to indicate that it is a share in the sense that it is a positive numver in (0,1) and $\sum_j s_j(p) = 1$.

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