<p>This index is derived solving the cost minimization problem</p>
<p><span class="math-container">$$\min_x \{ p^\top x \lvert \bar u(x) \geq \bar u\},$$</span></p>
<p>where <span class="math-container">$p=(p_1,...,p_J)$</span> is price vector and <span class="math-container">$x=(x_1,...,x_J)$</span> is the consumption bundle. The problem can therefore also be written as</p>
<p><span class="math-container">$$\min_{x_1,...,x_J} \sum_{j=1}^J p_j x_j \\[8pt]
s.t. \ \ u(x_1,...,x_J)\geq \bar u.$$</span></p>
<p>One interpretation of this problem is that the underlying question asked and answered by solving the problem is:</p>
<blockquote>
<p><strong>How much money is the consumer required to spend at the price <span class="math-container">$p$</span> in order to achieve the utility <span class="math-container">$\bar u$</span>?</strong></p>
</blockquote>
<p>Solving the problem you get the Hicks demand functions</p>
<p><span class="math-container">$$x^\star(p,\bar u),$$</span></p>
<p>and inserting these in the budget equation <span class="math-container">$p^\top x$</span> you get the expenditure function</p>
<p><span class="math-container">$$E(p,\bar u) = p^\top x^\star(p,u) = \sum_j p_j x_j^\star(p,u)$$</span></p>
<blockquote>
<p><strong>It is worth noting that the problem are analogous to the cost minimization problem for firms</strong></p>
</blockquote>
<p>Firms minimize cost of producing output <span class="math-container">$y$</span> given a production technology captured by production function <span class="math-container">$f(x)$</span>, hence the firm solve the problem</p>
<p><span class="math-container">$$\min_x \{p^\top x \lvert f(x) \geq y\},$$</span>
here prices are now factor prices. The solution here is conditional demand functions <span class="math-container">$x^\star(p,y)$</span> and the cost function <span class="math-container">$C(p,y) = p^\top x^\star(p,y) = \sum_j p_j x^\star_j(p,y)$</span>.</p>
<h2>Deriving the unit cost of utility with CES preferences</h2>
<p>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.</p>
<p>So the problem to be solved is</p>
<p><span class="math-container">$$\min_x \sum_j p_j x_j \lvert u(x) \geq u$$</span>
where</p>
<p><span class="math-container">$$(1) \ \ u(x) = \left(\sum_j x_j^\alpha\right)^{1/\alpha}$$</span></p>
<p>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</p>
<p><span class="math-container">$$\mathcal L(x,\lambda) = \sum_j p_j x_j - \lambda(\bar u - u(x)),$$</span></p>
<p>with FOC</p>
<p><span class="math-container">$$\frac{\partial \mathcal L(x,\lambda)}{\partial x_k} = p_k + \lambda \frac{\partial u(x)}{\partial x_k} = p_k - \lambda \left(\sum_j x_j^\alpha\right)^{\frac{1}{\alpha}-1} x^{\alpha - 1} =  0,$$</span></p>
<p>this implies - using that constraint is binding so <span class="math-container">$\lambda &gt; 0$</span> - that</p>
<p><span class="math-container">$$\frac{p_k}{p_j} = \frac{x_k^{\alpha - 1}}{x_j^{\alpha-1}}\phantom{xxx} \Leftrightarrow \phantom{xxx} x_k = \left(\frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{1}{\alpha - 1}},$$</span></p>
<p>which is inserted in the constraint <span class="math-container">$u(x) = \bar u$</span> to get</p>
<p><span class="math-container">$$\bar u = \left(\sum_k \left( \left(\frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{1}{\alpha - 1}}\right)^\alpha\right)^{1/\alpha} = \left(\sum_k \left( \frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha},$$</span></p>
<p>the sum index is running over <span class="math-container">$k$</span> so not binding <span class="math-container">$x_j$</span> and <span class="math-container">$p_j$</span> in the expression and these can therefore freely be moved outside the sum to get</p>
<p><span class="math-container">$$\bar u = \left(\sum_k \left( \frac{p_k}{p_j}x_j^{\alpha-1}\right)^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} = \frac{x_j}{p_j^{\frac{1}{\alpha-1}}}\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha},$$</span>
which gives us the result that</p>
<p><span class="math-container">$$x_j^\star(p,\bar u) = \frac{p_j^{\frac{1}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u,$$</span></p>
<p>which is Hicks demand. To get expenditure function use</p>
<p><span class="math-container">$$E(p, u) = \sum_j p_j x_j^\star(p,u) =\sum_j p_j  \frac{p_j^{\frac{1}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u =  \frac{\sum_j p_j^{\frac{\alpha}{\alpha-1}}}{\left(\sum_k p_k^{\frac{\alpha}{\alpha-1}} \right)^{1/\alpha}} \cdot \bar u = \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}\cdot \bar u,$$</span></p>
<p>where it is easy to see that <span class="math-container">$\bar p:= \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}$</span> is the cost of each utility point since the derivative of <span class="math-container">$E(p,u)$</span> with respect to <span class="math-container">$u$</span> does not depend on the level of utility.</p>
<p>It is perhaps worth adding a final property, namely that the Hicks demand can be reexpressed in terms of the price index:</p>
<p><span class="math-container">$$x^\star_j(p,u) = \frac{p_j^{\frac{1}{\alpha-1}}}{\sum_j p_j^{\frac{\alpha}{\alpha - 1}}}u \bar p = s_j(p) u \bar p = s_j(p)E(p,u),$$</span></p>
<p>where <span class="math-container">$s_j(p):=\frac{p_j^{\frac{1}{\alpha-1}}}{\sum_j p_j^{\frac{\alpha}{\alpha - 1}}}$</span> is denoted with &quot;s&quot; to indicate that it is a share in the sense that it is a positive numver in (0,1) and <span class="math-container">$\sum_j s_j(p) = 1$</span>.</p>

CC BY-SA 4.0
2020-12-15T14:11:44.320Z
2020-12-15T16:42:11.477Z
2020-12-15T16:42:11.477Z
Jesper Hybel
<p>I have recently encountered (in a few applications), the Dixit-Stiglitz Pricing index. Consider a Constant elasticity of substitution form of utility $$U=\left(\sum_{i=1}^{N}c_{i}^{1-\frac{1}{\sigma}}\right)$$</p>

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

<macroeconomics>
Dixit-Stiglitz Pricing Index
ChinG
# Dixit-Stiglitz Pricing Index

(Asked by: username_0 on 2015-11-26)

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?

**Tags:** macroeconomics


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## 1 answers

### Answer by username_1 on 2020-12-15. Score: 4 | Accepted answer

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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