<p>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 <span class="math-container">$n$</span> goods, <span class="math-container">$x_1,\dots,x_n$</span>) takes the form 
<span class="math-container">\begin{equation}
  u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho},
\end{equation}</span>
where <span class="math-container">$\rho\in(-\infty,1]\setminus\{0\}$</span>, <span class="math-container">$\alpha_i\in[0,1]$</span> and <span class="math-container">$\sum_i\alpha_i=1$</span>. We interpret <span class="math-container">$\alpha_i$</span> as the consumption share of good <span class="math-container">$i$</span> and <span class="math-container">$\sigma\equiv\frac{1}{1-\rho}$</span> as the constant elasticity of substitution. Note also that when <span class="math-container">$\sigma=1$</span> (or <span class="math-container">$\rho\to0$</span>), we get the Cobb-Douglas utility form. </p>

<p>Solving utility maximization subject to the usual budget constraint, we get the demand for good <span class="math-container">$i$</span> as 
<span class="math-container">\begin{equation}
  x_i(p_1,\dots,p_n,M)=\frac{M(\alpha_i/p_i)^\sigma}{\sum_{j=1}^n\alpha_j^\sigma p_j^{1-\sigma}},\quad i=1,\dots,n.
\end{equation}</span>
Again, observe that when <span class="math-container">$\sigma=1$</span> we get the demand associated with Cobb-Douglas utility. </p>

<p>The elasticity of substitution governs how relative expenditures on different goods change as relative prices change. Take a two-good example. An increase in the relative price <span class="math-container">$p_1/p_2$</span>, i.e. good 1 becoming relatively more expensive, causes two effects simultaneously: </p>

<ol>
<li>per unit expenditure on good 1 rises, as good 1 now costs more in relative terms; and </li>
<li>quantity demanded for good 1 decreases due to law of demand.</li>
</ol>

<p>These are opposing effects on the expenditure on good 1 relative to that on good 2. It turns out that the elasticity of substitution determines which effect dominates. If <span class="math-container">$\sigma&gt;1$</span>, the second effect dominates, and if <span class="math-container">$\sigma&lt;1$</span>, the first effect dominates. When <span class="math-container">$\sigma=1$</span>, which is the case of Cobb-Douglas, the two effects cancel each other exactly, so that relative expenditure is independent of the relative prices, and depends only on preference parameters (the <span class="math-container">$\alpha_i$</span>'s). </p>

<hr>

<p>Of course, CES utility is not the only class of utility functions that generate demands dependent on prices of other goods. Another common form of utility function, the quasi-linear utility function, 
<span class="math-container">\begin{equation}
u(x_1,\dots,x_n)=x_1+v(x_2,\dots,x_n),
\end{equation}</span>
where <span class="math-container">$v(\cdot)$</span> is strictly increasing and strictly concave, also generates demand functions that depend on other prices. A common example is <span class="math-container">$u(x_1,x_2)=x_1+2\sqrt{x_2}$</span>. I trust that you can verify that both <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> depend on the two prices <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span>.</p>

CC BY-SA 4.0
2020-03-31T07:28:40.033Z
Thank you very much, I'll see if I can do the two case.
Jonathan Andrews
CC BY-SA 4.0
2020-03-31T20:15:47.747Z
If this adequately answers your question, please consider accepting the answer @JonathanAndrews
Brennan
CC BY-SA 4.0
2020-04-02T10:57:31.303Z
Gosh, I tried the two good case. Horrible, when you try substitution but pretty straight forward using Lagrangians. Thanks again for your help.
Jonathan Andrews
CC BY-SA 4.0
2020-04-02T19:37:25.093Z
@JonathanAndrews: Check out https://cameronharwick.com/blog/how-to-derive-a-demand-function-from-a-ces-utility-function/
Herr K.
CC BY-SA 4.0
2020-03-31T00:40:17.977Z
2020-03-31T00:40:17.977Z
Herr K.
<p>Probably a daft question but I derived an equation for a demand curve from a general Cobb-Douglas utility function <span class="math-container">$$U(x,y)=\beta x^{\alpha}y^{1-\alpha}$$</span> given a budget constraint <span class="math-container">$$M=xP_x+yP_y$$</span> and found that the quantity of <span class="math-container">$x$</span> demanded would be <span class="math-container">$$x=\frac{\alpha M}{P_x}$$</span>.</p>

<p>which surprised me in that this function is independent of the price of good <span class="math-container">$y$</span>. I see some references on these pages which suggests that I didn't make any spectacular  errors but, my elementary understanding is that the demand curve is a functions of many thing, not least the price  of other related goods. Now surely goods <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are substitutes to some extent?</p>

<p>Now, my question is this, is my understanding of this essentially correct and, what sort of utility functions would generate demand curves <span class="math-container">$x=f(P_x, P_y, M, etc)$</span>?</p>

<p>Thank you</p>

<utility><cobb-douglas><demand>
Deriving a demand curve from a Cobb-Douglas utility
Jonathan Andrews
# Deriving a demand curve from a Cobb-Douglas utility

(Asked by: username_0 on 2020-03-30)

Probably a daft question but I derived an equation for a demand curve from a general Cobb-Douglas utility function $$U(x,y)=\beta x^{\alpha}y^{1-\alpha}$$ given a budget constraint $$M=xP\_x+yP\_y$$ and found that the quantity of $x$ demanded would be $$x=\frac{\alpha M}{P\_x}$$.

which surprised me in that this function is independent of the price of good $y$. I see some references on these pages which suggests that I didn't make any spectacular errors but, my elementary understanding is that the demand curve is a functions of many thing, not least the price of other related goods. Now surely goods $x$ and $y$ are substitutes to some extent?

Now, my question is this, is my understanding of this essentially correct and, what sort of utility functions would generate demand curves $x=f(P\_x, P\_y, M, etc)$?

Thank you

**Tags:** utility, cobb-douglas, demand

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## 1 answers

### Answer by username_1 on 2020-03-31. Score: 5 | Accepted answer

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
\begin{equation}
u(x\_1,\dots,x\_n)=\bigl[\alpha\_1x\_1^\rho+\cdots+\alpha\_nx\_n^\rho\bigr]^{1/\rho},
\end{equation}
where $\rho\in(-\infty,1]\setminus\{0\}$, $\alpha\_i\in[0,1]$ and $\sum\_i\alpha\_i=1$. We interpret $\alpha\_i$ as the consumption share of good $i$ and $\sigma\equiv\frac{1}{1-\rho}$ as the constant elasticity of substitution. Note also that when $\sigma=1$ (or $\rho\to0$), we get the Cobb-Douglas utility form.

Solving utility maximization subject to the usual budget constraint, we get the demand for good $i$ as
\begin{equation}
x\_i(p\_1,\dots,p\_n,M)=\frac{M(\alpha\_i/p\_i)^\sigma}{\sum\_{j=1}^n\alpha\_j^\sigma p\_j^{1-\sigma}},\quad i=1,\dots,n.
\end{equation}
Again, observe that when $\sigma=1$ we get the demand associated with Cobb-Douglas utility.

The elasticity of substitution governs how relative expenditures on different goods change as relative prices change. Take a two-good example. An increase in the relative price $p\_1/p\_2$, i.e. good 1 becoming relatively more expensive, causes two effects simultaneously:

1. per unit expenditure on good 1 rises, as good 1 now costs more in relative terms; and
2. quantity demanded for good 1 decreases due to law of demand.

These are opposing effects on the expenditure on good 1 relative to that on good 2. It turns out that the elasticity of substitution determines which effect dominates. If $\sigma>1$, the second effect dominates, and if $\sigma<1$, the first effect dominates. When $\sigma=1$, which is the case of Cobb-Douglas, the two effects cancel each other exactly, so that relative expenditure is independent of the relative prices, and depends only on preference parameters (the $\alpha\_i$'s).

---

Of course, CES utility is not the only class of utility functions that generate demands dependent on prices of other goods. Another common form of utility function, the quasi-linear utility function,
\begin{equation}
u(x\_1,\dots,x\_n)=x\_1+v(x\_2,\dots,x\_n),
\end{equation}
where $v(\cdot)$ is strictly increasing and strictly concave, also generates demand functions that depend on other prices. A common example is $u(x\_1,x\_2)=x\_1+2\sqrt{x\_2}$. I trust that you can verify that both $x\_1$ and $x\_2$ depend on the two prices $p\_1$ and $p\_2$.

> **Comments:**
> **username_0**: Thank you very much, I'll see if I can do the two case.  
> **username_2**: If this adequately answers your question, please consider accepting the answer @JonathanAndrews  
> **username_0**: Gosh, I tried the two good case. Horrible, when you try substitution but pretty straight forward using Lagrangians. Thanks again for your help.  
> **username_1**: @JonathanAndrews: Check out https://cameronharwick.com/blog/how-to-derive-a-demand-function-from-a-ces-utility-function/  

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