<p>This might just be a misunderstanding. The textbook contains various passages like e.g. &quot;<em>An alternative approach holds real income (or utility) constant while examining reactions to changes in <span class="math-container">$p_x$</span></em>&quot; (p. 151). When this terminology is introduced, however, &quot;<em>real</em>&quot; stands in quotation marks. This is in the context of explaining the compensated as opposed to the uncompensated (standard) demand curve. In this context, &quot;real&quot; income is meant to be income adjusted to let you achieve the same utility as before the price change. See e.g. p. 146, where it says &quot;<em>we are conceptually holding “real” income (that is, utility) constant.</em>&quot;</p>

CC BY-SA 4.0
2021-04-09T11:58:15.257Z
2021-04-09T11:58:15.257Z
VARulle
<p>Real income and utility are not the same thing but utility can be expressed as a function of income because income is what allows you to consume goods and services.</p>
<p>For example, let us assume there are only two goods <span class="math-container">$x_1$</span> an <span class="math-container">$x_2$</span> and consumer is given budget <span class="math-container">$p_1 x_1 + p_2 x_2 = m$</span>, where <span class="math-container">$p_i$</span> are prices for good 1 and 2 respectively and <span class="math-container">$m$</span> is an income.  Let us assume Cobb-Douglas utility function:</p>
<p><span class="math-container">$$ u(x_1, x_2) = x_1^\alpha x_2^{1-\alpha}$$</span></p>
<p>Here after solving the consumer optimization problem we would get optimum consumption of <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> as (I skipped the steps but you can easily derive it by setting up an Lagrangian):</p>
<p><span class="math-container">$$x_1^*= \alpha \frac{m}{p_1};x_2^* = (1 -\alpha) \frac{m}{p_2}$$</span></p>
<p>Consequently, given optimal consumption of <span class="math-container">$x_1^*$</span> and <span class="math-container">$x_2^*$</span> is:</p>
<p><span class="math-container">$$u(x_1^*, x_2^*) = \left( \alpha \frac{m}{p_1} \right)^\alpha \left((1 -\alpha) \frac{m}{p_2} \right)^{1-\alpha} $$</span></p>
<p>So the utility will end up being function of nothing else than a function of income <span class="math-container">$m$</span> an prices <span class="math-container">$p_1$</span> (<span class="math-container">$\alpha$</span> is just parameter). A utility function expressed in terms of prices and income even has its own special name 'indirect utility' typically denoted as <span class="math-container">$v$</span> (e.g. <span class="math-container">$v(\textbf{p},m) =\max u(\textbf{x})$</span> such that <span class="math-container">$\textbf{px} = m$</span> see Varian Microeconomic Analysis 3rd ed pp 99.)</p>
<p>Consequently as shown above utility does depend on income. Utility is generally not equal to real income, and I don't have the copy of the textbook you mention and you did not gave us the page for that claim, but they might have just used a utility function where utility would be proportional to real income. For a didactic purposes even assuming <span class="math-container">$v(m,\textbf{p})=\frac{m}{\textbf{p}}$</span> can be reasonable depending on setting even if it is not realistic because it simplifies calculations for novices (e.g. supply and demand are also typically not linear but your textbook likely draws them as linear functions).</p>

CC BY-SA 4.0
2021-04-09T08:37:06.647Z
2021-04-09T08:44:37.733Z
2021-04-09T08:44:37.733Z
1muflon1
<p>The textbook I'm using, &quot;<a href="https://rads.stackoverflow.com/amzn/click/com/1305505794" rel="nofollow noreferrer" rel="nofollow noreferrer">Microeconomic Theory: Basic Principles and Extensions</a>&quot;, treats utility and real income as the same thing in the chapters on compensated and uncompensated demand. I would like to understand the logic behind it. To me real income always meant &quot;Nominal Income/CPI&quot; but I don't think this can also be translated to utility.</p>

CC BY-SA 4.0
2021-04-09T11:04:29.303Z
I haven't read your course text. However, your intuition is broadly correct. In particular, the link between utility and real income is important because we need a meaningful account of what happens to welfare when CPI changes.
EB3112
<utility><competition>
How are real income and utility the same thing?
reasonStore
# How are real income and utility the same thing?

(Asked by: username_0 on 2021-04-09)

The textbook I'm using, "[Microeconomic Theory: Basic Principles and Extensions](https://rads.stackoverflow.com/amzn/click/com/1305505794)", treats utility and real income as the same thing in the chapters on compensated and uncompensated demand. I would like to understand the logic behind it. To me real income always meant "Nominal Income/CPI" but I don't think this can also be translated to utility.

**Tags:** utility, competition

> **Comments:**
> **username_1**: I haven't read your course text. However, your intuition is broadly correct. In particular, the link between utility and real income is important because we need a meaningful account of what happens to welfare when CPI changes.  

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## 2 answers

### Answer by username_3 on 2021-04-09. Score: 4 | Accepted answer

Real income and utility are not the same thing but utility can be expressed as a function of income because income is what allows you to consume goods and services.

For example, let us assume there are only two goods $x\_1$ an $x\_2$ and consumer is given budget $p\_1 x\_1 + p\_2 x\_2 = m$, where $p\_i$ are prices for good 1 and 2 respectively and $m$ is an income. Let us assume Cobb-Douglas utility function:

$$ u(x\_1, x\_2) = x\_1^\alpha x\_2^{1-\alpha}$$

Here after solving the consumer optimization problem we would get optimum consumption of $x\_1$ and $x\_2$ as (I skipped the steps but you can easily derive it by setting up an Lagrangian):

$$x\_1^\*= \alpha \frac{m}{p\_1};x\_2^\* = (1 -\alpha) \frac{m}{p\_2}$$

Consequently, given optimal consumption of $x\_1^\*$ and $x\_2^\*$ is:

$$u(x\_1^\*, x\_2^\*) = \left( \alpha \frac{m}{p\_1} \right)^\alpha \left((1 -\alpha) \frac{m}{p\_2} \right)^{1-\alpha} $$

So the utility will end up being function of nothing else than a function of income $m$ an prices $p\_1$ ($\alpha$ is just parameter). A utility function expressed in terms of prices and income even has its own special name 'indirect utility' typically denoted as $v$ (e.g. $v(\textbf{p},m) =\max u(\textbf{x})$ such that $\textbf{px} = m$ see Varian Microeconomic Analysis 3rd ed pp 99.)

Consequently as shown above utility does depend on income. Utility is generally not equal to real income, and I don't have the copy of the textbook you mention and you did not gave us the page for that claim, but they might have just used a utility function where utility would be proportional to real income. For a didactic purposes even assuming $v(m,\textbf{p})=\frac{m}{\textbf{p}}$ can be reasonable depending on setting even if it is not realistic because it simplifies calculations for novices (e.g. supply and demand are also typically not linear but your textbook likely draws them as linear functions).


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### Answer by username_2 on 2021-04-09. Score: 3

This might just be a misunderstanding. The textbook contains various passages like e.g. "*An alternative approach holds real income (or utility) constant while examining reactions to changes in $p\_x$*" (p. 151). When this terminology is introduced, however, "*real*" stands in quotation marks. This is in the context of explaining the compensated as opposed to the uncompensated (standard) demand curve. In this context, "real" income is meant to be income adjusted to let you achieve the same utility as before the price change. See e.g. p. 146, where it says "*we are conceptually holding “real” income (that is, utility) constant.*"


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