<p>Consider an economy with <span class="math-container">$n$</span> consumers with an identical utility function, which is quasilinear in <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$ u_i(x,y) = x+u(y) $$</span></p>
<p>If we want to represent all consumers by a single <a href="https://en.wikipedia.org/wiki/Representative_agent" rel="nofollow noreferrer">representative consumer</a>, what would his utility function look like?</p>
<p>Here is my current solution.</p>
<p>For every consumer, the demand for <span class="math-container">$y$</span> depends only on the price. It is determined by the equality:</p>
<p><span class="math-container">$$ u'(y_i) = p_y $$</span></p>
<p>so:</p>
<p><span class="math-container">$$ y_i(p, m_i) = (u')^{-1}(p) $$</span>
(where <span class="math-container">$m_i$</span> is the income of consumer <span class="math-container">$i$</span>).</p>
<p>Hence, the aggregate demand is:</p>
<p><span class="math-container">$$ Y(p, M) = n \cdot (u')^{-1}(p) $$</span>
(where <span class="math-container">$M$</span> is the aggregate income of all consumers).</p>
<p>Let's assume that a representative consumer has utility function <span class="math-container">$U$</span>, which is also quasilinear in <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$$ U(x,y) = x+U(y) $$</span></p>
<p>Then, his demand will be:</p>
<p><span class="math-container">$$ Y(p, M) = (U')^{-1}(p) $$</span></p>
<p>Equating the above expressions for the aggregate demand gives:</p>
<p><span class="math-container">$$ (U')^{-1}(p) = n \cdot (u')^{-1}(p) $$</span></p>
<p><span class="math-container">$$ U'(y) = u'({y \over n}) $$</span></p>
<p><span class="math-container">$$ U(y) = n\cdot u({y \over n}) $$</span></p>
<p>So, the utility function of the rep. agent is:</p>
<p><span class="math-container">$$ U(x,y) = x + n\cdot u({y \over n}) $$</span></p>
<p>Is this calculation correct?</p>

CC BY-SA 3.0
2015-12-02T18:47:12.957Z
You treat the case where consumers have identical preference structures (including any parameters), and they differ only in their level of income?
Alecos Papadopoulos
CC BY-SA 3.0
2015-12-03T18:00:18.587Z
Yes. I think a "representative consumer" exists also when the consumers differ in their function $u$, but the expression for the rep.consumer's utility function will probably be much more complicated. So I start with the simpler case.
Erel Segal-Halevi
<utility><consumer-theory>
Utility function of a representative agent with quasilinear utility
Erel Segal-Halevi
# Utility function of a representative agent with quasilinear utility

(Asked by: username_0 on 2015-12-02)

Consider an economy with $n$ consumers with an identical utility function, which is quasilinear in $x$:

$$ u\_i(x,y) = x+u(y) $$

If we want to represent all consumers by a single [representative consumer](https://en.wikipedia.org/wiki/Representative_agent), what would his utility function look like?

Here is my current solution.

For every consumer, the demand for $y$ depends only on the price. It is determined by the equality:

$$ u'(y\_i) = p\_y $$

so:

$$ y\_i(p, m\_i) = (u')^{-1}(p) $$
(where $m\_i$ is the income of consumer $i$).

Hence, the aggregate demand is:

$$ Y(p, M) = n \cdot (u')^{-1}(p) $$
(where $M$ is the aggregate income of all consumers).

Let's assume that a representative consumer has utility function $U$, which is also quasilinear in $x$:

$$ U(x,y) = x+U(y) $$

Then, his demand will be:

$$ Y(p, M) = (U')^{-1}(p) $$

Equating the above expressions for the aggregate demand gives:

$$ (U')^{-1}(p) = n \cdot (u')^{-1}(p) $$

$$ U'(y) = u'({y \over n}) $$

$$ U(y) = n\cdot u({y \over n}) $$

So, the utility function of the rep. agent is:

$$ U(x,y) = x + n\cdot u({y \over n}) $$

Is this calculation correct?

**Tags:** utility, consumer-theory
> **Comments:**
> **username_1**: You treat the case where consumers have identical preference structures (including any parameters), and they differ only in their level of income?  
> **username_0**: Yes. I think a "representative consumer" exists also when the consumers differ in their function $u$, but the expression for the rep.consumer's utility function will probably be much more complicated. So I start with the simpler case.  

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