# Utility function of a representative agent with quasilinear utility

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, 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?

• You treat the case where consumers have identical preference structures (including any parameters), and they differ only in their level of income? – Alecos Papadopoulos Dec 2 '15 at 18:47
• 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 Dec 3 '15 at 18:00