# Quasilinear Utility Functions

We know if the utility function is quasilinear (QL) w.r.t good 1, then the demand for other goods is independent of income (no income effect for goods $(2,\dots, N)$).

But is the reverse implication also true: i.e can we say if all the goods except one have demand functions independent of income then the utility function must be quasilinear?

I have been looking up all the standard micro grad level text books but haven't got an answer yet. I see books defining (rather characterizing) QL by just one side of implication (i.e QL implies No income effect) but are silent about the other.

Any reference in this regard will be highly useful.

# Is your claim 1 true?

Let's look at an agent who's Utility is quasi linear in money, $m$, and goods $c$.

The utility function is given by

$$U(m,c) = m + f(c)$$ for some concave function $f$.

The maximization problem, given wealth $w$, prices of money and consumption $p, p_c$ is then

$$\max_{m,c} U(m,c) \text{ s.t. } pm + p_c c = w\\ = \max_c f(c) + \frac{w}{p} - \frac{p_c}{p}c\\ \Rightarrow f'(c) = \frac{p_c}{p}$$

So the optimal amount for consumption $c$ is independent of the wealth level $w$, if the interior solution qualifies as the global solution - which it does for sufficiently high wealth level. I'm not from a micro background but I suppose this qualifies as no income effect for the nonlinear good.

The intuition is, that, beyond a threshold of income, you have your optimal consumption locus $c^*$ which is identified by the previous FOC. Having a higher income will make you want to save that money, not put it into consumption.

Now that we got the intuition, we can try to work it backwards. What do we need? That

$$\exists c^* : U(m+\epsilon, c^*) > U(m, c^* + \frac{p}{p_c}\epsilon) \,\forall \epsilon > 0$$

One can show (do it!) that if $U$ was concave in $m$, eventually the marginal value of increasing $m$ (for a large enough $\epsilon$) was smaller than investing it into $c$.

On the other hand, if $U$ was additively separable in $(m,c)$ and increasing but convex in $m$, we would - after a threshold - always invest all into $m$ and set $c=0$. Also here, after that threshold, the optimal locus for $c$ is trivially independent of the wealth level: no income effect.

Another example would be decreasing utility in $c$, which would make us set it independently of the wealth level to its global minimum (often 0).

You see, unless you give us a class of utility functions, there can be many valid (but economically stupid) cases of utility functions where the inverse does not hold.

• If someone knows how to make headlines on stack exchange, I'd be much obliged. – FooBar Feb 15 '15 at 16:18
• I edited your post to have a title. Use <h#> tags from HTML – BKay Feb 15 '15 at 16:48
• Thanks FooBar. Yes, I can see we will need more restrictions for the inverse. Yes, I could think of a decreasing utility scenario but in my question I meant to ask given other usual assumptions on the utility function (such that increasing, quasi-concavity) apply. Sorry, I should have been more precise. – user200947 Feb 15 '15 at 17:24

Is that first claim right? I guess it depends what you mean by an income effect.

Say your quasi-linear utility form is: $$U(x,v) = x + \ln(v)$$ Say $P_v = P_x=1$ and so for wealth $w = P_v\cdot v + P_x \cdot x = x + v$. If $0<w\leq 1$ the household strictly prefers to buy $v$ to $x$ and buys no $x$ but $w>1$ is then exclusively $x$ is purchased beyond that. So demand for $x$ looks like this: $$w<1: x^* = 0$$ $$w<1: v^* = w$$ $$w\geq1: x^* = w-1$$ $$w\geq1: v^* = 1$$

So the marginal demand for $x/v$ depends on $w$.

• Yes, you are right. I should have been more precise in my question. I meant over the income ranges (that is at sufficiently high income) where we get an interior solution.Thanks! – user200947 Feb 15 '15 at 17:01

Blockquote Say your quasi-linear utility form is: U(x,v)=x+ln(v) Say Pv=Px=1 and so for wealth w=Pv⋅v+Px⋅x=x+v. If 01 is then exclusively x is purchased beyond that. So demand for x looks like this: w<1:x∗=0 w<1:v∗=w w≥1:x∗=w−1 w≥1:v∗=1

How does that make sense. If you allocate v=w and x=0 when w<1 then U(0,v=w<1) < 0 as ln(v) for v<1 is negative.