# Assumption of sufficient wealth in quasi-linear preferences

Whenever we talk about quasi-linear preferences, we assume that the consumer is sufficiently wealthy. As far as I understand is that we need that assumption in order to obtain an interior solution. But what would the economic intuition behind that premise be? Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?

Is it that the consumer would not consume anything of the nonlinear good in case of insufficient wealth?

No, it is exactly the other way around.

The utility function in the two goods case will have the following form:

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

We assume that $$u'(x)>0$$ and $$u''(x)<0$$ or marginal utility of $$x$$ is decreasing in $$x$$. $$MU_y$$ meanwhile is constant. Another assumption that is often not explicitly stated is that $$\frac{MU_x}{p_x}>\frac{MU_y}{p_y}$$ for some $$x$$. In words: There is some range of $$x$$ values at which the marginal utility of a dollar spent on $$x$$ exceeds the marginal utility of a dollar spent on $$y$$. Now because we know $$MU_x$$ is decreasing in $$x$$, we know that the preceding inequality will hold true when the consumer is consuming a relatively small quantity of $$x$$.

Now if we assume that the budget is very large, we know that the left side of the inequality gets smaller and smaller while the right side remains constant as the consumer consumes more and more of $$x$$. Because of decreasing $$MU_x$$ a moment will come where a dollar spent on $$y$$ will generate at least as much utility as a dollar spent on $$x$$. When this moment comes (the consumption of $$x$$ at which $$\frac{MU_x}{p_x}=\frac{MU_y}{p_y}$$ ), the consumer will switch to consuming $$y$$ with the rest of his budget.

When the consumer has a small budget however, he will never attain the point where the preceding equality holds true, that is, he is left consuming $$x$$ in a quantity for which $$\frac{MU_x}{p_x}>\frac{MU_y}{p_y}$$ This is the case of insufficient wealth. In this case the consumer only consumes the nonlinear good, in this case denoted by $$x$$.

• Awesome, that was a very intuitive answer ;) thanks a lot – SimonDude Aug 7 at 14:32
• happy to be of help – user18214 Aug 8 at 9:31

This depends on the exact preferences, but usually the utility function $$U(x,y) = v(x) + y$$ is such that $$\lim_{x \to 0} |MRS(x,y)| = \lim_{x \to 0} \frac{\text{d}v(x)}{\text{d} x} = \infty.$$ In this case it is the consumption of an additional marginal unit of the nonlinear good $$x$$ that is infinitely useful compared to the consumption of an additional marginal unit of the linear good $$y$$. Hence the consumer will consume some units of $$x$$ before buying any of $$y$$. If the consumer is not 'wealthy enough', the corner solution will be such that $$x > 0$$, $$y = 0$$.