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In this section, there are two parts which I do not quite understand.

The first one is

In general the price at which a consumer is willing to purchase some amount of good 1 will depend on how much money he has for consuming other goods. But in the special case of quasilinear utility the reservation prices are independent of the amount of money the consumer has to spend on other goods

What I understand is that when the utility function is quasilinear, $x_1^* = x_1(p_1,p_2)$ and $x_2^* = x_2(p_2, m)$. Is it becaues $x_1^*$ does not depend on the income $m$, we say that "reservation prices are independent of the amount of money the consumer has to spend on other goods"?

The second one is

Economists say that with quasilinear utility there is “no income effect” since changes in income don’t affect demand. This is what allows us to calculate utility in such a simple way

I know that quasilinear utitliy implies that there is no income effect on Good 1. But why does this allow us to calculate utility in such a simple way? If Good 1 can be affected by income effect, what will happen?

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Is it becaues $x_1^*$ does not depend on the income m, we say that "reservation prices are independent of the amount of money the consumer has to spend on other goods"?

Exactly.

But why does this allow us to calculate utility in such a simple way? If Good 1 can be affected by income effect, what will happen?

The point of quasilinear utility is that the consumer's decision on how much to buy of $x_1$ depends on the price $p1$ only, which simplifies the optimality condition. Eventually, this makes calculating the utility in the optimum also easier. I find the quote a bit strange, though, because one is usually interested in the utility-maximizing choice and not the level of utility associated with this choice.

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