# Quasilinear Preference and MRS

Do quasi-linear indifference curves have MRS's that depend only on the nonlinear variable? For example, for a U(x) = √(x) + y, I calculated that the y would not be in the MRS. Does that mean that a consumer does not factor the consumption of a "linear" good y into their utility maximization?

If this is the case, can someone explain to me, practically, what it means for a good to be "linear" and for an MRS to exclude one variable?

• The MRS is a function of $x$ and $y$. The fact that $y$ doesn't appear in the formula simply means that this function is constant in $y$. The linearity does not imply indifference. It means that an additional value of 1 is valued equally by the agent irrespective of his baseline consumption. In your example, this contrasts with the marginal utility of consuming $x$ which is decreasing with the baseline consumption. – Oliv Feb 1 '16 at 6:59
• @Oliv I would vote for this if posted as an answer. – Giskard Feb 1 '16 at 15:14

The MRS is a function of $x$ and $y$. The fact that $y$ doesn't appear in the formula simply means that this function is constant in $y$.

The linearity does not imply indifference. It means that an additional value of $y$ is valued equally by the agent irrespective of his baseline consumption. In your example, this contrasts with the marginal utility of consuming $x$ which is decreasing with the baseline consumption since the function $u$ is concave in $x$.

• A promise is a promise :) – Giskard Feb 1 '16 at 18:23
• @denesp I was not sure it was worth an answer but your opinion lays down the law here ;-) Thanks! – Oliv Feb 1 '16 at 18:28

Assume that your utility function is

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

$$p_xx+ p_yy = M$$

The Lagrangean is

$$\Lambda = \sqrt x + y + \lambda[M-p_xx+ p_yy]$$

and the first-order condition with respect to $y$ is

$$1 =\lambda p_y$$

We Know that $\lambda$ is the marginal effect of an increase in Income on the maximized utility function. So (considering discrete changes) if your income increases by $1$ kudo, maximized utility will increase by $1/p_y$.

It makes sense then to treat $y$ as the numeraire good, in which case its price is undetermined, our budget constraint is $p_xx+ y = M$, and the effect on the utility function by an increase in $M$ by one unit is $1$... as is the effect of increasing $y$ by one unit! But that makes $y$ equivalent to income, it is not "quantity" anymore, but value.

In other words, a good entering linearly the utility function is a natural modelling choice for a composite good that represents "all other goods", i.e. residual income after trading for the good $x$ we want to focus on.