<p>Assume that your utility function is</p>

<p>$$U(x,y) = \sqrt x + y$$</p>

<p>and your budget constraint is</p>

<p>$$p_xx+ p_yy = M$$</p>

<p>The Lagrangean is</p>

<p>$$\Lambda = \sqrt x + y + \lambda[M-p_xx+ p_yy]$$</p>

<p>and the first-order condition with respect to $y$ is</p>

<p>$$1 =\lambda p_y$$</p>

<p>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 $1/p_y$.  </p>

<p>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 <em>income</em>, it is not "quantity" anymore, but <em>value</em>.</p>

<p>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.</p>

CC BY-SA 3.0
2016-02-02T12:56:52.017Z
2016-10-01T21:39:02.150Z
2016-10-01T21:39:02.150Z
Alecos Papadopoulos
<p>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 $y$. </p>

<p>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 $x$.</p>

CC BY-SA 3.0
2016-02-01T18:23:41.470Z
A promise is a promise :)
Giskard
CC BY-SA 3.0
2016-02-01T18:28:27.597Z
@denesp I was not sure it was worth an answer but your opinion lays down the law here ;-) Thanks!
Oliv
CC BY-SA 3.0
2016-02-01T18:21:12.777Z
2016-02-01T18:21:12.777Z
Oliv
<p>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? </p>

<p>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?</p>

CC BY-SA 3.0
2016-02-01T06:59:47.250Z
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
CC BY-SA 3.0
2016-02-01T15:14:11.583Z
@Oliv I would vote for this if posted as an answer.
Giskard
<utility>
Quasilinear Preference and MRS
bloopton
# Quasilinear Preference and MRS

(Asked by: username_0 on 2016-02-01)

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?

**Tags:** utility

> **Comments:**
> **username_1**: 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.  
> **username_2**: @username_1 I would vote for this if posted as an answer.  

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## 2 answers

### Answer by username_1 on 2016-02-01. Score: 7 | Accepted answer

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 $x$.

> **Comments:**
> **username_2**: A promise is a promise :)  
> **username_1**: @denesp I was not sure it was worth an answer but your opinion lays down the law here ;-) Thanks!  

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### Answer by username_3 on 2016-02-02. Score: 4

Assume that your utility function is

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

and your budget constraint is

$$p\_xx+ p\_yy = M$$

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.


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