<p>The indirect utility function is homogeneous of degree zero in prices and income because if you multiply all prices and income by the same positive constant, then this has no effect on the budget constraint, and so must have no effect on the solution to the utility maximization problem. Since multiplying all prices and income by a positive constant does not affect the maximized utility, the indirect utility function is homogeneous of degree zero.</p>
<p>For example, with two goods <span class="math-container">$1$</span> and <span class="math-container">$2$</span> with prices <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span> the budget constraint is</p>
<p><span class="math-container">$$p_1x_1+p_2x_2\leq m \tag{BC1}$$</span></p>
<p>where <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are the quantities of goods <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, and where <span class="math-container">$m$</span> is income. Let <span class="math-container">$V(p_1,p_2,m)$</span> denote the indirect utility (the maximized utility given prices and income). To show <span class="math-container">$V$</span> is homogeneous of degree zero, we need to show that for any <span class="math-container">$\lambda&gt;0$</span> we have</p>
<p><span class="math-container">$$V(\lambda p_1,\lambda p_2, \lambda m)=V(p_1,p_2,m) \tag{1}$$</span></p>
<p>If we multiply all prices and income by <span class="math-container">$\lambda&gt;0$</span>, then the budget constraint for  becomes</p>
<p><span class="math-container">$$(\lambda p_1)x_1+(\lambda p_2)x_2\leq (\lambda m) \tag{BC2}$$</span></p>
<p>But dividing by <span class="math-container">$\lambda$</span> shows that BC2 is equivalent to the BC1, and so the utility maximization problem is unchanged. Hence (1) holds.</p>
<hr />
<p>A level curve of the indirect utility function is the set of points <span class="math-container">$(p_1,p_2,m)$</span> satisfying the equation <span class="math-container">$V(p_1,p_2,m)=U$</span> for some level of utility <span class="math-container">$U$</span>.</p>
<p>If we hold <span class="math-container">$m$</span> constant, then we can plot level curves in in <span class="math-container">$p_1p_2$</span>-space. Since <span class="math-container">$V$</span> is decreasing in prices (assuming monotonic preferences), the level curves will be downward sloping and level curves to the northeast will represent lower values of utility.</p>

CC BY-SA 4.0
2023-08-24T12:06:19.170Z
2023-08-24T12:06:19.170Z
smcc
<p>the indirect utility function states that  higher indirect indifference curves carry lower utility. however it also assumes homogenity of degree zero in price and income. how?</p>
<p>please explain it to me in terms of simple language as i am not familiar with linear programming techniques.</p>

CC BY-SA 4.0
2023-08-23T06:41:09.670Z
"*higher indirect indifference curves*" Higher in what coordinate system?
Giskard
CC BY-SA 4.0
2023-08-23T06:41:37.040Z
"*the indirect utility function states*" Can you please provide a reference for this claim, where exactly did you read it?
Giskard
CC BY-SA 4.0
2023-08-26T08:44:15.333Z
M.L. Jhingan's "Advanced economic theory"
sruthipriya mahesh
<utility>
Why is the indirect utility function state that higher indirect indifference curves has lower utility when it assumes homogenity in prices and income?
sruthipriya mahesh
# Why is the indirect utility function state that higher indirect indifference curves has lower utility when it assumes homogenity in prices and income?

(Asked by: username_0 on 2023-08-23)

the indirect utility function states that higher indirect indifference curves carry lower utility. however it also assumes homogenity of degree zero in price and income. how?

please explain it to me in terms of simple language as i am not familiar with linear programming techniques.

**Tags:** utility
> **Comments:**
> **username_1**: "*higher indirect indifference curves*" Higher in what coordinate system?  
> **username_1**: "*the indirect utility function states*" Can you please provide a reference for this claim, where exactly did you read it?  
> **username_0**: M.L. Jhingan's "Advanced economic theory"  

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## 1 answers

### Answer by username_2 on 2023-08-24. Score: 0

The indirect utility function is homogeneous of degree zero in prices and income because if you multiply all prices and income by the same positive constant, then this has no effect on the budget constraint, and so must have no effect on the solution to the utility maximization problem. Since multiplying all prices and income by a positive constant does not affect the maximized utility, the indirect utility function is homogeneous of degree zero.

For example, with two goods $1$ and $2$ with prices $p\_1$ and $p\_2$ the budget constraint is

$$p\_1x\_1+p\_2x\_2\leq m \tag{BC1}$$

where $x\_1$ and $x\_2$ are the quantities of goods $1$ and $2$, and where $m$ is income. Let $V(p\_1,p\_2,m)$ denote the indirect utility (the maximized utility given prices and income). To show $V$ is homogeneous of degree zero, we need to show that for any $\lambda>0$ we have

$$V(\lambda p\_1,\lambda p\_2, \lambda m)=V(p\_1,p\_2,m) \tag{1}$$

If we multiply all prices and income by $\lambda>0$, then the budget constraint for becomes

$$(\lambda p\_1)x\_1+(\lambda p\_2)x\_2\leq (\lambda m) \tag{BC2}$$

But dividing by $\lambda$ shows that BC2 is equivalent to the BC1, and so the utility maximization problem is unchanged. Hence (1) holds.

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A level curve of the indirect utility function is the set of points $(p\_1,p\_2,m)$ satisfying the equation $V(p\_1,p\_2,m)=U$ for some level of utility $U$.

If we hold $m$ constant, then we can plot level curves in in $p\_1p\_2$-space. Since $V$ is decreasing in prices (assuming monotonic preferences), the level curves will be downward sloping and level curves to the northeast will represent lower values of utility.


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