<p>I do not understand your query about "what about $a$ and $b$?"</p>

<p>The answer to your question at large is yes. Utility functions represent the same preferences when monotonically transformed. That is,</p>

<blockquote>
  <p>If $u(x)$ represents $\succcurlyeq$ on $X$, for $x \in X$, then for any strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, then the function $v(x) = f(u(x))$ also represents $\succcurlyeq$.</p>
</blockquote>

<p>You can try generalizing this result for more dimensions or whatever. It should be easy to apply this result to your utility function in question.</p>

CC BY-SA 3.0
2017-02-08T04:14:13.550Z
2017-02-08T06:13:25.127Z
2017-02-08T06:13:25.127Z
Kitsune Cavalry
<p>Consider a typical car, which is made of one car-body (for lack of a better word) and four tires. Let $x_1$ denote the number of car-bodies and $x_2$ denote the number of tires. Most consumers would prefer cars in the form of one car-body with four tires, and not in any other combinations. So we may represent their preference as 
$$U(x_1,x_2)=\min\{4x_1,x_2\}.\tag{1}$$
Thus two cars are preferred to one car can be expressed as 
$$
(2,8)\succ(1,4)\quad\Leftrightarrow\quad U(2,8)=8&gt;4=U(1,4). \tag{2}
$$
One car with two extra tires does not increase one's utility:
$$
(1,4)\sim(1,6)\quad\Leftrightarrow\quad U(1,4)=4=4=U(1,6).  \tag{3}
$$
Equation $(1)$ is the utility function associated with a perfect complement preference. In particular, it is a special case of the general form $\min\{ax_1,bx_2\}$ where $a=4$ and $b=1$. </p>

<p>Let's now apply an affine positive transformation to $U(x_1,x_2)$. To avoid notational confusion, let's use $\alpha\in\mathbb R$ and $\beta&gt;0$ as the parameters:
$$
V(x_1,x_2)=\alpha+\beta U(x_1,x_2).
$$
For concreteness, suppose $\alpha=1$ and $\beta=0.5$. And we can verify that the preference illustrated in $(2)$ and $(3)$ is preserved:
$$
(2,8)\succ(1,4)\quad\Leftrightarrow\quad V(2,8)=5&gt;3=V(1,4). 
$$
$$
(1,4)\sim(1,6)\quad\Leftrightarrow\quad V(1,4)=3=3=V(1,6).  
$$</p>

<p>Therefore, while $\min\{ax_1,bx_2\}$ represents some perfect complement preference, its monotonic transformation, $\alpha+\beta\min\{ax_1,bx_2\}$, also represents the same preference.</p>

CC BY-SA 3.0
2017-02-08T06:11:33.163Z
2017-02-08T06:11:33.163Z
Herr K.
<p>It is possible to have a monotonic transformation on this type of utility function, but what about $a$ and $b$? Usually a function with perfect complements is $U(x_1, x_2) = \min \{a x_1, b x_2 \}$</p>

<microeconomics>
Let the utility function $U = a + b \min\{x,y\}$. Are $x$ and $y$ perfect complements?
TheLight OI
# Let the utility function $U = a + b \min\{x,y\}$. Are $x$ and $y$ perfect complements?

(Asked by: username_0 on 2017-02-08)

It is possible to have a monotonic transformation on this type of utility function, but what about $a$ and $b$? Usually a function with perfect complements is $U(x\_1, x\_2) = \min \{a x\_1, b x\_2 \}$

**Tags:** microeconomics


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## 2 answers

### Answer by username_2 on 2017-02-08. Score: 3 | Accepted answer

Consider a typical car, which is made of one car-body (for lack of a better word) and four tires. Let $x\_1$ denote the number of car-bodies and $x\_2$ denote the number of tires. Most consumers would prefer cars in the form of one car-body with four tires, and not in any other combinations. So we may represent their preference as
$$U(x\_1,x\_2)=\min\{4x\_1,x\_2\}.\tag{1}$$
Thus two cars are preferred to one car can be expressed as
$$
(2,8)\succ(1,4)\quad\Leftrightarrow\quad U(2,8)=8>4=U(1,4). \tag{2}
$$
One car with two extra tires does not increase one's utility:
$$
(1,4)\sim(1,6)\quad\Leftrightarrow\quad U(1,4)=4=4=U(1,6). \tag{3}
$$
Equation $(1)$ is the utility function associated with a perfect complement preference. In particular, it is a special case of the general form $\min\{ax\_1,bx\_2\}$ where $a=4$ and $b=1$.

Let's now apply an affine positive transformation to $U(x\_1,x\_2)$. To avoid notational confusion, let's use $\alpha\in\mathbb R$ and $\beta>0$ as the parameters:
$$
V(x\_1,x\_2)=\alpha+\beta U(x\_1,x\_2).
$$
For concreteness, suppose $\alpha=1$ and $\beta=0.5$. And we can verify that the preference illustrated in $(2)$ and $(3)$ is preserved:
$$
(2,8)\succ(1,4)\quad\Leftrightarrow\quad V(2,8)=5>3=V(1,4).
$$
$$
(1,4)\sim(1,6)\quad\Leftrightarrow\quad V(1,4)=3=3=V(1,6).
$$

Therefore, while $\min\{ax\_1,bx\_2\}$ represents some perfect complement preference, its monotonic transformation, $\alpha+\beta\min\{ax\_1,bx\_2\}$, also represents the same preference.


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### Answer by username_1 on 2017-02-08. Score: 0

I do not understand your query about "what about $a$ and $b$?"

The answer to your question at large is yes. Utility functions represent the same preferences when monotonically transformed. That is,

> If $u(x)$ represents $\succcurlyeq$ on $X$, for $x \in X$, then for any strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, then the function $v(x) = f(u(x))$ also represents $\succcurlyeq$.

You can try generalizing this result for more dimensions or whatever. It should be easy to apply this result to your utility function in question.


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