Let the utility function $U = a + b \min\{x,y\}$. Are $x$ and $y$ perfect complements?

Question

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 $U(x_1, x_2) = \min \{a x_1, b x_2 \}$

Answer 1

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.

Answer 2

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.