# Does $U(x,y) = x^2 + y^2 + 2xy$ represent transitive, monotonic preferences?

I'm a monitor for a microeconomics course and a student came up with this question. That this utility function represents monotonic preferences I think it's clear. Both goods have positive and constant marginal utilities. What I think is less clear is if this preference relation is transitive. How can I assess it?

• How about applying the definitions of "utility function represents preferences" and of "transitive preferences"? – Giskard Oct 4 '18 at 21:06
• A preference relation is transitive if $A \succ B, B \succ C \implies A \succ C$. So if there's $p,q, r \in \mathbb{R}^2$ and $U(p) > U(q), U(q) > U(r) \implies U(p) > U(r)$ then $U()$ represents a transitive preference relation. Hmm... Yeah, positive and constant derivatives should do the trick to guarantee it. Thanks, man. – Pedro Cavalcante Oct 4 '18 at 21:41
• $U(x,y) = (x+y)^2$ is a monotonic transformation of $U^{'}(x,y) = x+y$.(the transformation is $f(z) = z^2$) – superhulk Oct 5 '18 at 1:56
• @PedroCavalcanteOliveira Why do you need anything about derivaties? Why is $$U(A) > U(B) \text{ and } U(B) > U(C) \Rightarrow U(A) > U(C)$$ not enough? – Giskard Oct 5 '18 at 4:21

Transitivity.

Definition. If $$A \succsim B$$ and $$B \succsim C$$, then $$A \succsim C$$.

Proof. Suppose $$A \succsim B$$ and $$B \succsim C$$. Then by definition of a utility representation, $$U(A) \geq U(B)$$ and $$U(B) \geq U(C)$$. By the transitivity of $$\geq$$, we have $$U(A) \geq U(C)$$. And so again by definition of a utility representation, $$A \succsim C$$, so that $$\succsim$$ is transitive.

Monotonicity.

Definition. If $$x_1\geq x_2$$ and $$y_1\geq y_2$$, then $$\left(x_1,y_1\right)\succsim\left(x_2,y_2\right)$$.

Proof. Suppose $$x_1\geq x_2 \left(\geq 0\right)$$ and $$y_1\geq y_2\left(\geq 0\right)$$. Then $$U\left(x_1,y_1\right) = x_1^2 +y_1^2 +2x_1y_1 \geq x_2^2 +y_2^2 +2x_2y_2 = U\left(x_2,y_2\right),$$ so that definition of a utility representation, $$\succsim$$ is monotonic.

Any preference relation represented by an utility function is transtive.

Suppose $$x \succsim y$$ and $$y \succsim z$$ and $$\succsim$$ is represented by $$U$$. Then $$U(x) \geq U(y)$$ and $$U(y) \geq U(z)$$, so $$U(x) \geq U(z)$$ and then $$x \succsim z$$.