# Diminishing Nature of Utility Function From Convexity of Preference $\succsim$?

Is diminishing nature of utility function in consumer theory stemming from the assumption of convexity on the preference relation $\succsim$?

My understanding is:

$\succsim$ is convex. Then, $u(.)$ is quasiconcave. By the definition of quasiconcavity, the indifference sets are convex. Also, by another definition of quasiconcavity, the cross-section of the utility function versus a commodity shows a concave function indicating the diminishing nature of utility in any good $x$.

• Perhaps it might be useful to include the definitions you refer to because I think I don't understand this one: "by another definition of quasiconcavity, the cross-section of the utility function versus a commodity shows a concave function". If you mean for any set value of $y$, the function $U(x,y)$ is concave in $x$, I don't think that is true. A counterexample would be $U(x,y) = x^2 \cdot y^2$. This function is quasiconcave but $x^2 \cdot y^2$ is convex in $x$. – Giskard Jul 5 '16 at 6:35

To see this, consider preferences defined over $\mathbb{R}_{++}$, perhaps thought of as monetary outcomes. If we take $x \succeq y$ if and only if $x \geq y$, we have a convex preference. Notice that both $U(x) = ln(x)$ and $U(x) = e^x$ represent our preference. Both are quasi-concave functions, although only $ln(x)$ is also concave.
If you are willing to entertain additional functional restrictions on $U$, then you can find axiomatic restrictions on $\succeq$ guaranteeing the concavity of $U$ over $\mathbb R$. The obvious example is the expected utility model, which assumes that $U$ is linear over the set of simple lotteries over $\mathbb R$, i.e., takes the form: $$U(p) = \sum_{supp(p)} p_i u(x_i)$$ where $p$ is a lottery over $\mathbb R$. In this case, the risk aversion axiom---that the certainty equivalent of $p$ is less than the expected value of $p$---will ensure the concavity of $u$.
This is possible only because we have taken a stand on the cardinal normalization of $U$. In other words, any EU representation of $\succeq$ will have a concave $u$ but there still exist other convex non-expected utility representations.