<p>Diminishing marginal utility is a <strong>cardinal</strong> property. In other words: it is not invariant to arbitrary monotonic transformations. This indicates that it cannot be associated with any restriction on preferences (i.e., over a binary relation over consumption), since restrictions on preferences are by nature <strong>ordinal</strong>. </p>

<p>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.</p>

<p>If you are willing to entertain additional functional restrictions on $U$, then you can find axiomatic restrictions on $\succeq$ guaranteeing the <em>concavity</em> 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 $u$. </p>

<p>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 <em>there still exist other <strong>convex</strong> non-expected utility representations</em>. </p>

CC BY-SA 4.0
2018-07-09T20:10:35.207Z
2018-07-09T20:10:35.207Z
201p
<p>Is diminishing nature of utility function in consumer theory stemming from the assumption of convexity on the preference relation $\succsim$?</p>

<p>My understanding is:</p>

<blockquote>
  <p>$\succsim$ 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 $x$.</p>
</blockquote>

CC BY-SA 3.0
2016-07-05T06:35:28.753Z
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
<microeconomics><utility><consumer-theory><preferences>
Diminishing Nature of Utility Function From Convexity of Preference $\succsim$?
Frank Swanton
# Diminishing Nature of Utility Function From Convexity of Preference $\succsim$?

(Asked by: username_0 on 2016-07-05)

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 $x$.

**Tags:** microeconomics, utility, consumer-theory, preferences
> **Comments:**
> **username_1**: 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 $x$.  

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## 1 answers

### Answer by username_2 on 2018-07-09. Score: 1 | Accepted answer

Diminishing marginal utility is a **cardinal** property. In other words: it is not invariant to arbitrary monotonic transformations. This indicates that it cannot be associated with any restriction on preferences (i.e., over a binary relation over consumption), since restrictions on preferences are by nature **ordinal**.

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*.


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