When can one safely talk about decreasing marginal utility? - Economics Stack Exchange most recent 30 from economics.stackexchange.com 2019-07-17T11:09:03Z https://economics.stackexchange.com/feeds/question/370 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://economics.stackexchange.com/q/370 8 When can one safely talk about decreasing marginal utility? Ubiquitous https://economics.stackexchange.com/users/108 2014-11-30T08:58:56Z 2014-11-30T18:44:09Z <p>One thing I hear a lot is talk of decreasing marginal utility—the idea being that additional units of a good become progressively less attractive the more units of that good one has already.</p> <p>However, this always made me a little uncomfortable because of the ordinality of utility. If we take the trivial case of a world in which there is only one good with utility $u(x)$ satisfying $u'(x),\ u''(x)&lt;0$ (decreasing marginal utility) then it is clearly possible to construct an increasing function $f$ such that $(f\circ u)$ is linear in $x$. Moreover, since utility functions are invariant to monotone-increasing transformations, $(f\circ u)$ is a utility function that represents the same preferences as $u$ (but now has constant marginal utility). Thus, in a world with a single good it seems that it never makes sense to talk about diminishing marginal utility.</p> <p>My question is this: consider a market with $L&gt;1$ goods. Is there a formal condition under which we can safely talk about decreasing marginal utility? That is to say, is there a class of preferences such that <strong><em>every</em></strong> valid utility representation, $u(\mathbf{x})$, has $u_{ii}(\mathbf{x})&lt;0$ for some $i$?</p> <p>Alternatively, is there some simple proof that, for $L&gt;1$, the existence of a utility representation with $u_{ii}(\mathbf{x})&lt;0$ for some $i$ necessarily implies that all utility representations have $u_{ii}(\mathbf{x})&lt;0$?</p> https://economics.stackexchange.com/questions/370/-/371#371 4 Answer by jmbejara for When can one safely talk about decreasing marginal utility? jmbejara https://economics.stackexchange.com/users/59 2014-11-30T13:19:56Z 2014-11-30T15:21:59Z <p>The fact that you ask about "safety" implies that you believe that some result is in jeopardy. This answer can be improved if you can specify a result that you might have in mind. Otherwise, take as an example the first and second welfare theorems. They do not rely on decreasing marginal utility.</p> <p>If you're concerned about results about preferences over uncertainty (ideas about risk aversion, etc.) then recall that although a standard utility function representation of preferences without uncertainty is unique up to a positive monotonic transformation, a Von Neumann-Morgenstern utility function representation of preferences over uncertainty is unique only up to positive <em>affine</em> transformations.</p> <p>EDIT: Extra Notes.</p> <p>The definition of a utility function is given as follows (from <em>Advanced Microeconomic Theory</em> by Jehle and Reny, 2011): <img src="https://i.stack.imgur.com/iaCcu.png" alt="enter image description here"></p> https://economics.stackexchange.com/questions/370/-/376#376 6 Answer by Alecos Papadopoulos for When can one safely talk about decreasing marginal utility? Alecos Papadopoulos https://economics.stackexchange.com/users/61 2014-11-30T17:20:20Z 2014-11-30T18:22:37Z <p>The concept of "marginal utility" (and therefore of decreasing such) has meaning only in the context of <em>cardinal</em> utility.</p> <p>Assume we have an ordinal utility index $u()$, on a single good, and three quantities of this good, $q_1&lt;q_2&lt;q_3$, with $q_2-q_1 = q_3-q_2$.<br> Preferences are well behaved and satisfy the benchmark regularity conditions, so</p> <p>$$u(q_1)&lt; u(q_2) &lt; u(q_3)$$</p> <p>This is <em>ordinal</em> utility. Only the ranking is meaningful, not the distances. So the distances $u(q_2) - u(q_1)$ and $u(q_3) - u(q_2)$ <em>have no behavioral/economic interpretation</em>. If they don't, neither do the ratios</p> <p>$$\frac {u(q_2) - u(q_1)}{q_2-q_1},\;\; \frac {u(q_3) - u(q_2)}{q_3-q_2}$$</p> <p>But the limits of these ratios as the denominator goes to zero would be the definition of the derivative of the function $u()$. So the derivative is devoid of economic/behavioral interpretation, and so comparing two instances of the derivative function would not produce any meaningful content. </p> <p>Of course this does not mean that the derivatives of $u()$ do not exist as mathematical concepts. They can exist, if $u()$ satisfies the conditions needed for differentiability. So one can ask the purely mathematical question "under which condition the function representing ordinal utility has <strong>strictly negative second derivative</strong>" (or negative definite Hessian for the multivariate case), trying not to interpret it as "decreasing marginal utility" with economic/behavioral content, but as just a mathematical property that may play some role in the model he examines. </p> <p>In such a case, we know that:<br> 1) If preferences are convex, the utility index is a quasi-concave function<br> 2) If preferences are strictly convex, the utility index is strictly quasi-concave</p> <p>But quasi-concavity is a <em>different kind of property</em> than concavity: quasi-concavity is an "ordinal" property in the sense that it is preserved under an increasing transformation of the function. </p> <p>On the other hand, <strong>concavity is a "cardinal" property, in the sense that it won't necessarily be preserved under an increasing transformation.</strong><br> Consider what this implies: assume that we find a characterization of preferences such that they can be represented by <em>a</em> utility index which is concave as a function. Then we can find and implement some increasing transformation of this utility index, that will eliminate the concavity property.</p>