# Diminishing marginal utility role in downward sloping demand curve

One of the reasons people generally denote the downward sloping demand curve is diminishing marginal utility, which says that each incremental unit brings less and less utility. But, how does it affect the price to go down or up as quantity demanded changes? I don't understand why it causes the demand curve to go down.

• Out of interest, where did you read that diminishing marginal utility is connected to whether demand is downward sloping? Commented May 28, 2022 at 9:59

As it turns out, the concept of 'diminishing marginal utility' has nothing to do with whether demand curves slope upwards or downwards. For instance, consider the utility function $$u = x^2 y^2$$ defined over a pair of goods $$x$$ and $$y$$. As written, the utility function is convex, so implies 'increasing marginal utility'. However, using standard methods, one can check that this yields the demand curves $$x^* = \frac{m}{2 p_x}; y^* = \frac{m}{2 p_y}$$ where $$m$$ is the consumer's income, $$p_x$$ is the price of good $$x$$, and $$p_y$$ is the price of good $$y$$. Clearly, both demand curves are downward sloping; so downward sloping demand does not need decreasing marginal utility.

The more general point is that maximising a utility function $$u$$ is equivalent to maximising $$f(u)$$, where $$f$$ is a strictly increasing function. By choosing $$f$$ appropriately, we can often obtain a transformed utility function $$f(u)$$ that displays decreasing marginal utility; but we can also obtain a transformed utility function $$f(u)$$ that displays increasing marginal utility! So the question of whether the utility function has diminishing marginal utility or not can't be relevant to computing the consumer's demand.

Update: If you do want to understand whether demand slopes downwards or upwards, the key insight comes from the Slutsky Equation. Informally, this says that the effect of changing the price on demand can be decomposed into two things:

1. The substitution effect
2. The income effect

The substitution effect is always negative, suggesting that higher prices should lead to lower demand. Meanwhile, the income effect can be positive or negative, depending on whether a higher income leads to higher or lower demand. As a result, demand is upward sloping if and only if 1) the income effect is positive 2) the income effect is larger than the substitution effect (in absolute size).

• Would this argument still be valid in the real world case of many goods, or in the "simplified real world" case of 2 goods with one of them taken to be representative of all other goods? Commented May 28, 2022 at 11:09
• Yes, this all extends to an arbitrary number of goods. For example, consider the utility function $u = x_1^2 x_2^2 . . . x_n^2$. This function has 'increasing marginal utility', but leads to downward sloping demand. Commented May 28, 2022 at 16:54
• The problem with the extension to many goods is that the multiplicative form of the utility function becomes less plausible. Whether or not the terms are squared, the multiplicative form implies that utility is zero if consumption of any one good is zero. A more realistic many goods utility function might be expected to have a functional form combining multiplication (for complements) and addition (for substitutes). Commented May 28, 2022 at 20:36
• You miss an important point. With quasilinear preferences, the income effect is always zero, so the change in demand is totally attributed to the substitution effect. Assuming a quasilinear function with diminishing marginal utility on the non-nummeraire good leads to a downward -sloping demand curve as explained in my post. Your general assertion "'diminishing marginal utility' has nothing to do with whether demand curves slope upwards or downward" is therefore not true and missleading. There cases/models in which diminishing utility can lead to a downward-sloping demand curve. Commented Oct 7, 2023 at 12:37

The partial equilibrium analysis taught in Micro 101 with the downward-sloping demand function is derived from a representative consumer $$i$$ with a quasilinear utility function of the form $$u_i\left(x_{1}, x_{2}\right)=v_i(x_{1})+x_{2}$$ with $$x_1$$ being the good under study and $$x_2$$ the numeraire good. That is $$p_2$$ is set to 1 which is why we can rewrite the budget constraint as $$x_2=m-p_1x_1$$. Good $$x_2$$ is hence the money which is not spend for $$x_1$$.

Also, it is assumed that $$v_i^{\prime}>0, v_i^{\prime \prime}<0$$ which implies diminishing marginal utility as well as $$v(0)=0$$

Using our reformulated budget constraint, we can write $$u_i\left(x_{1}, x_{2}\right)$$ as $$u_i(x_{1})=v_i(x_1)+m-p_1x_1$$. Maximizing $$u_i(x_{1})$$ leads to $$p_1=v_i^{\prime}(x_1)$$. This is the individual inverse demand function of the representative consumer $$i$$ which is strictly monotone decreasing (recall that $$v_i^{\prime}>0, v_i^{\prime \prime}<0$$). Consequently, individual demand $$x_i(p_1)$$ and the aggregate demand $$x(p_1)$$ is also strictly monotone-decreasing or downward sloping.

In this setting, the assumption of marginal decreasing utility hence results in a downward sloping demand function. Importantly, $$p_1=v_i^{\prime}(x_1)$$ can be interpreted as the representative consumer i´s marginal willingness to pay which is important for the concept of consumer surplus which is typically applied in the context of partial equilibrium analysis.

• But we could also have strictly decreasing demand when $v$ is convex! For instance, consider an example where you choose to spend all your money on $x_1$, e.g. $u = 10x_1^2 + x_2$ considered over a suitable region of prices. So I don't see why the curvature of $v$ is critical even in this environment. Commented Jun 6, 2022 at 13:51
• No. When maximizing $u(x1,x2)=v(x1)+x_2$ on $m=p_1 * x_1 + p_2*x_2$ with $p_2=1$ you get $p_1=v′(x1)$ which is the inverse demand function. It is downward sloping with $v′i>0,v′′i<0$ and so is aggregate demand. When specifying $v(x_1)={10x_1}^2$ in $u(x_1,x_2)=v(x_1)+x_2$ as in your case, the resulting inverse demand function $p_1=20x_1$ is upwards sloping and so is aggregate demand. So the curvature is critical here and the reason why this assumption is standard in many models (especially with quasilinear preferences), see e.g. p. 9 on lem.sssup.it/fagiolo/files/get_2.pdf Commented Jun 7, 2022 at 10:20
• "the resulting inverse demand function $p_1 = 20x_1$ is upwards sloping" -- this is incorrect. Fix $p_2$ at 1. If you maximise $u = 10x_1^2 + x_2$ in the neighbourhood of $p_1 = 1$, you get a corner solution, namely to spend all your income on good 1. This is pretty obvious, and leads to strictly downward sloping demand since $x_1^* = m/p_1$ in that neighbourhood of prices. Commented Jun 7, 2022 at 14:07
• In case it helps, I should add that this problem can't be solved by blindly taking a derivative and setting it equal to zero. That seems to have been your approach. Commented Jun 7, 2022 at 14:10
• Okay I see your point. However, when your aim is to derive the standard downward sloping demand function which is used in partial equilibrium analysis (with applications in many fields), you normally also want the individual demand function of consumer i to be interpretable as the marginal willingness to pay (WTP) which then makes it possible to determine consumer surplus. While the WTP interpretation is possible with the approach I have pointed out above, I don´t think you get there with $u=10x_1^2+x_2$, right? See also p. 23 on econ.ucla.edu/sboard/teaching/econ11_notes.pdf Commented Jun 7, 2022 at 16:13

When the utility function is homothetic (and not necessarily concave), then the micro demand functions are decreasing in their own price. Some examples of utility functions given in this post fall into this category.
Concavity of the utility function is not sufficient and not necessary either.
Another sufficient condition on the utility function to yield decreasing demand functions have been given by Mitjuschin and Polterovich (1978). All these results and many others are discussed by:
Kannai, Y., and L. Selden, 2014, "Violation of the Law of Demand," Economic Theory, 55, 1–28.

One of the reasons people generally denote the downward sloping demand curve is diminishing marginal utility, which says that each incremental unit brings less and less utility. But, how does it affect the price to go down or up as quantity demanded changes? I don't understand why it causes the demand curve to go down.

First, as you point out in the question itself, it is one of the reasons that standard textbooks often give for downward sloping demand. It is neither the only reason nor it is a sufficient reason to always result in downward sloping demand curve.

Marginal utility determines value people place on things. For example, a hungry person might value first slice of a pizza at 30\$(given their marginal utility of consuming pizza). Second slice might be valued only at 20\$ since marginal utility of the second slice will be lower. Third slice will have even lower marginal utility and hence person might value it only at 5\$. If value of good declines as you consume more of it, this will be contributing factor to downward sloping demand curve. Hence demand will usually be downward sloping because due to declining marginal utility people will be willing to consume larger quantities only when price drops (although in reality there are some other effects like income effect that could sometimes make demand upward sloping like Giffen goods, nonetheless the marginal utility itself makes demand downward sloping). ### Is the condition above sufficient? It is not in itself sufficient, this was actually proven by Samuelson (1947). However, this is due to presence of the other factors that determine whether demand is downward sloping such as the cases where income effect dominates substitution effect and so on. As a result demand can be even upward sloping despite marginal utility declining. This however, does not invalidate undergraduate textbook claims that diminishing marginal utility is one of the factors making utility downward sloping. Even if it is not generally sufficient condition, outside cases like Giffen goods. For example, consider following case of utility: $$U = x^\alpha + y^\beta$$ In case $$\alpha > 1; \beta >1$$ we have increasing marginal utility, in case $$\alpha <1; \beta <1$$ we have decreasing marginal utility. Subject to the budget constraint $$px + qy =m$$. Consider case with increasing marginal utility: $$U = x^2 + y^2$$ The optimal demands here will be given by: $$x^* = 0 \text{ if } p> q ; x^* = \frac{m}{p} \text{ if } p< q$$ and $$y^* = 0 \text{ if } p q$$ In such case we get demand that is not always decreasing in price (see simulation of demand for good x below for $$m=100$$ and $$q=1$$): Now if we switch to utility function with declining marginal utility: $$U = x^{0.5} + y^{0.5}$$ the demands will be given by: $$x^* = \frac{qm}{p^2 +pq}; y^* = \frac{pm}{q^2 +pq}$$ if we plot this function you will see we will get nice demand function that decreases everywhere (see simulation of demand for good x below for $$m=100$$ and $$q=1$$): Above is just an example, and this is not result that holds for any utility function, but it holds for very large number of various utility functions. Hence, when textbook says that one of the reasons why supply curves are declining is declining marginal utility, what they mean is that for wide range of utility functions, declining marginal utility guarantees nice monotonically declining demand. However, as proven by Samuelson declining marginal utility in itself is not sufficient, hence textbook will typically mention it as one of the reasons. • But the utility function$U = x^{0.5} + y^{0.5}$(diminishing MU) is equivalent to the utility function$U = (x^{0.5} + y^{0.5})^3$(increasing MU) in the sense that both utility functions represent exactly the same preferences, and generate exactly the same demands! That is why, in classical consumer theory, we don't talk about whether marginal utility is increasing or decreasing (it is totally meaningless). Commented May 28, 2022 at 16:58 • @afreelunch its not completely meaningless, also yes it is not discussed in graduate level consumer theory but in undergraduate texts DMU is given as one of the reason why usually demands are downward sloping. My point is that it is not charitable to interpret the undergraduate texts to be stating general result, how I interpret that claim from undergraduate text is that if you would try to empirically find factors that contribute to demand being downward sloping like running$y = \textbf{x}\beta$where$y$is slope of demand$x$are various covariates one of which is DMU, the DMU would have – 1muflon1 Commented May 28, 2022 at 21:18 • @afreelunch I can see that$U=x^{0.5}+y^{0.5}$and$U=(x^{0.5}+y^{0.5})^3$represent the same preferences, but can't the latter exhibit either increasing or decreasing MU in$x$, depending on the value of$y$? The expansion of the cube contains the term$3x^{0.5}y$whose derivative is$1.5y/x^{0.5}$which is decreasing in$x\$. Commented May 30, 2022 at 10:27
• @freelunch In chapter 14 of Varian (Intermediate Microeconomics) the idea is roughly stated in terms of quasilinear preferences. It is more rigorous on these slides by Giorgio Fagiolo (p. 9 onwards): lem.sssup.it/fagiolo/files/get_2.pdf Here, the assumption of decreasing marginal utility (p. 9) leads to a downward sloping demand function. Commented Jun 3, 2022 at 7:45
• @afreelunch: Unfortunately, many introductory micro textbooks at least suggest this connection, even though not explicitly (and wrongly) stating it. E.g. Pindyck and Rubinfeld start their chapter 3.5 with an "intutitive" explanation of marginal utility, adding the "principle of diminishing marginal utility" and thereby conflating the ordinal and the cardinal concepts of utility. Commented Jun 5, 2022 at 2:45