# Demand curve is same as Marginal Benefit curve?

Demand curve is same as Marginal Benefit curve [Source: Microeconomics by Pindyck & Rubinfeld, Chapter 10, Section 5 - Monopsony]. I have only seen intuitive explanations for this statement. Can anyone prove it mathematically?

I am confused because for example is the curve is P = 10 - Q, then for Q = 1 to Q = 2, the total utility level changes for BOTH the first consumed product and the second one but the term "marginal benefit" seems to capture only the change due to second item consumed.

• You have many answered questions, how about accepting some answers? – Giskard May 17 at 5:44
• @Giskard done.. – reasonStore May 17 at 19:27
• Thank you kind sir ;) – Giskard May 17 at 21:26

Nuance matters:

1. The demand curve represents marginal benefit. The vertical distance at each quantity shows the mount consumers are willing to pay for that unit. Willingness to pay reflects the benefit derived from each unit.

So the actual claim is not that the demand curve is the same as the marginal benefit curve, but that it represents it in some way. To be even more precise, the inverse demand curve represents marginal benefit. The demand and inverse demand curves are often conflated as these mappings are represented by the same graph.

I do not know where you heard that but it is generally not true that demand curve is equivalent to marginal utility/benefit curve.

Consider trivial counter-example:

$$U = x^ay^b \text{ s. t. } px+qx=m$$

Where $$U$$ is the utility of consuming good $$x$$ and good $$y$$, $$p$$ and $$q$$ are their respective prices and $$m$$ is the consumer budget.

Here clearly the marginal utility (benefit) of consuming $$x$$ and $$y$$ is given by:

$$U_x' = a x^{a-1} y^b \tag{1}$$

$$U_y'= b x^a y^{b-1} \tag{2}$$

Whereas demands for $$x$$ and $$y$$ are given by:

$$x^*=x(p,q,m) = \frac{a}{a+b}\frac{m}{p} \tag{3}$$

$$y^*=y(p,q,m) = \frac{b}{a+b}\frac{m}{q} \tag{4}$$

1 and 3 and 2 and 4 are clearly generally not the same functions. This does not mean there is no relationship between marginal utility and demand function. In fact 3 and 4 are derived from constrained optimization problem that incorporates 1 and 2 but just because there is relationship between marginal utility and demand you can't just claim they are equivalent.

It is possible to make some special utility functions where demand and marginal utility will be equivalent, but you really need to 'fudge the numbers' to make them equal, and you would also have to impose some special restrictions on marginal utility function (which can be negative whereas demand typically can't). Thus as proven above the premise of the question is false, marginal utility/benefit curve is generally not equal to the demand curve.

• I have added the source in my question – reasonStore May 16 at 23:33
• Here's another: www3.nd.edu/~cwilber/econ504/504book/…. – reasonStore May 16 at 23:34
• @reasonStore 1 don’t have the textbook you use so I can’t see what is written there. 2. I proven mathematically above that does not generally hold (outside special cases), book might have a stylized case where it holds but it’s not the same. You can verify that by consulting MGW - bible of microeconomics which has many such example, or texts such as Varian Microeconomic Analysis etc all graduate level well known and cited books. 3. I doubt that the textbook literally states they are equivalent – 1muflon1 May 16 at 23:59
• @reasonStore I mean just another trivial example of utility function U=x at such utility marginal utility is given by 1. That does not mean your demand will be 1 in fact with such marginal utility the demand will be infinity at price equal or less to 1, and above 1 it will drop to zero. Again plainly showing that marginal utility is not equal to demand function- again there is a relationship there since to determine demand you need to know marginal utility but save special cases they won’t be equal – 1muflon1 May 17 at 0:05

### Quasi-linear functions

There is one special case where the inverse demand curve equals the marginal utility function. This is when the utility function is quasi-linear, i.e. it takes the form: $$u(x,y) = v(x) + y.$$ In this case, the first order condition is equal to: $$v'(x) = p_x.$$ Here $$v'(x)$$ is indeed the inverse demand function and it is also equal to the marginal utility curve.

The area under the inverse demand curve will be given by: $$\int_0^q v'(x) dx = v(q) - v(0).$$ So it equals the increase in utility from increasing $$x$$ from $$0$$ to $$q$$ while keeping $$y$$ fixed.

In general (without) quasi-linearity we have: $$\frac{\partial u}{\partial x} = \lambda p_x,$$ Now, $$\lambda$$ in general depends on prices and income, so it is not straightforward to invert this function.

In other words, the inverse demand will usually differ from the marginal utility function due to income effects. If income effects are constant (e.g. $$\lambda$$ does not change with prices and income) then the inverse demand function will be simple re-scaling of the marginal utility function. $$p_x = \frac{1}{\lambda} \frac{\partial u}{\partial x}.$$

This connects to the known fact that the consumer surplus will only be an exact measure of utility when there are no income effects, i.e. when utility is quasi-linear.

### Homothetic utility functions

There is another interesting approximate connection between the inverse demand and the marginal utility curve when utility functions are homothetic.

Let's multiply all first order conditions with $$x_i$$ and then sum over $$i$$ to obtain: $$\sum_i \frac{\partial u}{\partial x_i} x_i = \lambda \sum_i p_{x_i} x_i = \lambda m.$$ where $$m$$ is income.

By Euler's theorem, the left hand side equals the utility level $$u(x_1, \ldots, x_n)$$ so: $$\lambda = \frac{u}{m}.$$ As such the inverse demand takes the form: $$p_i = \frac{1}{u} \frac{\partial u}{\partial x_i}.$$ This is what you have for the Cobb-Douglas case:

The area under the inverse demand curve will then be equal to: $$\int_0^q \frac{1}{u} \frac{\partial u}{\partial x_i} dx_i = \ln(u(x_1, \ldots, x_{i-1}, q, x_{i+1},\ldots, x_n)) - \ln(u(x_1, \ldots, x_{i-1}, 0, x_{i+1},\ldots, x_n)).$$ Which is measures the (log) gain in utility from increasing consumption of good $$i$$ from $$0$$ to $$q$$.