I'm currently reading chapter 4 of Mas-Colell, Whinston, and Green (1995). I have a problem with the way integrals are treated. For instance, Proposition 4.C.4 (p.113) states that when wealth is uniformly distributed on $[0,W]$, the aggregate and the average demand function are: $$x(p) = \int_0^W x(p,w) dw $$

Where $x(p,w)$ is an individual demand (every individual has the same preferences, their demands only differ in the amount of wealth).

However, I think the right way to write the average would be: $$x(p) = \int_0^W \frac1W x(p,w) dw $$ because this is the definition of the expected value.

Am I wrong? When using integrals, can I just assume that averages are equal to aggregates? This a big problem to me because in exercise 4.C.10 it is asked to prove that $$C(p,w)=\int_0^W S(p,w)dw - S(p,W)=-D_w x(p).x^T(p)+\int_0^WD_w x(p,w).x^T(p,w)dw$$ is positive definite, where $S(p,w)$ are individual Slutsky matrices, and $S(p,W)$ is the Slutsky matrix of aggregate demand.

I thought to define $w=aW$ with $a$ uniformly distributed on some interval. Therefore, I could re-define $x(p)=x(p,W)$ and differentiate $x(p)=x(p,W)$ and $x(p,w)=x(p,aW)$ with respect to $W$. However, I'm not sure how to define an interval for $a$ such that average wealth is equal to aggregate wealth using $w=aW$.

Note: Proposition 4.C.4 is based on an example given by Hildebrand (1983) ("On the Law of Demand"). I read the article and he assumes wealth is distributed on $[0,1]$. With this interval, aggregate demand is equal to average demand. Does this mean Proposition 4.C.4 is misspelled? If it is so, I still have a problem with exercise 4.C.10 because I need an interval for a such that aggregate wealth is equal to average wealth.


1 Answer 1


I talked to my teacher to try and get a handle on the question.

You are correct that usually

$$\overline {f(x)} = \frac{1}{b-a}\int^b_a f(x) dx$$

But for the demand function there are two arguments, $w$ and $p$.

$$x(p) = \int_0^\bar W \tilde x(p,w) dw$$

  • The left hand side is no longer a function of $w$. The integral is to simplify wealth out of the picture, but we aren't averaging over wealth.
  • All consumer preferences are identical. For the sake of argument, suppose there were $n$ consumers. We aren't multiplying by $n$ on the inside of the integral, so we don't need to divide out anything (like $n$) through the outside of the integral. The "aggregation" is occurring over all consumers.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.