# How can I find the sum of an infinite series of products?

## Background

Today in my macroeconomics class my teacher taught us three concepts.

The first is very simple: consumption $c$ is a linear function of national income $y$. Mathematically, $$c = My + b$$

We will call $M$ the marginal propensity to consume, and refer to it accordingly as MPC. MPC can also be thought of as the fraction of income that is spent rather than saved. So MPC is going to be some number less or equal to one but greater than or equal to zero ($0 \leq M \leq 1$).

The second concept we learned is also quite simple. As income $y$ increases, MPC decreases.

The third concept is a little bit more complicated. Let Person 1 has $5$ dollars. Let MPC be universally constant and equal to $\frac{3}{4}$. Now, since MPC $= \frac{3}{4}$, Person 1 spends $\frac{3}{4}$ of his or her income on Person 2. So Person 2 has an income of $\frac{3}{4} * 5 = 3.75$ dollars. Person 2 will now spend $\frac{3}{4}$ of his or her income on Person 3. And so on to infinity. I instantly recognized this as the sum of a geometric series: $$5*\sum_{n=0}^\infty\left(\frac{3}{4}\right)^{n} = 20$$ This makes sense, but only when disregarding the second concept (MPC decreases as $y$ increases). When I asked about it my teacher said that since calculus isn't a prerequisite for the class we won't go any further in depth; we'll just let MPC be a constant to simplify things in an introductory-level macroeconomics class. So I thought about the problem for a while today and realized that if we did account for MPC decreasing as y increases, or in this case, MPC increasing as $n$ increases, then we could write the total money spent as an infinite series of kind of infinite product (??). My model can be found below.

## The Main Question

I just finished Calculus III last fall so I've never formally learned anything about infinite products. I'm not even sure if these are infinite products since many of them are finite. Anyways, can the following infinite series that I came up with be solved for convergence? If so, how can I figure out the number it converges too? Long explanations will be appreciated. Keep in mind I have a very coarse knowledge of mathematics beyond Calculus III.

$$\sum_{i=0}^{\infty} \left(\prod_{n=0}^{i} \frac{n + 2}{\sqrt{n^2 + 5n + 7}}\right)$$

Perhaps it would be better if I knew how to solve the infinite product first.

Also, I came up with the function $\frac{n + 2}{\sqrt{n^2 + 5n + 7}}$ as a model of MPC because I figured MPC should start around $\frac{3}{4}$ and increase asymptotically towards $1$ as $n$ approaches $\infty$.

• @Shahar Woops! I made a mistake. I've fixed it now. – user168210 Jan 22 '15 at 2:52
• The n+2 term gives (i+2)!. The denominator is harder (duh!). You can ignore the square root and then take the square root of the product, if you can find it. Somehow gamma or theta or q-theta functions come to mind. – marty cohen Jan 22 '15 at 2:59
• I think it may not converge because the asymptotic behaviour of $\sum_{k=0}^n(\log(1+2/k)-\log(1+5/k+7/k^2)/2)$ seems like the harmonic series, which has $-\log(n)$ behaviour, and then $\sum e^{-\log(n)}$ gives a harmonic series again. – user1537366 Jan 22 '15 at 3:13

Not really an answer to your question, but since you seem interested in modelling, I wanted to put some ideas to you. When trying to understand an idea, its good to start simple.

Perhaps you may want to start with a simpler model of a dynamic $M$ as a function of $y$: $M(y):=\min\{\frac{a}{y},1\}\; 0<a$, which implies that people spend roughly $\text{\$} a$regardless of income, unless they can't afford to (hence decreasing propensity). Then, if person$1$has$X$dollars, they spend either$a$or$X$dollars it on person$2$, who subsequently pays person$3$either$a$or$X$dollars. This is a very simple model, and you end up getting that everyone, except person 1, is living "hand-to-mouth"...not fun for the people...but man, what a great multiplier! (i.e.,$\infty\$) ;-)...good example of why maximizing GDP doesn't necessarily maximize psychological well being of a people.

Now, this model is obviously deficient, but in what way? Is the constant propensity model you were presented in class really all that unrealistic for relevant salary ranges (certainly many in the middle class would say its realistic). As an extreme example, consider philanthropy...its done by those with ridiculous incomes and they can spend ridiculous sums on their pet projects. What fraction would we be talking about? Given the small number of such people, are they relevant to economics? Or do we try to model the middle?

Anyway, its great that you are pushing beyond what you've been taught formally...that's the kind of curiosity that leads to new insights! Good luck with your studies.