Yes, this seems like a similar approach to prove your expression, replacing part of the infinite expression with itself. Not familiar with the economics part of it though ;)

south_korea_ln

science

[Daily puzzle] Exact value of continued fraction

By the way, I was wondering about this proof method.  Would the following proof work?

$$
x = 1 - (1 - (1 - (1 - \dots 
$$

$$
x = 1 - x
$$

$$
x = 0.5
$$

To shed some light, I'm an economist, not a mathematician.  But I have used this method to prove the formula for a discounted stream of infinite cash flows:

$$
NPV = 1 + \delta + \delta^2 + \ldots
$$

$$
NPV = 1 + \delta (1 + \delta + \ldots 
$$

$$
NPV = 1 + \delta NPV
$$

$$
NPV = \frac{1}{1-\delta}
$$

I guess it has to do with whether a series converges or not.  If it doesn't you probably can't assume that $$x \in \mathbb{R}$$, or something like that.
