What are the assumptions made about fixed points in the dynamics equations of Recursive macroeconomics?

Question

I am new to Macroeconomics, but I understand the basics of Recursive Macroeconomic models--following the Ljungqvist and Sargent book. So I get the basic recursive problem to find a vector of consumption $c_t$ and labor $h_t$ that optimizes the objective function over and infinite time horizon.

$$ max_{c, h} \quad \mathbb{E}(\sum^{\infty}_{t=0} \beta^tu(c_t, h_t)) $$

This is a common setup for the problem, and a similar setup seems possible for the production function over an infinite time horizon as well. The usual solution method is to apply Linear Quadratic Control, in other words solve a Riccati equation to obtain the values of $c_t, h_t$ or some control inputs.

My question has to do with fixed points. In order for LQ control to work, we have to assume that there is some fixed point for the utility function $u(\cdots)$, around which we want to optimize. That is, the consumption and labor choices to achieve some maximum at that fixed point.

But it is not clear that only 1 fixed point should exist in this model? Seems like it is possible for 2 or more fixed points to potentially exist, depending on the complexity of the system. So I was trying to understand what are the assumptions or intuitions surrounding the idea of essentially 1 fixed point in the fundamental recursive models in macro. I say essentially 1 fixed point, because while Ljungqvist and Sargent use the idea of fixed points in their book, I did not find an discussion on the uniqueness of fixed points, or the existence of only 1 fixed point, etc. Hence I was a bit confused.

Is there a good resource or explanation about what the assumptions or intuition is around fixed points in Recursive macro? Any help would be appreciated. It seems like this is a fundamental but subtle issue that can easily get lost when a lot of equations and derivations start flying. But if there are 2 different fixed points, given a utility function, then that means that there are 2 different vectors $c_t, h_t$ that I can optimize around--namely one for each fixed point.

Thanks.

Answer 1

On pages 53-55 of the Stokey, Lucas, with Prescott (1989) book they discuss the Contraction Mapping Theorem. This theorem guarantees existence and uniqueness of the solution (one fixed point).

The most convenient way to show the theorem holds is by making assumptions on the problem such that Blackwell’s Sufficiency Conditions are met (page 54). Namely, these are

1. Monotonicity (of the utility function in your example)
2. Discounting (given through $\beta$ in your problem)