## dynamic stochastic economics

The above equation summarizes the theory of "optimal control", which is how Von Neumann and others figured out how to continually adjust the path of a missile so it connects with its moving target.

What does the missile equation have to do with economics? The analogy is to "the savings problem".

Money comes in at variable rates, it benefits you differently at different spending levels, investing can make you money, but spending now is fun; money is complicated!  So you have to continually balance the inflow and outflow of cash.

Here’s what the letters mean.

• z is income.
• c is consumption, aka spending.
• U is utility, aka satisfaction you derive from spending — not necessarily the same for all people, nor the same at all levels of spending! Going from starving to a sandwich is better than going from a sandwich to a steak, for example. Also you had better eat at least something this week, or you won’t be around next week to enjoy that cake you’d been saving up for. Dynamicalself-correlated utility.
• β is the time value of money.
• t is time, and most of these variables are indexed by time.
• k is capital, aka money in the bank or investment money.
• π is the probability of making z dollars.
• δ is the depreciation rate of capital. Everything falls apart, you know.

λ is the Lagrangian, something that’s used to balance the equation. It’s an equation-solving tool. Here’s the gist.

$\large \dpi{200} \bg_white \begin{matrix} &[A] &= &[B] \\ \quad &x &= &\lambda \cdot y \\ x \; - \, \lambda\ \cdot \!\!\!\! &y &= &0 \\ x \; - \, \lambda\ \cdot \!\!\!\! &y &= &L \end{matrix}$

You can also think of λ being the sensitivity parameter (in a slack vector sense). Maybe that’s not the best gist. Sorry. But the main point is—isn’t it weird that missile control theory would be used in describing savings behavior?

hi-res

8 notes

1. isomorphismes posted this