I’m in a feud with my previous landlord at the moment. For a bit of background Philadelphia is a very old city, so most of the housing options are in older buildings that have been continually renovated over time, rather than larger apartment complexes that other cities might have. So finding housing can be a challenge, and where I used to live belonged to a company that managed several properties in the area.
The company is highly disorganized, and only has 3 people. The feud stems from their disorganization, where I should have received the security deposit weeks ago but between either the blatant lies to me or miscommunication amongst the 3 it hasn’t landed in my new mailbox.
So I’m not sure what to do. I know that small claims court is an option if the issue persists, but I just don’t want to have to deal with it anymore and put the old apartment and all its headaches behind me.
One could say my state, both financially and mentally, is worse off because of this. And so because of it maybe my decision for other purchases, like maybe an extra afternoon coffee as a reward for the day, are affected by it. Implicitly, I’d like to maximize my utility through maximizing my state. If my state changed, and I got the security deposit back, maybe I would choose differently.
This is maximization of utility and state is what macroeconomists attempt to study on the aggregate level through a class of models called DSGE. That is, Dynamic Stochastic General Equilibrium models. They’re dynamic since they vary over time, stochastic because they may or may not involve uncertainty, and general equilibrium because under the solution, we get an optimal value.
The models can be relatively simple and solvable by hand, usually where a representative consumer faces a decision on how much to spend or save in a given time period, to relatively complex involving many constraints and differentiation in the types of goods offered, and how capital is accumulated, usually requiring an iterative solution using a computer.
One way to solve these problems is analytically. You set up an objective function, your utility based on consumption, constraints, and take derivatives to find how you should spend / save in a given period.
If the problem is complex and taking derivatives isn’t something you want to do, another is through the recursive method, involving setting up something called a value function. What this value function does is measure the value of your current state. In a previous post I explained the difference between state and control variables, where the control variables can be used to optimize state variables which transfer through time.
So at its core, to measure the value of the state is to measure to sum total of all the variables you’ve carried through time to get to this point. Take the simple example below with 3 states and 1 state variable U
In the 3 states, Left, middle, and right, there is a different level of utility, equalling either 1, 2 or 3. Say you start in the middle state, and want to maximize the value of your state. In order to do so, you need a rule or set of rules that tells you how you move between states. So a rule could be “always left” or “always right” or “left half the time, right half the time”, or “don’t move." The expected value for each of these rules for 2 time periods, say now and later, is
Always left: U(now) + E(U(later)) = 2+1 = 3
Always right: U(now) + E(U(later)) = 2+3 = 5
Left half the time, right half the time = U(now) + E(U(later)) = 2+ 1*0.5 + 3*0.5 = 4
Don’t move: U(now) + E(U(later)) = 2 + 2 = 4
Clearly some policies here might lead to better outcomes, and picking the “Always right” policy function leads to the highest expected utility. Here the value of the state over the two periods, given your starting place at t=0 being in the middle, might be maximized then by following this rule.
What’s amazing about the recursive approach, is that by maximizing the value of your state- you also maximize the value of your objective function. This is an application of the envelope theorem, where by maximizing the value of one function that “envelopes,” or contains another function, you maximize the value of the inner function as well. So if V(U(c)) the value of the state is maximized, then the value of utility is also maximized.
Here the value of the state is derived from the potential of that state to translate to inputs into the objective function. For economics, and the save / consume trade off, this translates to inputs (consumption) into the objective function (utility). You can only get utility from consumption, and can only consume from spending capital (the state variable).
This relates to physics in the sense of potential energy.
The value of the state is only valuable due to the potential to convert from potential energy to kinetic, where the total energy for the rock at point A is the same at point D, when the rock strikes the ground. Here potential energy transitions to kinetic energy, following the policy rule of gravity.
For economics, this looks like capital being converted into consumption. The value of the state of your bank account is only valuable because of the ability to transition dollars into goods and services. But unlike the rock, money can be created from other money. So total energy doesn’t stay the same, the economy grows over time! This doesn’t change the principle, just the math a little bit.
Since we want to study the conditions where we maximize this utility from consumption, we want a policy that, like gravity transitions potential energy to kinetic energy, transitions capital to consumption. And ideally that policy will maximize our utility from consumption.
Setting up the recursive value function might look something like this then.
And with a given budget constraint we can find a rule that maximizes not only the utility over time, but maximizes the value of the state over time.
So, in the end, economics is telling us that if I want to maximize my utility I need to maximize my state. And hopefully that happens when I get this security deposit back.