3. Economies as Dynamical Systems¶
Is there a common theme underlying the different models that we will study, models that spanned so many decades and modelling philosophies?
3.1. Reduced-form representation¶
Fortunately, the answer is yes.
The similarity is in the “final” mathematical or statistical form, i.e., the “reduced-form” representation of a model’s dynamics.
What may differ within the generic form are the different theories’ implied magnitudes and directions of causal relationships.
Even if the models vary in their underlying assumptions or economic-political philosophies, at the end of the day, after we have solved our agents’ decision problems and their interactions through markets, our model’s (dynamic) equilibrium can be represented by a recursive or Markovian state-space form.
This vital recursive mapping will suffice to describe the evolution of the model economy. End users can then use their preferred model and its resulting recursive system to do things like, fitting the model to the data and testing the model empirically; or performing counterfactual policy analyses and simulations.
3.2. Recursive maps by example¶
Throughout this course we will focus on models that will imply a linear probability model form.
Let’s begin with a simple one. Suppose for now, we let a macroeconomy be represented by a simple probability model.
Warning
Note that there is no explicit economic theory behind this model (yet)! The (implicit) workflow here, as was the case of old-fashioned policy modelling, is
- Write down a statistical model
- Obtain values for unconstrained parameters. (Hold this thought for now; we’ll come to this point later).
- Then weave an economic/policy narrative around the statistical relationship.
So, for the moment, we are undertaking a modelling exercise like that of the policy modellers from the 1950s-1960s. A reincarnation of such policy modelling practice still survives today as statistical Vector Auto Regression models.
To make sure we all start from the same page, consider an linear probability model,
where
- \(\beta\) and \(\gamma\) are given parameters;
- \(Y_{t}\) is some endogenous variable, say, real GDP (let’s call this the state of the economy);
- \(X_{t}\) is some variable that is observable—to both the allegorical agent in the model economy and to the statistician/modeller—at data location \(t\). Suppose the index variable \(t\) keeps track of natural time. Assume time is countable: \(t \in \mathbb{N} := \{0, 1, 2, ...\}\); and
- \(\varepsilon_{t}\) is an exogenous shifter to the state of the economy which is governed by a given probability distribution \(\varphi\).
This linear probability model should be familiar to the reader with a first-course in statistics or econometrics training.
What if we make the content of \(X_{t}\), explicit?
Example 1
\(X_{t} := Y_{t-1}\).
Plug this into the previous linear probability model to get
This is a scalar first-order linear stochastic difference equation (LSDE). Let’s keep working with the reduced-form model of an economy for now. Consider:
Example 2
\(X_{t} := Y_{t-1}\) and \(\varepsilon_{t} = 1\) for all \(t\).
Plug this into the previous linear probability model to get
As long as \(\beta \neq 1\), a particular solution to this deterministic difference equation exists, and takes the form of a constant solution:
for all \(t \in \mathbb{N}\).
We often call this solution \(Y\) a stationary point, or, a steady state.
Exercise
Show that the general solution to this difference equation is
- Explain in words what this function says.
- Pick some numerical values for the parameters \((\beta,\gamma)\). Write a Python code and plot the general solution as a graph in \((t, Y_{t})\)-space.
- Show what happens to your graph if \(| \beta | > 1\)?
- Show what happens to your graph if \(| \beta | < 1\)?
3.3. Linear stochastic difference equation systems¶
Consider now the generalization of the scalar (univariate) example above. We often call these linear stochastic difference equations (LSDE):
Note
- READ:
PDF notes on LSDE
Many empirical and policy models with have (approximately) solutions of this form. In this course, we will begin with models where their individual solution is either an exact, or an approximate, linear recursive self-map. As in all the toy examples above.
What if we have a model with arbitrary dependency of its current state on an arbitrarily long record of its past? Not a problem. We will see that in general, we can re-define the problem to make it Markovian or recursive again. The trick will be in expanding the notion of the model’s state space and re-defining appropriate “dummy” or auxiliary state variables.
3.4. Postscript¶
Now take a look back to what we’ve done here; then look ahead to the rest of this course. Many models will generally have solutions to their respective decision-making or equilibrium concept in the form of a recursive function:
where \((x,w) \mapsto F(x,w)\) is some general (possibly nonlinear) recursive map, \(x_{t}\) represents the state vector of the model economy, and, \(w_{t}\) is some forcing process exogenous to the model system. To reiterate, we will only focus on the case that the mapping \(F\) is of a linear form.
You might ask: What is the point of all this “mathiness”? The goal is use this resulting recursive map to simulate and study the dynamic behavior of the model economy, under various economic policy scenarios. In short, the model is a laboratory for potential policy experiments, albeit an imperfect or inaccurate one. Most economists have to resort to inaccurate models to study policy counterfactuals simply because randomized control trials are difficult, especially when the subjects under study is a who economy.