Economic Uncontrollability

Economic control is a fantasy politicians and economists sell between crashes.

The economy is often described as a complex adaptive system: its behaviour emerges from the interactions of millions of agents, such as individuals, firms, and institutions, each with their own expectations, constraints, and abilities. It exhibits characteristics of second-order chaos: the economy reacts to predictions about itself. This raises a fundamental question: Is the economy even controllable through policies?

We shall look at controllability from the perspective of systems engineering and dynamical systems theory. From it, we shall learn that reflexivity hinders the controllability of even highly simplified economic systems. Since the real economy is vastly more complex, it stands to reason that control is all but out of reach.

Background concepts

Controllability in systems engineering

In control theory, a system is controllable if for any initial state \(\mathbf{x}(t_{\mathrm{i}})\) and any desired final state \(\mathbf{x}(t_{\mathrm{f}})\), there exists a control input \(\mathbf{u}(t)\) over a finite time interval \([t_{\mathrm{i}}, t_{\mathrm{f}}]\) that can steer the system from the initial to the final state.

For linear time-invariant systems, controllability is determined by the rank of the controllability matrix \(\mathcal{C}\), which for discrete-time systems with \(n\) state variables is \(\mathcal{C} = [B | AB | A^2B | \dots | A^{n-1}B]\). Here, \(A\) and \(B\) are defined through the difference equation: \(\mathbf{x}_{t+1} = A \mathbf{x}_t + B \mathbf{u}_t\). If \(\text{rank}(\mathcal{C}) = n\), then the system is said to be controllable.

Levels of chaos

In the context of dynamical systems, chaos is the result of tiny initial perturbations that grow exponentially over time and therefore can lead to vastly different outcomes.

Yuval Noah Harari popularized a useful distinction for chaotic systems:

• Level-1 chaos: dynamical systems that exhibit chaotic behaviour yet do not react to predictions about said systems, such as the weather and planetary orbits. For instance, predicting sunshine does not alter the likelihood of rain. These systems can be complex and hard to predict, but the prediction does not change the system’s behaviour unless humans intervene based on said forecasts.
• Level-2 chaos: dynamical systems that react to predictions about them, such as financial markets, political elections, and the economy. For example, forecasting a stock market crash can trigger panic selling, potentially fulfilling the prophecy. Economic forecasts influence decisions by firms, consumers, and policymakers, thereby changing the economic trajectory the forecast was trying to predict.

The economy is fundamentally a level-2 chaotic system due to the forward-looking, adaptive nature of its constituent agents.

The argument for economic uncontrollability

A formal proof of uncontrollability for the real economy is likely impossible due to its sheer scale and the inability to precisely model it. Nevertheless, a strong case can be made based on control theory principles and certain observed economic characteristics:

1. Model uncertainty and complexity: There is no universally accepted model that captures the high-dimensional, non-linear, time-varying dynamics of the entire economy. All models are vast simplifications.
2. State observation: Key state variables (e.g. consumer confidence) are unobservable or measured imperfectly with significant delays.
3. Control limitations: Policy tools (e.g. fiscal expenditure, taxation, interest rates) are blunt instruments. Their effects are indirect, involve long and variable delays, and the magnitude of their impact is often uncertain and state-dependent.
4. Reflexivity: Economic agents adapt their behaviour based on perceived policy rules, announcements, and forecasts. The system’s dynamics are therefore not fixed but change in response to control attempts.

A linear macroeconomic model

Let’s look at a highly stylized, discrete-time model. The output gap \(y_{t}\) at time \(t\) is the difference between the current economic output and the full sustainable potential output without causing inflation to rise. The aim of central banks is to reach an output gap of zero, which means the economy is neither overheating nor underperforming but at full capacity. The output gap in the next period \(y_{t+1}\) depends on the current one \(y_{t}\), and it is reduced by inflation \(\pi_{t}\). If the output gap is positive, firms can raise prices because of increased demand; this causes inflation. Likewise, if the output gap is negative, firms lower prices because of weak demand, which in turn lowers inflation. Adding a policy control \(u_{t}\), we have the following set of linear equations:

\[\left\lbrace\begin{eqnarray} y_{t+1} &=& a y_{t} - b \pi_{t} + cu_{t}, \\ \pi_{t+1} &=& d\pi_{t} + ey_{t}. \end{eqnarray}\right.\]

All constants \(a,b,c,d,e\in \mathbb{R}\) are positive.

With the state vector \(\mathbf{x}_{t} = \begin{bmatrix} y_{t} \\ \pi_{t} \end{bmatrix}\), we can rewrite the equations as the difference equations from before with \(A = \begin{bmatrix} a & -b \\ e & d \end{bmatrix}\) and \(B = \begin{bmatrix} c \\ 0 \end{bmatrix}\).

Since we have only \(n=2\) state variables (i.e. the output gap and the inflation), we must compute the controllability matrix:

\[AB = \begin{bmatrix} a & -b \\ e & d \end{bmatrix} \begin{bmatrix} c \\ 0 \end{bmatrix} = \begin{bmatrix} ac \\ ec \end{bmatrix},\]

from which it is easy to verify that

\[\mathcal{C} = \begin{bmatrix} B \mid AB \end{bmatrix} = \begin{bmatrix} c & ac \\ 0 & ec \end{bmatrix}.\]

Its determinant is \(\det(\mathcal{C}) = ec^2\). Since both \(e > 0\) and \(c > 0\), \(\det(\mathcal{C}) = ec^2 > 0\). The controllability matrix has full rank, and therefore this non-reflexive linear model is mathematically controllable.

A reflexive macroeconomic model

Let’s incorporate reflexivity. We shall assume that economic agents adapt their behaviour based on policy actions. A plausible mechanism is that the effectiveness of such actions \(s\) diminishes when policymakers often resort to large interventions. The parameter \(s\) becomes time-varying and dependent on control usage, \(s \mapsto s_{t}\). So, the policy effectiveness \(s_{t}\) decreases with the magnitude of the control input:

\[s_{t} = \frac{s_{0}}{1 + k \lVert \mathbf{u}_{t}\rVert},\]

where \(s_{0}\) is the baseline policy effectiveness and \(k > 0\) measures the adaptation strength. With that we have the same matrix \(A\), but \(B\) becomes dependent on time. Hence, the standard rank test of the controllability matrix does not apply, as the model is no longer time-invariant.

Large control inputs \(\lVert\mathbf{u}_{t}\rVert\) needed for significant changes reduce policy effectiveness \(s_{t}>0\), basically undermining control. More importantly, this mechanism bounds the set of reachable states, preventing arbitrary steering.

But why talk about reachable sets? For a more generic dynamical system, we can use the reachable set to talk about controllability. The reachable set is the set of possible states the system can be steered towards. A dynamical system is said to be globally controllable if for any initial state \(\mathbf{x}(t_{\mathrm{i}})\) and any desired final state \(\mathbf{x}(t_{\mathrm{f}})\), both in \(\mathbb{R}^{n}\), there exists a finite time \(T\) such that the final state is within the reachable set from the initial state within \(T\). Phrased differently, a system is globally controllable if and only if the reachable set from any point eventually covers the entire state space.

The key question to establish controllability, or the lack thereof, is whether we can prove whether the reachable set from an arbitrary initial state is the entire state space or not. Let’s define the control vector \(\mathbf{v}_{t}=B_{t}\mathbf{u}_{t} = \begin{bmatrix} \frac{s_{0}}{1+k \lVert\mathbf{u}_{t}\rVert} \\ 0 \end{bmatrix}\mathbf{u}_{t}\). The absolute value of the control vector is a monotonically increasing function, so we can compute its maximum value with L’Hôpital’s rule:

\[v_{\mathrm{max}} = \lim_{\lVert\mathbf{u}_{t}\rVert\to\infty}{\lVert \mathbf{v}_{t}\rVert} = \lim_{\lVert\mathbf{u}_{t}\rVert\to\infty}{\frac{s_{0}\lVert\mathbf{u}_{t}\rVert}{1+k\lVert\mathbf{u}_{t}\rVert}} = \frac{s_{0}}{k} > 0.\]

The control vector is therefore always bounded. We can use that in figuring out the reachable set after \(N\) iterations:

\[\begin{eqnarray} \mathbf{x}_{1} &=& A\mathbf{x}_{0} + \mathbf{v}_{0} \\ \mathbf{x}_{2} &=& A\mathbf{x}_{1} + \mathbf{v}_{1} \\ &=& A\left(A\mathbf{x}_{0} + \mathbf{v}_{0} \right) + \mathbf{v}_{1} \\ &=& A^{2}\mathbf{x}_{0} + A\mathbf{v}_{0}+\mathbf{v}_{1}\\ \vdots && \ldots \\ \mathbf{x}_{N} &=& A^{N}\mathbf{x}_{0} + \sum_{i=0}^{N-1}{A^{N-1-i}\mathbf{v}_{i}}. \end{eqnarray}\]

By applying the triangle inequality repeatedly and using the fact that the control vector is bounded (\(\lVert \mathbf{v}_{t}\rVert \leq v_{\mathrm{max}}\) for all \(t\)), we can show that the reachable set is a closed ball centred around the origin:

\[\begin{eqnarray} \left\lVert \mathbf{x}_{N} \right\rVert &=& \left\lVert A^{N}\mathbf{x}_{0} + \sum_{i=0}^{N-1}{A^{N-1-i}\mathbf{v}_{i}} \right\rVert \\ &\leq & \left\lVert A^{N}\mathbf{x}_{0} \right\rVert + \left\lVert\sum_{i=0}^{N-1}{A^{N-1-i}\mathbf{v}_{i}}\right\rVert \\ &\leq & \left\lVert A^{N}\right\rVert \left\lVert \mathbf{x}_{0} \right\rVert + \sum_{i=0}^{N-1}{\left\lVert A^{N-1-i}\mathbf{v}_{i} \right\rVert } \\ &\leq & \left\lVert A^{N}\right\rVert \left\lVert \mathbf{x}_{0} \right\rVert + \sum_{i=0}^{N-1}{\left\lVert A^{N-1-i}\right\rVert } \left\lVert \mathbf{v}_{i} \right\rVert \\ &\leq & \left\lVert A^{N}\right\rVert \left\lVert \mathbf{x}_{0} \right\rVert + \sum_{i=0}^{N-1}{\left\lVert A^{N-1-i}\right\rVert } \cdot v_{\mathrm{max}} \\ &=& \left\lVert A^{N}\right\rVert \left\lVert \mathbf{x}_{0} \right\rVert + v_{\mathrm{max}} \sum_{j=0}^{N-1}{\left\lVert A^{j}\right\rVert }. \end{eqnarray}\]

The sum of matrix products is a finite geometric sum, and thus bounded, provided \(A\) is Schur stable (i.e. its eigenvalues have magnitude less than unity), so that \(\left\lVert A^{N} \right\rVert \to 0\) as \(N\to\infty\). That is a fairly reasonable assumption, as otherwise the entire economy would diverge on its own, rendering any discussion of control moot. Anyway, the entire right-hand side is a finite value, say, \(M\), so that \(\lVert\mathbf{x}_{t}\rVert\leq M\). The system is therefore not globally controllable, as predicted, because the reachable set is bounded.

A more realistic macroeconomic model

A more realistic model of the economy is based on the Investment–Saving (IS curve), the Phillips curve (PC), and a monetary policy rule. The IS curve describes the demand side of the economy, whereas the PC pertains to the supply side. Such a model comes with reflexivity built in, as we shall see.

Demand

Aggregate demand is the total demand for final goods and services in an economy. This includes investments, corporate and government expenditure, net exports, and consumer spending. The Investment–Saving (IS) curve describes the aggregate demand: low interest rates boost consumption and investment, which increases output. We can represent the aggregate demand with the IS curve as follows:

\[y_{t} = \mathbb{E}_{t}[y_{t+1}] - \alpha \left( i_{t} - \mathbb{E}_{t}[\pi_{t+1}] -r_{t} \right).\]

Here, \(\mathbb{E}_{t}[y_{t+1}]\) is the current expectation of the output gap in the next period. Likewise, \(\mathbb{E}_{t}[\pi_{t+1}]\) encodes the current expectation of the inflation rate in the next period. The nominal interest rate at \(t\) is indicated by \(i_{t}\), and \(r_{t}\) is the assumed natural rate of interest. The whole second term, the real interest rate, says that if real inflation is high, it becomes expensive to borrow money, so that spending and output go down. In other words, current demand depends on expectations of demand, expectations of inflation rate, and the current interest rate.

Supply

When economic activity is high (\(y_{t}>0\)), the economy is booming. In that case, unemployment is low, and firms have to compete for workers by offering higher wages. To cover these higher costs, they raise prices on their products and services. This is described by the New Keynesian Phillips curve (PC), which is defined as follows:

\[\pi_{t} = \beta \mathbb{E}_{t}[\pi_{t+1}] + \kappa y_{t}.\]

It states that the market’s expectations of tomorrow are a part of the inflation rate of today: firms set prices based on what they think their costs and competitor’s prices will be in the near future. The second term shows that in a booming economy (\(y_{t}>0\)), costs are higher, so prices go up, and with it inflation.

Monetary policy

The Taylor rule explains how a central bank is supposed to set the nominal interest rate \(i_{t}\) based on the inflation rate and output gap:

\[i_{t} = r_{n} + \pi_{t} + \phi_{\pi}\left(\pi_{t}-\pi^{\ast}\right) + \phi_{y}y_{t}.\]

Any deviation from the target inflation rate \(\pi^{\ast}\) leads to a correction proportional to \(\phi_{\pi}\). In a similar way is there a correction proportional to \(\phi_{y}\) based on the output gap, which has a target of zero, as stated previously.

Aggregate demand

The aggregate demand (AD) curve combines all three ingredients. We can think through a common scenario:

1. Inflation (\(\pi_{t}\)) goes up.
2. The central bank reacts by raising the nominal interest rate (\(i_{t}\)).
3. The real interest rate increases.
4. Because of a higher real interest rate, businesses and households cut back on their spending.
5. Output in the economy goes down.

We can actually see reflexivity in this more realistic model, in which the nominal interest rate is the control variable. Any control \(i_{t}\) influences the output gap, inflation rate, and expectations thereof simultaneously. If the policy action is predictable, rational agents will factor that into their expectations, which neutralizes the effect on the current output gap and inflation rate. This is the essence of the Lucas critique. Policymakers must navigate a system that fights back, not with fists, but with foresight. Any attempt to control the system requires the policymaker to account for the fact that agents are trying to predict their actions. A policymaker must also steer the public’s expectations: reflexivity is therefore endogenous. In such cases, successful control depends on surprises.

We can treat the expectations as independent variables, so we have a four-dimensional state vector: \(\mathbf{x}_{t} = \begin{bmatrix} y_{t}, \pi_{t}, \mathbb{E}_{t}[y_{t+1}], \mathbb{E}_{t}[\pi_{t+1}] \end{bmatrix}^{^\intercal}\). The only tricky bit is that we need equations for the evolution of the expectations. We don’t if we assume perfect foresight: \(\mathbb{E}_{t}[y_{t+1}]=y_{t+1}\) and \(\mathbb{E}_{t}[\pi_{t+1}]=\pi_{t+1}\). This converts a four-dimensional forward-looking system into a standard two-dimensional one without reflexivity, from which we can derive the equations of state:

\[\left\lbrace\begin{eqnarray} y_{t+1} &=& \left(1+\frac{\alpha\kappa}{\beta}\right)y_{t} - \frac{\alpha}{\beta}\pi_{t} + \alpha i_{t}, \\ \pi_{t+1} &=& \frac{\pi_{t}-\kappa y_{t}}{\beta}. \end{eqnarray}\right.\]

For convenience, I have ignored the natural rate of interest \(r_{t}\). We can now then read off the matrices:

\[A = \begin{bmatrix} 1+\frac{\alpha\kappa}{\beta} & -\frac{\alpha}{\beta} \\ -\frac{\kappa}{\beta} & \frac{1}{\beta} \end{bmatrix}, \quad B = \begin{bmatrix} \alpha \\ 0 \end{bmatrix}.\]

The controllability matrix becomes:

\[\mathcal{C} = \begin{bmatrix} \alpha & \alpha\left(1+\frac{\alpha\kappa}{\beta}\right) \\ 0 & -\frac{\alpha\kappa}{\beta} \end{bmatrix}.\]

Its determinant is \(\det\left(\mathcal{C}\right) = -\frac{\alpha^{2}\kappa}{\beta}\), which is non-zero whenever \(\alpha\neq 0\) (i.e. the interest rate control), \(\kappa\neq 0\) (i.e. inflation reacts to the output gap), and \(\beta\neq 0\) (i.e. future inflation matters). It is therefore controllable under very reasonable assumptions in the special case of perfect foresight, which is, unfortunately, not a reflexive model.

In case you wish to verify these results with the open-source computer algebra system Maxima, you can copy-paste the following code:

eq1  : y[t] = y[t+1] - α*(i[t] - p[t+1]);
eq2  : p[t] = β*p[t+1] + κ*y[t];
sols : first( solve( [eq1, eq2], [ y[t+1], p[t+1] ] ) );
expand(sols);

y[rhs] : expand(rhs(sols[1]));
p[rhs] : expand(rhs(sols[2]));

A : matrix( [ coeff(y[rhs], y[t]), coeff(y[rhs], p[t]) ], 
            [ coeff(p[rhs], y[t]), coeff(p[rhs], p[t]) ] );

B : matrix( [ coeff(y[rhs], i[t]) ],
            [ coeff(p[rhs], i[t]) ] );

C : addcol(B, A . B);
determinant(C);

Conclusion

All in all, linear time-invariant models of the economy suggest economic control is theoretically achievable. Yet when we incorporate reflexivity, where agents adapt to policy rules as formalized by the Lucas critique, even basic models become uncontrollable in the mathematical sense. Does this prove the lack of controllability of the real economy? No, it does not.

That said, there are good reasons to believe it extends to the real economy:

1. The real economy is much more complex. It exhibits chaotic behaviour, in which small deviations can lead to large differences after a while.
2. For non-linear systems, controllability can usually only be guaranteed locally around a specific equilibrium. This is fine as long as the economy is in a relatively steady state. Once we need to steer it between recessions or hyperinflation, such local controllability is no longer valid.
3. Reflexivity, or the second-order chaos it entails, means that control strategies depend not only on the state of the economy but also on predictions about it, which inevitably make control a fuzzy moving target.

Policies are not irrelevant, though. They can still influence, stabilize, or shape economic outcomes. The idea that we can control the economy in a strict sense is, however, folly.