ECON526: Quantitative Economics with Data Science Applications

Linear and Nonlinear Dynamics

Jesse Perla
jesse.perla@ubc.ca

University of British Columbia

Overview

Motivation and Materials

  • In this lecture, we will apply some of our tools to non-linear equations, which come up in macroeconomics, industrial organization, and econometrics
  • The primary example is a simple version of the growth models
  • We will introduce the idea of a fixed point, which has many applications across fields of economics
  • A special emphasis will be placed on analyzing stability - which connects to the eigenvalues of the dynamical system

Packages and Other Materials

import matplotlib.pyplot as plt
import numpy as np
from numpy.linalg import norm
from scipy.linalg import inv, solve, det, eig, lu, eigvals

Fixed Points

Fixed Points of a Map

Fixed Point

Let \(f : S \to S\) where we will assume \(S \subseteq \mathbb{R}^N\). Then a fixed point \(x^* \in S\) of \(f\) is one where

\[ x^* = f(x^*) \]

Fixed points may not exist, or could have multiplicity

Fixed Points for Linear Functions

  • We have already done this for linear functions.
  • Let \(f(x) = \begin{bmatrix} 0.8 & 0.2 \\ 0.2 & 0.8 \end{bmatrix} x\)
  • Then we know that \(x^* = \begin{bmatrix} 0 & 0 \end{bmatrix}^T\) is a fixed point
  • Are there non-trivial others?
    • Could check eigevectors as we did before, \(\lambda \times x = A x\)
    • If there is an \((\lambda, x)\) pair with \(\lambda = 1\) it is a fixed point
A = np.array([[0.8, 0.2], [0.2, 0.8]])
eigvals, eigvecs = eig(A)
print(f"lambda_1={eigvals[0]}, ||x* - A x*||={norm(A @ eigvecs[:,0] - eigvecs[:,0])}")
lambda_1=(1+0j), ||x* - A x*||=1.1102230246251565e-16

Fixed Points for Nonlinear Functions

  • Consider \(f(x) = \sqrt{x}\) and \(f(x) = x^2\) for \(x \geq 0\)
  • Trivially \(x^* = 0\) is a fixed point of both, but what about others?
  • Plot the 45-degree line to see if they cross! Seems \(x^* = 1\) as well?
    • As we will discuss, though. The shape at \(x^*=1\) and \(x^*=0\) is very different
    • Think about what happens if we “perturb” slightly away from that point?

Plot Against 45 degree line

  • Consider \(f(x) = \sqrt{x}\) and \(f(x) = x^2\) for \(x \geq 0\)

Interpreting Iterations with the 45 degree line

  • To use these figures:
    1. Start with any point on the x-axis
    2. Jump to the \(f(\cdot)\) for that point to see where it went
    3. Go across to the 45 degree line
    4. Then down to the new value
  • Repeat! Useful to interpret dynamics as well as various numerical methods
  • Gives intuition on speed of convergence/etc. as well

Evaluating the \(\sqrt{x}\) near \(x=0.05 > 0\)

Evaluating the \(\sqrt{x}\) near \(x=1.1 > 1\)

Evaluating the \(x^2\) for \(x = 0.6 < 1\)

Evaluating the \(x^2\) for \(x = 1.01 > 1\)

Linear Dynamics and Stability

Scalar Linear Model

\[ x_{t+1} = a x_t + b \equiv f(x_t),\quad \text{ given }x_0 \]

\[ \begin{aligned} x_1 &= a x_0 +b\\ x_2 &= a x_1 + b = a^2 x_0 + ab + b\\ \ldots & \\ x_t &= a^t x_0 + b \sum_{i=0}^{t-1} a^i = a^t x_0 + b \frac{1-a^t}{1-a}\\ x^* &\equiv \lim_{t\to\infty}x_t = \begin{cases}\frac{b}{1-a} & \text{ if } |a|<1\\ \text{diverges} & \text{ if } |a| > 1 \text{ or } a=1, b\neq 0\\ \text{indeterminate} & \text{ if } a=1, b=0 \end{cases} \end{aligned} \]

Stability and Jacobians

  • Given \(f(x_t) = a x_t + b\)
    • The Jacobian (derivative since scalar) \(\nabla f(x_t) = a\)
  • Eigenvalues of a scalar are just the value itself, so can write the condition as
    • Stable at fixed point \(x^*\) if \(\rho(\nabla f(x^*)) < 1\), where \(\rho(A) = \max_i |\lambda_i(A)|\) the spectral radius
    • Saw this as a condition for stability with higher-dimensional linear systems when looking at Present Discounted Values

Linearization and Stability

  • Important condition for stability with nonlinear \(f(\cdot)\)
  • Intuition: assume \(x^*\) exists and then
    • Linearize around the steady state and see if it would be locally explosive
    • Necessary but not sufficient. \(\rho(\nabla f(x^*)) > 1 \implies x^*\) can’t be a stable fixed point
  • You may see this when working with macro models in Dynare and similar methods in macroeconomics

Linearization

\[ \begin{aligned} x_{t+1} &= f(x^*) + \nabla f(x^*) (x_t - x^*) + \text{second order and smaller terms}\\ x_{t+1} - x^* &\approx \nabla f(x^*) (x_t - x^*)\\ \hat{x}_{t+1} &\approx \nabla f(x^*) \hat{x}_t \end{aligned} \]

  • Where the last formulation is common in macroeconomics and time-series econometrics. \(\hat{x}_t\equiv x_t - x^*\) is the deviation from the steady state
    • For the linear case, these would all be exact as there are no higher-order terms

Quality of Linearization

  • Gives approximate dynamics for a perturbation close to the steady state
    • May have good approximation far away from \(x^*\) if \(f(\cdot)\) is close to linear
    • May have terrible approximations close to \(x^*\) if \(f(\cdot)\) highly nonlinear/asymmetric
    • Often log-linearization is used instead, which expresses in percent deviation

Plot Against 45 degree line Reminder

Stability of \(\sqrt{x}\) and \(x^2\)

  • Recall that both had fixed points at \(x^*=0\) and \(x^*=1\)

  • Lets check derivatives! Let \(f_1(x) = \sqrt{x}\) and \(f_2(x) = x^2\)

    • \(\nabla f_1{x} = \frac{1}{2\sqrt{x}}\) and \(\nabla f_2(x) = 2x\)

  • Check spectral radius of the Jacobians (trivial since univariate) at the fixed points:

    • At \(x^* = 0\), \(\nabla f_1(0) = \infty\) and \(\nabla f_2(0) = 0\)
    • At \(x^* = 1\), find \(\nabla f_1(1) = \frac{1}{2}\) and \(\nabla f_2(1) = 2\)

  • Interpretation:

    • \(f_1(x)\) is locally explosive at \(x^*=0\) and locally stable at \(x^*=1\)
    • \(f_2(x)\) is locally stable at \(x^*=0\) and locally explosive at \(x^*=1\)

Solow-Swan Growth Model

Model of Growth and Capital

  • An early growth model of economic growth is the Solow-Swan model
  • Simple model. Details of the derivation for self-study/macro classes:
    • \(k_t\) by capital per worker and \(y_t\) is total output per worker
    • \(\alpha \in (0,1)\) be a parameter which governs the marginal product of capital
    • \(\delta \in (0,1)\) is the depreciation rate (i.e., fraction of machines breaking each year)
    • \(A > 0\) is a parameter which governs the total factor productivity (TFP)
    • \(s \in (0,1)\) is the fraction of output used for investment and savings

Capital Dynamics

Then capital dynamics follow a nonlinear difference equation with steady state

\[ \begin{aligned} y_t &= A k_t^{\alpha}\\ k_{t+1} &= s y_t + (1-\delta)k_t = s A k_t^{\alpha} + (1-\delta) k_t \equiv g(k_t)\quad \text{ given }k_0\\ k^* &\equiv \left(\frac{s A}{\delta}\right)^{\frac{1}{1-\alpha}} \end{aligned} \]

Implementing the Solow-Swan Model

A, s, alpha, delta = 2, 0.3, 0.3, 0.4
def y(k):
    return A*k**alpha
# "closure" binds y, A, s, alpha, delta    
def g(k):
    return s*y(k) + (1-delta)*k

k_star = (s*A/delta)**(1/(1-alpha))
k_0 = 0.25
print(f"k_1 = g(k_0) = {g(k_0):.3f},\
k_2 = g(g(k_0)) = {g(g(k_0)):.3f}")
print(f"k_star = {k_star:.3f}")
k_1 = g(k_0) = 0.546,k_2 = g(g(k_0)) = 0.828
k_star = 1.785

Plotting \(k_t\) vs. \(k_{t+1}\) verifies our \(k^*\)

Jacobian of \(g\) at the steady state

\[ \begin{aligned} \nabla g(k^*) &= \alpha s A k^{*\alpha-1} + 1-\delta,\quad \text{ substitute for } k^*\\ &= \alpha s A \frac{\delta}{s A} + 1-\delta = \alpha \delta + 1 - \delta\\ &= 1 - (1-\alpha)\delta < 1 \end{aligned} \]

  • Key requirements were \(\alpha \in (0,1)\) and \(\delta \in (0,1)\)
  • The spectral radius of a scalar is just that value itself.
  • The spectral radius of \(||\nabla g(k^*)|| < 1\), a necessary condition for \(k^*\) stable
  • Aside: macroeconomics, industrial organization, etc. this is related to contraction mappings and Blackwell’s condition

Simulation

# Generic function, takes in a function!
def simulate(f, X_0, T):
    X = np.zeros((1, T+1))
    X[:,0] = X_0
    for t in range(T):
        X[:,t+1] = f(X[:,t])
    return X
T = 10
X_0 = np.array([0.25]) # initial condition
X = simulate(g, X_0, T) # use with our g
print(f"X_{T} = {X[:,T]}")
X_10 = [1.70531835]

Capital Transition from \(k_0 < k^*\) and \(k_0 > k^*\)

X_1 = simulate(g, X_0, T) # use with our g
X_2 = simulate(g, np.array([2.1]) , T)
fig, ax = plt.subplots()
ax.plot(range(T+1), X_1.T,
  label=r"$k_t$ from $k_0 = 0.25$")
ax.plot(range(T+1), X_2.T,
  label=r"$k_t$ from $k_0 = 2.1$")
ax.set(xlabel="t", ylabel="Capital per Worker")
ax.axhline(y=k_star, linestyle=':',
  color='black',label=r"$k^*$")
ax.legend()
plt.show()

Trajectories Using the 45 degree Line

PageRank and Other Applications

Network of Web Pages

  • Consider \(A, B, C, D\) as a set of web pages with links given below
G A A B B A->B C C A->C B->C D D B->D C->A C->D D->A D->B

Create an Adjacency Matrix

  • We can summarize the network of web pages with \(1\) or \(0\) if there is a link between two pages. Pages won’t link to themselves
  • This is in (arbitrary) order: A, B, C, D

\[ M = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \end{pmatrix} \]

PageRank Algorithm

One interpretation of this is that you can

  • Start on some page
  • With equal probability click on all pages linked at that page
  • Keep doing this process and then determine what fraction of time you spend on each page

Probabilistic Interpretation

Alternatively,

  • Start with a probability distribution, \(r_t\) that you will be on any given page (i.e. \(r_{nt} \geq 0\) and \(\sum_{n=1}^4 r_{nt} = 1\))
  • Iterate the process to see the probability distribution after you click the next links
  • Repeat until the probability distribution doesn’t change.

Adjacency Matrix to Probabilities

  • To implement, we want to put the same probability on going to any link for a given page (i.e. each row)

\[ S = \begin{pmatrix} 0 & 0.5 & 0.5 & 0 \\ 0 & 0 & 0.5 & 0.5 \\ 0.5 & 0 & 0 & 0.5 \\ 0.5 & 0.5 & 0 & 0 \end{pmatrix} \]

Probabilities Evolution

  • Now, we can see what happens after we click on a page
  • For a given \(r_t\) distribution of probabilities across page, I can see the new probabilities distribution as

\[ r_{t+1} = S^{\top} r_t \]

  • Motivation to learn more probability and Markov Chains (next set of lectures)

Fixed Points and Eigenvectors

  • What is a fixed point of this process?
  • Eigenvector of \(S^{\top}\) associated with \(\lambda = 1\) eigenvalue!
  • The real PageRank is a little more subtle (adds in dampening) but the same basic idea
  • Learn numerical algebra to use in practice. It is infeasible to actually compute the eigenvector of a huge matrix with a decomposition.