Lecture 13: Differential Equations and Continuous-Time Dynamics

Learning objectives

Solve first-order ODEs; analyze second-order linear equations; study systems using eigenvalues; interpret steady states and stability in economic dynamics.

1. First-order differential equations

A differential equation relates an unknown function to its derivatives. A separable equation \(\dot x=a(x)b(t)\) can be rearranged and integrated when division is valid. A linear equation \(\dot x+p(t)x=q(t)\) uses integrating factor \(M(t)=e^{\int p(t)dt}\).

2. Linear constant-coefficient equations

For \(\ddot x+a\dot x+bx=0\), try \(x(t)=e^{rt}\). The characteristic equation \(r^2+ar+b=0\) determines exponential, repeated-root, or oscillatory solutions. Initial conditions determine constants.

3. Systems of differential equations

For \(\dot x=Ax\), solutions are governed by eigenvalues and eigenvectors of \(A\). If all eigenvalues have negative real parts, the origin is asymptotically stable; if any eigenvalue has positive real part, it is unstable. Zero real parts require further analysis.

4. Nonlinear steady states

A steady state \(x^*\) solves \(f(x^*)=0\). Local dynamics around it are approximated by \(\dot u=J_f(x^*)u\), where \(u=x-x^*\). Eigenvalues of the Jacobian determine local stability for hyperbolic steady states.

5. Economic applications

Capital accumulation may be written \(\dot k=s f(k)-(n+g+\delta)k\). A steady state equates investment per effective worker with break-even investment. Local stability follows from the derivative of the dynamic law at the steady state.

Worked example

For \(\dot k=sk^\alpha-\delta k\), with \(0<\alpha<1\), the positive steady state satisfies \(k^*=(s/\delta)^{1/(1-\alpha)}\). The derivative is \(s\alpha k^{\alpha-1}-\delta\), which is negative at the positive steady state, giving local asymptotic stability.

Problem set

Solve a separable first-order ODE with an initial condition.
Solve a linear ODE using an integrating factor.
Classify roots of a second-order characteristic equation.
Determine stability from a two-dimensional Jacobian.
Derive and assess a capital-accumulation steady state.

Computational lab

Simulate ODE paths using numerical solvers. Compare analytical and numerical solutions; plot phase diagrams and trajectories near stable and unstable steady states.

Coding exercises

These exercises apply the calculus of this lecture to real economic data in Python, mixing symbolic (sympy), numerical (scipy) and statistical (statsmodels) tools. Install once with pip install sympy scipy numpy pandas statsmodels wooldridge. Try the starter code first, then open the solution. Every printed result below comes from running the solution code exactly as shown.

Exercise 13.1 — The continuous-time Solow model calibrated to the U.S.

Data: macrodata — U.S. investment share and population growth (calibration as in Lecture 11).

Capital per worker follows \(\dot k=s\,k^{\alpha}-(n+\delta)k\).

Solve the ODE numerically with solve_ivp starting from half the steady-state capital.
Linearize around \(k^*\): \(\dot k\approx-\lambda(k-k^*)\) with \(\lambda=(1-\alpha)(n+\delta)\). Compute the implied half-life \(\ln2/\lambda\).
Measure the half-life in the numerical solution. Why is it close but not identical to the linear prediction?

import numpy as np, statsmodels.api as sm
from scipy.integrate import solve_ivp
m = sm.datasets.macrodata.load_pandas().data
s = (m["realinv"] / m["realgdp"]).mean(); n = np.log(m["pop"]).diff().mean() * 4; a, dl = 1/3, 0.05
kss = (s / (n + dl))**(1 / (1 - a))
# TODO: solve_ivp(lambda t, k: s*k**a - (n+dl)*k, (0, 150), [kss/2], dense_output=True)

Solution, output and discussion

import numpy as np, statsmodels.api as sm
from scipy.integrate import solve_ivp
from scipy.optimize import brentq
m = sm.datasets.macrodata.load_pandas().data
s = (m["realinv"] / m["realgdp"]).mean(); n = np.log(m["pop"]).diff().mean() * 4; a, dl = 1/3, 0.05
kss = (s / (n + dl))**(1 / (1 - a))
sol = solve_ivp(lambda t, k: s * k**a - (n + dl) * k, (0, 150), [kss / 2], dense_output=True, rtol=1e-10, atol=1e-12)
lam = (1 - a) * (n + dl)
print(f"k* = {kss:.4f}; convergence rate lambda = {lam:.4f} per year; linear half-life = {np.log(2)/lam:.2f} years")
gap = lambda t: (kss - sol.sol(t)[0]) - 0.5 * (kss - kss / 2)
print(f"numerical half-life (gap halves): {brentq(gap, 0.1, 100):.2f} years")
for t in (0, 10, 20, 40, 80):
    print(f"  t = {t:3d}: k = {sol.sol(t)[0]:.4f} ({sol.sol(t)[0]/kss:.1%} of k*)")

Output:

k* = 3.2142; convergence rate lambda = 0.0406 per year; linear half-life = 17.06 years
numerical half-life (gap halves): 18.50 years
  t =   0: k = 1.6071 (50.0% of k*)
  t =  10: k = 2.1024 (65.4% of k*)
  t =  20: k = 2.4561 (76.4% of k*)
  t =  40: k = 2.8695 (89.3% of k*)
  t =  80: k = 3.1453 (97.9% of k*)

Starting at half the steady state, capital per worker closes 90% of the gap after about 40 years. Linearization predicts a half-life of 17.1 years; the numerical solution takes about 18.5. The linear approximation is accurate only near \(k^*\): starting far below, the true dynamics are slower at the beginning in terms of the proportional gap, because \(\dot k/k=sk^{\alpha-1}-(n+\delta)\) is not linear in \(k\). The calibrated convergence rate of about 4% a year is in the range of empirical estimates (2–3%) only roughly; a higher capital share brings it down.

Exercise 13.2 — Exponential growth: fitting \(\dot y=gy\) to U.S. real GDP

Data: macrodata — U.S. quarterly real GDP, 1959–2009.

The solution of \(\dot y=gy\) is \(y(t)=y_0e^{gt}\), so \(\log y\) is linear in \(t\). Estimate \(g\) by regressing \(\log GDP\) on time (in years).
Compute the implied doubling time \(\ln2/g\) and compare with the actual doubling time in the data.
Plot (or tabulate) the deviation of log GDP from the fitted exponential path at business-cycle troughs (1975Q1, 1982Q4, 1991Q1, 2009Q2). Is the growth rate constant over 1959–2009? Compare \(g\) for 1959–1983 and 1984–2009.

import numpy as np, statsmodels.api as sm, statsmodels.formula.api as smf
m = sm.datasets.macrodata.load_pandas().data
m["t"] = m["year"] + (m["quarter"] - 1) / 4 - 1959
# TODO: smf.ols("np.log(realgdp) ~ t", m).fit()

Solution, output and discussion

import numpy as np, statsmodels.api as sm, statsmodels.formula.api as smf
m = sm.datasets.macrodata.load_pandas().data
m["t"] = m["year"] + (m["quarter"] - 1) / 4 - 1959
r = smf.ols("np.log(realgdp) ~ t", data=m).fit()
g = r.params["t"]
print(f"g = {100*g:.3f}% per year;  implied doubling time {np.log(2)/g:.1f} years")
y0 = m["realgdp"].iloc[0]; k = np.argmax(m["realgdp"].to_numpy() >= 2 * y0)
print(f"actual first doubling: {m['t'].iloc[k]:.2f} years (reached in {int(m['year'].iloc[k])}Q{int(m['quarter'].iloc[k])})")
for yq in ((1975, 1), (1982, 4), (1991, 1), (2009, 2)):
    row = m[(m.year == yq[0]) & (m.quarter == yq[1])]
    if len(row): print(f"  {yq[0]}Q{yq[1]}: log deviation from exponential trend {100*r.resid[row.index[0]]:+.2f}%")
for lo, hi in ((1959, 1983), (1984, 2009)):
    s = m[(m.year >= lo) & (m.year <= hi)]
    print(f"g over {lo}-{hi}: {100*smf.ols('np.log(realgdp) ~ t', data=s).fit().params['t']:.3f}% per year")

Output:

g = 3.161% per year;  implied doubling time 21.9 years
actual first doubling: 18.50 years (reached in 1977Q3)
  1975Q1: log deviation from exponential trend -1.32%
  1982Q4: log deviation from exponential trend -5.58%
  1991Q1: log deviation from exponential trend -1.34%
  2009Q2: log deviation from exponential trend -10.60%
g over 1959-1983: 3.433% per year
g over 1984-2009: 2.985% per year

Real GDP grew at about 3.2% a year on average, implying a doubling time of about 22 years. It actually doubled faster, in 18.5 years (by 1977), because growth was higher early in the sample: 3.4% a year in 1959–83 against 3.0% in 1984–2009. The constant-growth model is a first approximation. The deviations from trend at recession troughs grow over time, ending 10.6% below trend in 2009 — partly because of the deep recession and partly because the trend slowed, which a single exponential cannot capture.

Reading

Boyce, DiPrima, and Meade, Elementary Differential Equations; Chiang and Wainwright, Fundamental Methods of Mathematical Economics; Strogatz, Nonlinear Dynamics and Chaos.