Egwald Mathematics: Nonlinear Dynamics:
Trygve Haavelmo Growth Model
Continuous versus Discrete Time
by
Elmer G. Wiens
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introduction  continuous time  phase diagrams  discrete time  phase diagrams  bifurcation diagram continuous versus discrete  references
The Trygve Haavelmo Growth Model
The Trygve Haavelmo growth model in A Study in the Theory of Economic Evolution. provides an example of the different dynamical behaviours arising from equivalent models expressed as either differential or difference equations. While the solution trajectories of the continuous time version of the model converge to its fixed point, the solution trajectories of the discrete time version may exhibit chaotic behaviour.
As described by HansWalter Lorenz in Nonlinear Dynamical Economics and Chaotic Motion (141143), the onedimensional Haavelmo (2829) growth cycle model describes an economy's production of output as a function of the stock of capital (K) and the level of employment (N).
Continuous Time
Using a constant returns to scale CobbDouglas production function version of this model:
Y = c * K^{(1a)} * N^{a}, 0 < a < 1, c > 0, (1)
the differential equation governing the growth of employment is:
dN / dt = N * (α  β * N / Y), N(0) = N0, α, β > 0, (2)
whose solution is a function N(t) = N(t, N0)) of time, t, and the initial condition, N(t=0) = N0.
Thus, the rate of growth of employment, dN/dt / N, is an increasing function of per capita income (output), Y / N.
Combining equations (1) and (2) yields:
dN / dt = f(N, α) = α * N  β / (c * K^{(1a)}) * N^{(2  a)}, N(0) = N0. (3)
where α is the parameter of the differential equation (2) to be analyzed.
The Fixed Point
To find the fixed point, set f(N, α) = 0, and solve for N^{*}:
f(N, α) = α * N  β / (c * K ^{(1a)}) * N^{(2  a)} = 0, or
N^{*} = (α * c * K^{(1a)} / β)^{1 / (1  a)}. (4)
Linear Stability Analysis
Evaluating the partial derivative of f with respect to N at the fixed point N^{*} yields:
f_{N}(N, α) = α  (2  a) * (β / (c * K^{(1  a)} * N^{(1  a)}, and
f_{N}(N^{*}, α) = α * (a  1) < 0,
since α > 0 and 0 < a < 1. Thus the nonlinear process determined by the function f is stable with the fixed point at N^{*}.
Phase Diagrams and Solution Trajectories
Particularize the continuous growth model by setting:
K = 3, a = .3, β = 1, c = .9.
with parameter α and initial condition N(0) = N0.
The following diagrams show the phase diagrams and solution trajectories, N(t), for various initial conditions N(0), and two values of the parameter α.
α = 1.5, N^{*} = 4.6059, Y^{*} = 3.0706



α = 2.5, N^{*} = 9.5552, Y^{*} = 3.8221



Discrete Time
In the differential equation (3), replace the trajectory function N(t) by the trajectory orbit {N_{t}}, and the differential operator dN / dt with the finite difference N_{t+1}  N_{t} to yield the difference equation:
N_{t+1}  N_{t} = α * N_{t}  β / (c * K^{(1a)}) * N_{t}^{(2  a)}, N_{0} = N0, or
N_{t+1} = (1 + α) * N_{t}  β / (c * K^{(1  a)}) * N_{t}^{(2  a)}, N_{0} = N0. (5)
This difference equation can be transformed by the transformation:
N_{t} = (K^{(1  a)} * (1 + α) / β)^{1/(1  a)} * x_{t}
to produce the equation:
x_{t+1} = f(x_{t}, α) = (1 + α) * x_{t} * (1  x_{t}^{(1  a)}), x_{0} = x0 > 0. (6)
The dynamics of equation (6) are qualitatively equivalent to those of the logistics equation (set r = (1 + α) and let a → 0). Moreover, f(0, α) = f(1, α) = 0, and f is onehumped and noninvertible.
Fixed Points
Fixed points of the discrete f map satisfy:
f(x, α) = (1 + α) * x * (1  x^{(1  a)}) = x, (7)
which are:
x_{1} = 0,
x_{2} = [α/ (1 + α)]^{1/ (1  a)}. (8)
Linear Stability Analysis
The partial derivative of f with respect to x is:
f_{x} (x, α) = (1 + α) * (1  (2  a)) * x ^{(1  a)}. (9)
Evaluating f at its fixed points to obtain the multipliers:
λ_{1} = f_{x}(0, α) = (1 + α), and
λ_{2} = f_{x}(x_{2}, α) = (1 + α)*(1  (2  a) * α/ (1 + α) = 1  α * (1  a).
Since α > 0 → λ_{1} > 1 → x_{1} = 0 is a repeller. If 0 < α < 2 / (1  a) → 1 > λ_{2} > 1 → x_{2} is an attractor.
Phase Diagrams and Solution Trajectories
Particularize the discrete growth model by setting:
a = .3.
with parameter α and initial condition x_{0} = x0.
The fixed point x_{2} of f is an attractor in the range:
0 < α < 2 / (1  a) = 2 / .7 = 2.8571. (10)
At α = 2 / (1  a) = 2 / .7, the f map undergoes a flip bifurcation, where x_{2} switches from attractor to repeller.
The following diagrams show the phase diagrams and solution trajectory orbits, {x_{t}}, for various initial conditions x_{0} and values of the parameter α.
α = 2.5, x_{1} = 0 repeller, x_{2} = 0.6184 attractor.



α = 2 / .7 = 2.8571, x_{1} = 0 repeller, x_{2} = 0.651 attractor; flip bifurcation



α = 3.2, x_{1} = 0 repeller, x_{2} = 0.678 repeller
x_{3} = .4659 attractor, x_{4} =.81035 attractor



The Second Order Map of f, denoted by f^{2}, is x_{t+2} = f(x_{t+1}, α) = f (f(x_{t}, α), α ) = f^{2}(x_{t}, α) .
The following graphs display the phase diagrams of f in blue and f^{2} in red. The stable fixed points x_{3} and x_{4} of f^{2} emerge as α increases past 2/.7. At α = 2 / .7 = 2.8571, f^{2} has a fixed point of multiplicity three (ie x_{2} = x_{3} = x_{4} = 2 / .7). For α > 2/.7, x_{3} and x_{4} bracket x_{2}, and establish the period2 cycle seen in the above trajectory diagram for α = 3.2. Eventually, these period2 fixed points become unstable, and undergo flip bifurcations with respect to f^{4}, the period doubling map of f^{2}.
α = 2.5

α = 2.8571 flip bifurcation

α = 3.2




Second Order (f^{2}) Flip Bifurcation
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process. At α = 3.48365646, f^{2} undergoes a period doubling (flip) bifurcation with x_{3} and x_{4} switching from attractors to repellers. The trajectory x_{t} switches between the four attracting fixed points of f^{4}, creating a stable fourcycle.
α = 3.4837, x_{1} = 0 repeller, x_{2} = 0.697 repeller;
x_{3} = 0.4058 attractor, x_{4} =0.8517 attractor; f^{2}flip bifurcation



The Fourth Order Map of f, denoted by f^{4}, is x_{t+4} = f^{2} (f^{2}(x_{t}, α), α ) = f^{4}(x_{t}, α) .
The following graph displays the phase diagrams of f in blue, f^{2} in red, and f^{4} in black. The fixed points x_{3} and x_{4} of f^{2} flip from attractor to repeller as α increases past 3.48365646. For α > 3.48365646, four stable fixed points of f^{4} emerge bracketing x_{3} and x_{4}, and establish the period4 cycle. Eventually, these period4 fixed points become unstable, and undergo flip bifurcations with respect to f^{8}, the period doubling map of f^{4}.
α = 3.4837 f^{2} flip bifurcation


Fourth Order (f^{4}) Flip Bifurcation
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process. At α = 3.61271754, f^{4} undergoes a period doubling (flip) bifurcation with x_{5}, x_{6}, x_{7}, and x_{8} switching from attractors to repellers. The trajectory x_{t} switches between the eight attracting fixed points of f^{8}, creating a stable eightcycle.
α = 3.6127, x_{1} = 0 repeller, x_{2} = 0.705 repeller;
x_{3} = 0.38459 repeller, x_{4} = 0.86524 repeller;
x_{5} = 0.32709, x_{6} = 0.49343, x_{7} = 0.8187, x_{8} = 0.88789 attractors;
f^{4}flip bifurcation



The following graph displays the phase diagrams of f in blue, f^{4} in red, and f^{8} in black. The fixed points x_{5}, x_{6}, x_{7}, and x_{8} of f^{4} flip from attractor to repeller as α increases past 3.61271754. For α > 3.61271754, eight stable fixed points of f^{8} emerge bracketing x_{5}, x_{6}, x_{7}, and x_{8}, and establish the period8 cycle. Eventually, these period8 fixed points become unstable, and undergo flip bifurcations with respect to f^{16}, the period doubling map of f^{8}.
α = 3.6127 f^{4} flip bifurcation


Period Doubling Cascade
As α increases, period doublings occur as f, f^{2}, f^{4}, f^{8}, . . . bifurcate at α_{1} = 2/.7, α_{2} = 3.4837, α_{3} = 3.6127, α_{4} = . . ..
Bifurcation Diagram
The intermittent emergence of order and chaos is revealed in the orbit diagram below, for α (called r) in the interval [2.8, 4].
The orbit:
{x_{t+1} = f(x_{t} ,α), x_{0} = x0}
exhibits aperiodic — chaotic — behaviour, as α increases, with periodic windows appearing. These dynamics are qualitatively similar to those observed with the logistics map.
Continuous versus Discrete Time Dynamics.
The table below shows how the economy's levels of employment and output increases in a stable manner with α in the continuous time model. However, in the discrete time model, these levels of employment and output cycle and exhibit chaotic dynamics as α increases.
 Continuous Time  Discrete Time 
α  N  Y  x  N  Y 
2.5 
9.555 
3.822 
0.618 
11.107 
3.999 
2.857 
11.563 
4.047 
0.651 
13.442 
4.234 
3.2 
13.596 
4.249 
0.4659 0.8104 
10.859 18.886 
3.972 4.689 
3.484 
15.349 
4.406 
0.4058 0.8517 
10.383 21.793 
3.919 4.895 
3.613 
16.168 
4.475 
0.3271 0.4934 0.8187 0.8879 
8.716 13.148 21.815 23.659 
3.718 4.206 4.896 5.017 
References.
 Haavelmo, Trygve. A Study in the Theory of Economic Evolution. Amsterdam: NorthHolland, 1954.
 Lorenz, HansWalter. Nonlinear Dynamical Economics and Chaotic Motion. Berlin: SpringerVerlag, 1993.
