11.7 Controlling the Robinson-Crusoe Economy
191
dot-dashed red line assumes large negative values, which implies that a significant
portion of the profits is used to replenish the capital, shown by the solid black line.
The lower plot in Fig. 11.62 shows the situation when chosing the weights q = 1
and r = 5. Here the controller tries to use small excitations u, or equivalently, it tries
to keep the profits reasonably constant. We observe that it now takes much longer to
return to the steady state—the time constant τ is much larger—but the profits, shown
as the dash-dotted red line, are much less affected, when compared to the situation
depicted in the upper plot. We thus find that chosing suitable weights q and r allows
us to flexibly tailor the system’s behavior, depending on our preferences.
After having made a fortune with stocks and options and learnt how to control all
that wealth, it’s about time we talk about money. And, considering that we are living
in the twentyfirst century, we will talk about cryptocurrencies.
Exercises
1. Simulate the Robinson Crusoe economy, defined by the equations
k t+1 = (1 − δ)k t + i t , y t = k
t , i t = σ y t and p t = y t − i t
with δ = 0.1, , = 0.7, σ = 0.2. Here we assume a Cobb-Douglas production
function y t = k
t and that a fraction σ of the output y t is devoted to investments i t ,
the rest we take out as profit to goof off. Today, you derive a certain amount of joy
V [ p] from all future profits, which is characterized by V [ p] =
∞
j=0 β
j log( p j ),
where we discount future joy by a “discount factor” β = 0.9. Write a simulation
and vary the investment fraction σ to maximize V after 100 iterations.
2. Simulate the stochastic process
λ t+1 = γ λ t + (2 − γ ) + aε t
for 1 < t < 500 generations. Here ε t are random number drawn from a Gaussian
distribution with zero mean and an rms of unity. Use the parameters γ = 0.9 and
a = 0.2. Towards what value does λ tend on average? Find an analytic expression
for this asymptotic value.
3. Determine the Hamiltonian H (x, θ, p x , p θ ) for a mechanical system with the
Lagrange
function
L(x, θ, ˙
x, ˙
θ) = 3m ˙
x
2
/2 + mr ˙
x ˙
θ + mr
2 ˙
θ
2
/2 − kx
2
− mgrθ
2
/2, where x and θ are the coordinates, ˙
x, and ˙
θ are the corresponding velocities, and p x and p θ are the corresponding momenta. Moreover, and
m, r, k, and g are constants.
4. Consider the unstable one-dimensional process, defined by ˙
x = ax + bu, where
a and b are positive constants and the performance measure is given by (11.61).
Show that the “gain”κ of the feedback, which defines the optimal controller
through u = κ x is given by (11.62).
191
dot-dashed red line assumes large negative values, which implies that a significant
portion of the profits is used to replenish the capital, shown by the solid black line.
The lower plot in Fig. 11.62 shows the situation when chosing the weights q = 1
and r = 5. Here the controller tries to use small excitations u, or equivalently, it tries
to keep the profits reasonably constant. We observe that it now takes much longer to
return to the steady state—the time constant τ is much larger—but the profits, shown
as the dash-dotted red line, are much less affected, when compared to the situation
depicted in the upper plot. We thus find that chosing suitable weights q and r allows
us to flexibly tailor the system’s behavior, depending on our preferences.
After having made a fortune with stocks and options and learnt how to control all
that wealth, it’s about time we talk about money. And, considering that we are living
in the twentyfirst century, we will talk about cryptocurrencies.
Exercises
1. Simulate the Robinson Crusoe economy, defined by the equations
k t+1 = (1 − δ)k t + i t , y t = k
t , i t = σ y t and p t = y t − i t
with δ = 0.1, , = 0.7, σ = 0.2. Here we assume a Cobb-Douglas production
function y t = k
t and that a fraction σ of the output y t is devoted to investments i t ,
the rest we take out as profit to goof off. Today, you derive a certain amount of joy
V [ p] from all future profits, which is characterized by V [ p] =
∞
j=0 β
j log( p j ),
where we discount future joy by a “discount factor” β = 0.9. Write a simulation
and vary the investment fraction σ to maximize V after 100 iterations.
2. Simulate the stochastic process
λ t+1 = γ λ t + (2 − γ ) + aε t
for 1 < t < 500 generations. Here ε t are random number drawn from a Gaussian
distribution with zero mean and an rms of unity. Use the parameters γ = 0.9 and
a = 0.2. Towards what value does λ tend on average? Find an analytic expression
for this asymptotic value.
3. Determine the Hamiltonian H (x, θ, p x , p θ ) for a mechanical system with the
Lagrange
function
L(x, θ, ˙
x, ˙
θ) = 3m ˙
x
2
/2 + mr ˙
x ˙
θ + mr
2 ˙
θ
2
/2 − kx
2
− mgrθ
2
/2, where x and θ are the coordinates, ˙
x, and ˙
θ are the corresponding velocities, and p x and p θ are the corresponding momenta. Moreover, and
m, r, k, and g are constants.
4. Consider the unstable one-dimensional process, defined by ˙
x = ax + bu, where
a and b are positive constants and the performance measure is given by (11.61).
Show that the “gain”κ of the feedback, which defines the optimal controller
through u = κ x is given by (11.62).