CentER · Macroeconomics 1 · Problem Set 2 · Ramsey–Cass–Koopmans

Ramsey Saddle Lab

Graphical companion to the planner's problem. Move the parameters, press Play, and hover any chart to read values.

PlannerThe social planner's problem

\[ \max_{c(t)}\ \int_0^\infty e^{-(\rho-n)t}\,u(c(t))\,dt \quad\text{s.t.}\quad \dot k=f(k)-c-(n+\delta)k,\quad k(0)=k_0,\qquad u(c)=\tfrac{c^{1-\sigma}-1}{1-\sigma},\quad f(k)=\Omega k^{\alpha}. \]

Parameters for all figures
Fig. 1The objective is an area
Along the optimal path from \(k_0\).
Utility flow \(u(c(t))\)
\(u(c(t))\)\(u(c^*)\)
Discount factor
\(e^{-(\rho-n)t}\)
Their product: the shaded area is \(U\)
\(e^{-(\rho-n)t}u(c(t))\)\(u(c(t))\), undiscounted
What to notice
  • Discounting shrinks the far future, so the area is finite. Push \(\rho\) toward \(n\) and the tail fattens without bound.
  • \(u\) can be negative, and area below zero counts negatively.
Fig. 2The planner chooses a whole path \(c(t)\), and with it the shape of the area
The alternative is feasible: \(c=(1-s)f(k)\) and \(\dot k=sf(k)-(n+\delta)k\), from the same \(k_0\).
Consumption paths
optimalconstant \(s\)
Discounted utility flows
optimalconstant \(s\)optimal path higherconstant-\(s\) path higher
What to notice
  • Changing the path moves area between dates: saving more lowers the flow early and raises it later.
  • Whatever \(s\), the optimal path has the larger total area. Its saving rate changes over time (hover the left panel), which no single \(s\) can do.
Fig. 3The phase diagram in \((k, c)\)
Click anywhere to choose \((k_0, c_0)\). Then press the button: \(k_0\) cannot move, so \(c_0\) jumps vertically onto the saddle path.
\(\dot k=0\)\(\dot c=0\)saddle pathyour path\(k_{gold}\)
What to notice
  • From a given \(k_0\), only one \(c_0\) leads to \((k^*,c^*)\). Every other choice diverges: too high and \(k\) runs out, too low and \(c\) collapses toward zero.
  • \(k^*\lt k_{gold}\): the planner discounts, so it does not push capital all the way to the golden rule.
Fig. 4Why the transversality condition picks the saddle path
\(k(t)/k^*\) and \(c(t)/c^*\)
\(k\)\(c\)\(k\), saddle\(c\), saddle
\(\ln[\mu(t)k(t)]\): present value of the capital stock
chosensaddle
What to notice
  • Too low: capital piles up while \(c\to0\), so \(\mu k\) grows and the TVC fails: capital is left unused forever.
  • Too high: \(k\) hits zero in finite time, and the path is infeasible.
  • Saddle: \(\mu k\) falls to zero, so the TVC holds.
Fig. 5The infinite horizon as the limit of finite horizons
With a finite horizon the planner ends with \(k(T)=0\).
Optimal paths in \((k,c)\)
\(k(t)\)
What to notice
  • The longer the horizon, the longer the path stays near the steady state before running capital down.
  • As \(T\) grows the paths approach the saddle path, from below or above \(k^*\): the TVC is the limit of \(\mu(T)k(T)=0\).