Ramsey Growth

Ramsey Growth
State dimension:	1
Differential states:	1
Discrete control functions:	1

The Ramsey growth problem is a foundational benchmark in optimal control and macroeconomic theory. This description is taken from [1].

First formulated by Frank Ramsey in 1928 [2] and later extended by Cass [3] and Koopmans [4], the problem determines the socially optimal path of capital accumulation and consumption for a representative household economy. The single state is the capital stock per worker, and the single control is consumption per worker. Capital accumulates through production (described by a Cobb–Douglas function), net of depreciation and consumption. The objective is to maximise the present value of a stream of discounted utility of consumption over a finite horizon, which is equivalent to minimising its negative. The resulting optimal trajectories converge towards a steady-state balanced growth path, and the problem serves as a standard test for indirect shooting methods and continuation techniques in optimal control.

Mathematical formulation

$\begin{array}{lll}\underset{c}{\min }& & -\underset{0}{\overset{t_{\mathrm{f}}}{\int }}{e}^{-\rho t}\frac{c(t{)}^{1-\sigma }-1}{1-\sigma }\mathrm{d}t\\ \text{subject to}\\ \dot{k}(t)& =& Ak(t{)}^{\alpha }-(\delta +n)k(t)-c(t),\\ k(0)& =& k_0,\\ k(t)& \ge & 0\mathrm{\forall }t\in [0,t_{\mathrm{f}}],\\ c(t)& >& 0\mathrm{\forall }t\in [0,t_{\mathrm{f}}],\end{array}$

where $k(t)$ is capital per worker, $c(t)$ is consumption per worker, $A{k}^{\alpha }$ is the Cobb–Douglas production function, and $\frac{c^{1-\sigma }-1}{1-\sigma }$ is the constant-relative-risk-aversion (CRRA) utility function. As $\sigma \to 1$, the CRRA utility converges to $\ln c$ (logarithmic utility). The transversality condition at the terminal time is $e^{-\rho t_{\mathrm{f}}}k(t_{\mathrm{f}})\ge 0$.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$\rho$	0.05	Pure rate of time preference (discount rate)
$\sigma$	1.5	Inverse of the intertemporal elasticity of substitution
$\alpha$	0.33	Capital share in production (Cobb–Douglas exponent)
$\delta$	0.05	Capital depreciation rate
$n$	0.01	Population (labour) growth rate
$A$	1	Total factor productivity
$k_0$	0.5	Initial capital per worker
$t_{\mathrm{f}}$	100	Horizon of the control problem

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  Optimal capital and consumption paths for a reference solution.

Miscellaneous and Further Reading

The steady-state capital stock $k^{*}$ satisfies $A\alpha (k^{*}{)}^{\alpha -1}=\rho +\sigma n+\delta$ (the modified golden rule). The optimal consumption path satisfies the Euler equation $\dot{c}/c=(\alpha A{k}^{\alpha -1}-\delta -\rho -\sigma n)/\sigma$, known as the Keynes–Ramsey rule. The problem is solved analytically in the infinite-horizon limit and numerically for finite horizons using shooting or direct methods. This formulation and further discussion can be found in [1] and [2].

References

[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl
[2] Ramsey, F. P. (1928). A mathematical theory of saving. The Economic Journal, 38(152), 543–559.
[3] Cass, D. (1965). Optimum growth in an aggregative model of capital accumulation. The Review of Economic Studies, 32(3), 233–240.
[4] Koopmans, T. C. (1965). On the concept of optimal economic growth. The Econometric Approach to Development Planning. North-Holland.