View source for Ramsey Growth

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

First formulated by Frank Ramsey in 1928 [[#Ramsey|[2]]] and later extended by Cass [[#Cass|[3]]] and Koopmans [[#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 ==
<p>
<math>
 \begin{array}{lll}
 \displaystyle \min_{c} && -\int_0^{t_\mathrm{f}} e^{-\rho\, t}\, \frac{c(t)^{1-\sigma} - 1}{1 - \sigma}\, \mathrm{d}t \\[6pt]
 \text{subject to} \\
\quad \dot{k}(t) &=& A\, k(t)^\alpha - (\delta + n)\, k(t) - c(t), \\
\quad k(0) &=& k_0, \\
\quad k(t) &\geq& 0 \quad \forall\, t \in [0, t_\mathrm{f}], \\
\quad c(t) &>& 0 \quad \forall\, t \in [0, t_\mathrm{f}],
 \end{array}
</math>
</p>

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

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\rho</math> || align=right | 0.05 || Pure rate of time preference (discount rate)
|-
| align=center | <math>\sigma</math> || align=right | 1.5 || Inverse of the intertemporal elasticity of substitution
|-
| align=center | <math>\alpha</math> || align=right | 0.33 || Capital share in production (Cobb–Douglas exponent)
|-
| align=center | <math>\delta</math> || align=right | 0.05 || Capital depreciation rate
|-
| align=center | <math>n</math> || align=right | 0.01 || Population (labour) growth rate
|-
| align=center | <math>A</math> || align=right | 1 || Total factor productivity
|-
| align=center | <math>k_0</math> || align=right | 0.5 || Initial capital per worker
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 100 || Horizon of the control problem
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
 Image: Ramsey_Growth.png | Optimal capital and consumption paths for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The steady-state capital stock <math>k^*</math> satisfies <math>A\,\alpha\,(k^*)^{\alpha-1} = \rho + \sigma\,n + \delta</math> (the modified golden rule).
The optimal consumption path satisfies the Euler equation
<math>\dot{c}/c = (\alpha A k^{\alpha-1} - \delta - \rho - \sigma n)/\sigma</math>,
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 [[#OCPjl|[1]]] and [[#Ramsey|[2]]].

== References ==
<span id="OCPjl">[1]</span> 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<br>
<span id="Ramsey">[2]</span> Ramsey, F. P. (1928). A mathematical theory of saving. <i>The Economic Journal</i>, 38(152), 543–559.<br>
<span id="Cass">[3]</span> Cass, D. (1965). Optimum growth in an aggregative model of capital accumulation. <i>The Review of Economic Studies</i>, 32(3), 233–240.<br>
<span id="Koopmans">[4]</span> Koopmans, T. C. (1965). On the concept of optimal economic growth. <i>The Econometric Approach to Development Planning</i>. North-Holland.<br>


[[Category:ODE model]]
[[Category:Economic model]]