<p>Assuming that utility is forward discounted <span class="math-container">$ \rho $</span>, lifetime utility = <span class="math-container">$  \int_{0}^{\infty} e^{-\rho t}u(c) \,dt   $</span>.
And in continuous time, we also know that <span class="math-container">$ \dot{k} = y - (\delta + n)k - c$</span>.</p>
<p>Given these two equations, our problem is:</p>
<p><span class="math-container">$  \arg \max_{c}(\int_{0}^{\infty} e^{-\rho t}u(c) \,dt) $</span> such that <span class="math-container">$  c = y - (\delta + n)k - \dot{k} $</span></p>
<p>How is this solved in an optimal control context?</p>

CC BY-SA 4.0
2023-01-15T02:01:32.847Z
Economic Growth, (The MIT Press) 2nd Edition
by Robert J. Barro (Author), Xavier I. Sala-I-Martin (Author)
Alecos Papadopoulos
CC BY-SA 4.0
2023-01-15T21:47:11.650Z
Alecos, what part of the book? And is there a pdf format? Or maybe you could summarize it in an answer, that would help...
user43188
CC BY-SA 4.0
2023-01-16T10:42:29.490Z
What prior research have you done? Have you googled "ramsey problem continous time"?
BrsG
<optimization><optimal-control><ramsey>
How is the constrained maximization problem in the Ramsey model solved?
user43188
# How is the constrained maximization problem in the Ramsey model solved?

(Asked by: username_0 on 2023-01-15)

Assuming that utility is forward discounted $ \rho $, lifetime utility = $ \int\_{0}^{\infty} e^{-\rho t}u(c) \,dt $.
And in continuous time, we also know that $ \dot{k} = y - (\delta + n)k - c$.

Given these two equations, our problem is:

$ \arg \max\_{c}(\int\_{0}^{\infty} e^{-\rho t}u(c) \,dt) $ such that $ c = y - (\delta + n)k - \dot{k} $

How is this solved in an optimal control context?

**Tags:** optimization, optimal-control, ramsey
> **Comments:**
> **username_1**: Economic Growth, (The MIT Press) 2nd Edition by Robert J. Barro (Author), Xavier I. Sala-I-Martin (Author)  
> **username_0**: Alecos, what part of the book? And is there a pdf format? Or maybe you could summarize it in an answer, that would help...  
> **username_2**: What prior research have you done? Have you googled "ramsey problem continous time"?  

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## 0 answers

