<p>I recently learned about the neoclassical growth model in both discrete and continuous time. The intuitive meaning of the shadow price for both cases is that it represents the value of one additional unit of capital. However, in discrete time, this value is the discounted value of the marginal utility at time t consumption level. Whereas, in continuous time, the shadow price is the discounted value of marginal utility at time 0 consumption level. I was wondering why is there such a difference when changing from discrete to continuous time.</p>

<p>Moreover, I feel that the discrete-time seems more meaningful as the value of capital at time t should depend on the marginal utility at time t and not at time 0. </p>

<p>Thank you!</p>

<h2>Continuous time case</h2>

<p><a href="https://i.sstatic.net/6Ml3l.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6Ml3l.png" alt="enter image description here"></a></p>

<h2>Discrete time case</h2>

<p><a href="https://i.sstatic.net/uuDlp.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uuDlp.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-09-24T17:51:18.133Z
Hi, welcome to Stack Exchange. Could you please cite the source of the screenshot?
Student
CC BY-SA 4.0
2019-09-27T17:19:48.970Z
We need more information to answer your question. What is $r(v)$? What is $n$? What is $c$?
Angela Pretorius
CC BY-SA 4.0
2019-10-07T20:15:16.370Z
Where did you find the expression for the discrete time situation? We need to have a look at the surrounding material.
Alecos Papadopoulos
<macroeconomics><mathematical-economics><economic-growth>
Neoclassical Economic Growth Model Shadow Price for Discrete vs Continuous Time
Lim Kaizhuo
# Neoclassical Economic Growth Model Shadow Price for Discrete vs Continuous Time

(Asked by: username_0 on 2019-09-24)

I recently learned about the neoclassical growth model in both discrete and continuous time. The intuitive meaning of the shadow price for both cases is that it represents the value of one additional unit of capital. However, in discrete time, this value is the discounted value of the marginal utility at time t consumption level. Whereas, in continuous time, the shadow price is the discounted value of marginal utility at time 0 consumption level. I was wondering why is there such a difference when changing from discrete to continuous time.

Moreover, I feel that the discrete-time seems more meaningful as the value of capital at time t should depend on the marginal utility at time t and not at time 0.

Thank you!

## Continuous time case

[![enter image description here](https://i.sstatic.net/6Ml3l.png)](https://i.sstatic.net/6Ml3l.png)

## Discrete time case

[![enter image description here](https://i.sstatic.net/uuDlp.png)](https://i.sstatic.net/uuDlp.png)

**Tags:** macroeconomics, mathematical-economics, economic-growth

> **Comments:**
> **username_1**: Hi, welcome to Stack Exchange. Could you please cite the source of the screenshot?  
> **username_2**: We need more information to answer your question. What is $r(v)$? What is $n$? What is $c$?  
> **username_3**: Where did you find the expression for the discrete time situation? We need to have a look at the surrounding material.  

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## 0 answers

