<p>In the standard/canonical Ramsey model for discrete time, the Euler equation is of the form (notation is standard here, I won't write the whole model),</p>

<p>$$\beta (1+r_{t+1})u'(c_{t+1}) = u'(c_{t})$$</p>

<p>To obtain a zero-change locus for consumption, we must satisfy </p>

<p>$$c_{t+1} = c_t \implies u'(c_{t+1}) = u'(c_{t}) \implies \beta (1+r_{t+1}) = 1$$</p>

<p>$$ \implies r = (1/\beta)-1$$</p>

<p>-irrespective of the form of the utility function assumed (as long as it does not change, as a function, from period to period), and irrespective of the level of consumption. The locus  therefore cannot have a slope in the consumption-capital space but it has to be vertical to the capital axis at a level of capital for which $r$ (which equals the marginal product of capital) equals the above constant.</p>

CC BY-SA 3.0
2017-11-24T23:58:53.873Z
2017-11-24T23:58:53.873Z
Alecos Papadopoulos
<p>If the steady state curve of consumption depends on the physical capital, then the $\dot{c}$ will not be equal to zero. Especially, in models with endogenous discount, the steady state curve of consumption is not a vertical line. </p>

<p>Here is an example : </p>

<p><a href="https://hal.archives-ouvertes.fr/hal-00356233/document" rel="nofollow noreferrer">https://hal.archives-ouvertes.fr/hal-00356233/document</a></p>

CC BY-SA 3.0
2017-11-25T15:02:52.373Z
i think the case in the paper you suggested is continuous case, while in my case it's dicrete. anyway thank you very much!
user68863
CC BY-SA 3.0
2017-11-25T15:38:41.057Z
It is the same idea. In the paper, when $\dot{c}=0$, you have $c=c(k)$ In the same manner, in a discrete model, you will have the same thing. You can look at OLG models (in discrete time) with multiple equilibria. You can find phase diagrams where the steady state curve of consumption is not a vertical line. 
There is a very nice book of Phillippe Michel and David de la Croix explaining this kind of issues  : 

http://assets.cambridge.org/97805218/06428/sample/9780521806428ws.pdf
optimal control
CC BY-SA 3.0
2017-11-25T13:58:13.420Z
2017-11-25T13:58:13.420Z
optimal control
<p>In almost all documents I have read on the internet, the locus of consumption where $C_{t} = C_{t+1}$ seems to be a vertical line as shown below. However, my teacher said it could well be not a vertical line. I really don't know why it should be different. Thank you for all your help.
<a href="https://i.sstatic.net/EIbMT.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EIbMT.jpg" alt="enter image description here"></a></p>

CC BY-SA 3.0
2017-11-24T10:28:59.443Z
You need to make your question more clear. Are you asking about the consumption locus in the canonical Ramsey model, or in the context of other models also, that may be variants of the canonical model, or even different models altogether? If your teacher wasn't clear about this point, perhaps you should first ask your teacher.
Alecos Papadopoulos
CC BY-SA 3.0
2017-11-24T23:35:17.227Z
Thank you Alecos, i want to ask in the case of discrete infinite time Ramsey model. I guess it is canonical. The production function is $f=k_{t}^{\alpha}$, and the utility function is CEIS.
user68863
<economic-growth>
Is the locus of consumption in Ramsey growth model a vertical line?
user68863
# Is the locus of consumption in Ramsey growth model a vertical line?

(Asked by: username_0 on 2017-11-23)

In almost all documents I have read on the internet, the locus of consumption where $C\_{t} = C\_{t+1}$ seems to be a vertical line as shown below. However, my teacher said it could well be not a vertical line. I really don't know why it should be different. Thank you for all your help.
[![enter image description here](https://i.sstatic.net/EIbMT.jpg)](https://i.sstatic.net/EIbMT.jpg)

**Tags:** economic-growth
> **Comments:**
> **username_1**: You need to make your question more clear. Are you asking about the consumption locus in the canonical Ramsey model, or in the context of other models also, that may be variants of the canonical model, or even different models altogether? If your teacher wasn't clear about this point, perhaps you should first ask your teacher.  
> **username_0**: Thank you Alecos, i want to ask in the case of discrete infinite time Ramsey model. I guess it is canonical. The production function is $f=k_{t}^{\alpha}$, and the utility function is CEIS.  

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## 2 answers

### Answer by username_1 on 2017-11-24. Score: 1 | Accepted answer

In the standard/canonical Ramsey model for discrete time, the Euler equation is of the form (notation is standard here, I won't write the whole model),

$$\beta (1+r\_{t+1})u'(c\_{t+1}) = u'(c\_{t})$$

To obtain a zero-change locus for consumption, we must satisfy

$$c\_{t+1} = c\_t \implies u'(c\_{t+1}) = u'(c\_{t}) \implies \beta (1+r\_{t+1}) = 1$$

$$ \implies r = (1/\beta)-1$$

-irrespective of the form of the utility function assumed (as long as it does not change, as a function, from period to period), and irrespective of the level of consumption. The locus therefore cannot have a slope in the consumption-capital space but it has to be vertical to the capital axis at a level of capital for which $r$ (which equals the marginal product of capital) equals the above constant.


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### Answer by username_2 on 2017-11-25. Score: 1

If the steady state curve of consumption depends on the physical capital, then the $\dot{c}$ will not be equal to zero. Especially, in models with endogenous discount, the steady state curve of consumption is not a vertical line.

Here is an example :

<https://hal.archives-ouvertes.fr/hal-00356233/document>

> **Comments:**
> **username_0**: i think the case in the paper you suggested is continuous case, while in my case it's dicrete. anyway thank you very much!  
> **username_2**: It is the same idea. In the paper, when $\dot{c}=0$, you have $c=c(k)$ In the same manner, in a discrete model, you will have the same thing. You can look at OLG models (in discrete time) with multiple equilibria. You can find phase diagrams where the steady state curve of consumption is not a vertical line.  There is a very nice book of Phillippe Michel and David de la Croix explaining this kind of issues  :   http://assets.cambridge.org/97805218/06428/sample/9780521806428ws.pdf  

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