# Is the locus of consumption in Ramsey growth model a vertical line?

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.

• 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 Nov 24 '17 at 10:28
• 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 Nov 24 '17 at 23:35

## 2 Answers

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.

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

• 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 Nov 25 '17 at 15:02
• 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 : assets.cambridge.org/97805218/06428/sample/9780521806428ws.pdf – optimal control Nov 25 '17 at 15:38