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.
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.