What is the use of the Euler equation in the Ramsey growth model?

I apologise for being brief, but I don't understand how is Euler equation used in the Ramsey growth model. I am reading a textbook "Dynamic General Equilibrium Modeling" and there is mentioned about it.

I am modeling the Ramsey model using the Kuhn-Tucker theorem, and I have a set of first-order conditions. I think this is good enough, as I can use a standard constrained optimization method to solve this Ramsey model.

Why do we need the Euler equation, which is a second-order different equation? It is not used in the constrained optimization method.

• Hi Michael. Your question is really an economics question, and unfortunately, the economics stack exchange was recently closed due to lack of participation. While some econ questions also belong on this site, I am not sure you will find the right audience for your question here. – Tal Fishman Jul 18 '12 at 14:06
• Hi Tal, thank you for your reply. I think stack exchange is a really good platform, so I am trying my luck by posting my question here. =) – Michael Jul 18 '12 at 14:40
• There is a second-order equation in the wiki article en.wikipedia.org/wiki/… -- it's just a guess on my part, but your model may have already incorporated a solution to that equation by the time you're doing the optimization you mention. – JL344 Jul 27 '12 at 4:53

Consider the Lagrangian:

$$L = \sum_{t=0}^{\infty} \beta ^t \{ U(c_t) + \lambda _t [(1-\delta)k_t+f(k_t) - k_{t+1}-c_t]\}$$

The FOC of the Lagrangian with respect to $$c_t$$ gives:

$$\frac{\partial L}{\partial c_t} = 0 \Rightarrow U'(c_t) = \lambda_t$$

The FOC with respect to $$k_{t+1}$$ gives:

$$\frac{\partial L}{\partial k_{t+1}} = 0 \Rightarrow \lambda_t = \beta[1-\delta+f'(k_{t+1})]\lambda_{t+1}$$

These two equations give you an alternative representation of the Euler equation:

$$\frac{U'(c_t)}{\beta U(c_{t+1})} = 1 - \delta + f'(k_{t+1})$$

The Euler condition imposes equality between the marginal rate of intertemporal substitution in consumption and the corresponding marginal rate of transformation, which is simply the marginal cost of capital net of depreciation (plus one).