Deriving the Ramsey Model Euler Equation from First-Order Conditions

Summary

The document shows how the Euler condition in the Ramsey growth model follows from the first-order conditions of a discounted lifetime-utility problem. In the stated Lagrangian, differentiating with respect to consumption equates marginal utility to the multiplier on the resource constraint. Differentiating with respect to next period’s capital links that multiplier across adjacent periods through the discount factor, depreciation, and the marginal product of capital.

Combining these relationships expresses the intertemporal condition in terms of consumption marginal utilities and the return to carrying capital forward. The economic interpretation is that the household’s marginal rate of substitution between consumption across periods must match the marginal return from postponing consumption through investment. The response clarifies that the Euler equation is not separate from constrained optimization; it combines its first-order conditions. Its displayed final formula appears to use marginal utility at next-period consumption, though the transcription should be checked if exact notation matters.

Key ideas

• The consumption first-order condition equates marginal utility with the resource-constraint multiplier.
• The capital first-order condition relates multipliers across periods through the return on capital.
• Combining the conditions produces the intertemporal Euler equation.
• The Euler condition matches the tradeoff between consumption across time to capital’s net return.

Tags

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


# 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.

## Answer by python_enthusiast (score 1)

https://quant.stackexchange.com/a/44428

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.