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Use of the Euler equations to estimate consumption appears to have advantages over traditional models. First, using Euler equations is simpler than conventional methods. This avoids the need to solve the consumer's optimization problem and is the most appealing element of using Euler equations to some economists. [4]
The Keynes–Ramsey rule is named after Frank P. Ramsey, who derived it in 1928, [3] and his mentor John Maynard Keynes, who provided an economic interpretation. [4] Mathematically, the Keynes–Ramsey rule is a necessary first-order condition for an optimal control problem, also known as an Euler–Lagrange equation. [5]
In contrast, a recursive model involves two or more periods, in which the consumer or producer trades off benefits and costs across the two time periods. This trade-off is sometimes represented in what is called an Euler equation. A time-series path in the recursive model is the result of a series of these two-period decisions.
The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: + =, or simply in Einstein notation: + =, where the conservation quantity in this case is a vector, and is a flux matrix. This can be simply proved.
Consumption smoothing is an economic concept for the practice of optimizing a person's standard of living through an appropriate balance between savings and consumption over time. An optimal consumption rate should be relatively similar at each stage of a person's life rather than fluctuate wildly.
The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has + on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2] When x = π, Euler's formula may be rewritten as e iπ + 1 = 0 or e iπ = −1, which is known as Euler's identity.
In the published version of Koopmans' paper, he imposes a new Condition Alpha in addition to the Euler equations, stating that the only admissible trajectories among those satisfying the Euler equations is the one that converges to the optimal steady-state equilibrium of the model.