A Consideration for the Euler Equation inMacroeconomics

Taro Ikeda
February 2019
Discussion Paper No.1903

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANA Consideration for the Euler Equation in Macroeconomics
Taro Ikeda
Graduate School of Economics, Kobe University
February 4, 2019
Abstract
This paper considers the problem of the Euler equation in the infinite horizon
optimization problem in dynamic macroeconomics.
We show that the assumption of the two-period Euler equation in the infinite
horizon problem yields the Euler equation for the one-period optimization problem.

A. The Euler equation in the infinite horizon optimization problem
Consider the simple infinite horizon optimization problem as follows:∞max 𝑈(𝒄) = ∑ 𝛽 𝑡−1 𝑢(𝑐𝑡 ) = 𝑢(𝑐1 ) + 𝛽𝑢(𝑐2 ) + ⋯{𝑐𝑡 }∞
𝑡=1

𝑡=1

This problem stresses that total utility 𝑈(𝑐) is maximized by all elements of 𝑐 = (𝑐1 , 𝑐2 , … )as its control, and the instantaneous utility 𝑢(∙) satisfies all necessary conditions, especiallyconcavity, differentiability, and increasing (𝑢′ (𝑐𝑡 ) ≥ 0). Here, the constraint is not written
explicitly, but satisfies all necessary conditions, especially convexity.
The Euler equation then implies that the difference in total utility 𝑼(𝒄) with respect to all its
elements 𝒄 = (𝒄𝟏 , 𝒄𝟐 , … ) equals zero,𝑑𝑈(𝒄) =

𝜕𝑈(𝒄)
𝜕𝑈(𝒄)
𝜕𝑈(𝒄)
𝜕𝑈(𝒄)
𝑑𝑐1 +
𝑑𝑐2 + ⋯ +𝑑𝑐𝑡 +𝑑𝑐 + ⋯ = 0,
𝜕𝑐1
𝜕𝑐2
𝜕𝑐𝑡
𝜕𝑐𝑡+1 𝑡+1

where instantaneous utility is additive and discounted by 𝛽, and therefore,𝜕𝑢(𝑐1 )
𝜕𝑢(𝑐2 )
𝜕𝑢(𝑐𝑡 )
𝜕𝑢(𝑐)𝑑𝑐1 + 𝛽𝑑𝑐2 + ⋯ + 𝛽 𝑡
𝑑𝑐𝑡 + 𝛽 𝑡+1
𝑑𝑐 + ⋯ = 0.
𝜕𝑐1𝜕𝑐2𝜕𝑐𝑡𝜕𝑐𝑡+1 𝑡+1
We then obtain the Euler equation in the infinite horizon optimization problem as follows:
∞(A)

∑ 𝛽𝑡
𝑡=1

𝜕𝑢(𝑐𝑡 )
𝑑𝑐𝑡 = 0.
𝜕𝑐𝑡1

B. The Euler equation used in dynamic programming
The Euler equation used frequently in dynamic programming (see Acemoglu [1]) can be
written as:𝜕𝑢(𝑐𝑡 )
𝜕𝑢(𝑐𝑡+1 )𝑑𝑐𝑡 + 𝛽𝑑𝑐𝑡+1 = 0,
𝜕𝑐𝑡
𝜕𝑐𝑡+1(B)𝐟𝐨𝐫 ∀𝒕 ≥ 𝟏.

C. The problem with dynamic programming caused by the Euler equation (B)We assume that the Euler equation (B) holds for all 𝑡 in the model. We then show that thisassumption yields the Euler equation for the one-period optimization problem.

The basics of the logic
For example, consider a terminal date of three with the three-period optimization problem.
The Euler equation (B) for the terminal date 𝑡 = 𝑇 = 3 then becomes
𝛽3
However, 𝛽 4𝜕𝑢(𝑐4 )
𝜕𝑐4

𝜕𝑢(𝑐3 )
𝜕𝑢(𝑐4 )
𝑑𝑐3 + 𝛽 4𝑑𝑐4 = 0.
𝜕𝑐3
𝜕𝑐4

𝑑𝑐4 does not exist because 𝑇 = 3. By considering it as zero by taking agood intuition, the condition becomes𝛽3𝜕𝑢(𝑐3 )𝑑𝑐3 = 0.
𝜕𝑐3

By inserting the terminal condition into the Euler equation (B) for 𝑡 = 2,𝛽2

𝜕𝑢(𝑐2 )
𝜕𝑢(𝑐3 )
𝑑𝑐2 + 𝛽 3
𝑑𝑐3 = 0,𝜕𝑐2
𝜕𝑐3

we obtain𝛽2

𝜕𝑢(𝑐2 )
𝑑𝑐2 = 0.
𝜕𝑐2

By applying the same logic for the Euler equation (B) for 𝑡 = 1,𝛽1

𝜕𝑢(𝑐1 )
𝜕𝑢(𝑐2 )
𝑑𝑐1 + 𝛽 2𝑑𝑐2 = 0,𝜕𝑐1
𝜕𝑐2

we obtain𝛽1

𝜕𝑢(𝑐1 )
𝑑𝑐1 = 0.
𝜕𝑐1

Therefore, we obtain the condition2

(C)𝜕𝑢(𝑐𝑡 )
𝑑𝑐𝑡 = 0,𝜕𝑐𝑡𝐟𝐨𝐫∀𝒕 ≥ 𝟏.

This is the condition for the one-period optimization problem.

To infinity
For the above discussion, increase 𝑇 sequentially by one. We thus recognize that thisproblem continues forever.

D. Consideration
In this section, we discuss additive consideration after assuming the two-period overlapping
generations (OLG) model.
In the two-period OLG model, the physical environment and the reproduction of thepopulation continue forever and each generation optimizes over their two periods of life.
Therefore, we do not encounter the terminal problem considered here.

E. Summary
We considered the problem caused by the Euler equation in the infinite optimization problem.
For dynamic programming and its Euler equation, refer to Ch. 6 (especially Sect. 6) ofAcemoglu [1]. See Ikeda [2] for the basics of monetary policy analysis using the OLG model.

F. References[1] Acemoglu, Daron, Introduction to Modern Economic Growth, Princeton University Press,Princeton, NJ, 2009.
[2] Ikeda, Taro, “An introduction of a simple monetary policy with savings taxation in theoverlapping generations model,” Kobe University Discussion Paper 1810, 2018.

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