Euler's equation of consumption

The Euler equation of consumption describes the optimal intertemporal consumption allocation of a utility-maximizing household. The equation is u. a. a core component of optimal solutions in macroeconomic dynamic models. The equation is intended to take into account a positive time preference (i.e. the fact that consumption today is preferred over consumption tomorrow).

situation

A representative household makes its consumption decision in such a way that it wants to maximize its intertemporal utility function . Overall, the consumer has, for example, a so-called value function that is the sum of individual weighted utility functions. For example, this problem could look like this:

{\ displaystyle V_ {t} = \ sum _ {s = 0} ^ {\ infty} \ beta ^ {s} U (C_ {t, s})}

assuming a positive but decreasing marginal utility ( ). There is also a discount factor . The discount factor reflects a positive time preference , i.e. H. the consumption at later points in time should appear less valuable to the individuals than the consumption in the current period. In addition, the following budget restrictions should be adhered to: ${\ displaystyle U '(c)> 0, U' '(c) <0}$ ${\ displaystyle 0 <\ beta <{\ frac {1} {1+ \ theta}} <1}$

{\ displaystyle \ Delta a_ {t + s + 1} + c_ {t + s} = x_ {t + s} + r_ {t + s} a_ {t + s}}

where stands for the respective period and is an index . Also includes net worth, household income, and the interest rate paid on financial assets. Since this problem involves variables in different periods, it can be solved with methods of dynamic programming . It is also possible to arrive at a solution using the calculus of variations or using Pontryagin's maximum principle . Since there is no stochastic problem, it is also possible to use the simpler method of the Lagrange multipliers . ${\ displaystyle t}$ ${\ displaystyle s}$ ${\ displaystyle a_ {t}}$ ${\ displaystyle x_ {t}}$ ${\ displaystyle r_ {t}}$

Solution and interpretation in the 2-period case

Solution in the 2-period case.

For the sake of simplicity, consider the two-period case ( ). The question here is how much consumption will increase in the next period ( ) if the current consumption ( ) is reduced by a small amount ( ) so that the total value ( ) remains unchanged. ${\ displaystyle t = 0, s = [0,1]}$ ${\ displaystyle c_ {t + 1}}$ ${\ displaystyle c_ {t}}$ ${\ displaystyle dc_ {t}}$ ${\ displaystyle V_ {t}}$

The Euler equation in this case results in:

{\ displaystyle \ beta {\ frac {U '(c_ {2})} {U' (c_ {1})}} (1 + r_ {2}) = 1}

.

Further simplifications were made (transversality conditions) after which z. B. no transfer of assets should have taken place before the first period and neither assets nor debts should exist after the last period. The interest rate can also be identical in both periods.

In the graphic opposite you can see the solution as the tangential intersection of the advertising function and the intertemporal resource limitation . The axes of intersection of this resource constraint are to be understood as maximum consumption values per period. At the point of intersection with the abscissa is maximal, i. This means that nothing at all would be consumed in the following period, but the entire future income is already being spent (via borrowing and paying in the next period). Analogously, the intersection of the ordinates means a situation in which everything would be saved in the first year, in order to then consume both incomes and the interest in the second period. The rise in the restriction line is independent of income. A change in income would lead to a parallel shift in the straight line, a change in interest rates would lead to a rotation. ${\ displaystyle c_ {t} ^ {*}, c_ {t + 1} ^ {*}}$ ${\ displaystyle V_ {t}}$ ${\ displaystyle c_ {t}}$ ${\ displaystyle c_ {t + 1}}$ ${\ displaystyle - (1 + r)}$

use

Robert E. Hall devised a random walk model of consumption using the Euler equation.

Individual evidence

↑ Euler equation of consumption - article at the Gabler Wirtschaftslexikon
^ Stochastic Implications of the Life Cycle-Permanent Income Hypothesis. Theory and Evidence. In: Journal of Political Economy. Volume 86, No. 6, December 1978, pp. 971-987.

literature

Wickens, Michael. Macroeconomic theory: a dynamic general equilibrium approach. Princeton University Press, 2012. pp. 17-20, 55-59

Web links

Euler equation of consumption - article at the Gabler Wirtschaftslexikon

[1] Euler equation of consumption - article at the Gabler Wirtschaftslexikon

[2] Stochastic Implications of the Life Cycle-Permanent Income Hypothesis. Theory and Evidence. In: Journal of Political Economy. Volume 86, No. 6, December 1978, pp. 971-987.