Economic growth · Derivation
The Euler equation, derived: from two periods to continuous time
The Euler equation is the heart of the Ramsey–Cass–Koopmans model and of modern macroeconomics. This page derives it twice: first in two periods with a Lagrangian, which shows the economics, then in continuous time with a Hamiltonian, which is what the growth model uses. Every symbol is defined before it is used.
New to this? The few things you need first
- Utility and marginal utility
- u(c) is the satisfaction from consuming c. Marginal utility u′(c) is the extra satisfaction from one more unit; it falls as c rises (u″ < 0).
- Discounting
- Utility next period is weighted by β = 1/(1 + ρ), where ρ is the rate of time preference (impatience).
- The Ramsey model
- The lesson this derivation supports, where the equation is used with the capital accumulation equation to find the saddle path.
Start with two periods
Two periods: the Lagrangian
A household has lifetime wealth W (in today’s units) to spend on consumption c₁ today and c₂ tomorrow. Any saving earns a net return r, so a unit not consumed today buys 1 + r units tomorrow. The budget constraint, in present value, is
The household maximises with β = 1/(1 + ρ). Form the Lagrangian with multiplier μ on the budget:
Derivation
Find the first-order conditions and eliminate μ.
- Differentiate with respect to c₁ and set to zero.
- Differentiate with respect to c₂ and set to zero.
- Divide the first by the second to eliminate μ.
- Rearrange.
Answer
That is the Euler equation. In words: giving up one unit of consumption today costs u′(c₁) of utility; saving it yields 1 + r units tomorrow, each worth u′(c₂), discounted by β. At the optimum the two are equal.
A specific utility function
With CRRA utility
Take (log utility when θ = 1). Its marginal utility is u′(c) = c−θ, so the Euler equation reads c₁−θ = β(1 + r)c₂−θ, or
Take logs and use ln(1 + x) ≈ x for small x: the growth rate of consumption is approximately (r − ρ)/θ. The exponent 1/θ is the elasticity of intertemporal substitution: how much consumption growth responds to the gap between the return and impatience. For r = 7% and ρ = 3%, log utility (θ = 1) gives growth of about 4%, and θ = 2 gives about 2%.
The growth-model version
Continuous time: the Hamiltonian
In the Ramsey model time is continuous and the household’s problem is to choose a path of consumption per head c(t) to maximise
The factor e−(ρ−n)t combines impatience ρ with the growth of the household’s own size n. The tool is the current-value Hamiltonian, with the costate μ(t), the value of an extra unit of capital in utility terms:
Derivation
Find the first-order conditions and derive ċ/c.
- Optimality in c: ∂H/∂c = 0.
- Evolution of the costate: μ̇ = (ρ − n)μ − ∂H/∂k.
- Differentiate u′(c) = μ with respect to time, for CRRA u′ = c^(−θ).
- Equate the two expressions for μ̇/μ.
- Solve for ċ/c.
Answer
The continuous-time Euler equation, with r = f′(k) − δ. It matches the two-period result: consumption grows when the net return exceeds impatience.
The transversality condition
Without it, the first-order conditions allow plans that pile up capital for ever. This condition excludes them, and with the Euler equation selects the saddle path.
What it says
What the Euler equation says
The equation compares two rates: the return that saving earns and the household’s impatience. If the return is higher, a household willing to wait is rewarded by consumption that rises over time; if lower, consumption declines. The equation says nothing about the level of consumption. That is set by the budget constraint, and the two together, in the growth model, pin down the saddle path of Lesson 9.
Why this equation is everywhere
Almost every modern macro model has a household whose saving is described by an Euler equation. In the New Keynesian model it becomes the dynamic IS curve, linking output to the real interest rate. The same logic appears in consumption-smoothing and permanent-income theory.
Practice
Check yourself
Numbers
Problem 1
Two periods, log utility (θ = 1), β = 0.95, r = 0.10. By what factor does consumption grow from period 1 to period 2? (three decimals)
Problem 2
Same β and r, but θ = 2. Find c₂/c₁ (three decimals).
Problem 3
Continuous time, CRRA with θ = 2, ρ = 3%, δ = 5%, and f′(k) = 0.13. Find the growth rate of consumption (per cent a year).
Concepts
1If the net return on saving exceeds the rate of time preference, the Euler equation says consumption
2In ċ/c = (r − ρ)/θ, a larger θ means
Quick answers
Frequently asked questions
What is the Euler equation in macroeconomics?
The consumption Euler equation links consumption in adjacent periods along an optimal plan. In two periods it is u′(c₁) = β(1 + r)u′(c₂): the utility lost by saving one unit today equals the discounted utility of the 1 + r units it yields tomorrow. In continuous time with CRRA utility it becomes ċ/c = (r − ρ)/θ.
How do you derive the Euler equation?
Write the household's problem as maximising lifetime utility subject to a lifetime budget constraint. In two periods, form the Lagrangian u(c₁) + βu(c₂) + μ(W − c₁ − c₂/(1 + r)) and take first-order conditions. In continuous time, form a current-value Hamiltonian with a costate for capital, take the first-order conditions for consumption and capital, and eliminate the costate.
What does θ mean in ċ/c = (r − ρ)/θ?
θ is the coefficient of relative risk aversion, and 1/θ is the elasticity of intertemporal substitution. A larger θ means the household dislikes consumption that changes over time, so a given gap between the return and impatience tilts consumption growth less.
What is the transversality condition?
It is the end-of-time condition that rules out over-saving: the present value of the capital stock must go to zero, lim e^(−(ρ−n)t) μ(t) k(t) = 0. A plan that keeps capital accumulating at a rate that outpaces discounting leaves resources unconsumed and cannot be optimal.
Read next
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