Economic Theory 22, 671–674 (2003)

A new proof of the maximum principle

Ngo Van Long1 and Koji Shimomura2

1 Department of Economics, McGill University, 855 Sherbrooke Street West,Montreal, H3A 2T7 Quebec, CANADA (e-mail: ngo.long@mcgill.ca)

2 Research Institute for Economics and Business Administration, Kobe University,2-1 Rokkodai-cho, Nada-ku, Kobe, 657-8501 JAPAN (e-mail: simomura@rieb.kobe-u.ac.jp)

Received: April 11, 2002; revised version: June 26, 2002

Summary. We offer a new proof of the maximum principle, by using the envelopetheorem that is frequently used in the standard microeconomic theory.

Keywords and Phrases: Maximum principle, Envelope theorem.

JEL Classification Numbers: C 61.

1 Introduction

The purpose of this note is to offer a new and simple proof of the maximumprinciple, using the envelope theorem.1 A number of authors have provided proofsof the maximum principle using the idea that a small deviation from the optimal pathof the control variable must have a negative impact on the value of the objectivefunction (see, for example, Kamien and Schwartz, 1991; pp. 124–127; Leonardand Long, 1992, pp. 161–163). The proof by Kamien and Schwartz relies on theassumption of a “free value of the state variable at the terminal point” (p.124)while Leonard and Long, dealing with a fixed value of the state variable at theterminal point, assume the existence of a class of deviation function with somespecial properties. The proof that we offer below does not rely on either of theseassumptions.2

Correspondence to: K. Shimomura1 After completing this paper, we discovered that a similar proof is offered in Dana and Le Van

(2001). The main feature that distinguishes our approach from theirs is our direct application of theenvelope theorem, a tool that many economic researchers have found useful.

2 Other economists, such as Arrow and Kurz (1970); Silberberg (1990, pp. 620–621), proved themaximum principle by using the Hamilton-Jacobi-Bellman equation. Such an approach relies on theassumption of a twice-differentiable value function. Our proof does not rely on that assumption.

672 N. Van Long and K. Shimomura

2 A proof based on the envelope theorem

Let k = (k1, ..., kn) denote the vector of state variables and z = (z1, ..., zm) denotethe vector of control variables. The transition equation, in vector notation, is

k̇ = g(k, z, t) (1)

where g(.) have continuous partial derivatives. Given k and t, the control vector isrestricted to belong to some set A(k, t). We assume that this set can be describedby

A(k, t) = {z : φ(k, z, t) ≥ 0}where φ = (φ1, ..., φs) is a vector of s differentiable functions. The objectivefunction is

max∫ T

0U(k, z, t)dt (2)

subject to (1), z ∈ A(k, t), and the boundary conditions k(0) = k0 (given) andk(T ) = kT (fixed).

In the special case where it is possible to invert (1) to obtain z = h(k, k̇, t),one can substitute for z and apply the calculus of variations. In general, however,such inversion is not possible (for example, when the number of control variablesexceed the number of state variables). In this note we deal with the general case.

Let (k∗(t), z∗(t)) be an optimal time path, i.e., a solution to problem (2). Thenat any time t, given k∗(t) and k̇∗(t), it must be the case that z∗(t) solves the staticproblem

maxz

U(k∗(t), z, t) (3)

subject to the equality constraint

g(k∗(t), z, t) − k̇∗(t) = 0 (4)

and the inequality constraint

φ(k∗(t), z, t) ≥ 0. (5)

We apply the Kuhn-Tucker theory to the static problem (3). Here, we omit theasterisk to lighten notation.We assume that the constraint qualifications hold (see,for example, Takayama, 1985, p 101). Then there exists a vector (λ, µ) with whichwe can form the Lagrangian function

L(z, λ, µ; k(t), k̇(t), t) =

U(k(t), z, t) + λ[g(k(t), z, t) − k̇(t)

]+ µφ(k(t), z, t) (6)

and obtain the necessary conditions:

Lz = Uz + λgz + µφz = 0 (7)

Lµ = φ(k(t), z, t) ≥ 0, µ ≥ 0, µφ(k(t), z, t) = 0 (8)

Lλ = g(k(t), z, t) − k̇(t) = 0 (9)

A new proof of the maximum principle 673

The solution to this system of equation is denoted by z = z(k(t), k̇(t), t), λ =λ(k(t), k̇(t), t), µ = µ(k(t), k̇(t), t). Now, by appealing to the Envelope Theorem(see, for example, Takayama, 1985, p. 138) we have

∂

∂k(t)U(k(t), z(k(t), k̇(t), t), t) =

∂

∂k(t)L(z, λ, µ; k(t), k̇(t), t)

= Uk + λgk + µφk (10)

∂

∂k̇(t)U(k(t), z(k(t), k̇(t), t), t) =

∂

∂k̇(t)L(z, λ, µ; k(t), k̇(t), t)

= −λ(k(t), k̇(t), t) (11)

Now, let us return to the dynamic optimization problem (2), with z = z(k(t),k̇(t), t). That is, we seek to maximize

∫ T

0U(k(t), z(k(t), k̇(t), t), t)dt ≡

∫ T

0F (k(t), k̇(t), t)dt

subject to the boundary conditions are k(0) = k0 (given) and k(T ) = kT (fixed).The following Euler-Lagrange condition is necessary for an optimal solution:

d

dt

[Fk̇

]=

∂F

∂k(12)

Substituting (11) into the left-hand side of (12) and (10) into the right-hand side of(12), we get:

− d

dtλ(k∗(t), k̇∗(t), t) = Uk + λgk + µφk (13)

Let us we define

λ∗(t) = λ(k∗(t), k̇∗(t), t) and µ∗(t) = µ(k∗(t), k̇∗(t), t)

andH(k, z, λ∗, t) = U(k, z, t) + λ∗g(k, z, t) (14)

L(k, z, λ∗, µ∗, t) = H(k, z, λ∗, t) + µ∗φ(k, z, λ∗, µ∗, t) (15)

Then, from the definitions (14)–(15), the equations (7)–(9) and (13) constitute themaximin principle:

∂L∂z

=∂H∂z

+ µ∗ ∂φ

∂z= 0

λ̇ = −∂L∂k

k̇ =∂L∂λ

This concludes our proof of the maximum principle.

674 N. Van Long and K. Shimomura

References

Arrow, K., Kurz, M.: Public investment, the rate of return and optimal fiscal policy. Baltimore: JohnsHopkins Press 1970

Dana, R.-A., Le Van, C.: Optimal control in infinite horizon. Typescript (2001)Kamien, M., Schwartz, N.: Dynamic optimization: the calculus of variations and optimal control in

economics and management, 2nd edn. Amsterdam: North-Holland 1991Leonard, D., Van Long, N.: Optimal control theory and static optimization in economics. Cambridge

New York: Cambridge University Press 1992Silberberg, E.: The structure of economics: a mathematical analysis. New York: McGaw-Hill 1990Takayama, A.: Mathematical economics, 2nd edn. Cambridge New York: Cambridge University Press

1985