Transversality Conditions andDynamic Economic Behavior∗

Takashi Kamihigashi
RIEB, Kobe University
Rokkodai, Nada, Kobe 657-8501, Japantkamihig@rieb.kobe-u.ac.jp

January 31, 2006

Abstract
Transversality conditions are optimality conditions often used along with Euler equations to characterize the optimal paths of dynamic economic models.
This article explains the foundations of transversality conditions using a geometric example, a finite horizon problem, and an infinite horizon problem.
Their relationships to asset bubbles, hyperdeflations, and no-Ponzi-game conditions are also discussed.
∗Prepared for The New Palgrave Dictionary of Economics, 2nd Edition.

Transversality conditions and dynamic economic
behavior
Transversality conditions are optimality conditions often used along with Euler equations to characterize the optimal paths (plans, programs, trajectories,etc) of dynamic economic models.
An Euler equation is a local condition that no gain be achieved by slightly
deviating from an optimal path for a short period of time. In many cases
an Euler equation is equivalent to the property that no gain be achieved
by deviating from an optimal path and eventually returning to it. If theterminal (or initial) point is not fixed, there may be many paths satisfying
the Euler equation. A transversality condition enables one to single out theoptimal path among those satisfying the Euler equation, or at least to ruleout some non-optimal paths. Along with the Euler equation, it requires thatno gain be achieved by deviating from an optimal path and never returning
to it. Such deviations are possible only if the terminal point is not fixed.
A simple geometric example best illustrates the roles of an Euler equation
and a transversality condition. What is the shortest path from a point A to astraight line L infinitely long in both directions? The answer is of course thestraight line from point A to line L that is perpendicular to line L. There are
two conditions involved here. The first condition is that the shortest path bea straight line: one cannot make the path shorter by deviating from it andeventually returning to it. This is the implication of the Euler equation forthis problem. But there are infinitely many straight lines from point A toline L. In fact, a straight line from point A to line L can be arbitrarily long,so that even very bad choices satisfy the Euler equation. This is why one
needs the second condition, that the shortest path be perpendicular to line
L. This additional condition ensures that one cannot make the path shorterby deviating from it and never returning to it.
The condition of perpendicularity in this example and similar conditions
on end points in other problems are called transversality conditions in dynamic optimization theory (Hestenes, 1966, p. 87). According to Bolza (1904,p. 106), the term was first introduced by Kneser (1900).

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The finite horizon caseThough both Euler equations and transversality conditions were initiallydeveloped for continuous time models (more precisely, calculus of variations
problems), the basic arguments can be better understood in discrete timemodels, to which we restrict ourselves. All arguments below apply to bothcases.
Consider the following maximization problem:maxT
X+1{xt }Tt=0

β t v(xt , xt+1 )(1)

t=0s.t. x0 = x0 ,xt+1 ≥ 0, t = 0, 1, 2, . . . , T,(2)where β ∈ (0, 1) is called the discount factor, v is called the return function,and x0 is a given initial condition. To be concrete, we interpret xt as thestock of wealth (or capital) at the beginning of period t.
There may be other constraints, but we assume that they are not bindingat the optimum. We assume for simplicity that the nonnegativity constraint
is not binding at the optimum except for xT +1 . This can be ensured byassuming the Inada condition v1 (0, x) = ∞ for all x. Note that xT +1 isfree except for the nonnegativity constraint. We also assume that the return
function v is differentiable and concave. It may be allowed to depend on t
though we do not assume so here for notational simplicity.
The Euler equation for this problem is simply the first order condition
with respect to xt+1 for t < T :v2 (xt , xt+1 ) + βv1 (xt+1 , xt+2 ) = 0.

(3)This condition means that no gain can be achieved by deviating from an
optimal path for one period.
The first order condition with respect to xT +1 consists of two cases:
β T v2 (xT , xT +1 ) = 0 or β T v2 (xT , xT +1 ) ≤ 0, xt+1 = 0.

(4)

In most economic problems, it is costly to accumulate wealth. Hence weassume that v2 (x, y) ≤ 0 for all x, y. Then the two cases in (4) can becombined into
β T [−v2 (xT , xT +1 )]xT +1 = 0.
(5)2This is the transversality condition for this problem. It means that nothing should be saved in the last period unless it is costless to do so (i.e.,−v2 (xT , xT +1 ) = 0). Alternatively it can be interpreted as saying that thepresent discounted value of the terminal stock must be zero.
Since the Euler equation (3) and the transversality condition (5) are firstorder conditions, they are necessary for optimality. It is easy to verify thatthey are sufficient as well by concavity of the return function v. There is notechnical issue in the finite horizon case.
Equation (5) is a typical transversality condition in economics, but it is
not the only possibility in other models. The proper choice of transversality
condition depends on the exact constraint on the terminal stock.

The infinite horizon caseLet us now consider the infinite horizon case. The maximization problem ismax{xt }∞t=0∞
X

β t v(xt , xt+1 )

(6)

t=0

s.t. x0 = x0 ,xt+1 ≥ 0, t = 0, 1, 2, . . . .

(7)

We assume for simplicity that the nonnegativity constraint is never bindingat the optimum. This can be ensured by the Inada condition mentionedabove.
The first order condition with respect to xt+1 remains the same, so thatthe Euler equation remains the same. Unfortunately, unlike in the finitehorizon case, one cannot directly derive the transversality condition here.
Instead one needs to derive the transversality condition for the finite horizoncase first ((5) in this case) and then take the limit:
lim β T [−v2 (xT , xT +1 )]xT +1 = 0.

T →∞

(8)

This can be interpreted as saying that the present discounted value of wealthat infinity must be zero, or wealth (xT +1 ) should not grow too fast compared
to its marginal value (β T [−v2 (xT , xT +1 )]). In other words, the transversality condition (8) rules out overaccumulation of wealth. The idea is that if
one saves too much and spends too little forever, then one is not behaving
optimally.

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There is an alternative transversality condition that is often used:lim β T v1 (xT , xT +1 )xT = 0.

T →∞

(9)

Though this condition is equivalent to (8) under the Euler equation (3), it
has no counterpart in the finite horizon case and in the continuous time case.
It is well known (and easy to show) that the Euler equation (3) and thetransversality condition (8) are sufficient for optimality (e.g., Stokey andLucas, 1989, p. 89). This result is often credited to Mangasarian (1966), whoshowed the finite horizon version of the result for a continuous time model.
Since the Euler equation is simply the first order condition with respect
to xt+1 , it remains to be a necessary condition in the infinite horizon case.
On the other hand, necessity of the transversality condition in the infinite
horizon case is often considered to be a difficult issue. But there are two
simple ways to prove it if the objective function is assumed to be finite for all
feasible paths (Kamihigashi, 2002, 2005). If this assumption is not assured,
one can try the following test. Shift the entire optimal path downward bya small fixed proportion. Does it reduce the value of the objective function
by only a finite amount? If so, the transversality condition is necessary.
See Kamihigashi (2001, 2003) for precise assumptions and statements. See
Weitzman (1973), Benveniste and Scheinkman (1982), and Michele (1982)
for earlier results and arguments, and Kamihigashi (2001) for a literature
review.

Asset bubbles and transversality conditions
Transversality conditions are often used to rule out asset bubbles. To bespecific, consider a deterministic version of the Lucas (1978) asset pricing
model. There are many homogeneous agents, a single good, and a single
asset that pays a dividend of dt units of the good in each period t. The
population of agents is normalized to one; so is the supply of the asset. Eachagent solves
max{ct ,xt }∞t=0∞
X

β t u(ct )

(10)

t=0

s.t. x0 = 1,
ct + pt xt+1 = (pt + dt )xt , xt+1 ≥ 0, t = 0, 1, 2, . . . ,
4

(11)
(12)

where ct is consumption, pt is the price of the asset, and xt is shares in
the asset at the beginning of period t. In equilibrium, ct = dt and xt =1. We assume that the utility function u is concave, differentiable, andstrictly increasing. The Euler equation and the transversality condition inequilibrium areu0 (dt )pt = βu0 (dt+1 )(pt+1 + dt+1 ),T0lim β u (dT )pT = 0.

T →∞

(13)
(14)

It is easy to see that the sequence {p∗t } given byp∗t =∞Xi=1

βiu0 (dt+i )
dt+iu0 (dt )(15)satisfies the Euler equation (13). The right-hand side of (15) is called thefundamental value of the asset. Let {bt } be any nonnegative sequence satisfyingu0 (dt )bt = βu0 (dt+1 )bt+1 .
(16)
Then the sequence {p∗t +bt } also satisfies the Euler equation. Hence there areinfinitely many paths satisfying the Euler equation. The extra component bt ,which grows at a gross rate of u0 (dt )/[βu0 (dt+1 )], is interpreted as a bubble.
Notice that the bubble component bt , if positive, violates the transversality condition (14) (with pT = bT ). Therefore, if the transversality condition isnecessary, the bubble component must vanish, so that the price must always
be equal to the fundamental value. This is indeed the case here (Kamihigashi,2001, p. 1007).
In stochastic models, bubbles can be ruled out under standard assumptions, but there are pathological cases in which bubbles are possible (Kamihigashi, 1998; Montrucchio and Privileggi, 2001).

Hyperdeflations and transversality conditions
Transversality conditions are often used to rule out hyperdeflationary paths
in money-in-the-utility-function models of the type studied by Brock (1974)
and Obstfeld and Rogoff (1986). In these models, agents derive utility fromreal money balances in addition to consumption. As in the Lucas asset pricing
model, there are many paths satisfying the Euler equation. A solution to the5Euler equation with a positive bubble is often called a hyperdeflationay path,in which the nominal price keeps declining toward zero. If the nominal pricefalls, the value of real balances rises. Hence in a hyperdeflationary path, thevalue of real balances grows unboundedly. Under reasonable assumptions,such paths are ruled out by an appropriate transversality condition, which
once again rules out overaccumulation of wealth.
However, there are cases in which the transversality condition does not
rule out hyperdeflationary paths (Obstfeld and Rogoff, 1986, p. 356). This isbecause agents, who derive utility from real balances, directly benefit fromaccumulating wealth.

No-Ponzi-game conditions and transversality conditions
In formulating a consumer’s problem, one must include some constraint ondebt, since otherwise the consumer would never pay back his debt, letting
it grow unboundedly. One way to rule out this behavior is to prohibit debt
entirely, i.e., to require wealth to be always nonnegative. A more lenient
way is to require only the present discounted value of wealth at infinity to
be nonnegative. This type of condition is known as a no-Ponzi-game condition (Blanchard and Fischer, 1989, p. 49), but often called a transversality
condition as well. A no-Ponzi-game condition is a constraint that prevents
overaccumulation of debt, while a transversality condition is an optimality
condition that rules out overaccumulation of wealth. They place opposite
restrictions, and should not be confused.

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