Transversality Condition

UNDERSTANDING TRANSVERSAILITY CONDITIONS IN ECONOMICS THIBAUT LAMADON

Abstract. The goal of this document is to clarify the notion of “transversality condition”. On the one hand the transervality condition is sometimes happily ignored when not directly serving the purpose of the presenter, and the other hand it can be dropped in the middle of a derivition to rule out unwanted solution. We present here the different forms of transversality condition as well as the contexts in which they must be satisfied. We’ll also provide some intuition to why they are what they are. Most of the content of those note are based the book from Stokey Lucas and the excelent notes of Nicola Pavoni.

The content of this document is unfortunately not extremely rigourous since the goal is to provide intuitions. It is highly recommended to read Stokey Lucas, Other1, Other2 for a detailed coverage of teh topic. On the other hand you probably only have 30 minutes to spend on this and that’s why I made this!

1. Introduction Transversality conditions appear in a lot of different places in economics (and probably in other fields). In general it is refered to in three different cases: • when solving a classical sequence problem using Euler first order conditions • when solving a classical sequence problem using the Bellman functional equation • when solving a classical sequence problem with one-period bond and assuming no ponzy game We will look at each case and describe what is meant by "transversality condition" in each context. Finally we will compare those three cases and show why they are conceptually very different. All the notations here and theorems are consistent with Stokey-Lucas "Recursive Methods in Economic Dynamics". Please refer to their chapter 4 for detailed derivations.

2. Euler - First Order Conditions let’s define the sequential equation to the problem: P∞ v ∗ = sup t=0 β t F (xt , xt+1 ) st xt+1 ∈ Γ(xt ) (2.1) (SE) x0 given This is the less restricitve representation for the problem of the agent maximization. Later section will explain how to go from this representation to a functional formulation. contact: thibaut.lamadon@gmail.com.

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UNDERSTANDING TRANSVERSAILITY CONDITIONS IN ECONOMICS

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2.1. Sufficiency Theorem. assume l X ∈ R+ , convex and Γ is non empty, compact-valued and continous F is bounded and continous 0<β<1 F (·, y) is striclty increasing in each of the first l arguments F is concave and continously differentialbe  0 = Fy (x∗t , x∗t+1 ) + βFx (x∗t+1 , x∗t+2 )  lim β t Fx (x∗t , x∗t+1 )x∗t = 0 ⇒ {x∗t }∞ t=0 is optimal for (SE)  ∗ ∗ xt+1 ∈ intΓ(xt )

• • • • •

2.2. Transversality condition for the Euler Equation. Our transversality condition here is lim β t Fx (x∗t , x∗t+1 )x∗t = 0 and needs to hold only for the optimal path

3. Bellman - Principle Of Optimality 3.1. Notations. Let’s first define the functional equation (FE) form to the problem: (F E)

v(x) = sup [F (x, y) + βv(y)] y∈Γ(x)

It is important to not that this rewritting is possible because of the structure of the constraint in the SE. In the class of problem considered here, the history can be represented using a set that does not change with time. In this class of problem we can use state variables. If we had written F (x0 , ..., xt , xt+1 ) instead of F (xt , xt+1 ) without further restrictions on F , the problem could not even be represented as a Bellman equation.

3.2. Sufficiency condtion for the Bellman equation solution. Assume • Γ(x) is non empty for all x P • for all x0 ∈ X and x ∈ Π(x0 ), lim β t F (xt , xt+1 ) exists (can be infinity) Given those 2 assumptions, we have the existence of v ∗ and v ∗ is defined by (SE)

v solves (FE) and lim β n v(xn ) = 0 ∀(x0 , x1 , ..) ∈ Π(x0 ) ∀x0 ∈ X

v ∗ solves (FE)

⇒    

⇒

v = v∗

  

3.3. Transversality condition for the Bellman Equation solution. Our transversality condition here is lim β n v(xn ) = 0 and must hold for all feasible path and not only for the optimal one!

UNDERSTANDING TRANSVERSAILITY CONDITIONS IN ECONOMICS

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3.4. The intuition behind the theorem. Suppose that v is a solution to F E, then for any feasible sequence {xt } v(x0 ) ≥

F (x0 , x1 ) + βv(x1 )

≥

F (x0 , x1 ) + βF (x1 , x2 ) + β 2 v(x2 )

≥

un (x) + β n+1 v(xn+1 )

From this simple algebra, we see that we need the last term to go to zero. We also see that this term is constructed using any feasible sequence.

3.5. Counter exemple to the transversality condtion.

4. One-period Bond - No Ponzy Game 5. The Optimal Growth Problem and the Confusion E-mail address: thibaut.lamadon@gmail.com