Transversality Conditions for Higher Order Infinite Horizon Discrete Time Optimization Problems

where N ∈ N, U is a real-valued Nth-order continuously differentiable function, and c ≡ (c 1 , c 2 , • • • , c n ) is Nth-order continuously differentiable. 1 Notice that the objective functional of (1) can be infinite. [7] considers the continuous time first order differential problems: v (x (t) , ẋ (t) , t). It generalizes the results of [5,6,10,12]. So far, the most general form of the transversality conditions for continuous time version of problem (1) is presented in [11], which extends the first order case considered in [7](Theorem 3.2) to higher order cases. [7] was later extended to the discrete time stochastic case by [8]. In this paper, we aim to extend these results to deterministic higher order difference problems, using the “squeezing” argument.

The application of higher order difference problems can be widely found in economics. In particular, they appear in the discussion concerning the overlapping generations models. A satisfactory examination of the individuals’ marriage and fertility decisions would necessitate the division of the representative agent’s lifetime to multiple periods, instead of only two periods, young and old. However, as argued in [4], the properties of a model with two-period-lived agents cannot be readily extended to n-period-lived agents. To consider the n-periodlived agents case, transversality conditions for higher order difference problems would be imperative.

We first use the “squeezing” argument to derive both Euler’s condition and the transversality condition for higher order difference problems, showing the argument needs two imperative assumptions. These two assumptions constitute the discrete time version of Assumption 1 and 2 in [11]. We then provide a counterexample, in which the transversality condition is not satisfied without the two assumptions. Because Assumption 1 and 2 are satisfied when a discounting factor is incorporated into the model, our transversality conditions also generalize the results obtained in the presence of discounting. For approaches on

how to explicitly construct the optimal solutions to the undiscounted infinite horizon optimization problems, see [2,3].

Suppose that the optimal path to (1) exists and is given by c * (t), optimal in the sense of an overtaking criterion to be defined below. We perturb it with Nth-order continuously differentiable curves q (t), ( 2)

We define

In this paper, [1]’s notion of weak maximality is used as our optimality criterion. We assume that there exists an optimal path that satisfy the weak maximality criterion, which is defined as: an attainable path (c * (t)) is optimal if no other attainable path overtakes it2 : lim

Let lim

). We assume Assumption 1. Assume Ω converges uniformly for ε when T → ∞ .

Assume Assumption 1, we can then restate (5) as

We also assume Assumption 2. We assume for any T > 0, inf

converges uniformly for ε.

As in [11], a precise interpretation of Assumption 2 can be given as follows: Let

Then there exists a sequence A (T ′ n , ε) for each ε > 0, so that lim

, uniformly for ε, that is, the sequence is uniformly convergence for ε.

Assumptions 1 and 2 extend Assumption 3.1 in [7]. When Assumptions 1 and 2 are satisfied, then lim ε→ + 0 and inf T T ′ can be interchanged, and equality (6) can then restated as

Because T ′ is finite uniformly for ε, if

exists, ( 7) is then rewritten as

From the differentiability of U, we have lim

Hence,

We derive

Hence, Euler’s condition is

which extends the standard Euler’s condition, and the transversality condition is given by lim

Note that when ε → -0, the argument is the same: lim

Next, we consider the linkage between our result and that in [7]. We fix 0 < ᾱ < 1 and α : R

Because ᾱ > 0, we then have lim

which is an extension of [7]’s transversality condition.

We proceed to show that Assumption 1 and 2 are imperative in the sense that (11) becomes invalid if one of them is violated. We consider the following simple counterexample: (14) U (c(t), c(t + 1), c(t + 2), t) = (c(t)α) 2 + βc(t + 1) + γc(t + 2), where α > 0, β > 0, γ > 0, and the initial values c(0) = c 0 , c(1) = c 1 are given. From (10), we see that Euler’s condition is given by

which implies

Choosing a p so that p(0) = 0 and p(t) > 0, there exists T 0 > 0, p(t) is a constant p ∞ > 0 when t ≥ T 0 .

From (14), we see that

Hence, we have arrived at a contradiction to (11).

Next, we show that Assumption 1 is violated, which causes this contradiction. We consider U (c * (t) + εp(t), c * (t + 1) + εp(t + 1), c * (t + 2) + εp(t + 2))

Hence, inf

Ω is the limit of (18) when T → ∞, ε → 0. However, because lim

This paper gives the two assumptions that would be imperative when examining infinite horizon discrete time optimization problems in which the objective functions are unbounded. Our results generalizes the results of [5,6,7,10,12](N = 1) to higher order difference problems. Specifically, when N

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