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Article Excerpt
Abstract We adapt the classic one-sector optimal growth model to include an endogenous rate of time preference along the lines of Becker and Mulligan (1997). The resulting model is both time-consistent and analytically tractable. Capital sequences are shown to be globally monotone and stable under very general circumstances using lattice programming techniques and value orders. We analyze a series of examples that exhibit a variety of behaviors, including closed-form solutions, unique steady-states, multiple steady-states, and conditionally sustained growth. The endogenous rate of discount preserves monotonicity and stability while allowing for the possibility of non-global convergence.
Keywords Endogenous preferences * Time preference * Discounting * Optimal Growth * Monotonicity * Lattice programming
JEL Classification Numbers C61 * D99 * O41
1 Introduction
Patience has long been recognized as an important virtue. It was Saint Augustine who said that patience is the companion of wisdom and Emerson who said, 'Adopt the pace of nature, her secret is patience.' Economists have also long recognized the importance of patience in regards to wealth accumulation and economic growth. In fact, it was conjectured as far back as Ramsey (1928), and later shown by Becker (1980), that the most patient individual in a competitive economy would eventually own the entire capital stock. One can argue that patience and technology are the two cornerstones of economic growth.
There has been renewed interest in recent years in the way we treat consumer impatience or time preference. Constant rates of geometric discounting have long been the norm, but recent work, such as Laibson (1997), Barro (1999), and Harris and Laibson (2000), have brought to the forefront the issue of non-geometric discounting. A long-noted consequence of non-geometric discounting is that of time inconsistency. The potential impact of such structures on even relatively simple optimal growth frameworks can be profound. Krusell and Smith (2003) recently showed that quasi-geometric discounting in a basic one-sector optimal growth model creates an entire continuum of Markov equilibrium savings rules. We seek to analyze the same basic optimal growth framework of Krussel and Smith, but with a differing assumption regarding time preference. In this paper, we will adapt the classic optimal growth model to include an endogenous rate of time preference along the lines of Becker and Mulligan (1997).
A very redeeming aspect of the Becker and Mulligan framework is that it allows variation in the rate of time preference while at the same time maintaining time consistency. Thus we hope to provide a model of economic growth with a more flexible framework in regards to the discounting of time, while at the same time maintaining much of the analytic simplicity of the classic optimal growth model. We will show that the model exhibits globally monotone and stable optimal capital sequences while also yielding a richer set of potential dynamics than the classic model with constant geometric discounting. We provide examples of well-behaved, closed-form solutions that exhibit global convergence to a unique stationary point, examples of multiple steady-states with mere local convergence, and also a case of conditionally sustained growth.
The basic layout of the paper is relatively straightforward. We formally define the model and provide interpretations in the next section. We shall also briefly mention the recursive properties of the model and a few needed properties of the value function and optimal policy correspondence. In Section 3, we shall analyze the monotonicity and stability of optimal capital sequences on a very general level with lattice programming methods and a new value order from Mirman and Ruble (2004). Section 4 examines the issue of global convergence through a series of examples. We start with a case of closed-form solutions that yield a unique steady-state with global convergence. Next, we exhibit a case where there exist multiple steady-states and the point of eventual convergence is conditional on the initial state. Lastly, we shall examine the issue of sustained growth and its potential dependence on the initial state.
2 The general model
The classic optimal growth model is characterized by an infinitely-lived agent that chooses consumption [c.sub.t] each period in order to maximize a discounted sum of instantaneous utilities. The real valued function u gives the instantaneous utility from consumption. The amount of current resources not consumed by the agent is saved as capital, [k.sub.t], to be used in production the following period. The total amount of resources that can be produced from a given level of capital is given by the production function, f(k). The model is formally specified as follows:
[max.[([c.sub.t],[k.sub.t])[.sub.t=1.sup.[infinity]]]] [[infinity].summation over (t=1)] [[beta].sup.t-1]u([c.sub.t])
subject to
[c.sub.t] + [k.sub.t] [less than or equal to] f([k.sub.t-1]), [c.sub.t] [greater than or equal to] 0, [k.sub.t] [greater than or equal to] 0,
for t = 1, 2,..., [infinity] and [k.sub.0] given.
The level of discount on future utility, [beta], is generally taken to be a fixed real number between zero and one. The only feature of this classic model that we are going to alter is the nature of discounting. Rather than assuming that [beta] is an exogenous parameter, we will assume that the agent can engage in some activity or sacrifice in order to alter the discount factor each period. At the end of this section we will provide two different interpretations of these patience altering actions. We quantify the amount of this activity chosen in period i by the new control variable [s.sub.i]. The discount on the future in period i will be a function of [s.sub.i] and we will use [beta] to denote this function. We also assume that [s.sub.i] will cost the planner an amount [pi] [s.sub.i] in terms of current resources. The parameter [pi] merely acts as a price that converts [s.sub.i] into the units of consumption and capital. (1) The formal specification of the model is given below:
[max.[([c.sub.t],[s.sub.t],[k.sub.t])[.sub.t=1.sup.[infinity]]]] [[infinity].summation over (t=1)] ([t-1.[product].[i=1]] [beta]([s.sub.i]))u([c.sub.t])
subject to [c.sub.t] + [pi] [s.sub.t] + [k.sub.t] [less than or equal to] f([k.sub.t-1]),
[c.sub.t] [greater than or equal to] 0, [s.sub.t] [greater than or equal to] 0, [k.sub.t] [greater than or equal to] 0,
for t = 1, 2,..., [infinity] and [k.sub.0] given. (1)
Before we proceed to discuss exactly what [s.sub.i] may represent, we will make a few basic assumptions regarding the discount function, [beta], the utility function, u, and the production function, f.
Assumption 1 u : [R.sub.+] [right arrow] [R.sub.+] and [beta] : [R.sub.+] [right arrow] [R.sub.++] are continuous, concave, and strictly increasing. (2)
Assumption 2 f : [R.sub.+] [right arrow] [R.sub.+] is continuous, strictly increasing, and there exists a [k.sub.max] such that [beta]([k.sub.max]/[pi]) [k.sub.max].
We make no assumption regarding the curvature of the production function. Very much like the classical optimal growth model, concavity of the production function is not necessary in order to establish the monotonicity and stability of the optimal capital sequence. Our assumption that [beta] is strictly increasing implies that an increase in [s.sub.t] results in greater current appreciation of future utility. The assumed concavity of the discount function promises a diminishing return to investment in [s.sub.t] and is also important in establishing monotone properties for our model.
The maximum sustainable capital stock together with the limits on discounting will yield a finite value function. One should also note that the felicity function, u, is nonnegative. This assumption is critical since a negative value function would make investment in a higher discount factor a 'bad' rather than a 'good'. We now turn our attention to interpreting the model and the new choice variable [s.sub.t].
2.1 Interpretations
There are two main ways of interpreting our model and the endogenous discount factor. The difference between the two interpretations depends on whether or not you view the optimal growth model as a single individual with an infinite lifetime or as a dynastic family. In the case of a dynasty, the optimal growth model is obtained through the optimizing behavior of an infinite sequence of generations with adult individuals exhibiting altruism towards their immediate offspring. Each generation lives for two periods, a childhood and an adulthood. The generations of individuals are indexed by the time period of adulthood, and childhood utility is suppressed. The adult generation during time period t will attempt to maximize the following objective function:
u([c.sub.t]) + [beta]([s.sub.t])[U.sub.child]([k.sub.t]). (2)
When the optimal growth model is viewed as a dynasty, then the discount factor is really the degree of intergenerational altruism. [beta]([s.sub.t]) will be the degree to which generation t cares for generation t + 1. Hence, we are endogenously modeling the strength of the relationship between the parent and the child. The variable [s.sub.t] would therefore represent actions that the parent could take in order to strengthen the relationship with his child. The degree to which the parent involves and dedicates himself to the rearing of the child would certainly be a determinant in the strength of the relationship. Thus, spending more time with the child as well as the proverbial "taking the kids to the ballpark" could be represented and quantified by the new variable 's'. (3) An investment in the parent-child relationship would certainly cost the parent current resources either in terms of actual expenditures or forgone production. Hence the factor [pi][s.sub.t] is included in the budget constraint. (4)
If the optimal growth model is interpreted as a single individual with an infinite lifetime then the interpretation of our model is very similar to Becker and Mulligan (1997). The discount factor represents the degree to which the individual appreciates future utility when making current decisions. Each period the individual maximizes the following objective function where [U.sub.future] denotes the utility of the remainder of the individual's life.
u([c.sub.t]) + [beta]([s.sub.t])[U.sub.future]([k.sub.t]).
In this case, the variable [s.sub.t] would represent actions that the individual could take to increase his appreciation of the future. Becker and Mulligan argue that such things as education,...
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