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Multiple BGPs in a growth model with habit persistence: a comment.

Article Excerpt
RECENTLY IN THIS JOURNAL, Chen (2007) analyzes a model of an economy without distortions in which rational forward-looking g households care about how their current consumption compares with their own past consumption. The habit formation mechanism considered by Chen is a generalization of the mechanism introduced by Ryder and Heal (1973) and studied by Carroll, Overland, and Weil (1997, 2000) in a growing environment, and a special case of the mechanism considered by Iannaccone (1986) in a stationary environment. Chen claims to prove the existence of two saddle-point stable balanced growth paths. In this comment, I show that his claim is incorrect: there is a unique saddle-point stable balanced growth path and no global indeterminacy.

I summarize Chen's model, correct his propositions, and offer a simple proof that the balanced growth path is unique. In the interest of concision, I have relegated details to a longer version of the article, available upon request (Toche 2008).

1. CHEN'S MODEL

1.1 The Optimization Problem

I use Chen's notation throughout. I also find it convenient to introduce the following new notation. Let r = A - [[delta].sub.k] > denote the rate of interest. Let g(c, S) stand for the input in the habit formation mechanism, and let U(c, S) denote the utility function, both defined explicitly below.

The problem is to select a consumption path to maximize an intertemporal utility function defined in terms of consumption and a reference habit stock,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

subject to a budget constraint:

[[??].sub.t] = r[k.sub.t] - [c.sub.t]. (2a)

and a habit formation mechanism:

[[??].sub.t] = g([c.sub.t], [S.sub.t]) - [[delta].sub.S][S.sub.t]. (2b)

The optimal control program consists in maximizing the objective function (1) subject to the dynamic constraints (2a) and (2b), where [c.sub.t] is a control variable, and [S.sub.t] and [k.sub.t] are two state variables--[c.sub.t] is the flow of consumption, [k.sub.t] is the capital stock, with [k.sub.0] > given, [S.sub.t] is the reference index to which consumption is compared, with [S.sub.0] > given, [rho] is the discount rate, and [[delta].sub.S] is the depreciation rate of the habit stock [S.sub.t].

1.2 The Assumptions

ASSUMPTION (HARMFUL HABITS). The utility function is U(c, S)=u(c/[S.sup.[gamma]])=[(1 - [sigma]).sup.-.1] [(c/[S.sup.[gamma]]).sup.1-[sigma]], where [sigma] > 1...

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