Dynamic Models of Portfolio Behavior: Comment on Purvis
Douglas Purvis' discussion of an integrated approach to consumption and portfolio decisions is an attractive extension of the framework advocated by William Brainard and James Tobin. The pitfalls is concerned with the portfolio allocation of a level of wealth which is predetermined by beginning of period asset holdings and current period saving and capital gains. One of the innovative features of this is the inclusion of all asset yields and lagged asset holdings as explanatory variables in the asset demand equations. Purvis supplements the BrainardTobin asset demands with a consumptionsaving relationship that includes a similar list of explanatory variables and reinterprets this system as a of integrated rather than sequential decision making. Despite his observation that, when combined with a consumption-savings relationship such as (2), the Brainard-Tobin will in principle give rise to exactly the same shortand long-run behavior as the integrated model (p. 407), most of Purvis' discussion is concerned with alleged dissimilarities between the two approaches. This is apparently due to his implicit coupling of a simple consumption function and sequential decision making. In particular most of his comments on the BrainardTobin approach are actually concerned with whether or not lagged asset holdings should be included in a consumption function. This is rather unfair to Brainard and Tobin since there is no consumption function in the pitfalls model, and the two issues are really conceptually distinct. An integrated approach does not preclude, and a sequential approach does not require, a simple consumption function. The spirit of Brainard and Tobin's work is in fact that the inherited composition of wealth is very important to consumption, but consumption decisions precede asset demand decisions. The substance of their sequential approach is not that the composition of wealth is unimportant to consumption but rather that there are some variables which influence consumption and yet do not separately affect asset demands; only the net amount of saving motivated by these influences is important. In this paper I have consequently tried to separate these two issues: the use of an integrated or sequential framework and the imposition of parametric assumptions. One of the reasons for the merging of these two issues in Purvis' discussion is that he uses a deterministic scenario which makes the distinction between integrated and sequential decisions unimportant. In Purvis' integrated model, consumption and asset demands are constrained by lagged asset holdings plus income. In the relevant sequential interpretation of this model, consumption is first determined, setting the amount of saving and the level of end of period wealth. Asset demands are then decided upon, subject to the budget constraint that they sum to the predetermined end of period wealth. Thus the integrated asset demands include income as an explanatory variable while the sequential asset demands instead include end of period wealth. In a deterministic world there are no substantive differences between these approaches as long as income and wealth are related through a consumption-saving equation. This equivalence breaks down if the marginal propensity to save out of income is zero (since wealth is then no longer related to income) or if there is an unobserved disturbance term in the consumption *Yale University. Note that equations numbered (1) through (11) are in Purvis' paper. My equations are numbered in the same sequence.
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