RDP 9209: Financial Liberalisation and Consumption Behaviour 2. The Hall Model and Modifications

Over the last decade or so most research on aggregate consumption has taken the Hall (1978) model as a starting point of analysis. Hall argued that the PIH implies that consumption behaviour should obey the first-order conditions for life-time utility maximisation of a representative individual. He begins with the conventional consumer model of life-cycle consumption under uncertainty:

where:

Intuitively, an individual consumer, seeking to maximise his utility, is faced with the decision of whether to consume today or at some time in the future. This decision will depend upon his rate of time preference, the opportunity cost of interest foregone on income consumed today, and his expectation of the utility he would derive from consuming this income in the future. This may be written algebraically as:

If one is prepared to maintain the following somewhat strong assumptions regarding individual consumers:

1. they have identical, time-separable preferences with a quadratic representation for instantaneous utility:

   
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2. they cannot die in debt;
3. they have access to perfect capital markets in which the constant real rate of interest is equal to the subjective rate of time discount; and
4. they form expectations of future income rationally,

the first-order condition for an optimum can be shown to be:

where:

Under these conditions, consumption follows a random walk. The present level of consumption is the optimal forecast of its future level or, alternatively, changes in consumption are unforecastable.

Alternatively, if the utility function is a power function of the form:

the behaviour of consumers can be approximated by:

where a drift term θ is included to represent the long-run rise in aggregate consumption.

If disposable income is assumed to be generated by a process of the form:

where:

then the coefficient on any variable belonging to X_t−1 should not be significantly different from zero in a regression of consumption on a constant, its own first lag and X_t−1 (equation (5)), or in a regression of the rate of growth of consumption against a constant term and X_t−1 (equation (7)).

Using variants of the Hall model, several researchers have found that current aggregate consumption is significantly more sensitive to changes in current disposable income than the PIH predicts. This excess sensitivity is frequently rationalised as arising from the presence of liquidity constraints. In terms of the life-cycle model, individuals make labour supply and consumption decisions over a known lifetime. Income will typically fall short of desired consumption in youth, exceed it in middle age, and again fall short of it in retirement. With perfect capital markets, individuals should be able to smooth consumption relative to income by borrowing when they are young and lending in middle age. In the presence of liquidity constraints, however, consumption cannot be fully smoothed because, for example, households cannot borrow when they are young against their future labour income.

Clearly, a breakdown of one or other of the abovementioned maintained hypotheses underlying the derivation of the random walk model, such as perfect capital markets; rational expectations of future labour income; additive time-separable preferences; separability between consumption, leisure and other goods (Mankiw, Rotemberg and Summers (1985)); or constant real rates of interest and discount rates (Mankiw (1981)), could cause current income to be sensitive to current consumption. Furthermore, the pattern of consumption for non-durables will be affected if durables and non-durables are non-separable in consumption, and if durables are subject to gradual adjustment to optimal levels. Bernanke (1985) has suggested that the illusion of excess sensitivity could consequently be created by the failure to account properly for durables expenditures. Another factor which could potentially explain the observed “excess” sensitivity has been emphasised by Zeldes (1989a and b). If there is uncertainty about future labour income, then consumers will self-insure by engaging in precautionary savings. An increase in such uncertainty will increase savings and reduce consumption relative to income. In other words, relative to a world of certainty, individuals' current consumption is “too” low and expected consumption growth “too” high, again creating an impression of “excess” sensitivity.