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The Life-Cycle Permanent-Income Model: Extensions of the Ss Model

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The Life-Cycle Permanent-Income Model: Extensions of the Ss Model The Life-Cycle Permanent-Income Model: Extensions of the Ss Model Lucas Zalduendo MMSS Senior Thesis Adviser Robert J. Gordon ______________________________ Many people were instrumental in the development of this thesis. To my friends and family I owe thanks for the love and support they have shown me at every step of this process. To Bruno Strulovici and Martin Eichenbaum I owe thanks for helpful comments that guided me along my journey. To Mark Witte and Bob Gordon I owe thanks for all of the above plus special thanks for their role in my intellectual development as budding macroeconomist. And to Abigail Jacobs, thank you for 4 wonderful years and counting. Spending on durable goods has long been identified as a key feature of business cycle dynamics. Comprising a significant percentage of GDP and being highly correlated with business cycle fluctuations, its not difficult to see why a large body of literature has risen up to the task of better understanding durable goods consumption. Yet despite large strides elucidating some aspects, many problems remain. Chief among these problems is the difficulty in replicating the empirical dynamics of durable goods spending in theoretical macroeconomic models. Many frameworks have been used to address this modeling problem, but two of the most prominent bodies of literature build off the so called stock adjustment models and the Ss models. After highlighting some of the pros and cons of these two frameworks, we opt for the greater flexibility granted by the Ss model framework. From this flexible foundation, we present and examine two extensions. The first extension will allow us to model the consumption of multiple goods of varying durability and presents a strong endogenous force propagating the co-movement of nondurables and durables. This endogenous force could go a long way in fixing the so called co-movement problem, prevalent in the modeling literature. The second extension will allow us to model the consumption of durable goods under risk. Some simple empirics will reveal that this extension correctly predicts, contra to the random walk theory of consumption, that factors such as current unemployment and uncertainty play a significant role in future durable goods consumption. Section 1 introduces a brief overview of consumption optimization theory starting with the LCH/PIH. Section 2 focuses on the stock adjustment literature attempting to incorporate a significant durable goods component. Section 3 looks at the Ss literature addressing the same topic by re-deriving a simple, more flexible model developed by Blinder and Bar-Ilan. Section 4 examines some problems with the standard Ss models and Stock Adjustment models and will highlight the greater flexibility that the Ss model allows in addressing modeling problems. Section 5 expands the model to account for multiple durable goods. Section 6 expands the model to account for risk and tests the implications empirically. Section 7 concludes. 2 1 Introduction The permanent income-life cycle hypothesis (PIH) developed by Franco Modigliani and Richard Brumberg (Modigliani & Brumberg, 1954) and Milton Friedman (Friedman, 1957) postulate how a rational economic agent should behave when making consumption decisions. The primary assumption underpinning the models is that consumers take into account their life-time income and decide how much to consume each period based on this life-time income and a desire to keep their utility flows “smooth.” The PIH has been frequently put to empirical test, with evidence both for and against put forward. Despite the wide appeal of its logic, it still has a few apparent shortcomings when tested empirically. One critical limitation of this theory is that it applies only to consumption in a utility sense whereas the macroeconomic data available is typically of expenditures. The distinction is important because it implies that consumers can have “smooth” utility paths over time, but “unsmooth” expenditure paths. For example, take an economy that experiences a large negative wealth shock. As an extreme example, assume no one buys a new car during the month following the large wealth shock. This reduction in expenditure is contra to the PIH, but the consumption path is not. This is because people still “consume” or “use” their old vehicles. In this way, expenditures can be expected to be more volatile than true “consumption” or “utility.” When working with non-durable goods, this distinction is of no great concern, since goods purchased in one period are typically consumed in that same period. For this reason, expenditures on non-durables tend to be a very good proxy for consumption of non-durables. Thus, empirical evaluation of the PIH with regards to non-durable goods consumption is fairly straight forward and often favorable to the PIH (Hall, 1978). Unfortunately, the evaluation of these theories with regards to durable goods consumption is less straight forward and, typically, unfavorable to the basic PIH (Mankiw, 1982). This problem arises in large part from the fact that even if a person doesn’t buy a durable good in a given period, they still use (i.e.- consume) goods bought in previous periods. Thus, expenditure on durable goods tends to be a poor proxy for consumption of durable goods. In empirical evaluation, this can be problematic because the macroeconomic data we observe is expenditure on goods, not consumption of goods. These modeling problems are magnified by their importance in understanding business cycles due to the strong, procyclical nature of durable goods consumption (Black & Cusbert, 2010). The prominent shortcoming of many models of durable goods consumption, coupled with its prominence in business cycles, has resulted in a rich literature attempting to supplement PIH models with improved dynamics for durable goods expenditures. The next section reviews one of the initial models that attempted to reconcile the PIH with a durable goods component: the stock adjustment model. 2 Literature Review: Stock Adjustment Model One standard model of consumption is the stock adjustment model. These models arose long initially in theories of private firm inventories. The most famous and comprehensive review and development of these methods is the famous book Planning Production, Inventories, and Work Force, by 3 Holt, Modigliani, Muth, and Simon (1960). The parrallels between firm investments and consumer purchases of durable goods were quickly realized and the stock adjustment models were soon used to model consumption. When dealing with durable goods, these models assume that consumers own a stock of durable good that depreciates, each period, by some constant factor, δ. Each period, the consumer replaces the depreciated stock by buying new stock until reaching the desired level of durable good stock. The transition equation for the stock of goods looks something like: Where K is the realized stock, K* is the desired stock; the subscripts, t and t+1, denote the time period. Another assumption that is often made is that consumers only close the gap between their current stock and desired stock by a constant fraction, γ. This γ makes the model dynamics better mimic real world data. The simplicity of these models, both in intuitive appeal and implementation, made them fairly popular; however, despite the wide appeal, there remain a number of questionable properties. First, it is often assumed that the speed of adjustment, γ, is constant. The motivation for including the γ term, is to create “habit formation.” In practice, this means that after a large shift in permanent income, the rate of change in consumption is large and then declines as we return to the steady state. Intuitively, this is what allows the nice “hump-shaped response of consumption;” however, though it is a justifiable term in order to create desirable consumption dynamics that mimic real world data, it is admittedly “arbitrary” (Christiano, Eichenbaum, & Evans, 2005). Ideally, we would have a stronger justification for the dynamics of the speed of adjustment. Second, and more problematic, is the counter-intuitive nature of the model with regards to the dynamics for individual consumers. Holt, Modigliani, Muth and Simon (1960) show that the SA model can be justified rigorously by quadratic costs of adjustment; however, as Bliner and Bar-Ilan (1988) point out, this implies that consumers partially adjust their stocks over several periods. This is problematic because there is no apparent reason, either theoretical or empirical, why people should adjust holdings in several small steps rather than all at once. Some attempts to rationalize the assumption of quadratic adjustment costs can fall flat upon closer inspection. For example, Bernanke (1984) points out that “it takes time to shop for and acquire a new car”. But this, as blinder and Bar-Ilan point out, implies the existence of transactions costs, not adjustment costs. Third, a related problem is, at least on a micro level, the stock adjustment model of durable goods consumption seems infeasible because, more often than not, consumers cannot buy fractions of goods. For, example when a TV’s image brightness begins to depreciate or the neighbors the Joneses get a new 60 inch TV, a consumer cannot go and buy 1/10th of a new TV to restore the brightness or stretch out the now relatively lowly 52 inch TV.1 Yet, in the stock adjustment model, this is the only kind 1 One could point out that some depreciated goods could be repaired. If a TV controller breaks, one could easily go out and buy a new controller. This is a fair argument, but one can retort that many repairs are impractically costly (restoring brightness to a TV) or cost more than the good depreciated (e.g.- replacing one plate from a set of 4 o f purchase; agents wishing to raise their stock of durable goods can go out and buy whatever fraction of good they wish.
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