In this chapter, I present four models of stochastic growth emphasizing different aspects of the interaction between growth and uncertainty.
The first is the baseline neoclassical growth model (with complete markets) augmented with stochastic productivity shocks, first studied by Brock and Mirman (1972). This model is not only an important generalization of the baseline neoclassical growth of Chapter 8 but also provides the starting point of the influential Real Business Cycle models, which have been used widely to study a range of short- and medium-run macroeconomics questions.
I present this model and some of its implications in the next three sections. The baseline neoclassical growth model incorporates complete markets in the sense that households and firms can trade using any Arrow-Debreu commodity. In the presence of uncertainty, this implies that a full set of contingent claims is traded competitively. For example, an individual can buy an asset that will pay one unit of the final good after a pre-specified history. The presence of complete markets—or the set of contingent claims—implies that individuals can fully insure themselves against idiosyncratic risks. The source of interesting uncertainty in these models is aggregate shocks. For this reason, the standard neoclassical growth model under uncertainty does not even introduce idiosyncratic shocks (had they been present, they could have been easily diversified away). This shows the importance of contingent claims in the basic neoclassical model under uncertainty. Moreover, trading in contingent claims is not only sufficient, but it is essentially also necessary for the representative household assumption to hold in environments with uncertainty. This is illustrated in Section 17.4, which considers a model where households cannot use contingent claims and can only trade in riskless bonds. This model, which builds on Bewley’s seminal work in the 1970s and the 1980s, explicitly prevents risk-sharing across households and thus features “incomplete markets”—in particular, one of the most relevant type of market incompleteness for macroeconomic questions, which prevents the sharing or diversification of idiosyncratic risk. Households face a stochastic stream of labor income and can only achieve consumption smoothing via “self-insurance,” that is, by borrowing and lending at a market interest rate. Like the overlapping generations model of Chapter 9, the Bewley model does not admit a representative consumer. The Bewley model is not only important in illustrating the role of contingent claims in models under uncertainty, but also because it is a tractable model for the study of a range of macroeconomic questions related torisk, income fluctuations and policy. Consequently, over the past decade or so, it has become a workhorse model for macroeconomic analysis.
The last two sections, Sections 17.5 and 17.6 turn to stochastic overlapping generations models. The first presents a simple extension of the canonical overlapping generations model that includes stochastic elements.
Section 17.6 shows how stochastic growth models can be useful in understanding the process of takeoff from low growth and to sustained growth, which we discussed in Chapter 1. A notable feature of the long-run experience of many societies is that the early stages of economic development were characterized by slow or no growth in income per capita and by frequent economic crises. The process of takeoff not only led to faster growth but also to a more steady (less variable) growth process. An investigation of these issues requires a model of stochastic growth. Section 17.6 presents a model that provides a unified framework for the analysis of the variability of economic performance and take off. The key feature is the tradeoff between investment in risky activities and safer activities with lower returns. At the early stages of development, societies do not have enough resources to invest in sufficiently many activities to achieve diversification and are thus forced to bear considerable risk. As a way of reducing this risk, they also invest in low-return safe activities, such as a storage or safe technology and low-yield agricultural products.
The result is an equilibrium process that features a lengthy period of slow or no growth associated with high levels of variability in economic performance. The growth is truly stochastic and an economy can escape this stage of development and takeoff into sustained growth only when its risky investments are successful for a number of periods. When this happens, the economy achieves better diversification and also better risk management through more developed financial markets. Better diversification reduces risk and also enables the economy to channel its investments in higher return activities, increasing its productivity and growth rate. Thus this simple model of stochastic growth presents a stylistic account of the process of takeoff from low and variable growth and to sustained and steady growth. The model I will use to illustrate these ideas features both a simple form of stochastic growth and also endogenously incomplete markets. I will therefore use this model to show how some simple ideas from Markov processes can be used to characterize the stochastic equilibrium path of a dynamic economy and also to highlight potential inefficiencies resulting from models with endogenous incomplete markets. Finally, this model will give us a first glimpse of the relationship between financial development and economic growth, a topic to which we will return in Chapter 20.17.1.