Conclusion

In this chapter, we focused on equilibrium outcomes in competitive economies lasting for only two periods, to highlight some of the main themes of the intertemporal approach to macroeconomics.

We first analyzed the properties of general equilibrium in a one-period competitive macroeconomic model with exogenous capital and labor endowments. We then investigated intertemporal general equilibrium in a competitive two-period macroeconomic model with fixed capital and labor endowments. We also investigated extensions of the two-period model to examine intertemporal substitution in labor supply, the role of money, and the role of fiscal policy. In all models, we assumed full and competitive markets.

The two-period competitive model is the simplest possible intertemporal model, and we have used it to investigate some of the salient characteristics of the intertemporal approach when perfect competitive markets exist. As will become apparent in the remainder of this book, many of the properties of the two-period model carry over to competitive market models, in which economies last for more than two periods, such as the infinite-horizon competitive representative household model. However, models that are based on market distortions (such as transactions costs, externalities, uncertainty, increasing returns to scale, asymmetric information, or nominal wage and price rigidities) can be best understood in juxtaposition to competitive intertemporal models, such as the ones analyzed in this chapter. Hence, the models presented in this chapter should be understood as ideal types, against which more realistic models of absence of markets, market distortions, and gradual adjustment of nominal wages and prices are to be compared.

Many of the macroeconomic themes examined through the two-period models of this chapter will be revisited in the context of the intertemporal models without perfect markets used in the remainder of this book.

1. See Rodrik [2015], and in particular, chapter 1, for an informed discussion of the role of models and the role of simplifying assumptions in economics.

2. Appendix A contains a more general treatment of mathematical concepts related to mathematical models, such as variables, functions, and optimization.

3. Note that the household can consume all of its capital endowment, although it may choose to rent out part of it. This is why the budget constraint (2.1) has this form instead of the form c ≤ (1 + r)k + wl.

4. See appendix A for an introduction to utility maximization and the Lagrange method.

5. This production function was first put forward by Cobb and Douglas [1928] to fit empirical equations for production, employment, and the capital stock in US manufacturing. It has since been known as the Cobb-Douglas production function, although this functional form was first explored in the nineteenth century by economists such as von Thunen, Marshall, Clark, Wicksell, and Wicksteed. See Humphrey [1997] for a discussion of the evolution of algebraic production functions before Cobb and Douglas [1928]. More general production functions, such as the CES production function, are introduced in appendix A.

6. The assumption of perfect foresight will be maintained in all intertemporal models in which there is no uncertainty. In models with uncertainty about the future, it will be replaced by the assumption of rational expectations.

7. Time separability of utility means that past consumption does not influence current and future utility. This form of preferences does not restrict the size of intertemporal substitution effects. See Barro and King [1984] for the implications of time separability in more general models.

8. In models with uncertainty, (2.39) is also called the constant relative risk aversion utility function, and θ is interpreted as the coefficient of constant relative risk aversion. See appendix A for a more general discussion of this functional form.

9. The essence of L’Hopital’s rule is that if the first derivative of a function is defined at the limit θ = 1 and is the first derivative of another known function, then the original function tends to that other known function as θ tends to unity. See appendix A.

10. In fact, the assumption of a production function linear in employment is made for simplicity and to maintain constant returns to scale.

11. Intertemporal substitution in labor supply was put forward as an explanation for employment fluctuations in an important paper by Lucas and Rapping [1969], in the context of a competitive model of the labor market. It has since served as an important building block in “new classical” models of aggregate fluctuations.

12. The timing could, of course, be the opposite. Households receive money from firms in exchange for their labor services at the beginning of the period, and use the money to buy goods at the end of the period.

13. There are other ways of motivating the demand for money other than the particular way assumed in this chapter. The two main approaches used in the literature are the cash in advance approach, a version of which is used in this chapter, and the money in the utility function approach, where it is assumed that money yields utility because of its liquidity services. The latter approach is introduced in chapter 7, and we return to the similarities and differences between the two approaches in chapter 12, where we examine more general intertemporal models of the money market.

14. To quote from Fisher [1896, p. 58]: “When prices are rising or falling, money is depreciating or appreciating relative to commodities. Our theory would therefore require high or low interest according as prices are rising or falling, provided we assume that the rate of interest in the commodity standard should not vary.” The rate of interest in the commodity standard is the real interest rate, and rising or falling prices are expected inflation. The Fisher equation was further elaborated in Fisher [1930], where it was made even clearer that Fisher was referring to expected inflation.

15. Ricardian equivalence was brought to the forefront of macroeconomics by Barro [1974], who addressed the question of whether government bonds are considered as wealth by households who internalize the welfare of future generations. We return to this issue in chapter 6.