References and Literature

The model analyzed in this chapter was first developed in Solow (1956) and Swan (1956). Solow (1970) gives a nice and accessible treatment, with historical references. Barro and Sala-i-Martin’s (2004, Chapter 1) textbook presents a more up-to-date treatment of the basic Solow model at the graduate level, while Jones (1998, Chapter 2) presents an excellent undergraduate treatment.

The treatment in the chapter made frequent references to basic consumer and general equilibrium theory. These are prerequisites for an adequate understanding of the theory of economic growth. Some of the important results from dynamic general equilibrium theory are discussed in Chapter 5. Mas-Colell, Whinston and Green’s (1995) graduate microeconomics textbook contains an excellent treatment of most of the necessary material, including pro­ducer theory and an accessible presentation of the basic notions of general equilibrium theory, including a discussion of Arrow securities and the definition of Arrow-Debreu commodities.

Properties of homogeneous functions and Euler’s Theorem can be found, for example, in Simon and Blume (1994, Chapter 20). The reader should be familiar with the Implicit Func­tion Theorem and properties of concave and convex functions, which will be used throughout the book. A review is given in the Appendix Chapter A. The reader may also want to consult Simon and Blume (1994) and Rudin (1976).

Appendix Chapter B provides an overview of solutions to differential equations and dif­ference equations and a discussion of stability. Theorems 2.2, 2.3, 2.4 and 2.5 follow from the results presented there. In addition, the reader may want to consult Simon and Blume (1994), Luenberger (1979) or Boyce and DiPrima (1977) for various results on difference and differential equations. Knowledge of solutions to simple differential equations and stability properties of difference and differential equations at the level of Appendix Chapter B is as­sumed.

The golden rule saving rate was introduced by Edmund Phelps (1961). It is called the “golden rule” rate with reference to the biblical golden rule “do unto others as you would have them do unto you” applied in an intergenerational setting—i.e., presuming that those living and consuming at each different date form a different generation. While the golden rule saving rate is of historical interest and useful for discussions of dynamic efficiency it has no intrinsic optimality property, since it is not derived from well-defined preferences. Optimal saving policies will be discussed in greater detail in Chapter 8.

The balanced growth facts were first noted by Kaldor (1963). Figure 2.11 uses data from Piketty and Saez (2004). Homer and Sylla (1991) discuss the history of interest rates over many centuries and across different societies; they show that there is no notable upward or downward trend in interest rate. Nevertheless, not all aspects of the economic growth process are “balanced,” and the non-balanced nature of growth will be discussed in detail in Part 7 of the book, which also contains references to changes in the sectoral composition of output in the course of the growth process.

The steady state theorem, Theorem 2.6, was first proved by Uzawa (1961). Many different proofs are available. The proof given here is adapted from Schlicht (2006), which is also discussed in Jones and Scrimgeour (2006). A similar proof also appears in Wan (1971). Barro and Sala-i-Martin’s (2004, Chapter 1) also suggest a proof. Nevertheless, their argument is incomplete, since it assumes that technological change must be a combination of Harrod and Solow-neutral technological change, which is rather restrictive and not necessary for the proof. The theorem and the proof provided here are therefore more general and complete.

As noted in the text, the CES production function was first introduced by Arrow, Chenery, Minhas and Solow (1961).

Finally, the interested reader should look at the paper by Hakenes and Irmen (2006) for why Inada conditions can remove the steady state at k = O in continuous time even when f (0) = 0. Here it suffices to say that whether this steady state exists or not is a matter of the order in which limits are taken. In any case, as noted in the text, the steady state at k = 0 has no economic content and will be ignored throughout the book.

2.11.