Economics 314 Coursebook, 2008 Jeffrey Parker 3 GROWTH AND CAPITAL ACCUMULATION: THE SOLOW MODEL Chapter 3 Contents A. Topics and Tools ..................................................................................2 B. Growth in Continuous Time: Logarithmic and Exponential Functions ...................................................................................................2 Continuous-time vs. discrete-time models...........................................................2 Growth in discrete and continuous time..............................................................4 Exponentials, logs, and continuous growth..........................................................7 C. Some Basic Calculus Tools ..................................................................9 Derivatives of powers, sums, products, and quotients.........................................13 Derivatives and maximization........................................................................14 Other rules of differentiation...........................................................................15 An application: time derivatives ......................................................................16 Growth rates of products, quotients, and powers................................................17 Multivariate functions and partial derivatives .................................................18 Total differentials ...........................................................................................19 Multivariate maximization and minimization ................................................19 D. Understanding Romer’s Chapter 1....................................................20 Manipulating the production function..............................................................20 The Cobb-Douglas production function ............................................................22 The nature of growth equilibrium ...................................................................24 Basic dynamic analysis of k ..............................................................................25 Using Taylor series to approximate the speed of convergence..............................26 Growth models and the environment ...............................................................28 E. Suggestions for Further Reading .......................................................29 Expositions of the Solow model.........................................................................293 - 2 A. Topics and Tools Romer’s Chapter 1, covering the Solow growth model and related theories, presents several challenges that may be new to macroeconomics students. First and foremost, it may be the first time that you have used calculus and related mathematical methods to analyze economic models. Basic calculus concepts are reviewed in Section C of this chapter. If your calculus is shaky or rusty, this sec-tion may help, but you may also want to pursue remedial tutorial work through the Quantitative Skills Center. The second novelty of this chapter is the concept of a dynamic equilibrium growth path rather than a static point of equilibrium. We construct the Solow model in “continuous time,” which enables us to describe rates of change in terms of “time derivatives” and to make extensive use of the logarithmic and ex-ponential functions to model the movements of variables over time. These methods will be very familiar to you if you have taken a course in differential equations, but otherwise might be quite new. Section B introduces you to some of the concepts of continuous-time modeling that we will use extensively. The central element of growth theory is the feedback from current economic conditions to investment in new capital to increases in productive capacity that influence future economic conditions. This seems to suggest the possibility of self-sustaining growth through capital deepening. The Solow growth model ex-amines a simple proposition: Can an economy that saves and invests a constant share of its income grow forever? The answer is no. With a constant saving rate, such an economy will converge to an equilibrium capital-labor ratio, after which any growth that occurs must originate in a growing labor force or improving technology. B. Growth in Continuous Time: Logarithmic and Exponential Functions Continuous-time vs. discrete-time models When we construct a dynamic macroeconomic model, we must decide whether time should pass in discrete intervals or as a continuous flow. Discrete-3 - 3 time models assume that there is an interval of time–one period–during which the values of all variables remain unchanged. When a period ends, all variables may jump to different values for the next period, but they then remain un-changed through the duration of that period. Graphically, the time path of a typical variable in a discrete-time model looks like the step function in Figure 1. In continuous-time models, time flows continuously and variables can change to new values at any moment. A typical variable in a continuous-time model might have a time path like the smooth line in Figure 1. Although we usually think of time as flowing continuously, there are actually many examples of discrete time in real economies. The price of gold is fixed twice daily, for example, and banks reckon one’s deposit balances once a day at the close of business. Moreover, all macroeconomic data are published only at discrete intervals such as a day, month, quarter, or year, even when the underly-ing variables move continuously. In these cases, the single monthly value as-signed to the variable might be an average of its values on various days of the month (as with some time-aggregated measures of interest rates and exchange rates) or its value on a particular day in the month (as with estimates of the un-employment rate and the consumer price index). Since the world we are modeling has elements of both continuous and dis-crete time, neither type of model is obviously preferable. We usually choose the modeling strategy that is most convenient for the particular analysis we are per-forming. Empirical models are nearly always discrete because of the discrete availability of data, while many theoretical models are easier to analyze in con-yttimeFigure 1Discrete timeContinuous time3 - 4 tinuous time. We shall examine models of both kinds during this course. The first growth models we encounter are in continuous