Economics 314 Coursebook, 2009 Jeffrey Parker 4 OPTIMIZING THE SAVING DECISION IN A GROWTH MODEL Chapter 4 Contents A. Topics and Tools ............................................................................. 2 B. Discounting the Future in Discrete and Continuous Time .......................... 3 The idea of discounting ................................................................................................. 3 Frequency of compounding and present value .................................................................. 4 Discounting money vs. discounting utility ....................................................................... 6 Adding up values in continuous time using integrals ........................................................ 7 Discounting utility in continuous time ............................................................................ 9 C. Constrained Maximization: The Lagrangian .......................................... 9 D. Understanding Romer’s Chapter 2, Part A .............................................. 11 Family vs. individual utility ........................................................................................ 11 Choosing a functional form for the utility function ......................................................... 11 Consumption smoothing ............................................................................................. 12 Discounting with varying interest rates: R(t) and r(t) ..................................................... 14 The positivity restriction on ρ − n − (1 − θ)g ................................................................. 15 Understanding the Ramsey consumption-equilibrium equation ...................................... 16 The steady-state balanced-growth path in the Ramsey model .......................................... 18 Saddle-path convergence to the steady state ................................................................... 20 E. Understanding Romer’s Chapter 2, Part B ........................................... 21 Consumer behavior in Diamond’s overlapping-generations model ................................... 21 Steady-state equilibrium in the Diamond model ............................................................ 22 Welfare analysis in the Diamond model ....................................................................... 23 F. Government Spending in Growth Models ............................................. 24 The effects of government purchases .............................................................................. 24 G. Suggestions for Further Reading ........................................................ 28 Original expositions of the models ................................................................................ 28 Alternative presentations and mathematical methods ..................................................... 28 H. Works Cited in Text ........................................................................ 28 4 – 2 A. Topics and Tools One of our goals in approaching macroeconomic analysis is to make sure that our models are well-grounded in microeconomic behavior. The Solow model’s as-sumption that people save a constant share of their income is exactly the kind of ad hoc assumption that we are trying to avoid. A reasonable theory of saving should al-low people to decide how much of their income to save and consume. This choice should be influenced by such factors as the real interest rate, which is the market’s incentive for people to save, and the relationship between their current income and their expected future income. In microeconomics, we model saving and consumption choices using utility maximization. The Ramsey and Diamond growth models, which we study here in Romer’s Chapter 2, use the standard microeconomic theory of saving to make the saving rate endogenous. Because saving is a dynamic decision depending on past, present, and future income, we will need some new tools to analyze it. We use (at a fairly superficial level) tools of dynamic optimal control theory to examine the household’s optimal consumption/saving decision over time. Most macroeconomic models being developed today begin from the Ram-sey/Diamond framework of utility maximization, varying mainly in whether time is continuous (as in Ramsey) or discrete (as in Diamond) and whether households have infinite (Ramsey) or finite (Diamond) lifetimes. Endogenous saving adds considerable complication to the dynamics of growth. The marginal rate of return on capital (the equivalent of the real interest rate in this model) depends on the capital-labor ratio. As the capital-labor ratio changes during convergence toward the steady-state, the corresponding change in the return to capi-tal will cause changes in the saving rate. In order to track the dynamics of two va-riables as we move toward equilibrium, we will need a two-dimensional “phase plane” in which two variables simultaneously converge. Moreover, the nature of the equilibrium in this model is a “saddle point,” which has interesting dynamic proper-ties. Chapter 2 is one of the most challenging chapters in the Romer text. Don’t be discouraged if you don’t understand everything immediately. Rely on a combination of the text, class lectures, and this coursebook chapter to help you achieve and work-ing understanding of the model. As always, don’t hesitate to ask for help!4 – 3 B. Discounting the Future in Discrete and Continuous Time The idea of discounting Introductory economics teaches you that comparing values at different points in time requires discounting—expressing future and past quantities in terms of compara-ble present values. For example, if the market interest rate at which you can borrow or lend is 10 percent, then you get the same consumption opportunity from receiving $100 today as from receiving $110 dollars one year from today. Table 1. Consumption opportunities Table 1 shows this by examining four cases in a 2 × 2 table. The top-left and bot-tom-right cells show what happens if the individual consumes the income when it is received; the top-right and bottom-left cells illustrate the individual’s ability to per-form intertemporal substitution through borrowing or saving at an interest rate of 10 percent. The upper row shows your options if you receive $100 now. If you wish to con-sume now, you simply spend the $100. If you would rather