Economics 314 Coursebook, 2008 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 .....................................................5 Discounting money vs. discounting utility...........................................................6 Adding up values in continuous time using integrals...........................................8 Discounting utility in continuous time ...............................................................9 C. Constrained Maximization: The Lagrangian.....................................10 D. Understanding Romer’s Chapter 2, Part A .........................................12 Family vs. individual utility............................................................................12 Choosing a functional form for the utility function............................................12 Consumption smoothing ..................................................................................13 Discounting with varying interest rates: R(t) and r(t) ......................................15 The positivity restriction on ρ − n − (1 − θ)g ...................................................16 Understanding the Ramsey consumption-equilibrium equation .........................17 The steady-state balanced-growth path in the Ramsey model.............................20 Saddle-path convergence to the steady state.......................................................22 E. Understanding Romer’s Chapter 2, Part B .......................................23 Consumer behavior in Diamond’s overlapping-generations model .....................23 Steady-state equilibrium in the Diamond model...............................................24 Welfare analysis in the Diamond model...........................................................25 F. Government Spending in Growth Models.........................................26 The effects of government purchases .................................................................27 G. Suggestions for Further Reading.......................................................31 Original expositions of the models ....................................................................31 Alternative presentations and mathematical methods........................................31 H. Works Cited in Text ..........................................................................314 — 2 A. Topics and Tools One of our goals in approaching macroeconomic analysis was to make sure that our models were well-grounded in microeconomic behavior. The Solow model’s assumption 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 allow people to decide how much of their income to save and con-sume. This choice should be influenced by such factors as the real interest rate, which is the market’s incentive to save, and the relationship between their cur-rent 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 exam-ine the household’s optimal consumption/saving decision over time. Endogenous saving adds considerable complication to the dynamics of the model as well. The marginal rate of return on capital (the equivalent of the real interest rate in this model) depends on the capital-labor ratio, so rather than be-ing constant the saving rate varies with k as the economy moves toward the steady-state. In order to track the dynamics of two variables as we move toward equilibrium, we will need a two-dimensional “phase plane” in which two vari-ables simultaneously converge. Moreover, the nature of the equilibrium in this model is a “saddle point,” which has interesting dynamics that you are unlikely to have seen before. 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 com-bination of Romer, class lectures, and this 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 the future and past values in terms of comparable 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 opportu-nity from receiving $100 today as from receiving $110 dollars one year from to-day. Table 1. Consumption opportunities Table 1 shows this by examining four cases in a 2 × 2 table. The top-left and bottom-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 perform 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 consume now, you simply spend the $100. If you would rather spend the money next year, you lend the $100 out at 10 percent interest. Next year you receive $110 in principal and interest payments and spend it on $110 worth of goods. The lower row shows that you get the same consumption options from receiving $110 next year. If you wish to consume next year, you simply spend the money when it is received. If you wish to consume today, you borrow $100 and spend it Consume $100 today Consume $110 next year Receive $100 today Consume $100 today when received Lend $100 today at 10%, receive and consume $110 next year Receive $110 next year Borrow $100 today at 10% and consume;