Supplementary Notes on Chapter 5 of D. Romer s Advanced Macroeconomics Textbook (4th Edition)

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1 Supplementary Notes on Chapter 5 of D. Romer s Advanced Macroeconomics Textbook (4th Edition) Changsheng Xu & Ming Yi School of Economics, Huazhong University of Science and Technology This version: February 23, 208 Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes / 2

2 Section 5.3 Interpreting ρ A and ρ G Interpreting ρ A and ρ G (5.9) Ã t = ρ A Ã t + ε A,t, < ρ A < (5.) Gt = ρ G Gt + ε G,t, < ρ G < Merits of Stationary processes? Effects of a sudden shock keep diminishing. Extreme case: ρ = 0. E(Ãt) = E( G t ) = 0. Q: Considering the Endogenous Growth ingredients, ρ A 0? Q: Considering the Government spending rules, ρ G 0? Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 2 / 2

3 Section 5.4 Interpreting Equation (5.24) Interpreting Equation (5.24) (5.23) (5.24) [ ] = e ρ E t ( + r t+ ) c t c t+ [ ( = e ρ E t c t c t+ ) E t ( + r t+ ) + Cov ( )], + r t+ c t+ Euler Equation with uncertainty. Cov(X, Y ) = E[(X E(X))(Y E(Y ))]. Interpreted as how the patterns of the realization bundles of c t+ and r t+ (they are both random variables) would look like. If Cov < 0, households know that it is more likely (compared to the baseline case where Cov = 0) that if a higher next-period interest rate occurs, a lower next-period marginal utility (on consumption) also occurs. As a result, given other conditions fixed, i.e, given E t ( c t+ ) and E t ( + r t+ ), the households want to save less (higher c t ). What if Cov > 0? Economic intuitions. Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 3 / 2

4 Section 5.5 Interpreting Equation (5.3) Interpreting Equation (5.3) (5.23) [ ] = e ρ E t ( + r t+ ) c t c t+ (5.3) ln s t ln( s t ) = ρ + n + ln α + ln E t ( s t+ ) If households choose ŝ = ln α + n ρ in each period, then there is no uncertainty in s t+ and ŝ = s t = s t+ solves (5.3). Economic intuition? In this specific and oversimplified model (with δ = 0 and G t 0), the technological shock, say a positive one, leads to both a decrease in c t+ (given other things fixed) and an increase in r t+ (why?). Magically, in this model, the two effects offset each other: Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 4 / 2

5 Section 5.5 Interpreting Equation (5.3) Interpreting Equation (5.3) (Continued) + r t+ c t+ = ( ) α α At+ L t+ K t+ ( s t+ )Y t+ t+ = αy t+k. ( s N t+ t+ )Y t+ The two Y t+ s offset each other. According to (5.27), there is no uncertainty in K t+ at time t. As a result, if s t+ is fixed at ŝ, there is no need for the households to adjust c t according to the contingent technological shock Ãt+. Any other equilibrium paths? There may be, but isn t an equilibrium path with s t ŝ beautiful and intuitive? Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 5 / 2

6 Section 5.5 Interpreting Equation (5.3) Interpreting Equation (5.37) it is thus straightforward to understand equation (5.37), given = wt ŝ α b (5.26), c t = Y t N t, w t = Y t l tn t : First, l t does not need to c t l t adjust according to Ãt+. Second, shock Ãt does not change l t, a current-period positive technological shock tends to increase income and to decrease marginal utility on consumption, these two effects offset each other. Consequently, we have l t ˆl = α α+b( ŝ). Again, we get fixed ˆl and ŝ only in this oversimplified model! Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 6 / 2

7 Section 5.5 Interpreting Fluctuations in Y t Interpreting Fluctuations in Y t (5.39) ln Y t = α ln ŝ + α ln Y trend t where +αỹt + ( α)ãt = ln Y trend t + Ỹt, + ( α)(ā + gt + ln ˆl + N + nt) ln Y0 trend = α ln K 0 + ( α)(ā + ln ˆl + N) () Ỹ 0 = ( α)ã0 = ( α)ε A,0 (2) (5.42) Ỹ t = (α + ρ A )Ỹt αρ A Ỹ t 2 + ( α)ε A,t Numerical examples: α = 3, ρ A = 3, Ā = 0, g = 5, b = 2, n = 0.2, ρ = 0., N = 0, K 0 = 0, V ar(ε A,t ) = 0. Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 7 / 2

8 Section 5.5 Interpreting Fluctuations in Y t Interpreting Fluctuations in Y t (Continued) Y(t) Y trend (t) t Y ~ (t) t Figure : Numerical example with α = 3, ρ A = 3, Ā = 0, g = 5, b =, n = 0.2, ρ = 0., 2 N = 0, K 0 = 0, and V ar(ε A,t ) = 0. Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 8 / 2

9 Section 5.5 The oversimplified model, Pros and Cons The oversimplified model, Pros and Cons Pros Analytically solvable. Delivers the idea of Real Fluctuations under parsimonious settings. Cons Unrealistic predictions. Recall that s(t) ŝ, so consumptions and investments fluctuate at the same rate as the output does. Table Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 9 / 2

10 Equation (5.54), an example. An Example α = 3, g = 0.005, n = , δ = 0.025, ρ A = 0.025, ρ G = 0.95, r = 0.05, l = 3. a LA = 0.35, a LK = 0.3, a CA = 0.38, a CK = 0.59, b KA = 0.08, b KK = 0.95, a CG = 0.3, a LG = 0.5, b KG = For China, reset b KG = ( 0.004). (Why?) Ỹ t = ln Y t ln Yt trend = ln Y t ln Y t trend Y t Y t g Y (t) g fundamental Y (t) Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 0 / 2

11 Equation (5.54), an example. Figure 2 Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes / 2

12 Equation (5.54), an example Y ~ (t) Y ~ (t) Y ~2 (t) Y ~3 (t) t Figure 3: Period: 2nd quarter of st quarter of 205. Ỹ is the real outcome, Ỹ neglects the 2008 financial crisis and 2009 bailout plans, while Ỹ 2 considers the former and Ỹ 3 further considers the latter. Xu & Yi (HUST) Advanced Macroeconomics Supplementary Notes 2 / 2