Advanced Macroeconomics (2)

Transcription

1 Advanced Macroeconomics (2) Real-Business-Cycle Theory Alessio Moneta Institute of Economics Scuola Superiore Sant Anna, Pisa March-April 2015 LM in Economics Scuola Superiore Sant Anna - Università di Pisa Real-Business-Cycle Theory 1/34

2 Main features: Main features of the Real Business Cycle approach: 1 emphasis on models as artificial economies (cfr. Lucas 1980) 2 representative agent (household/firm) 3 maximization under constraints of utility/profits 4 rational expectations hypothesis 5 continuous equilibrium (market clearing) 6 dominant impulse factor: technological shocks (random changes) 7 propagation mechanism: consumption smoothing lags in the investment process (time to build ) intertemporal labour substitution 8 fluctuations in employment mainly voluntary due to the substitutability of work and leisure 9 downplaying monetary policy, neutrality of money 10 breaking down of the cycle/trend (short-run/long-run) dichotomy 11 calibration Real-Business-Cycle Theory 2/34

13 Contributors Contributors of the RBC approach: E. Prescott (Minnesota, CMU Pittsburgh, Nobel 2004) F. Kydland (U. California, CMU Pittsburgh, Nobel 2004) C. Plosser (U. of Rochester, Fed Penn) J. Long, S. Rebelo, R. King Major contributions: Kydland, F. and E. Prescott (1982) Time to build and aggregate fluctuations, Econometrica, Long, J. and C. Plosser (1983) Real business cycles, The Journal of Political Economy, Prescott, E. (1986) Theory Ahead of Business Cycle Measurement, Federal Reserve Bank of Minneapolis Quarterly Review. Kydland, F. and E. Prescott (1996) The computational experiment: an econometric tool, The Journal of Economic Perspectives, Real-Business-Cycle Theory 3/34

14 Theoretical background RBC can be seen as a development of the neoclassical growth theory of the 1950s (cfr. Solow and Ramsey models), in which: g y = αg l + (1 α)g k + z (1) where g y, g l and g k are the growth rate of output, labour, and capital (respectively) and z is the multifactor productivity growth (Solow residual) Rearranging: (g y g l ) = 1 α α (g k g y ) + 1 α z (2) RBC augments this model by incorporating stochastic fluctuations in the rate of technology progress Real-Business-Cycle Theory 4/34

15 Basic RBC model Assumptions: identical price-taking firms/ (infinitely lived) households Production (inputs: capital K, labour L, and technology A): Y t = K α t (A t L t ) 1 α, 0 < α < 1 (Cobb-Douglas prod. fun.) (3) Capital (N.B. Y C + I + G): K t+1 = K t + I t δk t = K t + Y t C t G t δk t (4) where C is consumption, I investment, G government purchases, δ rate of depreciation of capital Real-Business-Cycle Theory 5/34

16 Basic RBC model L and K are paid their marginal products. Thus real wages (w) and real interest rate (r) are: w t = Y ( ) α t Kt = (1 α) A t (5) L t A t L t r t = Y ( ) t At L (1 α) t δ = α δ (6) K t K t N.B. To the m.p.c. it has to be subtracted the depreciation rate δ. Real-Business-Cycle Theory 6/34

17 Basic RBC model Representative household optimization problem: max expected value of U = e ρt u(c t, 1 l t ) N t H t=0 (7) where u( ): instantaneous utility function of the repres. member of the household ρ: discount rate N t : population; H: # households; N t /H: # members of the household c: consumption per member, 1 l: leisure per member; l: amount each member works c C/N; l L/N Real-Business-Cycle Theory 7/34

18 Basic RBC model Population growth: log N t = N + nt n < ρ (8) log-linear u( ) u t = log c t + b log(1 l t ) b > 0 (9) technology path: log A t = A + gt + Ã t (10) Ã t = ρ A Ã t 1 + ɛ A,t, 1 < ρ A < 1 government purchases : log G t = G + (n + g)t + G t (11) G t = ρ G G t 1 + ɛ G,t, 1 < ρ G < 1 Real-Business-Cycle Theory 8/34

19 Household behaviour Case I: Hd. lives 1 period + no initial wealth + 1 member max(u) = log c + b log(1 l) sub c = wl Lagrangian: L = log c + b log(1 l) + λ(wl c) (12) f.o.c. for c and l: 1 c λ = 0 b 1 l + λw = 0 solving for c and l: c = w b+1 l = 1 b+1 N.B.: labour supply is independent of the wage (in this particular case) Real-Business-Cycle Theory 9/34

22 Household behaviour Case II: Hd. lives 2 periods + no initial wealth + 1 member + no uncertainty (about r and w 2 ) max(u) = log c 1 + b log(1 l 1 ) + e ρ [log c 2 + b log(1 l 2 )] sub c 1 + c 2 = w 1 l 1 + r(w 1 l 1 c 1 ) + w 2 l 2 Lagrangian: L = log c 1 + b log(1 l 1 ) + e ρ [log c 2 + b log(1 l 2 )] + λ[w 1 l 1 + r(w 1 l 1 c 1 ) + w 2 l 2 c 1 c 2 ] (13) f.o.c. for l 1 and l 2 : L l1 = 1 l b 1 + λw 1 (1 + r) = 0 L l2 = e ρ b 1 l 2 + λw 2 = 0 Real-Business-Cycle Theory 10/34

23 Household behaviour Case II: Hd. lives 2 periods + no initial wealth + 1 member + no uncertainty (about r and w 2 ) max(u) = log c 1 + b log(1 l 1 ) + e ρ [log c 2 + b log(1 l 2 )] sub c 1 + c 2 = w 1 l 1 + r(w 1 l 1 c 1 ) + w 2 l 2 Lagrangian: L = log c 1 + b log(1 l 1 ) + e ρ [log c 2 + b log(1 l 2 )] + λ[w 1 l 1 + r(w 1 l 1 c 1 ) + w 2 l 2 c 1 c 2 ] (13) f.o.c. for l 1 and l 2 : L l1 = 1 l b 1 + λw 1 (1 + r) = 0 L l2 = e ρ b 1 l 2 + λw 2 = 0 Real-Business-Cycle Theory 10/34

24 Intertemporal substitution in labour supply From the optimization problem (13), it follows: 1 l 1 1 l 2 = Two remarks: 1 w 2 e ρ (14) (1 + r) w 1 relative labour supply responds to relative wages N.B. elasticity of substitution between leisure 1 and leisure 2 = 1 relative labour supply responds to the real interest rate Cfr. intertemporal substitution in labour supply in Lucas and Rapping (1969) Real-Business-Cycle Theory 11/34

25 Intertemporal substitution in labour supply From the optimization problem (13), it follows: 1 l 1 1 l 2 = Two remarks: 1 w 2 e ρ (14) (1 + r) w 1 relative labour supply responds to relative wages N.B. elasticity of substitution between leisure 1 and leisure 2 = 1 relative labour supply responds to the real interest rate Cfr. intertemporal substitution in labour supply in Lucas and Rapping (1969) Real-Business-Cycle Theory 11/34

26 Household optimization under uncertainty Uncertainty about return to saving and future wages Consider household reducing c t by c in order to increase c t+1 : if it behaves optimally, marginal cost = expected marginal benefit: marginal cost = U c t c = e ρt N t H c c t [ ] m. expected benefit = E U t c t+1 increase in c t+1 N.B. c t+1 is consumption per member and there are in the household e n times as many members at t + 1 as at t, i.e. N t+1 = N t e n ] m. expected benefit = E t [e ρ(t+1) N t+1 1 H c t+1 e n (1 + r t+1 ) c Hence: 1 c t = e ρ E t [ ] 1 (1 + r c t+1 ) t+1 (15) Real-Business-Cycle Theory 12/34

27 Household optimization under uncertainty [ ] 1 = e ρ 1 E t (1 + r c t c t+1 ) t+1 1 c t = e ρ { E t [ 1 c t+1 ] ( )} 1 E t [1 + r t+1 ] + Cov, 1 + r c t+1 t+1 (16) 1 c t+1 = c. marginal utility constant, since CMU t+1 = U c t+1 = e ρ(t+1) N t+1 Hc t r t+1 = rate of return Hence: 1 c t = const E[CMU t+1 ]E[r.o.r] + const Cov[CMU t+1, r.o.r] Cov[CMU t+1, r.o.r] = expectation of co-movements between CMU t+1 and r.o.r tradeoff between present and future consumption Real-Business-Cycle Theory 13/34

31 Consumption and Labour Supply Tradeoff between present consumption and present labour supply LMU t = U l t = e ρt N t H 1 l b t (N.B. it is a disutility) Marginal cost = LMU t l = e ρt N t H b 1 l t l Marginal benefit (if resulting income is used for c t ) = CMU t w t l = e ρt N t 1 w t l H c t From m. cost = m. benefit: c t = w t 1 l t b (17) Real-Business-Cycle Theory 14/34

32 Solving the model Two simplifying assumptions: no government δ = 1, i.e. 100% depreciation each period K t+1 = Y t C t ( At L t 1 + r t = α ) 1 α K t Cfr. Long and Plosser (1983) (18) (19) Real-Business-Cycle Theory 15/34

33 s t : fraction of output that is saved. c t = (1 s t ) Y t N From eq. (16): ( [ ]) 1 c = e ρ 1 E t t c (1 + r t+1 ) t+1 [ log (1 s t ) Y ] t N t [ ] (1 + rt+1 )N = ρ + log E t+1 t (1 s t+1 )Y t+1 (20) Since 1 + r t+1 = αy t+1 K t+1 and K t+1 = Y t C t = s t Y t log s t log(1 s t ) = ρ + n + log α + log E t [ 1 1 s t+1 ] (21) If s is constant s = αe n ρ (22) Real-Business-Cycle Theory 16/34

34 From eq. (17): c t 1 l = w t t b l = 1 α (1 α) + b(1 s ) (23) Hence: both saving rate and labour supply are constant offsetting effects of technology on labour supply through wages and interest rates competitive equilibrium: unique solution Real-Business-Cycle Theory 17/34

35 In the RBC model: real shocks drive output movements the movements are optimal responses to the shocks Walrasian economy time-varying Pareto optimum fluctuations do not reflect any market failure government interventions in the RBC framework can only reduce welfare Cfr. Romer (2001: 181) Real-Business-Cycle Theory 18/34

36 Output deviations as AR(2) Recall: Y t = K α t (A t L t ) 1 α K t = s Y t 1 log A t = A + gt + Ã t log N t = N + nt We get: log Y t = α log s + α log Y t 1 + (1 α)(a + gt + Ã t + log l + N + nt) Call Ỹ t : deviations of log Y t from the normal path (difference between log Y t and value it would take if Ã t = 0) Ỹ t = αỹ t 1 + (1 α)ã t (24) Since Ã t = ρ A Ã t 1 + ɛ A,t : Ỹ t = (α + ρ A )Ỹ t 1 αρ A Ỹ t 2 + (1 α)ɛ A,t AR(2) process (25) hump-shaped responses to shocks Real-Business-Cycle Theory 19/34

37 Model s predictions vs. empirical evidence Good match: model: Ỹ t AR(2) evidence: hump-shaped linearly detrended US log output (Blanchard 1981) Bad match: model: constant s and constant l evidence: I more volatile than C; employment and hours strongly procyclical Partial match model: w strongly procyclical evidence: w moderately procyclical Real-Business-Cycle Theory 20/34

38 Model s improvements Smaller depreciation rate: 0 δ 1 Note: δ = 1 permitted analytical solution of the model l t is not constant and increases if Ã t > 0 (causing increases in I t and E t [r t+1 ]) Introduction of government purchases and shocks Ã t to G if G t > 0 (causing a decrease in households wealth, increase in l t and Y t ) tendency of w t to decrease (possibly offset by an increase in Ã t ). Real-Business-Cycle Theory 21/34

39 General model The full model (cfr. eq. 3-11) cannot be solved analytically Numerical solutions based on particular choices of parameters: study of variances and correlations of variables Campbell (1994) approach: first-order Taylor approximation of the balanced path (no shocks) study how the variables respond to shocks (technology and gov. purchases) empirical analogous: study of impulse-response functions in structural VAR Real-Business-Cycle Theory 22/34

40 Effects of technology shocks Source: Romer D. (4ed 2012) Advanced Macroeconomics, Mc Graw Hill, Figure 5.2. Real-Business-Cycle Theory 23/34

43 Effects of gov. purchases shocks Source: Romer D. (4ed 2012) Advanced Macroeconomics, Mc Graw Hill, Figure 5.5. Real-Business-Cycle Theory 26/34

46 Neutrality of Money Tobin, Friedman, and Lucas agreed that money supply has some real effects in the economy RBC: real as opposed to monetary theory of fluctuations monetary neutrality: changes in the stock of money affects only nominal (e.g. prices, wages, exchange rates) and not real variables (e.g. employment, inflation-adjusted GDP and consumption) Real-Business-Cycle Theory 29/34

47 Neutrality of Money How is the money-income correlation explained? monetarist view by Friedman and Schwartz (1963) Lucas (1972): inability of agents to discriminate perfectly in the short run between real and nominal sources of price shocks. credit view (financial market imperfections): shocks to credit markets cause output movements (cfr. Ben Friedman 1983, Bernanke-Blinder 1988) RBC view: agents increase their demand for transactions when (expected) output is high cfr. King R.G. and Plosser C.I. (1984), Money, Credit and Prices in a Real Business Cycle, AER Post-Keynesian view: endogeneity of money (cfr. Kaldor 1970, Robinson 1971) Real-Business-Cycle Theory 30/34

48 Neutrality of Money Empirical investigation on causality between money and income: Sims (1972), Granger-causality test: in favour of the monetarist view Sims (1980), VAR including interest rate: in favour of the RBC view Bernanke (1988), structural VAR: in favour of the credit view King-Plosser-Stock-Watson (1991), structural VAR: we are led to conclude that the US data are not consistent with the view that a single real permanent shock is the dominant source of business-cycle fluctuations (King et al., Stochastic trends and economic fluctuations, AER 1991) Real-Business-Cycle Theory 31/34

49 Neutrality of Money Kydland and Prescott (1990), Business Cycles: Real Facts and the Monetary Myth, Fed Minneapolis Q. Review: there is no evidence that either the monetary base or M1 leads the cycle although some economists still believe this monetary myth (KP 1990) Real-Business-Cycle Theory 32/34

50 Calibration RBC modellers reject conventional econometric testing and endorse instead the calibration approach as strategy to confront theory with data. Main tenets: model as set of instructions about how to build an artificial economy set the parameters of the model to values derived from pre-existing microeconomic studies and knowledge of the stylized facts, and/or select values for parameters so that the model is capable of mimicking the observed behaviour of variables simulate the effects of a shock (typically technology shock) Cfr. Snowdon and Vane 2005: 6.11, pp Real-Business-Cycle Theory 33/34

51 Criticism the calibration methodology...lacks any discipline as stern as that imposed by econometric methods...above all, it is not clear on what standards competing, but contradictory, models are to be compared and adjudicated (K. Hoover 1995, Facts and Artifacts: Calibration and the Empirical Assessment of Real-Business Cycle Models, Oxford Economic Papers) Real-Business-Cycle Theory 34/34