The Real Business Cycle Model

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1 The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov / 27

2 Introduction The RBC model explains the co-movements in the uctuations of aggregate economic variables around their trend It is a competitive model with perfect markets: No externalities Symmetric information Complete markets No other imperfections The real business cycle model builds up on the Solow growth model, which generates an economy which converges to a balanced growth path and then grows smoothly Ester Faia (Goethe University Frankfurt) RBC Nov / 27

3 Introduction We modify this model in order to generate: Fluctuations of aggregate output around trend Co-movements of output and other aggregate economic variables around their respective trends The two ingredients used are: 1 Shocks to the economy s technology (changes in the production function from period to period. Another possible source of shocks is the unexpected changes in government purchases). 2 An optimizing household that decide how much to consume and to work. The cost of work is the loss in leisure time. Therefore we follow the Brock and Mirman 72 idea that Growth and Fluctuations are not distinct phenomena, to be studied with separate data and di erent analytical tools. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

4 Introduction Note 1: since markets are perfect, there are no market failures, and uctuations are the optimal responses of agents to the exogenous shocks. Therefore: There is no deterministic cycle (as in the Mitchell sense). There is no scope for government intervention. Note 2: here we consider a walrasian model of the aggregate economy where uctuations are generated by real shocks. The current debate in the economic theory is about the fact that walrasian models with real shocks are insu cient in explaining aggregate economic uctuations. Later we will consider non-walrasian models of aggregate economic activity where uctuations are generated by nominal shocks. Other strands of macroeconomics consider models with real shocks and with non walrasian imperfections. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

5 The economy is populated by: A large number of identical, price-taking rms A large number of identical, price-taking households A government which each period purchases an amount of goods G t and nances itself using lump sum taxes Since all agents are identical and price taking, we can aggregate and consider an economy with one representative rm and one representative household. The Ricardian equivalence holds Note: The government is only a source of real shocks in this model. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

6 The Firm In each period the rm produces output Y t using capital K t and labour L t The units of labour L t are multiplied by A t, the labour augmenting technology Therefore A t L t is the e ective labour input The production function is a CRTS Cobb Douglas function: Capital depreciates at the rate δ: where I t+1 is investment. Y t = K α t (A t L t ) 1 α (1) 0 < α < 1 K t+1 = (1 δ)k t + I t+1 (2) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

7 The Firm (con t) The technology A t is determined by the following equation: ln A t = Ā t + gt + Ã t (3) Ā and g are positive constants. Therefore without the last term we would have an economy growing smoothly along the trend. The last term is the random disturbance: Ã t = ρã t 1 + ε t (4) 1 < ρ < 1 ε t is a white noise: E (ε t ) = 0 (5) cov (ε t, ε s ) = 0 for any t 6= s (6) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

8 The Firm (con t) The binomial process we considered in the example last week is an example of a stochastic process that satis es (5) and (6). If ρ = 0 then A t = ε t. The technological shock is a white noise. If ρ > 0, it means that the shock in technology disappears gradually over time. Ã t is persistent. In the case when ρ 1, A t is so persistent that it seem to have a cyclical pattern. This is the shock that determines the business cycle uctuations. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

9 The Firm (con t) The rm observes A t and chooses K t and L t in order to maximize the pro ts at time t. Labour L t is paid with the wage w t, while the opportunity cost of capital is (r t + δ), where r t is the real interest rate. max Π t = max [Y t w t L t (r t + δ) K t ] (7a) K t,l t K t,l t We use (1) to substitute Y t = Y t = Kt α (A t L t ) 1 Order Conditions (FOC): α in (7a). The First Π t K t = αk α 1 t (A t L t ) 1 α (r t + δ) = 0 (8) ) r t = α K α t (A t L t ) 1 α K t δ (9) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

10 The Firm (con t) Π t L t = (1 α) A t K α t (A t L t ) α w t = 0 (10) ) w t = (1 α) K α t (A t L t ) 1 α L t (11) The rm solves (8) and (10) with respect to K t and L t. We can substitute Y t back in (8) and (10) and derive the equilibrium interest rates and wages: r t = α Y t K t δ (12) w t = (1 α) Y t L t (13) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

11 The Household The representative household is in nitely lived It is endowed with a certain amount of time each period (normalized to one unit), which can be used either to work or as leisure time. Therefore with respect to the optimal consumption problem analyzed before, here labour supply is endogenous. The household maximizes the expected value of the intertemporal utility function: # " U 0 = max E 0 β t u (C t, 1 L t ) C t,l t t=0 (14) 0 < β 1 (15) C t is the level of consumption; L t is the amount of time worked; (1 L t ) is the amount of leisure time; β is the intertemporal discount factor; Ester Faia (Goethe University Frankfurt) RBC Nov / 27

12 The Household (con t) The lower is β, the less future consumption and leisure are valued with respect to present ones. The utility function is assumed to be strictly concave in both arguments: u 1 > 0, u 11 < 0, u 2 > 0, u 22 < 0 The household maximizes the intertemporal utility function subject to the budget constraint (we introduce, as before, the stock of net asset A t ): B t+1 = (1 + r t+1 ) (B t + w t L t C t ) (16) w t L t is the labor income of the household Note 1: now we consume C t at the beginning of period t Note 2: the household can consume more than his salary: if C t > w t L t then net wealth is reduced. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

13 The Household (con t) If C t > w t L t ) B t+1 can become negative, hence the household cannot borrow in nitely (transversality condition): lim t! t j=0 B t (1 + r j ) 0 (17) We can once again use the Lagrangian solution method: t=0 L = β t (u (C E t, 1 L t ) 0 λ t+1 [(1 + r t+1 ) (B t + w t L t C t ) B t+1 ]) (18) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

14 The Household: Optimal household choices with certainty In this case the Lagrangian is without the expectation term: t=0 L = β t (u (C t, 1 L t ) +λ t+1 [(1 + r t+1 ) (B t + w t L t C t ) B t+1 ]) F.O.C. w.r.t. C t : F.O.C. w.r.t. L t : F.O.C. w.r.t. B t : (19) L C t = β t [u 1,t (1 + r t+1 ) λ t+1 ] = 0 (20) L L t = β t [ u 2,t + w t (1 + r t+1 ) λ t+1 ] = 0 (21) L B t = β t λ t+1 (1 + r t+1 ) β t 1 λ t = 0 (22) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

15 The Household: Intertemporal substitution in consumption We consider (20) and (22): First we substitute (23) in (24): Then we forward by one period: u 1,t = (1 + r t+1 ) λ t+1 (23) λ t = β (1 + r t+1 ) λ t+1 (24) λ t = βu 1,t (25) λ t+1 = βu 1,t+1 (26) Finally, we substitute (26) and (25) back in (24): u 1,t = β (1 + r t+1 ) u 1,t+1 (27) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

16 The Household: Intertemporal substitution in consumption (con t) (27) is the Euler equation for consumption, which has also the following interpretation: u 1,t βu 1,t+1 = 1 + r t+1 (28) Subjective value of present consumption with respect to future consumption: u 1,t βu 1,t+1 (29) Market price of present consumption with respect to future consumption: 1 + r t+1 (30) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

17 The Household: Intertemporal substitution in labor supply We consider now equations (21) and (22): First we substitute (31) in (32): Update one period: u 2,t = w t (1 + r t+1 ) λ t+1 (31) Finally, substitute (33) and (34) back in (32): λ t = β (1 + r t+1 ) λ t+1 (32) λ t = β u 2,t w t (33) λ t+1 = β u 2,t+1 w t+1 (34) u 2,t w t = β (1 + r t+1 ) u 2,t+1 w t+1 (35) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

18 The Household: Intertemporal substitution in labor supply (35) has the following interpretation: u 2,t βu 2,t+1 = w t w t+1 / (1 + r t+1 ) (36) Subjective value of present leisure with respect to the one of future leisure: u 2,t βu 2,t+1 Opportunity cost of present leisure with respect to the one of future leisure: w t w t+1 / (1 + r t+1 ) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

19 Intra-temporal substitution between consumption and leisure We consider now (20) and (21), the F.O.C. s w.r.t. consumption and leisure: u 1,t = (1 + r t+1 ) λ t+1 (37) u 2.t = w t (1 + r t+1 ) λ t+1 (38) Interpretation of 37: λ t+1 is the increase in the value function (intertemporal utility) if we increase the net assets by one unit (postpone consumption to the next period) (38) means that the loss in utility when decreasing leisure by one unit is equal to the gain we have by: working and gaining w t saving and increasing our net assets by w t (1 + r t+1 ) Therefore the trade o between consumption and leisure is the following: u 2,t u 1,t = w t (39) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

20 : An example We consider the logarithmic utility function: Therefore: The Euler equation for consumption: u (C t, 1 L t ) = ln C t b ln (1 L t ) (40) b > 0 u 1,t = 1 C t ; u 2,t = b 1 L t (41) u 1,t βu 1,t+1 = C t+1 βc t = 1 + r t+1 (42) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

21 : An example The Euler equation for labour becomes: 1 L t βu 2,t+1 u 2,t = 1 L t 1 L t+1 = L t L t+1 1 w t+1 (43) β (1 + r t+1 ) w t If 1 L t+1 # it follows that ". We decrease our leisure at time t and increase our labor supply at time t (relatively to the next period). This corresponds to an increase in salary today (relatively to next period)! 1 w t+1 β(1+r t+1 ) w t # Two e ects: income and substitution. If w t by substitution you wish to reduce labour supply and increase (leisure). By income e ect the opposite. The second prevail. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

22 : An example (con t) We now examine the substitution e ect between consumption and leisure. There are both income and substitution e ects: depending on the parameters one of the two prevails. This is clear from the intra-temporal optimal condition: which becomes: u 2,t u 1,t = w t (44) C t = w t 1 L t b (45) When the salary goes up: income increases and the household can consume more (income e ect). On the others side it is more costly to increase leisure as the opportunity cost, w t, has increased (hence to reduce labour) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

23 : Optimal Household choices with uncertainty In this case the Lagrangian is with the expectation term: t=0 L = β t (u (C E t, 1 L t ) 0 +λ t+1 [(1 + r t+1 ) (B t + w t L t C t ) B t+1 ]) (46) The rst order conditions are similar to before: L C t = β t E t [u 1,t (1 + r t+1 ) λ t+1 ] = 0 (47) L L t = β t E t [ u 2,t + w t (1 + r t+1 ) λ t+1 ] = 0 (48) L B t = E t β t λ t+1 (1 + r t+1 ) β t 1 λ t = 0 (49) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

24 : Optimal Household choices with uncertainty Equations (47), (48), and (49) can be rewritten as: u 1,t = E t [(1 + r t+1 ) λ t+1 ] (50) λ t = βe t [(1 + r t+1 ) λ t+1 ] (51) u 2,t = w t E t [(1 + r t+1 ) λ t+1 ] (52) Now we can derive the intertemporal optimal conditions in the same way as before. Ester Faia (Goethe University Frankfurt) RBC Nov / 27

25 : Optimal Household choices with uncertainty Consider for example the Euler equation for consumption: u 1,t = βe t [(1 + r t+1 ) u 1,t+1 ] (53) Using again the logarithmic utility function we have: 1 1 = βe t (1 + r t+1 ) C t C t+1 From this point onwards things are di erent with respect to the certainty case, because we have that: 1 1 E t (1 + r t+1 ) = E t (1 + r t+1 ) E t C t+1 C t+1 1 +cov 1 + r t+1, C t+1 (54) (55) (56) Ester Faia (Goethe University Frankfurt) RBC Nov / 27

26 : Optimal Household choices with uncertainty Using (55) in (54): Ct E = 1 C 1 t βcov 1 + r t+1, C t+1 (57) C t+1 βe t (1 + r t+1 ) 1 If cov 1 + r t+1, C t+1 = 0, then we have the same result as in the certainty case: Ct 1 E t = (58) C t+1 βe t+1 (1 + r t+1 ) The ratio of current to expected consumption is equal to the relative prices Ester Faia (Goethe University Frankfurt) RBC Nov / 27

27 : Optimal Household choices with uncertainty Now suppose that cov 1 + r t+1, 1 C t+1 < 0!This means that marginal utility of consumption 1 C t+1 tends to be lower when the interest rate is higher. In this case the household has less incentive to save for future consumption 1 In fact from (57) it follows that if cov 1 + r t+1, C t+1 decreases (or becomes negative) then E Ct t increases. This implies that future consumption decreases vis-a-vis current consumption C t+1 Ester Faia (Goethe University Frankfurt) RBC Nov / 27

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