The Real Business Cycle Model

The Real Business Cycle Model

Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations 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. We modify this model in order to generate: Fluctuations of aggregate output around trend Comovements of output and other aggregate economic variables around their respective trends

Macroeconomics II 3 The two ingredients used are: I 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. ) II An optimising 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 different analytical tools.

Macroeconomics II 4 Note : since markets are perfect, there are no market failures, and fluctuations are the optimal responses of agents to the exogenous shocks. Therefore: There is no deterministic cycle (in the Mitchell sense). There is no scope for government intervention. Note 2: here we consider a walrasian model of the aggregate economy where fluctuations are generated by real shocks. The current debate in the economic theory is about the fact that walrasian models with real shocks are insufficient in explaining aggregate economic fluctuations. Later we will consider non-walrasiam models of aggregate economic activity where fluctuations are generated by nominal shocks. Other strands of macroeconomics consider models with real shocks and with non walrasian imperfections.

Macroeconomics II 5 2 The baseline real business cycle model The economy is populated by: I A large number of identical, price-taking firms II A large number of identical, price-taking households III A government which each period purchases an amount of goods G t and finances itself using lump sum taxes Since all agents are identical and price taking, we can aggregate and consider an economy with one representative firm and one representative household. The ricardian equivalence holds Note: The government is only a source of real shocks in this model.

Macroeconomics II 6 2. The firm In each period the firm 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 effective labour input. The production function is a CRTS Cobb Douglas function: Y t = K α t (A t L t ) α () 0 <α< Capital depreciates at the rate δ : Where I t+ is investment. K t+ = δk t + I t+ (2)

Macroeconomics II 7 The technology A t is determined by the following equation: ln A t = A + gt + Ãt (3) A 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: ε t is a white noise: Ã t = ρãt + ε t (4) <ρ< E (ε t )=0 (5) cov (ε t,ε s )=0for any t s (6) The binomial process we considered in the example last week is an example of a stochastic process that satisfies (5) and (6). If ρ =0then Ãt = 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 last week example we saw that if ρ is close to, Ãt is so persistent that it seem to have a cyclical pattern This is the shock that determines the business cycle fluctuations

Macroeconomics II 8 The firm observes Ãt and chooses K t and L t in order to maximise the profits 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 K t,l t Π t =MAX K t,l t Y t w t L t (r t + δ) K t (7) We use () to substitute Y t in (7). The First Order Conditions (FOC): Π t K t = αk α t (A t L t ) α (r t + δ) =0 (8) = r t = α Kα t (A t L t ) α K t δ (9) Π t L t =( α) A t K α t (A t L t ) α w t =0 (0) = w t =( α) Kα t (A t L t ) α () L t The firm solves (8) and (0) with respect to K t and L t. Instead we substitute Y t back in (8) and (0) and derive the equilibrium interest rates and wages: r t = α Y t K t δ (2)

Macroeconomics II 9 w t =( α) Y t L t (3) Also useful is to rearrange the two equations as follows: (r t + δ)k t Y t = α (4) w t L t Y t = α (5)

Macroeconomics II 0 2.2 The household The representative household is infinitely lived. It is endowed with a certain amount of time each period (normalised to one unit), which can be used either to work or as leisure time. Therefore with respect to the optimal consumption problem analysed last week, here labour supply is endogenous. The household maximises the expected value of the intertemporal utility function: [ ] U 0 =maxe 0 β t u (C t, L t ) (6) c t,a t,l t t=0 C t is the level of consumption. L t is the amount of time worked. L t is the amount of leisure time. β is the intertemporal discount factor. 0 <β (7) The lower is β, the less future consumption and leisure are valued with respect to present ones.

Macroeconomics II The utility function is assumed to be strictly concave in both arguments: u > 0; u < 0; u 2 > 0; u 22 < 0 (8) We use the following notation: u,t = u(c t, L t ) C t ; u 2,t = u(c t, L t ) L t (9) u,t = 2 u (C t, L t ) ( C t ) 2 ; u 22,t = 2 u (C t, L t ) ( L t ) 2 (20) The household maximises the intertemporal utility function subject to the budget constraint. We introduce, like last week, the notion of the stock of net assets A t : A t+ =(+r t+ )(A t + w t L t C t ) (2) w t L t is the labour income of the household. Note : now we consume C t at the beginning of period t. Note that you can consume more than your salary: if C t >w t L t then you reduce your net wealth. This means that A t+ can become negative, but you cannot borrow infinitely: lim t A t t j=0 ( + r j) 0 (22)

Macroeconomics II 2 We can once again use the lagrangean solution method: L { = E 0 β t {u (C t, L t )+λ t+ [( + r t+ )(A t + w t L t C t ) A t+ t=0 2.3 Optimal household choices with certainty (23) In this case the lagrangean is without the expectation term: L { = β t {u (C t, L t )+λ t+ [( + r t+ )(A t + w t L t C t ) A t+ t=0 (24) The first order conditions are given by the first derivatives of L with respect to C t,l t and A t equal to zero. L C t = β t [u,t ( + r t+ ) λ t+ ]=0 (25) L = β t [ u 2,t + w t ( + r t+ ) λ t+ ]=0 L t (26) L = β t λ t+ ( + r t+ ) β t λ t =0 A t (27)

Macroeconomics II 3 2.3. Intertemporal substitution in consumption Like we did last week, we consider (25) and (27): First we substitute (28) in (29): Then we forward by one period: u,t =(+r t+ ) λ t+ (28) λ t = β ( + r t+ ) λ t+ (29) λ t = βu,t (30) λ t+ = βu,t+ (3) Finally we substitute (30) and (3) back in (29): u,t = β ( + r t+ ) u,t+ (32) (32) is the euler equation for consumption, which has also the following interpretation: u,t βu,t+ =+r t+ (33) u,t βu,t+ = +r t+ = SUBJECTIVE VALUE of present consumption with respect to futute consumption MARKET PRICE of present consumption with respect to futute consumption

Macroeconomics II 4 2.3.2 Intertemporal substitution in labour supply We consider now equations (26) and (27): u 2,t = w t ( + r t+ ) λ t+ (34) λ t = β ( + r t+ ) λ t+ (35) First we substitute (34) in (35): λ t = β u 2,t w t (36) Then we forward by one period: λ t = β u 2,t+ w t+ (37) Finally we substitute (36) and (37) back in (35): u 2,t w t (38) has also the following interpretation: u 2,t = βu 2,t+ u 2,t βu 2,t+ = w t w t+ / ( + r t+ ) = β ( + r t+ ) u 2,t+ w t+ (38) w t w t+ / ( + r t+ ) SUBJECTIVE VALUE of present leisure with respect to future leisure (39) = OPPORTUNITY COST of present leisure with respect to the one of futute leisure

Macroeconomics II 5 2.3.3 Intratemporal substitution between consumption and leisure We consider now (25) and (26), the FOCs with respect to consumption and labour: u,t =(+r t+ ) λ t+ (40) u 2,t = w t ( + r t+ ) λ t+ (4) Interpretation: λ t+ is the increase in the value function (intertemporal utility) if we increase the net assets by one unit. (4) means that the loss in utility in decreasing leisure by one unit is equal to the gain we have by: working and gaining w t savingw t and increasing our net assets by w t ( + r t+ ) Therefore the trade off between consumption and leisure is the following: u 2,t u,t = w t (42) w t is the relative price of leisure with respect to consumption

Macroeconomics II 6 2.4 An example We consider the logarithmic utility function: u (C t, L t )=lnc t + b ln ( L t ) (43) b>0 Therefore: u,t = ; u 2,t = b (44) C t L t The euler equation for consumption: u,t =+r t+ (45) βu,t+ Becomes: And similarly: Becomes: C t C t+ = u 2,t βu 2,t+ = L t L t+ = β ( + r t+ ) w t w t+ / ( + r t+ ) (46) (47) w t+ (48) β ( + r t+ ) w t Therefore if w t increases with respect to w t+, we have that w t+ w t Therefore L t L t+ and hence L t L t+. We decrease our leisure

Macroeconomics II 7 and increase our labour supply at time t.

Macroeconomics II 8 This is because the substitution effect (higher opportunity cost of leisure) more than compensates the income effect (higher wage means that we are richer and want to consume more) This is clear from the intratemporal optimal condition: u 2,t u,t = w t (49) Which becomes: C t = w t L t b (50)

Macroeconomics II 9 2.5 Optimal household choices with uncertainty In this case the lagrangean is with the expectation term: L { = E 0 β t {u (C t, L t )+λ t+ [( + r t+ )(A t + w t L t C t ) A t+ t=0 (5) The first order conditions are similar to before: L C t = β t E t [u,t ( + r t+ ) λ t+ ]=0 (52) L = β t E t [ u 2,t + w t ( + r t+ ) λ t+ ]=0 L t (53) L = β t [ E t λt+ ( + r t+ ) β ] t λ t =0 A t (54) Which can be written as: u,t = E t [( + r t+ ) λ t+ ] (55) λ t = βe t [( + r t+ ) λ t+ ] (56) u 2,t = w t E t [( + r t+ ) λ t+ ] (57) Now we can derive the intertemporal optimal conditions in the same way as before.

Macroeconomics II 20 Consider for example the euler equation for consumption: u,t = βe t [( + r t+ ) u,t+ ] (58) Using again the logarithmic utility function we have: [ ] = βe t ( + r t+ ) C t C t+ (59) From this point onwards things are different with respect to the certainty case, because we have that: ] E t [( + r t+ ) C t+ = ( ) ( ) (60) E t ( + r t+ ) E t C t+ + cov +r t+, C t+ Using (60) in (59): ( ) Ct E t C t+ = C t βcov ( +r t+, C t+ ) βe t ( + r t+ ) (6)

Macroeconomics II 2 ( ) If cov +r t+, C t+ =0then we have the same result than in the certainty case: ( ) Ct E t = (62) C t+ βe t ( + r t+ ) The ratio of current to expected consumption is equal to the relative prices. ( ) Now suppose that cov +r t+, C t+ < 0 ( ) This means that marginal utility of consumption C t+ tends to be lower when the interest rate is higher. In this case the household is less incentive to save for future consumption: ( ) In fact from (6) it follows that if cov +r t+, C ( ) t+ decreases then E Ct t C t+ increases!