Advanced Macroeconomics (ECON 402) Lecture 8 Real Business Cycle Theory

Transcription

1 Advanced Macroeconomics (ECON 402) Lecture 8 Real Business Cycle Theory Teng Wah Leo Some Stylized Facts Regarding Economic Fluctuations Having now understood various growth models, we will now delve into the minute detail of fluctuations within an economy. The two leading theories in this facet of macroeconomics are Real Business Cycle models, Keynesian and Neo-Keynesian models. One of key observations you can get from looking at aggregate macroeconomic indicators such as real GDP, unemployment rate etc, are that fluctuations do not occur with distinct regularity, or cyclical patterns, or duration, or time between each occurance. It used to be much effort was expended in discovering durations between each fluctuation, though the general concensus is that it is futile, and needless. The fundamental agreement in macroeconomics is that fluctuations in the macroeconomy occurs due to random shocks that perturb the equilibrium within the economy, and resonate throughout. The primary differential between these theories is how these theories affect the economy, i.e. the mechanism. A second fact regarding economic fluctuations are that the components of economic indicators are not affected the same, for instance, consumer durables are more affected by fluctuations than non-durables. Thirdly, there are no substantive differences in the changes in output coinciding with the fluctuations, in the sense that output fluctuates about the mean. But the duration an economy spends above the mean path is longer than when it is below. Fourthly, it has been found that fluctuations both pre- and post depression era were relatively similar. The is an interesting fact since the components of

2 output, and roles of government were quite different. This then suggests that either there exists determinant(s) that has been relatively stable (which suggests then it fluctuations as something worth further examination) or that the opposing effects of the two eras cancel each other out. Finally, the key descriptive indicator about how the fall in output is being realized is output per worker hour, i.e. during a recession, worker productivity falls. This in turn implies that the fall in employment is usually lower than the fall in output itself. You might recall now Okun s Law which describes this relationship between output and unemployment rate movements. 2 Baseline RBC Model The RBC model is principally derived from a general equilibrium model with both optimizing households, and firms. Since Ramsey s model is a general equilibrium model, it is the basis of its development, albeit in order to model fluctuations, the model needs to be augmented by shocks, since if you recall, without these shocks, the Ramsey s model converges to a balanced growth path/bliss state. There are two venues through which shocks has been added, one in technology, while the other in government spending, both of which has real effects on the economy, as opposed to adding shocks monetary variables. It is for this reason such models are termed Real Business Cycle models. Unlike Ramsey s model however, time is discrete as opposed to continuous. 2. Model Setup We will first examine a baseline version of a RBC model. The assumptions are as follows:. Economy consists of a large number of price-taking firms, and infinitely lived households. 2. Inputs into production are capital (K), labour (L) and technology (A). The production function is assumed to take the Cobb-Douglas form: Y t = Kt α (A t L t ) α () where α (0, ). 3. In turn, output is either consumed (C), invested (I), and consumed by the government (G). The latter is financed through lump-sum taxes, though you should 2

3 recall, government spending financing has no effect on the model. This assumption therefore can be summarized as, K t = K t I t δk t (2) = K t Y t C t G t δk t (3) 4. Labour and capital are paid their marginal products since all firms are perfectly competitive price takers. ( ) α Kt w t = ( α) A t (4) A t L t ( ) α At L t r t = α δ (5) 5. The household maximizes their expected lifetime utility, K t U = t=0 e ρt u(c t, l t ) N t H (6) where u(.) is the instantaneous utility of a representative agent of the household, and ρ as usual is the discount factor. In any period, the population is N t, and the number of households in the economy is H, so that the number of members in a household in any period is N t /H. 6. The exogenous growth rate of population is n, so that N t in any period is, ln N t = N nt (7) N t = e Nnt (8) 7. Instantaneous utility in each period is log linear in average consumption, c = C/N, and average labour supplied, l = L/N (since all individuals are the same, i.e. homogeneous). u t = ln c t b ln( l t ) (9) where b > 0. Notice the difference in notation relative to the classical labour-leisure choice model, where L is typically reserved for leisure instead of labour. 3

4 8. Technology is mean reverting, and trends at a rate g, with a shock term Ãt, that is ln A t = A gt Ãt (0) Ã t = ρ A Ã t ɛ A,t () for ρ A (, ), so that the shock term is a first order autoregressive process, with ɛ A,t being white noise disturbance. 9. Similarly, government spending is mean reverting, with a trend, and shock G t, that is, ln G t = G (n g)t G t (2) G = ρ G Gt ɛ G,t (3) where ρ G (, ), and the shock term for government spending is likewise a first order autoregressive process, and ɛ G,t and ɛ A,t are uncorrelated. What you should then notice is that unlike Ramsey s model, this model now allows individuals to choose their leisure, and consequently labour supply, which allows us to examine into employment outcomes in the aggregate economy as a result of real shocks, which Ramsey does not examine, but is included here. This would therefore make the model rather interesting. 2.2 Some Insights into Assumption of Instantaneous Utility There is no close form solution to this setup as we will see. Nonetheless, we will see what we can glean. First we will examine the static portion of the model, in other words, examine the conditions should we solve the model as a single period model, where each household has only one agent, and each agent has only wages that it earns from labour supply, and it has no initial wealth to speak of. constraint would just be c = wl, so that for we would get the following first order conditions, In this case, the individual budget L = ln c b ln( l) λ(c wl) (4) c b l 4 = λ (5) = λw (6)

5 so that the equilibrium condition is described by l c = b w (7) but we know from your intermediate microeconomics, due to local non-satiation, c = wl, so that the above condition becomes, l l = b (8) which means that the labour choice is independent of w. The reason for this is due to the assumption of the log linear model, and that there is no initial wealth to speak of, a change in wage would have the income and substitution effect completely cancelling out, i.e. if wages increases, susbtitution effect would dictate that leisure is more expensive, and the individual supplies more labour. But with the wage increase, the income effect drives up choice of leisure, and this completely cancels the substitution effect, so that labour-leisure choice becomes independent of wages. However, this does not mean that in a dynamic model such as the RBC model, that the intertemporal choice of labour supply is independent of wages. To see that, assume that there is only one member per household, and that there is no initial wealth. The intertemporal budget constraint now is, c c 2 r = w l w 2l 2 r Assume that there is no uncertainty, which the actual model does, in wages or rates of return from savings/investments. The Lagrangian is thus, ( L = ln c b ln( l ) e ρ [ln c 2 b ln( l 2 )] λ c c 2 r w l w ) 2l 2 (20) r Therefore, the first order conditions with respect to l i, i = {, 2} are, (9) b = λw (2) l e ρ b = λw 2 (22) l 2 r Therefore, the equilibrium condition in this two period case is, l l 2 = w 2 e ρ ( r)w (23) 5

6 so that in this case now, should the relative wages rise in favour of period, labour supply would increase in period, raising savings towards the second period, as would be expected. Notice also that should interest rates rise, labour supply likewise would increase in period to take advantage of this, and we have intertemporal substitution in labour supply. 2.3 Household Optimization The key differential between the RBC models, and Ramsey s model is the uncertainty, which complicates issues. In the deterministic Ramsey s model, each household/individual has a deterministic steady state path. What this uncertainty means here for the household/agent is that their choices now are dependent on the entire history of shocks it has experienced to that date. Not to mention that time now is discrete. Nonetheless, without further discussion of Dynamic Programming, we can obtain a similar version of the Euler Equation. To obtain the Euler equation intuitively, all we ll be using is the fact that the marginal benefit from any alteration to equilibrium choice, must equate with the marginal lost. So therefore, realize that for any alteration in consumption in one period by a quantity of c must leave expected lifetime utility unaltered. Based on the previous section s discussion, a change of consumption will have a marginal utility in period t of e ρt (N t /H)(/c t ), and the cost in terms of marginal utility lost in period t from a change in consumption of c is e ρt (N t /H)( c/c t ). Next note that in a subsequent period t, the household would have now e n additional members, since the population growth rate is n. This therefore means that each member s share of consumption in period t must be e n ( r t ) c. In turn, each members marginal utility from consumption is e ρ(t) (N t /H)(/c t ). Taking all this together, the total expected gain in utility in period t is E t [ e ρ(t) e n (N t /H)(( r t ) c/c t ) ]. Therefore, the equilibrium condition is, e ρt N [ t c = E t e ρ(t) e n N t H c t H ] ( r t ) c c t (24) and since N t = N t e n, c t = e ρ E t rt c t (25) 6

7 and you have the Euler Equation. It is interestint to note that since there is uncertainty with consumption and interest rates, ( ) ( ) rt E t = E t E t ( r t ) Cov, ( r t ) (26) c t c t c t ( ) and since Cov c t, ( r t ) 0, the correlation between consumption and interest rates affect the optimal level of consumption. To be precise, since interest rates and consumption are negatively correlated, consumption is higher than if correlation is zero. We have focused on the relationship between labour supply across each period. However, as is clear, there are four variables, so that a second equilibrium condition relates the relationship between consumption and labour supply/leisure. Here the intuitive idea is that if an agent of the household raises its labour supply, then there will be a commiserating increase in consumption in that period so that the expected utility remains unchanged in that period. Firstly, note that the marginal (dis)utility from labour supply is as before e ρt (N t /H)[b/( l t )]. On the other hand, the marginal utility from this labour supply change is e ρt (N t /H)(/c t ), noting that this is from an increase in consumption of w t l, so that the equilibrium condition is, e ρt N t b l = e ρt N t w t l (27) H l t H c t c t = w t (28) l t b Notice that unlike the intertemporal labour supply choice, there is no uncertainty since the variables are all current. Nonetheless, it remains a key equation describing household behavior. 2.4 Obtaining Insights to the Model The model as we have set up, and as have been noted, has no closed form solution. The manner in which they are solved are commonly via numerical simulations. However, we can glean some insights by (.) examining a special case of the model, or (2.) Try to get insights via first order Taylor series approximations of the model around the Balanced Growth Path. Each of these will be examined in turn A Simpler Version of the RBC Model The reason there is no close form solution is that the utility function, and investment equations contains both linear, and log linear elements, which complicates the solution 7

8 in a huff. We can perhaps gain some insight into the mechanics by making simplifying assumptions.. Elimination of Government Sector (so that the equations that describe real shocks in the government sector are eliminated), and 2. 00% Depreciation in each period, so that the equations describing capital stock across time, and real interest rates are now, K t = Y t C t (29) ( ) α At L t r t = α (30) The elimination of the public sector allows us to focus on technology shocks, and see how any shocks are transmitted through the economy. The complete depreciation assumption allows to the model to be solved analytically. A key to note of the model setup is that since the markets are competitive (consequently has no externalities), and there are at any time, a finite number of individuals, the model s equilibrium will be Pareto Optimal. K t Given this, the model can either be solved from a social planner s perspective, or the flip-side, i.e. the finding the competitive equilibrium directly. Since the latter is more amenable to examinations of market failure, the latter is pursued. The solution will be focused on equations (25) and (28), and we will focus on the labour variable, l, and savings s. The latter focus so that we can examine how technology shocks will act via labour choices, and savings to affect ultimately consumption and investment decisions. Focusing first on equation (25), and noticing that C = ( s)y, and that c = C/N, taking log s on both sides of the equation, it can be rewritten as, [ ln ( s t ) Y ] [ ] t r t = ρ ln E t N t ( s t ) Y t N t We also know that, (3) ( ) α At L t r t = α δ K t r t = α Y t K t (32) 8

9 since δ =, i.e. that depreciation is 00%. Further, with full depreciation, capital stock in every period is, K t = Y t C t (33) = Y t ( s t )Y t = s t Y t (34) So that the Euler equation is now, [ ] αy t ln( s t ) ln Y t ln N t = ρ ln E t ( s t ) Y tk t (35) N [ t ] αn t = ρ ln E t (36) ( s t )s t Y t = ρ ln α (N n(t )) ln s t ln Y t ln E t s t = ρ ln α ln N t n ln s t ln Y t ln E t (37) s t Therefore, ln s t ln( s t ) = ρ ln α n ln E t s t (38) From the above equation, what is interesting is that the steady state/stationary s t, s, is actually not dependent on shocks at all. To see that, set s t = s t = s, ln s ln( s ) = ρ ln α n ln s ln s ln( s ) = ρ ln α n ln( s ) ln s = ln α n ρ s = αe n ρ (39) which is a constant, and not dependent on technology shocks. Considering now equation (28), first note that equilibrium competitive wages is, w t = ( α) Y t L t = ( α) Y t l t N t (40) 9

10 Taking the above together with equation (28), we have c t = w t l t b ( s )Y t ln ln( l t ) = ln w t ln b N t ln( s ) ln Y t ln N t ln( l t ) = ln( α) ln Y t ln l t ln N t ln b ln( s ) ln( l t ) = ln( α) ln l t ln b lt ln = ln( α) ln b ln( s ) l t l t α = l t b( s ) α l t = b( s ) ( α) = l (4) So that labour supply is likewise a constant that is not dependent on technology shocks! The reason for this is that movements in technology or capital cancels each other out through the effects they have on relative wages, and interest rate s effects on labour supply. To see the arguement. Consider a positive technology shock, which raises the current wages relative to expected future wages. This then raises current period s supply of labour. However, with the increase in labour supply, so too does savings, which in turn lowers expected interest rates. This in turn lowers current period labour supply, completely cancelling each other out, thereby rendering a positive technology shock rather impotent. Further, this solution is a unique solution. Relating this model thus far to what you have learned till now, with an emphasis on Keynesian economics, the model is a general equilibrium model (Walrasian), which allows for shocks to generate fluctuations within an economy. The key point here is that fluctuations are a results of unanticipated real shocks, as opposed to market failures. This means that any government intervention in an attempt to reduce the impact of the shock, particularly when the shock is negative, is completely futile, within the model s context. This therefore means that our observations of aggregate output movements are just time varying Pareto Optimum. We will now examine the form of output fluctuations within the RBC model. To see the dynamics, first recall that the production is of the Cobb-Douglas form, so that taking 0

11 log s of the production function, we have, ln Y t = α ln K t ( α)(ln A t ln L t ) (42) = α ln(s Y t ) ( α)(ln A t ln(l /N t )) = α ln(s Y t ) ( α)(ln A t ln l ln N t )) = α ln s α ln Y t ( α)(a gt) ( α)ãt ( α)(ln l N nt) (43) where the last line uses the technology shock equation, and the equation for the growth rate of the population. From equation (43), notice that there are two variables on the right handside of the equation that do not follow deterministic paths, namely Y t and à t. Thus, it must be possible to rewrite it into the following form, Ỹ t = αỹt ( α)ãt (44) where Ỹt is just the difference between ln Y t and the value it would have taken baring any shock. This then implies that, Ỹ t = αỹt 2 ( α)ãt (45) Ãt = Ỹt αỹt 2 (46) α Ỹt = αỹt ( α)(ρ A à t ɛ A,t ) (47) = (α ρ A )Ỹt αρ A Ỹ t 2 ( α)ɛ A,t (48) where we have used the fact that Ãt follows a first order autoregressive process. What the last equation thus says is that departures of the ln output from the norm follows a second order autoregressive process. Thus depending on the values that α and ρ A takes, it is therefore possible to have a initially surge in response to a shock, before the effects dissipate (see your textbook for a numerical example). Typically, α is not large, so that the dynamics of output is dependent very much on the value that ρ A takes. However, as noted in your text, the model does not have any mechanism that translates a transitory shocks to a longer term one. What is interesting is that the general model behaves very much the same. The principal shortfall of this basic model are as follows:. Since savings are unaffected by the shocks, this translates to a jointly equally volatile consumption and investment with any shock. However, based on the stylized facts, investments are typically more volatile than investments.

12 2. Since labour choice do not vary, it runs counter to the fact the employment and hours worked are strongly procyclical. 3. Based on equation (28), and the above arguements, real wage would be procyclical, since labour choice does not respond to shocks. But the fact is real wages are only mildly procyclical General RBC Model From the insights gained from the previous section, we can see the reason why the general RBC model might work. Firstly, by including a lower level of depreciation, a positive technology shock raising the marginal product of capital would make it worth the while of the household to raise investments, where previously it was none, which means savings rate rises. This in turn means that the expected future consumption rises. From the Euler equation (25), this would mean in turn that interest rates must rise. With this, current labour supply would increase. The net implication is then that with depreciation, investments and employment now would be affected by shocks. Secondly, by introducing the government sector, the strong link between output and wages is diminished. Consider a positive government shock, which in turn would raise future tax burden, and consequently lifetime wealth (remember households are infinitely lived). This then reduces leisure, i.e. labour supply rises, which in turn drives down real wages. Whereas previously output and real wage effects are positive, the inclusion of the government sector introduces a negative relationship, dissipating the connection, and thereby allowing us to get a lesser procyclical relationship between output and real wages. As noted prior, the RBC model has no analytic solution. However, we may glean some insights by applying Taylor series expansions of the key equations of the model (without recourse to any shocks), then examining how the model s variables respond to shocks. To do so, let us first find the balanced growth path values of Y/AL, K/AL, C/AL, and G/AL, denoting them by y, k, c and G. There are six endogenous variables that are constant on a balanced growth path, y, k, c, w, l, and r. These can be obtain by solving the following six equations derived from the assumption, and the equilibrium conditions. Before we can solve for them, first not the following identities. The technology equation 2

13 on the balanced growth path, without any shocks is just, For the population growth, note that ln A t = A gt ln A t = A g(t ) ln A t = ln A t g ln A t A t = g ln A t A t = e g ln N t = N nt ln N t = N n(t ) ( Nt N t ) = ln N t n = n N t N t = e n Now to obtain the six equations, from the production function assumption, Y t Y t = K α t (A t L t ) α ( Kt y = A t L t = A t L t y = (k ) α (49) From the investment equation, ) α K t = K t Y t C t G t δk t K t A t L t = ( δ) K t A t L t K t A t L t A t L t A t L t = ( δ) K t A t L t Y t C t G t A t L t A t L t A t L t Y t C t G t A t L t A t L t A t L t K t A t l t N t = ( δ) K t A t L t A t l t N t A t L t A t L t A t L t A t L t k e ng = ( δ)k y c G k ( e ng ( δ) ) = y c G (50) Y t C t G t 3

14 noting that G is just per capita governemnt expenditure, without shocks. competitive equilibrium, wages must be such that From the w = ( wt A t w t ) = ( ) α Kt ( α) A t A t L t = ( α)(k ) α (5), and for rents, ( ) α At L t r t = α δ K t ( ) α r = α δ (52) For the last two equations, we use the equilibrium conditions. Firstly, from the Euler equation, N t ( = e ρ rt )N t E t C t C [ t ] N t A t L t ( = e ρ rt )N t A t L t E t C t A t L t C t A t L t ( r = e ρ )e g c l c l k = e (ρg) ( r ) r = e ρg (53) And finally, the ratio between consumption and labour supply choice, we have C t N t ( L t /N t ) C t /A t L t N t /A t L t ( L t /N t ) c ( l l ) = w t b = w t b = w b l c = w l b Thus solving for the six variables balanced growth path values are straight forward. Based on the above six equations, what you should notice is that for the two endogenous variables of C t and L t, they are dependent on K t, A t, and G t. For C t, this may noted from equation (50) where C t is dependent on K t, and G t, and A t. For L t, this may (54) 4

15 be noted from equation (54), which has L t dependent on C t, which in turn is dependent itself on the prior noted variables. This then suggests that the rules guiding consumption and employment must follow the following form, C t a CK Kt a CA Ã t a CG Gt (55) L t a LK Kt a LA Ã t a LG Gt (56) where the a s are dependent on the parameters of the model, while. denotes the deviation of the log of the variable from its log value on the balanced growth path. As in the simpler model in the previous section, focus will be on equations (25) and (28). The method we are applying here is known as the method of undetermined coefficient, where we have used our theory to guess at the structure of the solution. To formally solve for the solution to the a s, first focus on the simpler intertemporal condition of equation (28). First log linearizing the equation, and then performing a first order Taylor series expansion for each variable around its balanced growth path value, in log s. ln c t = w t l t b ln c t ln( l t ) = ln w t ln b ( ) α = ln α ln K t ( α) ln A t α ln L t b ( ) ( Ct ln ln L ) ( ) t α = ln α ln K t ( α) ln A t α ln L(57) t N t N t b ) ln N t ln(n t L t ) = (58) ( Ct N t Performing the first order Taylor series expansion around the log of the balanced growth path values, ( ) ) C ln N C t ln ( L L N N L L t ( ) α = ln α ln K α b K t ( α) ln A ( α) (A) t α ln L α L t Therefore, ( ) L C t L N L t = α K t ( α) (A) t α L t ( ) L C t N t L α L t = α K t ( α) (A) t (59) 5

16 where N t = e Nnt from our assumption, and noting that, ln L t L t = L t L t ln L t = L t Alternatively, the coefficient of L t can be rewritten as l /( l ). We can now substitute our linear rules for C t and L t into equation (59) to obtain, ( ) L ( a CK Kt a CA Ã t a CG Gt N t L α a K LK t a LAÃt a G ) LG t = α K t ( α) (A) t (60) Therefore the coefficients must be, ( ) L a CK N t L α a LK = α (6) ( ) L a CA N t L α a LA = α (62) ( ) L a CG N t L α a LG = 0 (63) This of course not sufficient to identify the coefficients, that is the a s. We would still need to do the same for equation (25). However, before we do so, it is useful to examine what these three equations are telling us. The first and second conditions relate to describing how consumption and labour supply would be affected by changes in capital, and a technology shock respectively. In the first condition, a (positive) change in capital would lead to an increase in demand for labour, holding the production function unchanged. This increase in demand would inturn drive wages higher for a given labour supply. The condition thus says that in response to such a change in capital, the individual/household has the option of either increasing labour supply, or consumption or both, and the marginal change will equate with the elasticity of the wage with respect to capital (given labour), α. This is likewise true of the second condition, which describes the condition for a change in technology, and where the elasticity is α instead. The final condition relates how consumption, and labour supply is affected by changes in government expenditures. Note that although the public sector does not enter directly into (28), the mechanics operates through the decisions made by the household. When there is a change in government spending, households can change its labour supply choices in response, which then affects equilibrium wages, and consequently the utility/disutility it obtains from labour, 6

17 and this in turn affects consumption. Because the right hand side of the equation is zero, this means that the household response to a change in government expenditure, through consumption and labour supply, must move in opposing directions. To do the same for analysis for equation (25), first let ( ) ( ) rt r Z t = ln ln c t c l A = [ln( r t ) ln( r )] (ln c t ln c ) ln l ln A = [ln( r t ) ln( r )] (ln c t ln c ) ln L ln N t ln A = [ln( r t ) ln( r )] ln C t ln N t ln C ln A ln L ln L ln N t ln A = [ln( r t ) ln( r )] C t ( ) ( ) rt r Z t = ln ln c t c l A ( = ln r ) t c t c t Next, note that our conjectured solution can also be written as, C t a CK Kt a CA Ã t a CG Gt as well, from the equilibrium rent, ( ) α At L t r t = α δ K t ( At L t r t = ( δ) α K t ) α = ( δ) α Kα t(a t L t ) α K t = ( δ) α Y t K t = ( δ)k t αy t K t ln( r t ) = ln(( δ)k t αy t ) ln K t With this, we can now perform a first order Taylor series expansion, noting that for the 7

18 right hand side, we are doing it for the variable ln( r t ). ln( r ) [ln( r t ) ln( r )] = ln(( δ)k αy ) ln K [ ( δ)k α 2 Y α( α)y K r t r α( α)y L r t K t [ ( δ)k α 2 Y α( α)y r t = K r t r α( α)y L r t K t ] à t ] à t With this we can substitute the conjectured solution for L t, ( δ)k α 2 Y α( α)y r t = K r t à r t α( α)y (a r LK Kt a LA à t a LG Gt ) K t ( δ)k α 2 Y α( α)a LK Y = K r t α( α)y α( α)a LA Y à r t α( α)alg Y G r t r t β rk Kt β ra à t β rg Gt (64) However, before we can substitute this into Z t, realize that K t is an endogenous variable in the model. We can however substitute the K t equation of motion, after performing the requisite log linearization. K t = ( δ)k t Y t C t G t ln K t = ln(( δ)k t Y t C t G t ) ln K K ( δ)k t = ln(k αy ( α)y ) K K t à K t ( α)y C G L K t C K t G K t ( δ)k K αy ( α)y t = K K t à K t ( α)y C G L t C t G t K 8 K K

19 Next, substituting our conjectured solution to C t and L t into the equation of motion, we have, K tt = = = ( δ)k αy ( α)y K K t à K t ( α)y (a K LK Kt a LA à t a LG Gt ) C G (a K CK Kt a CA à t a CG Gt ) G K t { } ( δ)k [α ( α)a LK ]Y a CK C K K t { } [( α) ( α)ala ]Y a CA C à K t ( α)alg Y a CG C G G K t { } ( δ)k [α ( α)a LK ]y a CK c K k t { } [( α)( ala )]y a CA c à k t ( α)alg y a CG c G G t k K t b KK Kt b KA à t b KG Gt (65) Now we can substitute the above equation (65) into equation (64) to obtain, r t = β rk (b KK Kt b KA à t b KG Gt ) β ra à t β rg Gt = β rk b KK Kt β rk b KA à t β ra à t β rk b KG Gt β rg Gt (66) This equation (66) can now be substituted into Z t to obtain, Z t = β rk b KK Kt β rk b KA à t β ra à t β rk b KG Gt β rg Gt a CK Kt a CA à t a CG Gt = (β rk a CK )b KK Kt (β rk a CK )b KA à t (β rk a CK )b KG Gt (β ra a CA )Ãt (β rg a CG ) G t (67) = γ rck Kt γ rcka à t γ rckg Gt γ rca à t γ rcg Gt (68) Notice now that the Euler equation is in terms of the endogenous variable K, and the two stochastic variables, A and G. All that is left is for us to take expectations, and find the 9

20 coefficients of K t, Ãt, and G t. It should be noted that since we have log linearized equation (25) in getting thus far, the expectations should be E t e Z t. However, by assuming Z t, t, to be normally distributed, this imples that e Z t would in turn be log normally distributed. Letting Z t N(µ, σ 2 ), this thus means that ln E t (e Z t ) = E t (Z t ) σ2 2. This assumption is in line with the structure of the model since the white noise associated with technology and government spending are assumed to be normally distributed. Thus, E t ( Z t ) = E t ( γ rck Kt γ rcka à t γ rckg Gt γ rca à t γ rcg Gt ) ) ( ) = γ rck Kt γ rcka à t γ rckg Gt γ rca E t (Ãt γ rcg E t Gt = γ rck Kt γ rcka à t γ rckg Gt γ rca ρ A à t γ rcg ρ G Gt E t ( Z t ) = γ rck Kt (γ rcka γ rca ρ A ) Ãt (γ rckg γ rcg ρ G ) G t (69) And equation (69) provides the remaining three equations that allows us to solve for the a s, albeit will be a complicated endeavour. From the derivation of (69), it is clear that it is not possible to derive any qualitative intuition as was done in for the intratemporal equilibrium condition. The only way we can gain an understanding into the mechanics behind the two shocks is through numerically replication of the model, equating the parameters of the model with informed empirical estimates, thereby examining the effects on the a s, and b s coefficients. With the coefficients in hand, the three equations of (55), (56), and (65), thus define consumption s, employment s, and capital s response to the real shocks of the model respectively. We can also use the remaining equations to describe how the other variables of the model would respond to real shocks. For instance, by log linearizing the production function, Ỹ t = α K t ( α) (Ãt L ) t = α K ) t ( α) (Ãt a LK Kt a LA à t a LG Gt = (α ( α)a LK ) ( α)( a LA )Ãt ( α)a LG Gt we can see how output responds to real shocks. The same is true for wages, rents/interest rates, and investments. You should try to complete the rest of the model, which is very simple. You should also avail yourself of the predictions of model derived from numerical methods. 20