Lecture Notes 7: Real Business Cycle Zhiwei Xu (xuzhiwei@sjtu.edu.cn) In this note, we introduce the Dynamic Stochastic General Equilibrium (DSGE) model, which is most widely used modelling framework in modern macroeconomics especially in the business cycle eld. We rst introduce the workhorse model Real Business Cycle (RBC) theory developed by Kyland and Prescott (98, Econometrica) in early 98s. Introduction We now brie y look at the business cycle stylized facts regarding the macroeconomic variables including output, consumption and investment, etc. We use U.S. economy as an example. Figure a plots the raw series of real consumption, investment and GDP after log-transformation. It can be seen that the real-time series uctuate around stochastic trends (dashed line). Figure b plots the business cycle component of above three variables. All the series are detrended by the HP lter. It can be seen that during the recession all variables experience sharp decline. In particular, consumption, investment positively comove with output, or these two variables are procyclical. Moreover, consumption is less volatile than GDP indicating that households are smoothing their consumption. While, the investment is much volatile than the GDP. The business cycle moments are summarized in Table. Business cycle theory aims to explain the uctuations in aggregate economy. Speci cally, two central questions need to answer: (i) what are the sources (shocks) of business cycles; (ii) what is the propagation mechanism of these shocks. In the business cycle literature, there are two major schools of thoughts, which have opposite policy implications. The Classical School. Their doctrine is supply determines demand, i.e., supply shocks are the major source of business cycles. Supply shocks include shocks to technology (TFP), to endowment, to costs of production, etc. The propagation mechanisms include real rigidities, capital accumulation, time-to-build, etc. The policy recommendation is do nothing. RBC theory is the mainstream of this school. The Keyesian School. Their doctrine is demand determines supply, i.e., demand shocks are the major source of business cycles. Demand shocks include shocks to consumption, to investment, to government spending, to exports, and to money supply. Speci cally, shocks Part of this note is borrowed from Prof. Wen s lecture notes at Tsinghua University. I thank him for kindly sharing his materials.
to consumption or investment also include extrinsic uncertainty (expectations, animal spirits, sunspots). The propagation mechanisms include nominal rigidities, nancial accelerator. Their policy recommendation is intervene. Figure a. Raw Series of Consumption, Investment and GDP 9.5 Consumption Investment GDP 9 8.5 8 7.5 7 6.5 6 5.5 975 98 985 99 995 5
3 Figure b. Business cycle components (HP ltered)..5 Recession Consumption Investment GDP..5.5..5. 975 98 985 99 995 5 Table. Business cycle moments of HP ltered data Y C I Standard Deviation :58 :9 :539 std(x) std(y ) : :86 3:44 Correlation with Y :8698 :84 Autocorrelation :8784 :89 :96 The business cycle thoeries aim to explain the abve stylized facts. We rst introduce RBC theory and its various extensions. In the future lectures, we will introduce the monetary DSGE model with price stickiness (New Keyesian theory).
4 Real Business Cycle Theory The benchmark RBC model is essentially discrete-time Ramsey model with leisure decision and stochastic environment. The key assumptions includes: (i) Prices adjust instantaneously to clear markets; (ii) Individuals have rational expectations. There are other minor assumptions, including perfect competition (allowing monopolistic competition does not change the model dynamics), perfect risk sharing (market is complete), no asymmetric information (there is no principle-agent problem), no externalities.. A Benchmark Model As there is no frictions, we consider a social planner s optimization problem. The social planner chooses paths of consumption C t, capital stock K t+ ; and employment (hours worked) n t to solve subject to resource constraint max fc t;k t+ ;n tg t= X E t [log C t + a n log ( n t )] () t= C t + K t+ ( ) K t = A t K t n t ; () and K > is given, where A t denotes a random technology shock to total factor productivity (TFP). We assume that A t has two components where a t is the stochastic part that follows and X t is the deterministic part that follows A t = a t Xt ; (3) log a t log a = (log a t log a) + t ; t ~N ; ; (4) Note that all the randomness comes from the a t : X t = X t ; > : (5) Since there is a potential growing trend in the economy, before solving the model, we need to transform the economy to the stationary one. We use the little case to denote the detrended (stationary) variables. In particular, we de ne c t C t =X t ; k t+ K t+ =X t+ : We do not detrend the hours worked because n t is bounded (one person has only 4 hours at most for one day). The optimization problem () can be transformed to max fc t;k t+ ;n tg t= X E t [log c t + a n log ( n t )] (6) t=
5 subject to resource constraint where a t follows (4). Note that the investment i t is given by c t + k t+ ( ) k t = a t k t n t ; (7) i t = k t+ ( ) k t : (8) Denote Lagrangian multiplier for (7) as t ; then FOCs for fc t ; n t ; k t+ g are given by c t = t ; (9) a n = t ( ) at kt n t ; () n t t = E t t+ a t+ kt+ n t+ + : () The above three equations combining with (7) de ne the full dynamic system. As the system is highly nonlinear, to obtain the solution we have to linearize the system around the steady state. We now rst solve the steady state. Steady State. In the steady state, () implies = or we can obtain the capital-output ratio k y = The investment-output ratio is given by The resource constraint then implies To fully solve the steady state, from () and (9), we have which further implies that the steady-state labor is y k + ; () ( ) : (3) i y = ( + ) k y : (4) c y = ( + ) k y : (5) a n n = c ( ) y n ; (6) n ss = c y a n < : (7) +
6 Furthermore, we have which implies y n k = a ; (8) k k ss = y=k a n ss ; (9) y ss = a (k ss ) (n ss ) ; () c ss = y ss c y ; () i ss = y ss i y : () We now transform the system into linearized version. In principle, we can do Taylor expansion for each equation around the steady state. But the literature often linearize the system through so-called Log-linearization method. That is, doing linearization for the logtransformed variable log z t instead of z t : Log-linearization. Suppose that there is a equation h t = f (x t ; y t ) : (3) De ne log-transformed variables as ~x t = log x t ; ~y t = log y t ; ~ h t = log h t : Then we can rewrite f (:) as e ~ h t = f e ~xt ; e ~yt : (4) Do Taylor expansion w.r.t. f~x t ; ~y t g around the steady state ~x ss ; ~y ss ; we have e hss ~ ht ~ h ss = f x e ~xss ; e ~yss e ~x ss (~x t ~x ss ) + f y e ~xss ; e ~yss e ~y ss (~y t ~y ss ) : (5) De ne new variables as ^x t = (~x t ~x ss ) ; ^y t = (~y t ~y ss ) and ^h t = ~ht h ~ ss ; which indicate the percentage deviation of x t ; y t and h t from their steady-state values. Equation (5) can be reduced to Several examples for f (x; y) : ^h t = f x (x ss ; y ss ) x ss f (x ss ; y ss ). f (x; y) = ax + by: Then we have ^x t + f y (x ss ; y ss ) y ss f (x ss ; y ss ^y t : (6) ) ^h t = a xss f ss ^x t + b yss f ss ^y t: (7). f (x; y) = axy: Then we have ^h t = ^x t + ^y t : (8)
7 3. f (x; y) = g (x t ) g(x t ): Just do the linearization, and we have where ^h t = h t h ss ; ^x t = x t x ss : 4. f (x; y) = yt y ss x t x + y ss t : Then we have 5. f (x; y) = +g yt y ss x t : Then we have ^h t = = x ss ^h t = g x (^x t ^x t ) ; (9) ^h t = ^y t : (3) +g yt x t y ss x ss yt x t ; where g () = g () = ; g () > x ss g () + g () yss (x ss ) + y ss x ss x ss g () y ss^y t + g y ss () (x ss ) + g () [:::] g y ss y ss () (x ss ) x ss x ss^x t y ss 3 g () (^y t ^x t ) (3) x ss Now go back to the RBC model. The log-linearized system for (7) and (9) to () are c y ^c t + k y ^k t+ ( ) k y ^k t = ^a t + ^k t + ( ) ^n t ; (3) ^ t = E t^t+ + ^c t = ^ t ; (33) n ss n ss ^n t = ^ t + ^a t + ^kt ^n t ; (34) y=k y=k + E t h ^a t+ + ( ) ^kt+ ^n t+ i ; (35) ^a t = ^a t + t : (36) Note that the state variable is k t ; the control variable is fc t ; n t g ; and the co-state variable is t : We can write the above system in a more compact way as follows. For the state and co-state variables, we have = k y ( )y=k y=k+ ( ) k y + + + c y ^a t + E t ^kt+ y=k y=k+ ^ t+ ^kt ^ t ^c t + ^n t ( )y=k y=k+ E t ^c t+ ^n t+ E t^a t+ : (37)
8 For the control variables, we have n n + ^c t ^n t = ^kt ^ t + ^a t : (38) Rearrange the terms, we may rewrite above two equations as ^c t ^kt = A + A ^a t ; (39) ^n t ^ t E t ^kt+ ^ t+ = B ^kt ^ t + B ^a t + B 3 E t^a t+ : (4) To solve di erence equation system (4), we employ the similar method used in Ramsey model. In particular, we have P E t ^kt+ ^ t+ = P ^kt ^ t + P B ^a t + P B 3 E t^a t+ : (4) The existence of saddle path implies that the number of eigenvalues in that are larger than is equal to the number of co-state variables. In this case, must have one eigenvalue that ^x t ^kt is greater than. Rede ne = P p ^kt + p ^t = ; then we have ^x t ^ t p ^kt + p ^t E t^x t+ = ^x t + ~ B ^a t + ~ B 3 E t^a t+ ; (4) where ~ B = P B ; ~ B3 = P B 3 : Suppose > ; and the second equation in (4) can be expressed as ^x t = E t^x t+ ~B (; ) ^a t + ~ B 3 (; ) E t^a t+ To solve the above equation, we need to do forward iteration. That is, ^x t = = = X j= ~ B (; ) ^a t j E t ~B (; ) ^a t+j + ~ B 3 (; ) ^a t++j X j= j B ~ (; ) + B ~ 3 (; ) E t^a t+j (43) X j E t^a t+j ; (44) j= where = ~ B (; ) ; j = j ~B (; ) + B ~ 3 (; ) :
9 Recall the de nition of ^x t ; from which we have or p ^kt + p ^t = ^x t = ^ t = p X j= X j E t^a t+j ; (45) j= j E t^a t+j p p ^kt : (46) Putting last equation into the rst line in (4), we then get the policy function of ^k t+ ^k t+ = B (; ) ^k t + B (; ) ^ t + B (; ) ^a t + B 3 (; ) E t^a t+ = B (; ) B (; ) p ^k t + B (; ) ^a t + B 3 (; ) E t^a t+ + B (; ) = p B (; ) B (; ) p ^k t + p p j= X j E t^a t+j X ~ j E t (^a t+j ) ; (47) j= where ~ = B (; ) + B (; ) p ; ~ = B 3 (; ) B (; ) p ; ~ j = B (; ) p j for all j > : Once we have the policy function of ^k t ; we can solve the optimal paths for other variables such as c t ; n t ; t ; i t and y t ; etc. To summarize, the optimal path of our benchmark RBC model takes a general form X ^k t+ = M sk^kt + ~ j E t^a t+j ; (48) c t n t j= X = M ck^kt + M ca j E t^a t+j ; (49) j= where M ck and M ca are matrices, j is obtained by putting (47) and (46) into (39). Several important issues here.. The optimal solution (48) and (49) is indeed the rational expectation equilibrium if E t is mathematical expectation operator.. Any policy evaluation based on the DSGE model does not su er the Lucas Critique. 3. Individuals expectations about future may directly a ect their economic behaviors in current period. Hence, news about the future may cause uctuations in current period, even though there is no change in today s economic fundamental.
. Numerical Exercises.. Calibration The advantage of DSGE models is that one can use them to conduct quantitative analysis through the simuation. The optimal solution (48) and (49) implies that the dynamics of endogenous variables are function of deep parameters. Therefore, before we do simulation, the value of parameters should be set. This work is called calibration, meaning that the parameter value must be calibrated such that the model economy is basically consistant with the target economy. DSGE literature calibrate the deep parameters such that the values of structural parameters roughly t the counterpart in the target economy. We take the U.S. economy as an example. Denote the parameter set as : The benchmark RBC model has following deep parameters f; ; ; ; a n ; ; g : In particular, indicates the capital share in total income, for U.S. economy, the value is around.4. is the average growth rate for one-quarter. As it is very small, we set it to be. As = indicates the real interest rate, it is set to be.99 for one quarter according to U.S. 3-month real interest rate. is the depreciation rate, according to some independent studies, it is around.5. The average working hours for a typical U.S. worker is around /3 of the total time, implying that n ss is.33. (7) implies that n ss depends on a n : Thus we set a n such that n ss = :33: is the coe cient in AR() process, and is the std of technology shock t. In the literature, people use Solow residual as a proxy of a t ; they nd the is about.979, and = :7: The model implied steady-state ratios are k=y = :4; i=y = :849; c=y = :7:.. Simulation Once we set the values of parameters, we can write computer code to solve the model to get the solution (48) and (49). As a t is AR() process, the optimal solution takes a general form Or more generally, ^S t ^C t ^kt+ = M ^St + M t ; (5) ^a t 3 ^c t ^n t 6 4 ^y 7 t 5 = ^S t : (5) ::: y t = g y y t + g u t ; (5) h where y t = ^St ; ^C i t ; t N (; e ) is m (i.e., there are m shocks).
Impulse Responses. Impulse response function captures the dynamic e ects of one-unit of shock on the economy over time. Therefore, through the pattern of impulse responses, we can study the propagation mechanism of the model. In particular, the impulse response of t on the variable x t is de ned as IR (j) = @x t+j @ t ; for j : (53) For the state variables, (5), the impulse responses function of ^S t is For the control variables, the impulse responses is IR s (j) = M M j : (54) IR c (j) = M M j : (55) Population Moments. Another important exercise is to calculate the population moments (e.g. standard deviations, correlations, etc) as we did for the U.S. data. Given the process (5), to calculate the -th ( > ) moment (), we multiply both sides by ^S t and take expectation or h E ^St ^S t i h = M E ^St ^S t i h + M E t ^S t i ; (56) () = M ( ) = M () : (57) For () ; we have () = M ( ) + M M ; (58) where = V ar ( t ) : Since we have () = ( ) ; last equation implies Some linear algebra calculation gives () = M () + M M = M () M + M M : (59) vec f ()g = [I M M ] vec M M : (6) Variance Decompostions. The h period-ahead forecase error of y t is de ned as Xh y t+h E t y t+h = k t+h k ; (6) k= where k = g k yg u : The variance of forecast error y t+h E t y t+h is Xh Var (y t+h E t y t+h ) = k k : (6) k=
In particular, the variance of forecast error of the i-th variable in y t+h is Xh Var (y i;t+h E t y i;t+h ) = k (i; :) k (i; :) ; (63) where k (i; :) is the i-th row of matrix k : Remember that the ^ t contains m shocks, thus each shock contributes to the variance of y i;t+h E t y i;t+h. Let us take the rst shock as example. The fraction of Var (y i;t+h E t y i;t+h ) explained by the rst shock ( t ) is given by FEVD (h) = P h k= k= k (i; :) vv k (i; :) P h k= k (i; :) k (i; :) (64) where v is the vector whose i-th element is and other elements are zero. 3 Simulations in Dynare Dynare is a toolbox based on the MATLAB. Thus, rst you need to install the MATLAB. Second, download Dynare via : http://www.dynare.org/download and install it. Last step, add the dynare installation folder as MATLAB toolbox path. The Chapter 3 and 4 in the Dynare user guide provides an excellent introduction. Here I provide a Dynare example (see the course website) to solve the RBC model we de ned in the lecture notes. Detailed illustration will be presented in the class. The dynare codes can be downloaded from the course website: http://macrosjtu.wix.com/macrotopics. 3. Numerical Results Dynare saves all the results in a mat le with name Filename_results.mat. For instance, if your Dynare code s name is RBC.mod, the results will be stored in a mat le with name RBC_results.mat. Load the result le, you may nd a structural variable oo_. All the outcomes regarding the simulation are stored in this variable. In particular, the coe cient matrices for the decision rules (policy functions) are in oo_:dr_ghx and oo_:dr_ghu. That is, the decision rules are y t = oo_:dr_ghx y state t + oo_:dr_ghu t ; (65) where y t records all the endogenous variables including state variables, control variables, etc; y state t contains state variables; t are innovations of structural shocks. However, to get the decision rules in a correct order (the order that you de ne variables in var module), you have to re-order the coe - cient matrices by oo_:dr:ghx(oo_:dr:inv_order_var; :) and oo_:dr:ghu(oo_:dr:inv_order_var; :). Let s take the baseline RBC model presented in section. as a concrete example. The Dynare code is presented below (can be downloaded from the course website).
3 Example Dynare code for the baseline RBC model: %=========================================== % A Benchmark RBC model: Log-linearized case % Zhiwei Xu, Advanced Macro I, SJTU % in this code we set the growth rate to be zero %=========================================== var Ct Yt Kt Nt It Wt Rt At ; % de ne the varibles appear in dynamic system %=========================================== varexo e_at; % de ne the exogenous technology shock %=========================================== % initialize the deep parameters parameters alpha beta del rho_a sig_a n_ss; alpha=.4; % capital share in production function beta=.99; % subjective discounting rate del=.5; % depreciation rate rho_a=.97; % AR() coe cient of production technolgoy process sig_a=.7; % standard error of the tech shock n_ss=.33; % steady-state labor %=========================================== model(linear); % declare the full dynamic system, here we simply input the log-linearized version. KY=alpha*beta/(-beta*(-del)); % calculate great ratio using deep parameters de ned above IY=del*KY; CY=-IY; Yt-Nt-Ct=n_ss/(-n_ss)*Nt; % labor market equilibrium -Ct = -Ct(+)+alpha*beta/KY*(Yt(+)-Kt); % Euler eqution for capital stock
4 CY*Ct+IY*It=Yt; % resource constraint Kt=(-del)*Kt(-)+del*It; % capital accumulation Yt=At+alpha*Kt(-)+(-alpha)*Nt; % production function Wt = Yt-Nt; % labor demand Rt = Yt-Kt(-); % capital demand At=rho_a*At(-)+e_at; %tech shock end; %=========================================== initval;% input the initial guess of steady-state, for the log-linearized varibles, they are just zero. Yt=;Ct=;Kt=;It=;Nt=;At=;Rt=;Wt=; end; %=========================================== % declare the exogenous shock shocks; var e_at; stderr sig_a; % de ne the std of A shock end; steady; % report the steady state, in log-linearized case, all variables are zero check; % compute the eigenvalues and check the Blanchard-Kahn condition %=========================================== stoch_simul(periods=5,order=,hp_ lter=6,nograph,irf=4); % end of the code After running the above code, Dynare reports all the results. De ne coe cient matrices with the correct order: oo_:dr:ghx(oo_:dr:inv_order_var; :) and oo_:dr:ghu(oo_:dr:inv_order_var; :);
5 the decision rules is given by 3 3 3 ^C t :696 :43 :4353 ^Y t :59 :338 :3796 ^K t :9595 :99 :937 ^N t ^I = :349 :637 ^Kt t :69 3:637 + :636 ^A t 3:7497 t : ^W t :494 :745 :7469 6 4 ^R 7 6 t 5 4 :749 :338 7 6 5 4 :3796 7 5 ^A t :97 : {z } {z } oo_:dr:ghx(oo_:dr:inv_order_var;:) oo_:dr:ghu(oo_:dr:inv_order_var;:) (66) The impulse responses (IRF) are stored in oo_irfs. Dynare also directly report IRFs of each endogenous variable with the name VARIABLENAME_SHOCKNAME. For instance, the IRF of output to t in this case is Yt_e_at: The gure below reports the IRFs to a positive onestandard-deviation of the technology shock. 6 x 3 Ct.4 It 4.. 3 4 Yt 3 4 5 x 3 Nt.5. 3 4 Rt 5 3 4 8 x 3 Wt 6.. 3 4 Kt 4. 3 4 At.5.5 3 4 3 4 Figure. IRFs to the technology shocks. The IRF of output shows that under an AR() process of technology shock, the output presents very similar dynamics to an AR() process. This implies that the baseline RBC model does not have rich dynamics or equivalently it does not have propagation mechanism. To see this, from the
6 decision rules (66), the process of output is given by ^Y t = :59 ^K t + :3796 ^A t = :3873 :8985L :9595L ^A t ; where the second line is obtained by substituting ^K t with the decision rule of ^K t : Since the term :8985L :9595L approximates to, the output has almost the same dynamics as that of technology shock ^A t : Cogley and Nason (995, AER) criticize that the RBC model is lack of propagation channel to amplify the uctuations. In 99s, the one direction of the RBC literature is to nd ways to make the model have rich dynamics. investment adjustment costs, etc. For instance, introduce real rigidities such as habit formations, In the dynare code, we also simulate the series of endogenous variables with 5 periods (see the stoch_simul line in the code part). The simulated series are stored in oo_endo_simul. Dynare also directly report simulated series of each endogenous variable with the name VARI- ABLENAME. For instance, the simulated output in this case is Yt. One purpose to simulate a long period of variable is to compute the model-simulated moments of each variable. For example, to compute the standard deviation of ^Y t ; we just type std( Y t) in the matlab. Notice that this way may not be accurate. Dynare also reports the theoretical moments which are stored in oo:var_ and oo:autocorr_. Table below reports the business cycle moments obtained in Dynare. The table shows that in general, RBC model has good performance to account for the aggregate uctuations. However, the model does not explain the labor market dynamics very well. Table. Business cycle moments (Model v.s. Data) Y C I N Standard Deviations Relative to Y U.S. Data..545.467.539 Baseline Model..349.75.468 First Order Autocorrelations U.S. Data.93.93.867.955 Baseline Model.7366.7995.78.76 Correlation with Y U.S. Data..996.975.88 Baseline Model..995.993.983
7 3. Expectation Driven Business Cycles (EDBC) Some empirical studies (Beaudry and Portier, 6; Beaudry and Lucke, ; Schmitt-Grohe and Uribe, ) show that news shocks, or anticipated shocks account for a large fraction of the total variation in aggregate series. In particular, when positive news (e.g. TFP will increase in the future) hit the economy, the output, consumption, investment, hours worked all increase. The comovement of aggregate variables under the news shock is called expectation driven business cycles (EDBC), see Beaudry and Portier (7). However, the standard real business cycle model is incapable of generating an EDBC. Intuitively, this is because without any real frictions, a positive news about future total factor productivity (TFP) will increase future income and therefore induce forwardlooking households to raise their current consumptions. Meanwhile, the income e ect increases households leisure. As a result, equilibrium labor decreases, causing output fall as well because the capital stock is predetermined. Consequently, positive news about future TFP results in opposing movement in output and consumption. Conside the baseline RBC model we discussed in the previous section. Suppose there is a news about TFP that after four periods, the TFP will increase %. The following gure reports the IRFs. It can be seen that under the positive news, consumption goes up, while output, investment, hours worked go down. There is no positive comovement among aggregate variables. Therefore, the RBC model is unable to generate the EDBC.
8 C t I t.8 5.6.4 4 6 8 5 4 6 8 Y t N t 4 6 8 4 6 8 R t W t.5 4 6 8 4 6 8 K t A t.5.5 4 6 8 4 6 8 Figure. IRFs to the positive news shock about future TFP In the Dynare, it is fairly easy to simulate the news shocks. Just replace the shock process in Example At=rho_a*At(-)+e_at; with At=News(-4); News=News(-)+e_at;. Notice that the news shock de ned in this way is implicitly assuming that the news will realize for sure. However, some news may not be true or it will not realize in the future. Dynare cannot directly model the unrealized news shocks. It is a bit complicated to compute the IRFs for the unrealized news. Notice that the IRFs for the realized news and unrealized news are the same for those periods before the realization period. For instance, if the news says that the TFP will increase in the fourth period, then the impulse respones of the economy in periods -4 are the same for the realized news and the unrealized news. The di erences in impulse responses will occur after the period 4. Therefore, to compute the impulse responses to the unrealized news, you can use the Dynare by setting the news as if it is true, and obtain the impulse responses for periods -4. For those periods after 4, you need to compute the impulse responses as if the economy is hit by a surprising shock that o sets the news
shocks (this makes the news unrealized). 9