A User's Guide to Solving Real Business Cycle Models
Contents
1. 101k 0 06h 0 102 18 h 2 786 k 2 78Z A random number generator can next be used to produce a sequence of technology shocks The above equilibrium equations can then be used to produce time series for capital consumption labor and output V ANALYZING OUTPUT FROM THE ARTIFICIAL ECONOMY The solution to the model is characterized by eqs 16 18 given initial values for capital and next generating a path for the exogenous technology shock z these equations will produce time series for z k h Two other series that most macroeconomists are interested in namely output and investment can be generated by linearizing the production function and the resource constraint respectively Specifically for output linearizing the assumed Cobb Douglas production function i e y z k h and using the calibrated value that 0 36 yields the following equation 19 5 Z 0 36 k 064h Finally a linear approximation of the condition that in equilibrium output must equal the sum of 13 page 14 Hartley Hoover Salyer RBC Models A User s Guide consumption and investment can be expressed in the form as a percentage deviation from the steady state as 20 lt lt l saje al Ol 2 Using the steady state values employed in the numerical solution the investment equation becomes 21 a 1 06 5 079 tt 3 925 2 922 0 27 0 27 Hence equilibriu2. unpublished manuscript Kydland Finn E and Edward C Prescott 1990 Business Cycles Real Facts and a Monetary Myth Federal Reserve Bank of Minneapolis Quarterly Review 14 2 Spring 3 18 reprinted here in Chapter 21 McGratten Ellen R 1994 A Progress Report on Business Cycle Models Federal Reserve Bank of Minneapolis Quarterly Review 18 4 Fall 2 16 Prescott Edward C 1986 Theory Ahead of Business Cycle Measurement in Federal Reserve Bank of Minneapolis Quarterly Review 10 4 Fall 9 22 reprinted here in Chapter 4 17 page 18 Hartley Hoover Salyer RBC Models A User s Guide Sargent Thomas J 1987 Dynamic Macroeconomic Theory Cambridge Mass Harvard University Press 18
3. 12 A B QAQ where Qis a matrix whose columns are the eigenvectors of A Band Aisa diagonal matrix whose diagonal elements are the eigenvalues of A B Using this decomposition and premultiplying both sides of the resulting expression in eq 11 by Q yields 13 Q u d AE d A Q u Note that the elements of the defined 4 x 7 column vector d are constructed from a linear combination of the elements in the rows of the 4 x 4 matrix Q and the elements of the 4 x 1 page 10 Hartley Hoover Salyer RBC Models A User s Guide column vector u Since A is a diagonal matrix eq 13 implies four independent equations 14 d hE died Ft 1237 i t i Since the equations in 14 must hold every period it is possible to recursively substitute the expressions forward for T periods to yield 15 AACE 238 The are four distinct eigenvalues associated with the four equilibrium conditions eqs 5 8 Since one of these conditions is the law of motion for the exogenous technology shock eq 8 one of the eigenvalues will be p Also the first rows of the matrices A and B are determined by the intratemporal efficiency condition since this is not a dynamic relationship one of the eigenvalues will be zero The remaining two eigenvalues will bracket the value of unity as is typical for a saddle path equilibrium implied by the underlying stochastic growth framework As implied by eq 15 the sta
4. 1 05 Z Next decomposing A B into QAQ and then pre multiplying by Q7 yields 2 18 0 048 0 048 2426 p 0 0 0 23 01 k n oS 136 0056 110 la Ore T 2 62 0 94 094 262 Z 1062 0 0 0 218 0 048 0 048 2426 0 105 0 0 0 0 0 2301 k 0 O 093 0 250 136 0 056 110 h 0 0 0 0 262 0 94 094 262 Z The entries in the matrix A i e the eigenvalues of AB determine the solution Note that the second diagonal entry is accounting for rounding error p The fourth row of A is associated with the intratemporal efficiency condition These values are proportional to those given in the first row of the A matrix consequently dividing all entries by 2 62 returns the original intra temporal efficiency condition The remaining two entries in the A matrix are those related to the saddle path properties of the steady state solution Since a stable rational expectations solution is associated with an eigenvalue less than unity the third row of the Q matrix provides the linear restriction we are seeking That is the rational expectations solution is 12 page 13 Hartley Hoover Salyer RBC Models A User s Guide 2 50 1 36 0 056k 1 10Z 0 Or 16 Z 054h 0 02k 0 44Z The law of motion for the capital stock the parameter values are given in the third row of the A matrix and the intratemporal efficiency condition provides two more equilibrium conditions 17 k 0 07
5. ratio is implied by the al directly determines h Typical parameter values based on postwar U S data see Hansen and Wright 1992 4 are 0 36 implying labor s share is 64 B 0 99 implying an annual riskless interest rate of 0 04 56 0 025 implying the capital output ratio where output is measured on a quarterly basis of roughly 10 and A 3 which implies that roughly 30 of time is spent in work activity These values will be used later in Section IV below II LINEARIZATION The solution to the social planner s problem is characterized by a set of policy functions page 6 Hartley Hoover Salyer RBC Models A User s Guide for capital consumption and labor moreover the solution exists and is unique see Prescott 1986 4 There is however no analytical solution To make the model operational therefore an approximate numerical solution is found One of the simplest methods is to take a linear approximation i e a first order Taylor series expansion of the three equilibrium conditions and the law of motion of the technology shock around the steady state values z k h z Provided the stochastic behavior of the model does not push the economy too far from the steady state behavior the linear approximation will be a good one The discussion below follows closely that of Farmer 1994 This technique is demonstrated below Intratemporal efficiency condition The optimal labor leisur
6. with the correlations of these series with output Table 1 Descriptive Statistics for U S and RBC Model relative i 0 82 0 76 consumption model investment U S data model U S data gt Statistics for U S data are taken from Kydland and Prescott 1990 21 Tables I and II p 10 11 15 page 16 Hartley Hoover Salyer RBC Models A User s Guide Figure 1 Output Consumption and Investment in RBC Model 0 3 deviation from steady state investment 0 2 16 page 17 Hartley Hoover Salyer RBC Models A User s Guide REFERENCES Debreu Gerard 1954 Valuation Equilibrium and Pareto Optimum Proceedings of the National Academy of Science 40 588 92 Farmer Roger E A 1993 The Macroeconomics of Self fulfilling Prophecies Cambridge MA MIT Press Hamilton James D 1994 Time Series Analysis Princeton New Jersey Princeton University Press Hansen Gary D 1985 Indivisible Labor and the Business Cycle Journal of Monetary Economics 16 3 November 309 28 reprinted here in Chapter 8 Hansen Gary D and Randall Wright 1992 The Labor Market in Real Business Cycle Theory Federal Reserve Bank of Minneapolis Quarterly Review 16 2 Spring 2 12 reprinted here in Chapter 9 Hoover Kevin D and Kevin D Salyer 1996 Technology Shocks or Colored Noise Why Real Business Cycle Models Cannot Explain Actual Business Cycles
7. ble rational expectations solution to the expectational difference equation is associated with the eigenvalue with a value less than one That is if gt Z then iterating forward implies d gt which is not a permissible equilibrium Furthermore for eq 15 to hold for all T again taking the limit of the right hand side in the stable case when A lt 1 it must be the true that d 0 this restriction provides the desired solution That is d 0 imposes the linear restriction on z ee z which is consistent with a rational expectations solution Recall that d represents a linear combination between the elements of a particular row of Q and the elements of the vector u za 10 page 11 Hartley Hoover Salyer RBC Models A User s Guide IV A PARAMETRIC EXAMPLE In this section a parameterized version of the RBC model described above is solved The following parameter values are used B 0 99 0 36 6 0 025 A 3 These imply the following steady state values z 0 79 k 10 90 h 0 29 y 1 06 Note that these values imply that agents spend roughly 30 of their time in work activities and the capital output ratio is approximately 10 output is measured on quarterly basis both of these values are broadly consistent with US experience see McGrattan 1994 The remaining parameter values determine the behavior of the technology shock These are estimated by constructing the So
8. e choice is represented by condition N1 c 1 a A z ken Linearizing around the steady state values Z k h z c c a 1 a A k h k k a 1 0 A k h h h 1 a A k h z z y a ay aqenne E a fiae Ho ai Al z a 4 k 7 Note that in the last expression all variables have been expressed as percentage deviations from the steady state the first two terms modify the respective derivatives while the last term uses the fact that z in steady state Consumption can be expressed as a percentage deviation from steady state by using the steady state condition U a y Al on dividing both sides of the Recall that the general form for the Taylor series expansion of a function around a point x is P A pee LED pr E where N denotes factorial page 7 Hartley Hoover Salyer RBC Models A User s Guide equation by this expression and denoting percentage deviations from steady state as X eq 4 can be written as 5 ak ah z Intertemporal Efficiency Condition This efficiency condition is given by N2 T B Efex 0 z 407hice 1 8 Again linearizing around the steady state and expressing all variables as percentage deviations from steady state yields 1 Bea kh 1 8 E Bao a 1 kha Fa Bca 1 a k h E E h ha Bea kh E Z Multiplying each side of the equation by c and using the steady state condition SS2
9. ent and0 lt 6 lt 1 is the depreciation rate of capital The exogenous technology shock is assumed to follow the autoregressive process given in the last equation the autocorrelation parameter is 0 lt p lt 1 and the innovation to technology is assumed to have a mean of one and standard deviation The first two constraints in 1 is the economy wide resource constraint and the second is the law of motion for the capital stock Dynamic Programming Problem page 3 Hartley Hoover Salyer RBC Models A User s Guide This infinite horizon problem can be solved by exploiting its recursive structure That is the nature of the social planner s problem is the same every period given the beginning of period capital stock and the current technology shock choose consumption labor and investment Note that utility is assumed to be time separable that is the choices of consumption and labor at time t do not affect the marginal utilities of consumption and leisure in any other time period Because of this recursive structure it is useful to cast the maximization problem as the following dynamic programming problem for a discussion of dynamic programming see Sargent 1987 state variables at time f k z control variables at time t c h k v k z max f le 1h B E Vka Cy Kray oly 2 subject to c k z f k z k 1 8 oP and Za TZ E Note that investment has been eliminated by using the law o
10. f motion for the capital stock A solution to this problem must satisfy the following necessary conditions and resource constraint NI U Smee N2 U BE oe aa J RC hess 2 5 k sh k 1 8 c Where the notation U 1 t i 1 2 denotes the derivative of the utility function with respect to the ith argument evaluated at the quantities c l h f 1 1 2 has an analogous interpretation N7 represents the intra temporal efficiency condition the labor leisure tradeoff lit implies that the marginal rate of substitution between labor and consumption must equal the marginal product of labor The second condition N2 represents the intertemporal efficiency condition The left hand page 4 Hartley Hoover Salyer RBC Models A User s Guide side represents the marginal cost in terms of utility of investing in more capital while the right hand side represents the expected marginal utility gain at an optimum these costs and benefits must be equal To simplify the analysis again see Prescott 1986 4 for a justification assume the following functional forms U c J h Inc A 1 h f k z zk h The assumption that utility is linear in leisure is based on Hansen s 1985 8 model Then the three equilibrium conditions become c U zk h JA 3 c7 BE fer oz kesh 1 8 kai zkh T k 1 8 C A steady state equilibrium for this economy is one in wh
11. hile it is fairly straightforward to show that a competitive equilibrium exists it is difficult to solve for the equilibrium sequences directly Instead an indirect approach is taken in which the Pareto optimum for this economy is determined this will be unique given the assumption of representative agents As shown by Debreu 1954 the Pareto optimum as characterized by the optimal sequences for consumption labor and capital in this environment will be identical to that in a competitive equilibrium Furthermore factor prices are determined by the page 2 Hartley Hoover Salyer RBC Models A User s Guide marginal products of capital and labor evaluated at the equilibrium quantities For a detailed exposition of the connection between the competitive equilibrium and Pareto optimum in a real business cycle model see Prescott 1986 4 We now provide an example of solving such a model I DERIVING THE EQUILIBRIUM CONDITIONS The first step in solving for the competitive equilibrium is to determine the Pareto optimum To do this the real business cycle model is recast as the following social planner s problem max e p u e1 A subject to 1 c i z f k h y Kis k 1 8 i pe a k is given where E denotes expectations conditional on information at t 1 0 lt B lt 1 is agents discount factor c denotes consumption 1 h is leisure agents endowment of time is normalized to one i is investm
12. ich the technology shock is assumed to be constant so that there is no uncertainty that is z 1 for all t and the values of capital labor and consumption are constant k k h h C C forall t Imposing these steady state conditions in 3 the steady state values are found by solving the following steady state equilibrium conditions SS1 1 a A k h SS2 B 7 1 8 ak h a 5 k SS3 k k h c 7 T In the above expressions Y denotes the steady state level of output Calibration page 5 Hartley Hoover Salyer RBC Models A User s Guide The next step in solving the model is to choose parameter values for the model This is done through calibration the set of parameters 5 B A a are chosen so that the steady state behavior of the model match the long run characteristics of the data The features of the data which do not exhibit cyclical characteristics are 1 1 a labor s average share of output 2 B Ti average risk free real interest rate 3 Given a A B choose 6 so that the output capital ratio from SS2 is consistent with observation 4 The parameter A determines the time spent in work activity To see this multiply both sides of SS1 by h and rearrange the expression to yield h 0 a Al y c But the steady state resource constraint SS3 implies parameter values chosen in the previous three steps Hence the choice of A that so that the output consumption
13. low residual and then detrending that series linearly Specifically the Solow residual is defined as Z In y Q Ink 1 a In h The Z series can then be regressed on a linear time trend which is consistent with the assumption of constant technological progress and the residual is identified as the technology shock z Using this procedure on quarterly data over the period 60 1 94 4 resulted in an estimate of the serial correlation of z the parameter p to be 0 95 The variance of the shock to technology i e the variance of in eq 8 was estimated to be 0 007 Note that the variance of the technology shock is not relevant in solving the linearized version of the model however when the solution of the model is used to generate artificial time series in the simulation of the economy this parameter value must be stipulated These values generated the following entries into the A and B matrices The use of the Solow residual as a measure of technology shocks is discussed in Hoover and Salyer 1996 11 page 12 Hartley Hoover Salyer RBC Models A User s Guide 1 036 036 1 VZ 0 0 0 0 ej 1 0 0 O k _ 1 0 022 0 022 0 035 s kz 0 072 1 010 0 062 0 098 0 1 0 o T es 0 0 0 095 Z 0 0 0 1 A Following the steps described in the previous section pre multiplying by A r yields the following 1 0 022 0022 0035 Z k 023 094 00051 027 k h 255 087 0057 275 h A 0 0 0
14. m in this economy is described by the following set of equations Z 0 54h 0 02k 0447 k 0 07 101k 0 06h 0 10Z h 278 k 2787 E F 036k 0 64h i 3 92 9 2 92 2 Z 0 952_ To generate the time series implied by the model it is necessary to first generate a series for the innovations to the technology shock i e These are assumed to have a mean of zero and a variance that is consistent with the observed variance for the innovations which as mentioned above is roughly 0 007 Then initializing Z 0 and using a random number generator in order to generate the innovations a path for the technology shocks is created Next assuming that all remaining values are initially at their steady state which implies that all initial values are set to zero the system of equations above can be solved to produce the time path for the endogenous variables 14 page 15 Hartley Hoover Salyer RBC Models A User s Guide We generate artificial time paths for consumption output and investment 3000 observations were created and only the last 120 were examined These are shown in Figure 1 It is clear from Figure 1 as is also true in the actual data that the volatility of investment is greater than that of output which is greater than that of consumption To see this more precisely the standard deviation of consumption labor and investment relative to output is reported in Table 1 along
15. page 1 Hartley Hoover Salyer RBC Models A User s Guide A User s Guide to Solving Real Business Cycle Models The typical real business cycle model is based upon an economy populated by identical infinitely lived households and firms so that economic choices are reflected in the decisions made by a single representative agent It is assumed that both output and factor markets are characterized by perfect competition Households sell capital k to firms at the rental rate of capital and sell labor h at the real wage rate Each period firms choose capital and labor subject to a production function to maximize profits Output is produced according to a constant returns to scale production function that is subject to random technology shocks Specifically y z f k h Where y is output and z is the technology shock The price of output is normalized to one Households decisions are more complicated given their initial capital stock agents determine how much labor to supply and how much consumption and investment to purchase These choices are made in order to maximize the expected value of lifetime utility Households must forecast the future path of wages and the rental rate of capital Itis assumed that these forecasts are made rationallyA rational expectations equilibrium consists of sequences for consumption capital labor output wages and the rental rate of capital such that factor and output markets clear W
16. that 1 B akh 1 8 yields aa SSE Ac t B a Tak ihr B 1 a a kh E f Ba kh E Za Resource Constraint Following the same procedure as before linearizing the resource constraint around the steady state yields page 8 Hartley Hoover Salyer RBC Models A User s Guide En ak 1 a k 1 0 hh Fes 9 E F z Technology Shock Process The critical difference between the steady state model and the real business cycle model is the assumption that technology shocks are random the shocks follow the autoregressive process described in eq 1 Linearizing the auto regressive process for the technology shock results in 8 Zai p t E A Taking expectations of both sides 9 E Z 1 p Z II SOLUTION METHOD The equations that define a rational expectations equilibrium eqs 5 6 7 9 can be written as a k vector expectational difference equation Let u where bold print denotes a vector then t z the linear system of equations can be written as 10 Au BE u The matrices A and B are page 9 Hartley Hoover Salyer RBC Models A User s Guide 1 a 0 1 ree 1 0 0 0 efk ak tht 41 8 1 a k A keh 0 0 0 p 0 0 0 0 eel Bla lak h Bil a ak th Bak th 0 1 0 0 0 0 0 1 Premultiplying both sides of eq 10 by A yields 11 u A BE u The matrix A B can be decomposed as see Hamilton 1994 for details
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