The Stochastic Growth Model Koen Vermeylen Download free books at Contents The Stochastic Growth Model Contents Introduction The stochastic growth model The steady state Linearization around the balanced growth path Solution of the linearized model Impulse response functions 13 Conclusions 18 Appendix A A1 The maximization problem of the representative firm A2 The maximization problem of the representative household 20 20 20 Appendix B 22 Appendix C C1 The linearized production function C2 The linearized law of motion of the capital stock C3 The linearized first-order condotion for the firm’s labor demand C4 The linearized first-order condotion for the firm’s capital demand C5 The linearized Euler equation of the representative household C6 The linearized equillibrium condition in the goods market 24 24 25 26 26 28 30 References 32 360° thinking 360° thinking 360° thinking Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities Discover the truth 2at www.deloitte.ca/careers Click on the ad to read more © Deloitte & Touche LLP and affiliated entities D The Stochastic Growth Model Introduction Introduction This article presents the stochastic growth model The stochastic growth model is a stochastic version of the neoclassical growth model with microfoundations,1 and provides the backbone of a lot of macroeconomic models that are used in modern macroeconomic research The most popular way to solve the stochastic growth model, is to linearize the model around a steady state,2 and to solve the linearized model with the method of undetermined coefficients This solution method is due to Campbell (1994) The set-up of the stochastic growth model is given in the next section Section solves for the steady state, around which the model is linearized in section The linearized model is then solved in section Section shows how the economy responds to stochastic shocks Some concluding remarks are given in section Download free eBooks at bookboon.com The stochastic growth model The Stochastic Growth Model The stochastic growth model The representative firm Assume that the production side of the economy is represented by a representative firm, which produces output according to a Cobb-Douglas production function: Yt = Ktα (At Lt )1−α with < α < (1) Y is aggregate output, K is the aggregate capital stock, L is aggregate labor supply and A is a technology parameter The subscript t denotes the time period The aggregate capital stock depends on aggregate investment I and the depreciation rate δ: Kt+1 = (1 − δ)Kt + It with ≤ δ ≤ (2) The productivity parameter A follows a stochastic path with trend growth g and an AR(1) stochastic component: ln At = ln A∗t + Aˆt Aˆt = φA Aˆt−1 + εA,t A∗t = A∗t−1 (1 with |φA | < (3) + g) The stochastic shock εA,t is i.i.d with mean zero The goods market always clears, such that the firm always sells its total production Taking current and future factor prices as given, the firm hires labor and invests in its capital stock to maximize its current value This leads to the following first-order-conditions:3 (1 − α) Yt Lt = wt = Et (4) Yt+1 1−δ α + Et + rt+1 Kt+1 + rt+1 (5) According to equation (4), the firm hires labor until the marginal product of labor is equal to its marginal cost (which is the real wage w) Equation (5) shows that the firm’s investment demand at time t is such that the marginal cost of investment, 1, is equal to the expected discounted marginal product of capital at time t + plus the expected discounted value of the extra capital stock which is left after depreciation at time t + Download free eBooks at bookboon.com The stochastic growth model The Stochastic Growth Model The government The government consumes every period t an amount Gt , which follows a stochastic path with trend growth g and an AR(1) stochastic component: ˆt ln Gt = ln G∗t + G ˆ t = φG G ˆ t−1 + εG,t G with |φG | < (6) G∗t = G∗t−1 (1 + g) The stochastic shock εG,t is i.i.d with mean zero εA and εG are uncorrelated at all leads and lags The government finances its consumption by issuing public debt, subject to a transversality condition,4 and by raising lump-sum taxes.5 The timing of taxation is irrelevant because of Ricardian Equivalence.6 Increase your impact with MSM Executive Education For almost 60 years Maastricht School of Management has been enhancing the management capacity of professionals and organizations around the world through state-of-the-art management education Our broad range of Open Enrollment Executive Programs offers you a unique interactive, stimulating and multicultural learning experience Be prepared for tomorrow’s management challenges and apply today For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via admissions@msm.nl For more information, visit www.msm.nl or contact us at +31 43 38 70 808 the globally networked management school or via admissions@msm.nl Executive Education-170x115-B2.indd 18-08-11 15:13 Download free eBooks at bookboon.com Click on the ad to read more The stochastic growth model The Stochastic Growth Model The representative household There is one representative household, who derives utility from her current and future consumption: Ut = Et ∞ s=t 1+ρ s−t ln Cs with ρ > (7) The parameter ρ is called the subjective discount rate Every period s, the household starts off with her assets Xs and receives interest payments Xs rs She also supplies L units of labor to the representative firm, and therefore receives labor income ws L Tax payments are lump-sum and amount to Ts She then decides how much she consumes, and how much assets she will hold in her portfolio until period s + This leads to her dynamic budget constraint: Xs+1 = Xs (1 + rs ) + ws L − Ts − Cs (8) We need to make sure that the household does not incur ever increasing debts, which she will never be able to pay back anymore Under plausible assumptions, this implies that over an infinitely long horizon the present discounted value of the household’s assets must be zero: s 1 + rs s =t lim Et s→∞ Xs+1 = (9) This equation is called the transversality condition The household then takes Xt and the current and expected values of r, w, and T as given, and chooses her consumption path to maximize her utility (7) subject to her dynamic budget constraint (8) and the transversality condition (9) This leads to the following Euler equation:7 Cs = Es + rs+1 1 + ρ Cs+1 (10) Equilibrium Every period, the factor markets and the goods market clear For the labor market, we already implicitly assumed this by using the same notation (L) for the representative household’s labor supply and the representative firm’s labor demand Equilibrium in the goods market requires that Yt = Ct + It + Gt Equilibrium in the capital market follows then from Walras’ law Download free eBooks at bookboon.com (11) The steady state The Stochastic Growth Model The steady state Let us now derive the model’s balanced growth path (or steady state); variables evaluated on the balanced growth path are denoted by a ∗ To derive the balanced growth path, we assume that by sheer luck εA,t = Aˆt = ˆ t = 0, ∀t The model then becomes a standard neoclassical growth εG,t = G model, for which the solution is given by:8 Yt∗ Kt∗ = α ∗ r +δ α 1−α = α ∗ r +δ 1−α A∗t L (12) A∗t L (13) It∗ Ct∗ = = wt∗ = r∗ = 1−α α (g + δ) ∗ A∗t L r +δ α α − (g + δ) ∗ ∗ r +δ r +δ α 1−α α (1 − α) ∗ A∗t r +δ (1 + ρ)(1 + g) − (14) α 1−α A∗t L − G∗t (15) (16) (17) GOT-THE-ENERGY-TO-LEAD.COM We believe that energy suppliers should be renewable, too We are therefore looking for enthusiastic new colleagues with plenty of ideas who want to join RWE in changing the world Visit us online to find out what we are offering and how we are working together to ensure the energy of the future Download free eBooks at bookboon.com Click on the ad to read more Linearization around the balanced growth path The Stochastic Growth Model Linearization around the balanced growth path Let us now linearize the model presented in section around the balanced growth path derived in section Loglinear deviations from the balanced growth path ˆ = ln X − ln X ∗ ) are denoted by aˆ(so that X Below are the loglinearized versions of the production function (1), the law of motion of the capital stock (2), the first-order conditions (4) and (5), the Euler equation (10) and the equilibrium condition (11):9 ˆ t + (1 − α)Aˆt Yˆt = αK ˆ t+1 = − δ K ˆ t + g + δ Iˆt K 1+g 1+g ˆ Yt = w ˆt Et r∗ rt+1 − + r∗ = r∗ +δ ˆ t+1 ) Et (Yˆt+1 ) − Et (K + r∗ rt+1 − r ∗ ˆ ˆ Ct = Et Ct+1 − Et + r∗ ∗ ∗ Ct ˆ It ˆ G∗t ˆ C It + ∗ Gt Yˆt = + t Yt∗ Yt∗ Yt (18) (19) (20) (21) (22) (23) The loglinearized laws of motion of A and G are given by equations (3) and (6): Aˆt+1 = φA Aˆt + εA,t+1 ˆ t+1 = φG G ˆ t + εG,t+1 G Download free eBooks at bookboon.com (24) (25) Solution of the linearized model The Stochastic Growth Model Solution of the linearized model I now solve the linearized model, which is described by equations (18) until (25) ˆ t are known in the beginning of period t: K ˆ t depends ˆ t , Aˆt and G First note that K ˆ t are determined by current and past on past investment decisions, and Aˆt and G ˆ t , Aˆt and G ˆ t are values of respectively εA and εG (which are exogenous) K therefore called period t’s state variables The values of the other variables in period t are endogenous, however: investment and consumption are chosen by the representative firm and the representative household in such a way that they maximize their profits and utility (Iˆt and Cˆt are therefore called period t’s control variables); the values of the interest rate and the wage are such that they clear the capital and the labor market Solving the model requires that we express period t’s endogenous variables as functions of period t’s state variables The solution of Cˆt , for instance, therefore looks as follows: ˆ t + ϕCA Aˆt + ϕCG G ˆt Cˆt = ϕCK K (26) The challenge now is to determine the ϕ-coefficients First substitute equation (26) in the Euler equation (22): ˆ t + ϕCA Aˆt + ϕCG G ˆt ϕCK K ˆ t+1 + ϕCA Aˆt+1 + ϕCG G ˆ t+1 − Et = Et ϕCK K rt+1 − r ∗ + r∗ (27) Now eliminate Et [(rt+1 − r ∗ )/(1+ r ∗ )] with equation (21), and use equations (18), ˆ t+1 in the resulting expression This (24) and (25) to eliminate Yˆt+1 , Aˆt+1 and G Download free eBooks at bookboon.com Solution of the linearized model The Stochastic Growth Model ˆ t+1 : leads to a relation between period t’s state variables, the ϕ-coefficients and K ˆ t + ϕCA Aˆt + ϕCG G ˆt ϕCK K ∗ r +δ ˆ r∗ + δ ˆt K = ϕCK + (1 − α) + ϕ − (1 − α) φA Aˆt + ϕCG φG G t+1 CA + r∗ + r∗ (28) We now derive a second relation between period t’s state variables, the ϕ-coefficients ˆ t+1 : rewrite the law of motion (19) by eliminating Iˆt with equation (23); and K eliminate Yˆt and Cˆt in the resulting equation with the production function (18) and expression (26); note that I ∗ = K ∗ (g + δ); and note that (1 − δ)/(1 + g) + (αYt∗ )/(Kt∗ (1 + g)) = (1 + r ∗ )/(1 + g) This yields: ∗ C∗ ˆ t+1 = + r − ˆt K ϕCK K 1+g K ∗ (1 + g) (1 − α)Y ∗ G∗ C∗ C∗ ˆt − ˆt + A − ϕ + ϕCG G CA K ∗ (1 + g) K ∗ (1 + g) K ∗ (1 + g) K ∗ (1 + g) (29) ˆ t+1 yields: Substituting equation (29) in equation (28) to eliminate K ˆ t + ϕCA Aˆt + ϕCG G ˆt ϕCK K r∗ + δ + r∗ C∗ ˆt − ϕCK K = ϕCK + (1 − α) + r∗ 1+g K ∗ (1 + g) r ∗ + δ (1 − α)Y ∗ C∗ + ϕCK + (1 − α) − ϕCA + r ∗ K ∗ (1 + g) K ∗ (1 + g) r∗ + δ G∗ C∗ − ϕCK + (1 − α) + ϕCG + r ∗ K ∗ (1 + g) K ∗ (1 + g) r∗ + δ ˆt + ϕCA − (1 − α) φA Aˆt − ϕCG φG G + r∗ Aˆt ˆt G (30) ˆ t , we find the following ˆ t , Aˆt and G As this equation must hold for all values of K system of three equations and three unknowns: ϕCK ϕCA ϕCG r∗ + δ + r∗ r∗ + δ = ϕCK + (1 − α) + r∗ r∗ + δ + ϕCA − (1 − α) + r∗ r∗ + δ = − ϕCK + (1 − α) + r∗ = ϕCK + (1 − α) + r∗ C∗ − ∗ ϕCK 1+g K (1 + g) (1 − α)Y ∗ C∗ − ϕCA K ∗ (1 + g) K ∗ (1 + g) φA (31) (32) C∗ G∗ + ϕCG − ϕCG φG K ∗ (1 + g) K ∗ (1 + g) (33) Download free eBooks at bookboon.com 10 Conclusions The Stochastic Growth Model Conclusions This note presented the stochastic growth model, and solved the model by first linearizing it around a steady state and by then solving the linearized model with the method of undetermined coefficients Even though the stochastic growth model itself might bear little resemblance to the real world, it has proven to be a useful framework that can easily be extended to account for a wide range of macroeconomic issues that are potentially important Kydland and Prescott (1982) introduced labor/leisure-substitution in the stochastic growth model, which gave rise to the so-called real-business-cycle literature Greenwood and Huffman (1991) and Baxter and King (1993) replaced the lump-sum taxation by distortionary taxation, to study how taxes affect the behavior of firms and households In the beginning of the 1990s, researchers started introducing money and nominal rigidities in the model, which gave rise to New Keynesian stochastic dynamic general equilibrium models that are now widely used to study monetary policy - see Goodfriend and King (1997) for an overview Vermeylen (2006) shows how the representative household can be replaced by a large number of households to study the effect of job insecurity on consumption and saving in a general equilibrium setting Download free eBooks at bookboon.com 18 Click on the ad to read more Conclusions The Stochastic Growth Model Microfoundations means that the objectives of the economic agents are formulated explicitly, and that their behavior is derived by assuming that they always try to achieve their objectives as well as they can A steady state is a condition in which a number of key variables are not changing In the stochastic growth model, these key variables are for instance the growth rate of aggregate production, the interest rate and the capital-output-ratio 10 11 See appendix A for derivations This means that the present discounted value of public debt in the distant future should be equal to zero, such that public debt cannot keep on rising at a rate that is higher than the interest rate This guarantees that public debt is always equal to the present discounted value of the government’s future primary surpluses Lump-sum taxes not affect the first-order conditions of the firms and the households, and therefore not affect their behavior either Ricardian equivalence is the phenomenon that - given certain assumptions - it turns out to be irrelevant whether the government finances its expenditures by issuing public debt or by raising taxes The reason for this is that given the time path of government expenditures, every increase in public debt must sooner or later be matched by an increase in taxes, such that the present discounted value of the taxes which a representative household has to pay is not affected by the way how the government finances its expenditures which implies that her current wealth and her consumption path are not affected either See appendix A for the derivation See appendix B for the derivation See appendix C for the derivations The solution with unstable dynamics not only does not make sense from an economic point of view, it also violates the transversality conditions Note that these values imply that the annual depreciation rate, the annual growth rate and the annual interest rate are about 10%, 2% and 6%, respectively Download free eBooks at bookboon.com 19 Appendix A The Stochastic Growth Model Appendix A A1 The maximization problem of the representative firm The maximization problem of the firm can be rewritten as: Vt (Kt ) = max {Lt ,It } Yt − wt Lt − It + Et Vt+1 (Kt+1 ) + rt+1 (A.1) s.t Yt = Ktα (At Lt )1−α Kt+1 = (1 − δ)Kt + It The first-order conditions for Lt , respectively It , are: L−α − wt (1 − α)Ktα A1−α t t ∂Vt+1 (Kt+1 ) −1 + Et + rt+1 ∂Kt+1 = (A.2) = (A.3) ∂Vt+1 (Kt+1 ) (1 − δ) + rt+1 ∂Kt+1 (A.4) In addition, the envelope theorem implies that ∂Vt (Kt ) ∂Kt = αKtα−1 (At Lt )1−α + Et Substituting the production function in (A.2) gives equation (4): (1 − α) Yt Lt = wt Substituting (A.3) in (A.4) yields: ∂Vt (Kt ) ∂Kt = αKtα−1 (At Lt )1−α + (1 − δ) Moving one period forward, and substituting again in (A.3) gives: −1 + Et α−1 αKt+1 (At+1 Lt+1 )1−α + (1 − δ) + rt+1 = Substituting the production function in the equation above and reshuffling leads to equation (5): = Et Yt+1 1−δ + Et α + rt+1 Kt+1 + rt+1 A2 The Themaximization maximizationproblem problemofofthe therepresentative representativehousehold household A2 The maximization problem of the household can be rewritten as: Ut (Xt ) = max ln Ct + {Ct } Et [Ut+1 (Xt+1 )] 1+ρ s.t Xt+1 = Xt (1 + rt ) + wt L − Tt − Ct Download free eBooks at bookboon.com 20 (A.5) Appendix A The Stochastic Growth Model The first-order condition for Ct is: ∂Ut+1 (Xt+1 ) 1 Et − Ct 1+ρ ∂Xt+1 = (A.6) In addition, the envelope theorem implies that ∂Ut (Xt ) ∂Xt = ∂Ut+1 (Xt+1 ) Et (1 + rt ) 1+ρ ∂Xt+1 (A.7) Substituting (A.6) in (A.7) yields: ∂Ut (Xt ) ∂Xt = (1 + rt ) Ct Moving one period forward, and substituting again in (A.6) gives the Euler equation (10): 1 + rt+1 − Et Ct + ρ Ct+1 = Turning a challenge into a learning curve Just another day at the office for a high performer Accenture Boot Camp – your toughest test yet Choose Accenture for a career where the variety of opportunities and challenges allows you to make a difference every day A place where you can develop your potential and grow professionally, working alongside talented colleagues The only place where you can learn from our unrivalled experience, while helping our global clients achieve high performance If this is your idea of a typical working day, then Accenture is the place to be It all starts at Boot Camp It’s 48 hours that will stimulate your mind and enhance your career prospects You’ll spend time with other students, top Accenture Consultants and special guests An inspirational two days packed with intellectual challenges and activities designed to let you discover what it really means to be a high performer in business We can’t tell you everything about Boot Camp, but expect a fast-paced, exhilarating and intense learning experience It could be your toughest test yet, which is exactly what will make it your biggest opportunity Find out more and apply online Visit accenture.com/bootcamp Download free eBooks at bookboon.com 21 Click on the ad to read more Appendix B The Stochastic Growth Model Appendix B If C grows at rate g, the Euler equation (10) implies that Cs∗ (1 + g) = + r∗ ∗ C 1+ρ s Rearranging gives then the gross real rate of return + r∗ : + r∗ = (1 + g)(1 + ρ) which immediately leads to equation (17) Subsituting in the firm’s first-order condition (5) gives: α ∗ Yt+1 ∗ Kt+1 = r∗ + δ Using the production function (1) to eliminate Y yields: ∗α−1 (At+1 L)1−α αKt+1 = r∗ + δ ∗ Rearranging gives then the value of Kt+1 : ∗ Kt+1 = α ∗ r +δ 1−α which is equivalent to equation (13) Download free eBooks at bookboon.com 22 At+1 L Appendix B The Stochastic Growth Model Substituting in the production function (1) gives then equation (12): Yt∗ α 1−α α r∗ + δ = At L Substituting (12) in the first-order condition (4) gives equation (16): wt∗ (1 − α) = α 1−α α ∗ r +δ At Substituting (13) in the law of motion (2) yields: 1−α α ∗ r +δ At+1 L = (1 − δ) α ∗ r +δ 1−α At L + It∗ such that It∗ is given by: It∗ = α ∗ r +δ = α ∗ r +δ = (g + δ) 1−α 1−α At+1 L − (1 − δ) α ∗ r +δ 1−α At L [(1 + g) − (1 − δ)] At L α r∗ + δ 1−α At L .which is equation (14) Consumption C ∗ can then be computed from the equilibrium condition in the goods market: Ct∗ = = = Yt∗ − It∗ − G∗t α ∗ r +δ α 1−α g+δ 1−α ∗ r +δ At L − (g + δ) α ∗ r +δ α 1−α α ∗ r +δ 1−α At L − G∗t At L − G∗t Now recall that on the balanced growth path, A and G grow at the rate of technological progress g The equation above then implies that C ∗ also grows at the rate g, such that our initial educated guess turns out to be correct Download free eBooks at bookboon.com 23 Appendix C The Stochastic Growth Model Appendix C C1 The linearized production function The production function is given by equation (1): Yt = Ktα (At Lt )1−α Taking logarithms of both sides of this equation, and subtracting from both sides their ˆ t = 0), immediately yields values on the balanced growth path (taking into account that L the linearized version of the production function: ln Yt ln Yt − ln Yt∗ Yˆt = = = α ln Kt + (1 − α) ln At + (1 − α) ln Lt α(ln Kt − ln Kt∗ ) + (1 − α)(ln At − ln A∗t ) + (1 − α)(ln Lt − ln L∗t ) ˆ t + (1 − α)Aˆt αK .which is equation (18) The Wake the only emission we want to leave behind QYURGGF 'PIKPGU /GFKWOURGGF 'PIKPGU 6WTDQEJCTIGTU 2TQRGNNGTU 2TQRWNUKQP 2CEMCIGU 2TKOG5GTX 6JG FGUKIP QH GEQHTKGPFN[ OCTKPG RQYGT CPF RTQRWNUKQP UQNWVKQPU KU ETWEKCN HQT /#0 &KGUGN 6WTDQ 2QYGT EQORGVGPEKGU CTG QHHGTGF YKVJ VJG YQTNFoU NCTIGUV GPIKPG RTQITCOOG s JCXKPI QWVRWVU URCPPKPI HTQO  VQ  M9 RGT GPIKPG )GV WR HTQPV (KPF QWV OQTG CV YYYOCPFKGUGNVWTDQEQO Download free eBooks at bookboon.com 24 Click on the ad to read more Appendix C The Stochastic Growth Model C2 The linearized law of motion of the capital stock The law of motion of the capital stock is given by equation (2): Kt+1 (1 − δ)Kt + It = Taking logarithms of both sides of this equation, and subtracting from both sides their values on the balanced growth path, yields: ∗ ln Kt+1 − ln Kt+1 ∗ = ln {(1 − δ)Kt + It } − ln Kt+1 Now take a first-order Taylor-approximation of the right-hand-side around ln Kt = ln Kt∗ and ln It = ln It∗ : ∗ ln Kt+1 − ln Kt+1 ˆ t+1 K ϕ1 (ln Kt − ln Kt∗ ) + ϕ2 (ln It − ln It∗ ) ˆ t + ϕ2 Iˆt ϕ1 K = = (C.1) where ϕ1 = ∂ ln {(1 − δ)Kt + It } ∂ ln Kt ϕ2 = ∂ ln {(1 − δ)Kt + It } ∂ ln It ∗ ∗ ϕ1 and ϕ2 can be worked out as follows: ϕ1 = ∂ ln {(1 − δ)Kt + It } ∂Kt ∂Kt ∂ ln Kt = 1−δ Kt (1 − δ)Kt + It ∗ ∗ ∗ = = ϕ2 1−δ Kt Kt+1 1−δ as Kt grows at rate g on the balanced growth path 1+g = ∂ ln {(1 − δ)Kt + It } ∂It ∂It ∂ ln It = It (1 − δ)Kt + It = = Kt+1 g+δ 1+g ∗ ∗ ∗ It as It∗ /Kt∗ = g + δ and Kt grows at rate g on the balanced growth path Substituting in equation (C.1) gives then the linearized law of motion for K: ˆ t+1 K = 1−δ ˆ g+δˆ Kt + It 1+g 1+g .which is equation (19) Download free eBooks at bookboon.com 25 Appendix C The Stochastic Growth Model C3 The linearized first-order condotion for the firm’s labor demand The first-order condition for the firm’s labor demand is given by equation (4): (1 − α) Yt Lt = wt Taking logarithms of both sides of this equation, and subtracting from both sides their ˆ t = 0), immediately yields values on the balanced growth path (taking into account that L the linearized version of this first-order condition: ln (1 − α) + ln Yt − ln Lt (ln Yt − ln Yt∗ ) − (ln Lt − ln L∗ ) Yˆt = ln wt = ln wt − ln wt∗ = w ˆt .which is equation (20) C4 The linearized first-order condition for the firms’ capital demand C4 The linearized first-order condotion for the firm’s capital demand The first-order condition for the firm’s capital demand is given by equation (5): = Et [Zt+1 ] with Zt+1 = (C.2) Yt+1 1+rt+1 α Kt+1 + 1−δ 1+rt+1 (C.3) Now take a first-order Taylor-approximation of the right-hand-side of equation (C.3) ∗ ∗ , ln Kt+1 = ln Kt+1 and rt+1 = r∗ : around ln Yt+1 = ln Yt+1 Zt+1 ∗ ∗ = + ϕ1 (ln Yt+1 − ln Yt+1 ) + ϕ2 (ln Kt+1 − ln Kt+1 ) + ϕ3 (rt+1 − r∗ ) ˆ t+1 + ϕ3 (rt+1 − r∗ ) = + ϕ1 Yˆt+1 + ϕ2 K (C.4) Download free eBooks at bookboon.com 26 Appendix C The Stochastic Growth Model where ⎛ ϕ1 = ⎝ ⎛ ϕ2 = ⎝ ⎛ ϕ3 = ⎝ ∂ Yt+1 1+rt+1 α Kt+1 + 1−δ 1+rt+1 ∂ ln Yt+1 ∂ Yt+1 1+rt+1 α Kt+1 + 1−δ 1+rt+1 ∂ ln Kt+1 ∂ Yt+1 1+rt+1 α Kt+1 + 1−δ 1+rt+1 ∂rt+1 ⎞∗ ⎠ ⎞∗ ⎠ ⎞∗ ⎠ ϕ1 , ϕ2 and ϕ3 can be worked out as follows: ⎛ ϕ1 = ⎝ ∂ Yt+1 1+rt+1 α Kt+1 + ⎞∗ ∂Yt+1 ⎠ ∂ ln Yt+1 1−δ 1+rt+1 ∂Yt+1 ∗ = = ϕ2 = 1 α Yt+1 + rt+1 Kt+1 r∗ + δ ∗ ∗ using the fact that αYt+1 = (r∗ + δ)Kt+1 + r∗ ⎛ ⎞∗ Yt+1 1−δ ∂ 1+r1t+1 α K + 1+rt+1 ∂Kt+1 ⎠ t+1 ⎝ ∂Kt+1 ∂ ln Kt+1 ∗ = = ϕ3 = = Yt+1 α Kt+1 + rt+1 Kt+1 ∗ r +δ ∗ ∗ − using the fact that αYt+1 = (r∗ + δ)Kt+1 + r∗ − Yt+1 α +1−δ (1 + rt+1 ) Kt+1 − + r∗ ∗ − Substituting in equation (C.4) gives then: Zt+1 = 1+ r∗ + δ ˆ r∗ + δ ˆ rt+1 − r∗ − − Y K t+1 t+1 + r∗ + r∗ + r∗ Substituting in equation (C.2) and rearranging, gives then equation (21): Et rt+1 − r∗ + r∗ = r∗ + δ ˆ t+1 ) Et (Yˆt+1 ) − Et (K + r∗ Download free eBooks at bookboon.com 27 (C.5) Appendix C The Stochastic Growth Model C5 The linearized Euler equation of the representative household The Euler equation of the representative household is given by equation (10), which is equivalent to: = Et [Zt+1 ] (C.6) with Zt+1 = 1+rt+1 Ct 1+ρ Ct+1 (C.7) Now take a first-order Taylor-approximation of the right-hand-side of equation (C.7) ∗ around ln Ct+1 = ln Ct+1 , ln Ct = ln Ct∗ and rt+1 = r∗ : Zt+1 = = ∗ + ϕ1 (ln Ct+1 − ln Ct+1 ) + ϕ2 (ln Ct − ln Ct∗ ) + ϕ3 (rt+1 − r∗ ) + ϕ1 Cˆt+1 + ϕ2 Cˆt + ϕ3 (rt+1 − r∗ ) where ⎛ ϕ1 = ⎝ ⎛ ϕ2 = ⎝ ⎛ ϕ3 = ⎝ ∂ 1+rt+1 Ct 1+ρ Ct+1 ∂ ln Ct+1 ∂ 1+rt+1 Ct 1+ρ Ct+1 ∂ ln Ct ∂ 1+rt+1 Ct 1+ρ Ct+1 ∂rt+1 ϕ1 , ϕ2 and ϕ3 can be worked out as follows: ⎛ t+1 Ct ∂ 1+r 1+ρ Ct+1 ϕ1 = ⎝ ∂Ct+1 = − = −1 ⎛ ϕ2 = ⎝ = ∂ ∂Ct + rt+1 Ct + ρ Ct+1 = ϕ3 = = Ct 1 + ρ Ct+1 1+rt+1 Ct 1+ρ Ct+1 ∗ ∗ + rt+1 Download free eBooks at bookboon.com 28 ⎠ ⎞∗ ⎠ ⎞∗ ⎠ ⎞∗ ∂Ct+1 ⎠ ∂ ln Ct+1 + rt+1 Ct Ct+1 + ρ Ct+1 1+rt+1 Ct 1+ρ Ct+1 ⎞∗ ∗ ⎞∗ ∂Ct ⎠ ∂ ln Ct ∗ (C.8) Appendix C The Stochastic Growth Model 1 + r∗ = Substituting in equation (C.8) gives then: Zt+1 = − Cˆt+1 + Cˆt + rt+1 − r∗ + r∗ Substituting in equation (C.6) and rearranging, gives then equation (22): Cˆt = rt+1 − r∗ Et Cˆt+1 − Et + r∗ Brain power By 2020, wind could provide one-tenth of our planet’s electricity needs Already today, SKF’s innovative knowhow is crucial to running a large proportion of the world’s wind turbines Up to 25 % of the generating costs relate to maintenance These can be reduced dramatically thanks to our 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The Power of Knowledge Engineering Plug into The Power of Knowledge Engineering Visit us at www.skf.com/knowledge Download free eBooks at bookboon.com 29 Click on the ad to read more Appendix C The Stochastic Growth Model C6 The linearized equillibrium condition in the goods market The equilibrium condition in the goods market is given by equation (11): Yt = Ct + It + Gt Taking logarithms of both sides of this equation, and subtracting from both sides their values on the balanced growth path, yields: ln Yt − ln Yt∗ = ln (Ct + It + Gt ) − ln Yt∗ Now take a first-order Taylor-approximation of the right-hand-side around ln Ct = ln Ct∗ , ln It = ln It∗ and ln Gt = ln G∗t : ln Yt − ln Yt∗ Yˆt = = ϕ1 (ln Ct − ln Ct∗ ) + ϕ2 (ln It − ln It∗ ) + ϕ3 (ln Gt − ln G∗t ) ˆt ϕ1 Cˆt + ϕ2 Iˆt + ϕ3 G (C.9) where ϕ1 = ∂ ln {Ct + It + Gt } ∂ ln Ct ϕ2 = ∂ ln {Ct + It + Gt } ∂ ln It = ∂ ln {Ct + It + Gt } ∂ ln Gt ϕ3 ∗ ∗ ∗ ϕ1 , ϕ2 and ϕ3 can be worked out as follows: ϕ1 = ∂ ln {Ct + It + Gt } ∂Ct ∂Ct ∂ ln Ct = Ct Ct + It + Gt = Ct∗ Yt∗ Download free eBooks at bookboon.com 30 ∗ ∗ Appendix C The Stochastic Growth Model ϕ2 = ∂ ln {Ct + It + Gt } ∂It ∂It ∂ ln It = It Ct + It + Gt = ϕ3 = ∗ It∗ Yt∗ ∂ ln {Ct + It + Gt } ∂Gt ∂Gt ∂ ln Gt = = ∗ Gt Ct + It + Gt G∗t Yt∗ ∗ ∗ Substituting in equation (C.9) gives then the linearized equilibrium condition in the goods market: Yˆt = Ct∗ ˆ It∗ ˆ G∗t ˆ + + C I Gt t t Yt∗ Yt∗ Yt∗ Download free eBooks at bookboon.com 31 Click on the ad to read more References The Stochastic Growth Model References Baxter, Marianne, and Robert G King (1993), ”Fiscal Policy in General Equilibrium”, American Economic Review 83 (June), 315-334 Campbell, John Y (1994), ”Inspecting the Mechanism: An Analytical Approach to the Stochastic Growth Model”, Journal of Monetary Economics 33 (June), 463-506 Goodfriend, Marvin and Robert G King (1997), ”The New Neoclassical Synthesis and the Role of Monetary Policy”, in Bernanke, Ben S., and Julio J Rotemberg, eds., NBER Macroeconomics Annual 1997, The MIT Press, pp 231-83 Greenwood, Jeremy, and Gregory W Huffman (1991), ”Tax Analysis in a Real-BusinessCycle Model: On Measuring Harberger Triangles and Okun Gaps”, Journal of Monetary Economics 27 (April), 167-190 Kydland, Finn E., and Edward C Prescott (1982), ”Time to Build and Aggregate Fluctuations”, Econometrica 50 (Nov.), 1345-1370 Vermeylen, Koen (2006), ”Heterogeneous Agents and Uninsurable Idiosyncratic Employment Shocks in a Linearized Dynamic General Equilibrium Model”, Journal of Money, Credit, and Banking 38, (April), 837-846 Download free eBooks at bookboon.com 32 ...Contents The Stochastic Growth Model Contents Introduction The stochastic growth model The steady state Linearization around the balanced growth path Solution of the linearized model Impulse... on the ad to read more © Deloitte & Touche LLP and affiliated entities D The Stochastic Growth Model Introduction Introduction This article presents the stochastic growth model The stochastic growth. .. (1994) The set-up of the stochastic growth model is given in the next section Section solves for the steady state, around which the model is linearized in section The linearized model is then solved