The stochastic growth model koen vermeylen

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 Please click the advert The next step for top-performing graduates Masters in Management Designed for high-achieving graduates across all disciplines, London Business School’s Masters in Management provides specific and tangible foundations for a successful career in business This 12-month, full-time programme is a business qualification with impact In 2010, our MiM employment rate was 95% within months of graduation*; the majority of graduates choosing to work in consulting or financial services As well as a renowned qualification from a world-class business school, you also gain access to the School’s network of more than 34,000 global alumni – a community that offers support and opportunities throughout your career For more information visit www.london.edu/mm, email mim@london.edu or give us a call on +44 (0)20 7000 7573 * Figures taken from London Business School’s Masters in Management 2010 employment report Download free ebooks at bookboon.com 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 Please click the advert Teach with the Best Learn with the Best Agilent offers a wide variety of affordable, industry-leading electronic test equipment as well as knowledge-rich, on-line resources —for professors and students We have 100’s of comprehensive web-based teaching tools, lab experiments, application notes, brochures, DVDs/ CDs, posters, and more See what Agilent can for you www.agilent.com/find/EDUstudents www.agilent.com/find/EDUeducators © Agilent Technologies, Inc 2012 u.s 1-800-829-4444 canada: 1-877-894-4414 Download free ebooks at bookboon.com 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 lim Et s→∞ 1 + rs s =t 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 (11) Equilibrium in the capital market follows then from Walras’ law Download free ebooks at bookboon.com 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) Ct∗ = = wt∗ = Please click the advert 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) You’re full of energy and ideas And that’s just what we are looking for © UBS 2010 All rights reserved It∗ Looking for a career where your ideas could really make a difference? UBS’s Graduate Programme and internships are a chance for you to experience for yourself what it’s like to be part of a global team that rewards your input and believes in succeeding together Wherever you are in your academic career, make your future a part of ours by visiting www.ubs.com/graduates www.ubs.com/graduates Download free ebooks at bookboon.com 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∗ = (18) (19) (20) 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 (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 (24) (25) Download free ebooks at bookboon.com 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 your chance Please click the advert to change the world Here at Ericsson we have a deep rooted belief that the innovations we make on a daily basis can have a profound effect on making the world a better place for people, business and society Join us In Germany we are especially looking for graduates as Integration Engineers for • Radio Access and IP Networks • IMS and IPTV We are looking forward to getting your application! To apply and for all current job openings please visit our web page: www.ericsson.com/careers Download free ebooks at bookboon.com 18 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 (A.5) s.t Xt+1 = Xt (1 + rt ) + wt L − Tt − Ct Download free ebooks at bookboon.com 20 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 = e Graduate Programme for Engineers and Geoscientists I joined MITAS because I wanted real responsibili Please click the advert Maersk.com/Mitas Real work Internationa al opportunities International wo or placements ree work Month 16 I was a construction supervisor in the North Sea advising and helping foremen he ssolve problems Download free ebooks at bookboon.com 21 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 At+1 L which is equivalent to equation (13) Download free ebooks at bookboon.com 22 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 Please click the advert .which is equation (18) We will turn your CV into an opportunity of a lifetime Do you like cars? Would you like to be a part of a successful brand? We will appreciate and reward both your enthusiasm and talent Send us your CV You will be surprised where it can take you Send us your CV on www.employerforlife.com Download free ebooks at bookboon.com 24 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∗ ∗ − (C.5) 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 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 ⎞∗ ⎠ ⎞∗ ⎠ ⎞∗ ⎠ ⎞∗ ∂Ct+1 ⎠ ∂ ln Ct+1 + rt+1 Ct Ct+1 + ρ Ct+1 1+rt+1 Ct 1+ρ Ct+1 (C.8) ∗ ⎞∗ ∂Ct ⎠ ∂ ln Ct ∗ = ϕ3 = = Ct 1 + ρ Ct+1 1+rt+1 Ct 1+ρ Ct+1 ∗ ∗ + rt+1 Download free ebooks at bookboon.com 28 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∗ Please click the advert Are you remarkable? Win one of the six full tuition scholarships for International MBA or MSc in Management register now rode www.Nyen ge.com n le al h MasterC Download free ebooks at bookboon.com 29 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∗ Please click the advert Budget-Friendly Knowledge-Rich The Agilent InfiniiVision X-Series and 1000 Series offer affordable oscilloscopes for your labs Plus resources such as lab guides, experiments, and more, to help enrich your curriculum and make your job easier Scan for free Agilent iPhone Apps or visit qrs.ly/po2Opli See what Agilent can for you www.agilent.com/find/EducationKit © Agilent Technologies, Inc 2012 u.s 1-800-829-4444 canada: 1-877-894-4414 Download free ebooks at bookboon.com 31 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 32 [...]... also Note that the real wage w again follows the time path of Y Eventually, all variables converge back to their steady state values Download free ebooks at bookboon.com 17 Conclusions The Stochastic Growth Model 7 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... Conclusions The Stochastic Growth Model 1 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 2 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. .. 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... we can compute the endogenous variables in quarter 1 and the state variables for quarter 2 in the same way as in the case of a technology shock which leads to the values for quarter 2’s endogenous variables and quarter 3’s state variables, and so on until the infinite future Download free ebooks at bookboon.com 16 Impulse response functions The Stochastic Growth Model Figure 2 shows the economy’s reaction... surpluses Lump-sum taxes do not affect the first-order conditions of the firms and the households, and therefore do 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,... functions The Stochastic Growth Model Figure 1 shows how the economy reacts during the first 40 quarters Note that Y jumps up in quarter 1, together with the technology shock As a result, the representative household increases her consumption, but as she wants to smooth her consumption over time, C increases less than Y Investment I therefore initially increases more than Y As I increases, the capital... 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:... MasterC Download free ebooks at bookboon.com 29 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... 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... after period 1 The expected rate of return, E(r), is at first higher than on the balanced growth path (thanks to the technology shock) However, as the technology shock dies out while the capital stock builds up, the expected interest rate rapidly falls and even becomes negative after a few quarters The real wage w follows the time path of Y Note that all variables eventually converge back to their steady ... bookboon.com 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. .. 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...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