The Stochastic Growth Model Koen Vermeylen Download free books at Koen Vermeylen The Stochastic Growth Model BusinessSumup Download free eBooks at bookboon.com The Stochastic Growth Model © 2008 Koen Vermeylen & BusinessSumup ISBN 978-87-7681-284-3 Download free eBooks at bookboon.com Contents The Stochastic Growth Model Contents Introduction The stochastic growth model The steady state Linearization around the balanced growth path 10 Solution of the linearized model 11 Impulse response functions 15 Conclusions 20 Appendix A A1 The maximization problem of the representative firm A2 The maximization problem of the representative household 22 22 22 Appendix B 24 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 26 26 27 28 28 30 32 References 34 Fast-track your career Masters in Management Stand out from the crowd Designed for graduates with less than one year of full-time postgraduate work experience, London Business School’s Masters in Management will expand your thinking and provide you with the foundations for a successful career in business The programme is developed in consultation with recruiters to provide you with the key skills that top employers demand Through 11 months of full-time study, you will gain the business knowledge and capabilities to increase your career choices and stand out from the crowd London Business School Regent’s Park London NW1 4SA United Kingdom Tel +44 (0)20 7000 7573 Email mim@london.edu Applications are now open for entry in September 2011 For more information visit www.london.edu/mim/ email mim@london.edu or call +44 (0)20 7000 7573 www.london.edu/mim/ Download free eBooks at bookboon.com Click on the ad to read more Koen Vermeylen The Stochastic Growth Model Introduction 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 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 ≤ 1 Download free eBooks at bookboon.com (2) 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 The Stochastic Growth Model The stochastic growth model The The stochastic growth modelmodel stochastic growth 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 a1stochastic 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 + g) with |φA | < (3) 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 + 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 G∗t = G∗t−1 (1 Download free eBooks at bookboon.com + g) with |φG | < (6) εA and εG are uncorrelated The stochastic shock εG,t is i.i.d with mean zero 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 The government The government consumes every period t an amount Gt , The stochastic growth model which follows a stochastic path with trend growth g and an AR(1) stochastic component: The Stochastic Growth Model ˆt ln Gt = ln G∗t + G ˆ t = φG G ˆ t−1 + εG,t G G∗t = G∗t−1 (1 + g) with |φG | < (6) 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 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 ∞ 1+ρ s=t 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 Equilibrium in the capital market follows then from Walras’ law Download free eBooks at bookboon.com (11) The Stochastic Growth Model The steady state The steady state 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 ∗ r +δ α 1−α 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) 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 your chance to change 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 the world ˆ t + (1 − α)Aˆt Yˆt = αK 1−δ ˆ g+δˆ Here at Ericsson we have a deep ˆ t+1rooted = belief thatK It K t+ the innovations we make on a daily basis can have a 1+g 1+g profound effect on making the world a better place ˆt =us w ˆt for people, business and society.YJoin r∗ + δ rt+1 − r ∗ ˆt+1 ) − Et (K ˆ t+1 ) In Germany we E are for graduates Et (Y t especially ∗looking = for +r + r∗ as Integration Engineers (18) (19) (20) (21) • 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 Click on the ad to read more α 1−α α A∗t ∗ r +δ = (1 + ρ)(1 + g) − wt∗ = (1 − α) r∗ (16) (17) The Stochastic Growth Model Linearization around the balanced growth path Linearization Linearization around the balanced growth path 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 + g + δ Iˆt ˆ t+1 = − δ K K 1+g 1+g ˆt Yˆt = w Et r∗ rt+1 − + r∗ = +δ ˆ t+1 ) Et (Yˆt+1 ) − Et (K + r∗ r∗ rt+1 − r ∗ Cˆt = Et Cˆt+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 + εG,t+1 ˆ t+1 = φG G G (24) (25) 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 ˆ on past investment decisions, and At and Gt are determined by current and past 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: Download free eBooks at bookboon.com Cˆ = ϕ t ˆ + ϕCA Aˆt + ϕCG G ˆt CK Kt 10 The challenge now is to determine the ϕ-coefficients First substitute equation (26) in the Euler equation (22): (26) result, C and I recover - and as I recovers, K and Y recover also Note that the real wage w again follows the time path of Y Eventually, all variables converge back to their steady state values The Stochastic Growth Model Conclusions Conclusions 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 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 See appendix A for derivations This means that the present discounted value of public debtelectricity in the needs distant future should Already today, SKF’s innovative knowbe equal to zero, such that public debt cannot keep on rising a rate that aislarge higher how is at crucial to running proportion of the wind turbines than the interest rate This guarantees that public debt isworld’s always equal to the present Up to 25 % of the generating costs relate to maintediscounted value of the government’s future primary surpluses Lump-sum taxes not affect the first-order conditions of the firms thecondition households, systems for and on-line monitoring and automatic and therefore not affect their behavior either lubrication We help make it more economical to create cleaner, cheaper energy of thin Ricardian equivalence is the phenomenon that - given certain assumptions - it out turns outair By sharing our experience, expertise, and creativity, to be irrelevant whether the government finances its expenditures by issuing public debt industries can boost performance beyond expectations or by raising taxes The reason for this is that given the time path of government expenTherefore we need the best employees who can ditures, every increase in public debt must sooner or later be matched by an increase in meet this challenge! 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 The Power of Knowledge Engineering which implies that her current wealth and her consumption path are not affected either Brain power By 2020, wind could provide one-tenth of our planet’s nance These can be reduced dramatically thanks to our See appendix A for the derivation See appendix B for the derivation See appendix C for the derivations 10 The solution with unstable dynamics not only does not make sense from an economic point of view, it also violates the transversality conditions 11 Plug into Power of Knowledge Engineering Note thatThe these values imply that the annual depreciation rate, the annual growth rate Visit us at www.skf.com/knowledge and the annual interest rate are about 10%, 2% and 6%, respectively Download free eBooks at bookboon.com 20 Click on the ad to read more 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 The Stochastic Growth Model and saving in a general equilibrium setting 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 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 10 The solution with unstable dynamics not only does not make sense from an economic point of view, it also violates the transversality conditions 11 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 21 Conclusions The Stochastic Growth Model Appendix A Appendix A Appendix A Appendix A A1 The maximization problem of the representative firm A1 The maximization problem of the representative firm The problem of the firm can be as: A1.maximization The maximization problem of rewritten the representative firm max Yt − wt Lt − It + Et Vt+1 (Kt+1 ) Vt (Kt ) = + rt+1 {Lt ,It } The maximization problem of the firm can be rewritten as: (A.1) s.t Yt = Ktα (At Lt )1−α = max Yt − wt Lt − It + Et Vt+1 (Kt+1 ) + rt+1 {L t ,It }= (1 − δ)Kt + It Kt+1 (A.1) Vt (Kt ) 1−α The first-order conditions respectively It , are: s.t Ytfor =L Kttα, (A t Lt ) α 1−α −α − wt = (A.2) (1(1 −− α)K Kt+1 = δ)K t+ t A t It Lt ∂Vt+1 t+1 ) The first-order conditions Ltt , respectively It ,(K are: = (A.3) −1for +E + rt+1 ∂K t+1 α 1−α −α (A.2) (1 − α)Kt At Lt − wt = In addition, the envelope theorem implies that ∂Vt+1 (Kt+1 ) = (K0t+1 ) (A.3) −1 + Et ∂Vt (Kt ) ∂Vt+1 α−1 1−α + r (1 − δ) (A.4) = αKt (At Lt ) t+1 + Et ∂Kt+1 ∂K + rt+1 ∂Kt+1 In addition, tthe envelope theorem implies that Substituting the production function in (A.2) gives equation (4): ∂Vt (Kt ) ∂Vt+1 (Kt+1 ) (1 − δ) (A.4) = αKtα−1 (At Lt )1−α + Y Ett ∂Kt ∂Kt+1 1=+ rwt+1 (1 − α) t Lt Substituting the production function in (A.2) gives equation (4): Substituting (A.3) in (A.4) yields: Yt = wt ∂Vt (Kt ) (1 − α)α−1 L = αKt t (At Lt )1−α + (1 − δ) ∂Kt Substituting (A.3) in (A.4) yields: Moving one period forward, and substituting again in (A.3) gives: ∂Vt (Kt ) = αKtα−1 (At Lt )1−α + (1 − δ) α−1 ∂K t αKt+1 (At+1 Lt+1 )1−α + (1 − δ) = −1 + Et + rt+1 Moving one period forward, and substituting again in (A.3) gives: Substituting the production function in the equation above and reshuffling leads to equa1 α−1 tion (5): αKt+1 (At+1 Lt+1 )1−α + (1 − δ) = −1 + Et + rt+1 Yt+1 1−δ +E = function Et α equation Substituting the production in the above and reshuffling leads to equat + rt+1 Kt+1 + rt+1 tion (5): = Et A2 The maximization Yt+1 1−δ + Et α K + r + rt+1 t+1 t+1 problem of the representative household The maximization problem of the household can be rewritten as: A2 The The maximization maximization problem the representative household A2 problemofof the representative household Et [Ut+1 (Xt+1 )] Ut (Xt ) = max ln Ct + 1+ρ {Ct } The maximization problem of the household can be rewritten as: (A.5) s.t Xt+1 = Xt (1 + r1t ) + wt L − Tt − Ct Et [Ut+1 (Xt+1 )] = max ln Ct + 1+ρ {Ct } (A.5) Ut (Xt ) s.t Xt+1 = Xt (1 + rt ) + wt L − Tt − Ct Download free eBooks at bookboon.com 22 The Stochastic Growth Model Appendix A 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 = 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 : The financial industry needs a strong software platform That’s why we need you 1−α α At+1 L ∗ SimCorp is a leading provider of software solutions for industry We work together to reach a common goal: to help our clients + financial δ r the ∗ Kt+1 = succeed by providing a strong, scalable IT platform that enables growth, while mitigating risk and reducing cost At SimCorp, we value which commitment is equivalent to equation and enable you to make (13) the most of your ambitions and potential Are you among the best qualified in finance, economics, IT or mathematics? Find your next challenge at www.simcorp.com/careers www.simcorp.com MITIGATE RISK REDUCE COST ENABLE GROWTH Download free eBooks at bookboon.com 23 Click on the ad to read more Moving one period forward, and substituting again in (A.6) gives the Euler equation (10): The Stochastic Growth Model 1 + rt+1 − Et Ct + ρ Ct+1 = Appendix Appendix BB 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 24 Appendix B The Stochastic Growth Model Appendix B Substituting in the production function (1) gives then equation (12): α 1−α α r∗ + δ = Yt∗ At L Substituting (12) in the first-order condition (4) gives equation (16): wt∗ = α 1−α α ∗ r +δ (1 − α) At Substituting (13) in the law of motion (2) yields: 1−α α r∗ + δ = (1 − δ) At+1 L 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 1−α α 1−α At L − (g + δ) g+δ r∗ + 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 Appendix C C1 The linearized production function The production function is given by equation (1): Yt = Ktα (At Lt )1−α Download free eBooks at bookboon.com 25 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 The Stochastic Growth Model Appendix C Appendix C Appendix C C1 productionfunction function C1 The The linearized linearized production 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) 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 = = ∗ ∗ ∗ 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 Download ϕ2 = free eBooks at bookboon.com ∗ 26 Click on the ad to read more ln Yt − ln Yt∗ Yˆt = 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 Stochastic Growth Model Appendix C C2 The The linearized linearized law of of thethe capital stock C2 lawofofmotion motion 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) C3 The linearized first-order condition for the firm’s labor demand Download free eBooks at bookboon.com The first-order condition for the firm’s labor demand27 is given by equation (4): (1 − α) Yt Lt = wt Taking logarithms of both sides of this equation, and subtracting from both sides their Substituting in equation (C.1) gives = then the linearized law ˆt + ˆ t+1 K Iˆt of motion for K: K 1+g 1+g ˆ t + g + δ Iˆt ˆ t+1 = − δ K K which is equation (19) 1+g 1+g The Stochastic Growth Model which is equation (19) Appendix C C3 The linearized first-order condition for the firm’s labor de- mand C3 The linearized first-order condotion for the firm’s labor demand C3 The linearized first-order condition for the firm’s labor demand The first-order condition for the firm’s labor demand is given by equation (4): Yt demand is given by equation (4): The first-order condition for the firm’s = wt (1 − labor α) Lt Yt = and wt subtracting from both sides their − α) Taking logarithms of both sides of(1this equation, Lt ˆ t = 0), immediately yields values on the balanced growth path (taking into account that L Taking logarithms of both of this condition: equation, and subtracting from both sides their the linearized version of thissides first-order ˆ t = 0), immediately yields values on the balanced growth path (taking into account that L ln (1 − α) + ln Y − ln L = ln w t t t the linearized version of this first-order condition: (ln Yt − ln Yt∗ ) − (ln Lt − ln L∗ ) = ln wt − ln wt∗ ln (1 − α) + ln Yt − ln Lˆt = ln wt ˆt Yt = w (ln Yt − ln Yt∗ ) − (ln Lt − ln L∗ ) = ln wt − ln wt∗ which is equation (20) ˆt Yˆt = w which is equation (20) C4 The linearized first-order condition for the firms’ capital demand C4 The linearized first-order condition for the firms’ capital deC4 The linearized first-order condotion for the firm’s capital demand mand The first-order condition for the firm’s capital demand is given by equation (5): for = the Et [Z t+1 ] capital demand is given by equation (5): The first-order condition firm’s Yt+1 1−δ with Zt+1 = 1+r1t+1 α K + 1+r t+1 t+1 = Et [Zt+1 ] (C.2) (C.3) (C.2) Yt+1 1−δ Now take a first-order Taylor-approximation of α the right-hand-side of equation (C.3) with Zt+1 = 1+r1t+1 + 1+r Kt+1 t+1 ∗ ∗ ∗ around ln Yt+1 = ln Yt+1 , ln Kt+1 = ln Kt+1 and rt+1 = r : Now take a first-order Taylor-approximation of the right-hand-side of equation (C.3) ∗1 (ln Yt+1 − ln Y ∗ ) ∗ + ϕ2 (ln Kt+1 −∗ln K ∗ ) + ϕ3 (rt+1 − r∗ ) = =1ln+Yϕt+1 Zt+1 , ln Kt+1 = lnt+1 Kt+1 and rt+1 = r : t+1 around ln Yt+1 ˆ ˆ = + ϕ Yt+1 + ϕ2 Kt+1∗+ ϕ3 (rt+1 − r∗ ) (C.4) ∗ Yt+1 − ln Yt+1 ) + ϕ2 (ln Kt+1 − ln Kt+1 ) + ϕ3 (rt+1 − r∗ ) Zt+1 = + ϕ11 (ln ˆ t+1 + ϕ3 (rt+1 − r∗ ) = + ϕ1 Yˆt+1 + ϕ2 K (C.4) Download free eBooks at bookboon.com 28 The Stochastic Growth Model Appendix C where ϕ1 ϕ2 ϕ3 = = =  � Yt+1 ∂ 1+r1t+1 α K + t+1  ∂ ln Yt+1 �  ∂ α Yt+1 +  1+rt+1 Kt+1 ∂ ln Kt+1  � Yt+1 ∂ 1+r1t+1 α K + t+1  ∂rt+1 1−δ 1+rt+1 � ∗ 1−δ 1+rt+1 � ∗ 1−δ 1+rt+1 � ∗    ϕ1 , ϕ2 and ϕ3 can be worked out as follows: �  � ∗ Yt+1 1−δ ∂ 1+r1t+1 α K + 1+rt+1 ∂Yt+1  t+1 ϕ1 =  ∂Yt+1 ∂ ln Yt+1 �∗ � 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 ϕ2 =  ∂Kt+1 ∂ ln Kt+1 �∗ � 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 ϕ3 = − + − δ (1 + rt+1 )2 Kt+1 = − + r∗ Substituting in equation (C.4) gives then: Zt+1 = 1+ r∗ + δ ˆ r∗ + δ ˆ rt+1 − r∗ Yt+1 − Kt+1 − ∗ ∗ 1+r 1+r + r∗ Substituting in equation (C.2) and rearranging, gives then equation (21): � � � rt+1 − r∗ r∗ + δ � ˆt+1 ) − Et (K ˆ t+1 ) = E ( Y Et t + r∗ + r∗ Download free eBooks at bookboon.com 29 (C.5) The Stochastic Growth Model Appendix C C5 The linearized Euler equation of the representative household 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 = = where ∗ + 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∗ ) ϕ1 ϕ2 ϕ3 � ∗  � 1+r t+1 Ct ∂ 1+ρ Ct+1  =  ∂ ln Ct+1 � ∗  � 1+r t+1 Ct ∂ 1+ρ Ct+1  =  ∂ ln Ct � ∗  � 1+r t+1 Ct ∂ 1+ρ Ct+1  =  ∂rt+1 ϕ1 , ϕ2 and ϕ3 can be worked out as follows: �  � 1+r ∗ t+1 Ct ∂ 1+ρ Ct+1 ∂C t+1  ϕ1 =  ∂Ct+1 ∂ ln Ct+1 �∗ � + rt+1 Ct = − Ct+1 + ρ Ct+1 = −1 ϕ2 ϕ3 �  � 1+r ∗ t+1 Ct ∂ 1+ρ Ct+1 ∂Ct  =  ∂Ct ∂ ln Ct �∗ � + rt+1 = Ct + ρ Ct+1 = = = � Ct 1 + ρ Ct+1 � 1+rt+1 1+ρ �∗ Ct Ct+1 + rt+1 �∗ Download free eBooks at bookboon.com 30 (C.8) The Stochastic Growth Model Appendix C 1 + r∗ = Substituting in equation (C.8) gives then: Zt+1 rt+1 − r∗ − Cˆt+1 + Cˆt + + r∗ = Substituting in equation (C.6) and rearranging, gives then equation (22): Cˆt rt+1 − r∗ Et Cˆt+1 − Et + r∗ = C6 The linearized equilibrium 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 ∗ ϕ3 = ∂ ln {Ct + It + Gt } ∂ ln Gt ∗ 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 31 Click on the ad to read more Substituting in equation (C.6) and rearranging, gives then equation (22): The Stochastic Growth Model Cˆt rt+1 − r∗ Et Cˆt+1 − Et + r∗ = Appendix C C6 The The linearized linearized equilibrium condition in the goods market C6 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 ∗ ϕ3 = ∂ ln {Ct + It + Gt } ∂ ln Gt ∗ 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 32 ∗ The Stochastic Growth Model ϕ2 = ∂ ln {Ct + It + Gt } ∂It ∂It ∂ ln It = It Ct + It + Gt = ϕ3 Appendix C = ∗ 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 33 The Stochastic Growth Model References References 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 34 Click on the ad to read more ... Stochastic Growth Model Introduction Introduction Introduction This article presents the stochastic growth model The stochastic growth model is a stochastic version of the neoclassical growth model. .. concluding remarks are given in section The Stochastic Growth Model The stochastic growth model The The stochastic growth modelmodel stochastic growth The representative firm Assume that the production... The Stochastic Growth Model Contents Introduction The stochastic growth model The steady state Linearization around the balanced growth path 10 Solution of the linearized model 11 Impulse