随机增长模型

Download free ebooks at bookboon.com Pl ea se c lic k th e ad ve rt The Stochastic Growth Model 2 Contents 1. Introduction 2. The stochastic growth model 3. The steady state 4. Linearization around the balanced growth path 5. Solution of the linearized model 6. Impulse response functions 7. Conclusions Appendix A A1. The maximization problem of the representative fi rm A2. The maximization problem of the representative household Appendix B Appendix C C1. The linearized production function C2. The linearized law of motion of the capital stock C3. The linearized fi rst-order condotion for the fi rm’s labor demand C4. The linearized fi rst-order condotion for the fi rm’s capital demand C5. The linearized Euler equation of the representative household C6. The linearized equillibrium condition in the goods market References Contents 3 4 7 8 9 13 18 20 20 20 22 24 24 25 26 26 28 30 32 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 3 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. Masters in Management The next step for top-performing graduates * Figures taken from London Business School’s Masters in Management 2010 employment report Download free ebooks at bookboon.com The Stochastic Growth Model 3 Introduction 1. 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 3 solves for the steady state, around which the model is linearized in section 4. The linearized model is then solved in section 5. Section 6 shows how the economy responds to stochastic shocks. Some concluding remarks are given in section 7. Download free ebooks at bookboon.com The Stochastic Growth Model 4 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 = Kαt (AtLt) 1−α with 0 < α < 1 (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 depreci- ation rate δ: Kt+1 = (1− δ)Kt + It with 0 ≤ δ ≤ 1 (2) 2. The stochastic growth model The productivity parameter A follows a stochastic path with trend growth g and an AR(1) stochastic component: lnAt = lnA∗t + Aˆt Aˆt = φAAˆt−1 + εA,t with |φA| < 1 (3) A∗t = A ∗ t−1(1 + 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 pro- duction. 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 (4) 1 = Et [ 1 1 + rt+1 α Yt+1 Kt+1 ] + Et [ 1− δ 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 + 1 plus the expected discounted value of the extra capital stock which is left after depreciation at time t + 1. The stochastic growth model Download free ebooks at bookboon.com Pl ea se c lic k th e ad ve rt The Stochastic Growth Model 5 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: lnGt = lnG∗t + Gˆt Gˆt = φGGˆt−1 + εG,t with |φG| < 1 (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 The stochastic growth model © Agilent Technologies, Inc. 2012 u.s. 1-800-829-4444 canada: 1-877-894-4414 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 do for you.www.agilent.com/find/EDUstudents www.agilent.com/find/EDUeducators Download free ebooks at bookboon.com The Stochastic Growth Model 6 The representative household There is one representative household, who derives utility from her current and future consumption: Ut = Et [ ∞∑ s=t ( 1 1 + ρ )s−t lnCs ] with ρ > 0 (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 Xsrs. She also supplies L units of labor to the representative firm, and therefore receives labor income wsL. 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 + 1. This leads to her dynamic budget constraint: Xs+1 = Xs(1 + rs) + wsL− 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: lim s→∞Et [( s∏ s′=t 1 1 + rs′ ) Xs+1 ] = 0 (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 1 Cs = Es [ 1 + rs+1 1 + ρ 1 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. The stochastic growth model Download free ebooks at bookboon.com Pl ea se c lic k th e ad ve rt The Stochastic Growth Model 7 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 = εG,t = Gˆt = 0, ∀t. The model then becomes a standard neoclassical growth model, for which the solution is given by:8 Y ∗t = ( α r∗ + δ ) α 1−α A∗tL (12) K∗t = ( α r∗ + δ ) 1 1−α A∗tL (13) I∗t = (g + δ) ( α r∗ + δ ) 1 1−α A∗tL (14) C∗t = [ 1− (g + δ) α r∗ + δ ]( α r∗ + δ ) α 1−α A∗tL−G∗t (15) w∗t = (1− α) ( α r∗ + δ ) α 1−α A∗t (16) r∗ = (1 + ρ)(1 + g)− 1 (17) 3. The steady state The steady state © U B S 20 10 . A ll ri g h ts r es er ve d . www.ubs.com/graduates 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. You’re full of energy and ideas. And that’s just what we are looking for. Download free ebooks at bookboon.com The Stochastic Growth Model 8 Let us now linearize the model presented in section 2 around the balanced growth path derived in section 3. Loglinear deviations from the balanced growth path are denoted by aˆ(so that Xˆ = lnX − lnX∗). 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 Yˆt = αKˆt + (1− α)Aˆt (18) Kˆt+1 = 1− δ 1 + g Kˆt + g + δ 1 + g Iˆt (19) Yˆt = wˆt (20) Et [ rt+1 − r∗ 1 + r∗ ] = r∗ + δ 1 + r∗ [ Et(Yˆt+1)− Et(Kˆt+1) ] (21) 4. Linearization around the balanced growth path Cˆt = Et [ Cˆt+1 ] − Et [ rt+1 − r∗ 1 + r∗ ] (22) Yˆt = C∗t Y ∗t Cˆt + I∗t Y ∗t Iˆt + G∗t Y ∗t Gˆt (23) The loglinearized laws of motion of A and G are given by equations (3) and (6): Aˆt+1 = φAAˆt + εA,t+1 (24) Gˆt+1 = φGGˆt + εG,t+1 (25) Linearization around the balanced growth path Download free ebooks at bookboon.com The Stochastic Growth Model 9 I now solve the linearized model, which is described by equations (18) until (25). First note that Kˆt, Aˆt and Gˆt are known in the beginning of period t: Kˆt depends on past investment decisions, and Aˆt and Gˆt are determined by current and past values of respectively εA and εG (which are exogenous). Kˆt, Aˆt and Gˆt are 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: Cˆt = ϕCKKˆt + ϕCAAˆt + ϕCGGˆt (26) The challenge now is to determine the ϕ-coefficients. First substitute equation (26) in the Euler equation (22): ϕCKKˆt + ϕCAAˆt + ϕCGGˆt = Et [ ϕCKKˆt+1 + ϕCAAˆt+1 + ϕCGGˆt+1 ] − Et [ rt+1 − r∗ 1 + r∗ ] (27) Now eliminate Et[(rt+1−r∗)/(1+r∗)] with equation (21), and use equations (18), (24) and (25) to eliminate Yˆt+1, Aˆt+1 and Gˆt+1 in the resulting expression. This 5. Solution of the linearized model Solution of the linearized model Download free ebooks at bookboon.com The Stochastic Growth Model 10 leads to a relation between period t’s state variables, the ϕ-coefficients and Kˆt+1: ϕCKKˆt + ϕCAAˆt + ϕCGGˆt = ( ϕCK + (1− α)r ∗ + δ 1 + r∗ ) Kˆt+1 + ( ϕCA − (1− α)r ∗ + δ 1 + r∗ ) φAAˆt + ϕCGφGGˆt (28) We now derive a second relation between period t’s state variables, the ϕ-coefficients and Kˆt+1: rewrite the law of motion (19) by eliminating Iˆt with equation (23); 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) + (αY ∗t )/(K∗t (1 + g)) = (1 + r∗)/(1 + g). This yields: Kˆt+1 = [ 1 + r∗ 1 + g − C ∗ K∗(1 + g) ϕCK ] Kˆt + [ (1− α)Y ∗ K∗(1 + g) − C ∗ K∗(1 + g) ϕCA ] Aˆt − [ G∗ K∗(1 + g) + C∗ K∗(1 + g) ϕCG ] Gˆt (29) Substituting equation (29) in equation (28) to eliminate Kˆt+1 yields: ϕCKKˆt + ϕCAAˆt + ϕCGGˆt = [ ϕCK + (1− α)r ∗ + δ 1 + r∗ ] [ 1 + r∗ 1 + g − C ∗ K∗(1 + g) ϕCK ] Kˆt + [ ϕCK + (1− α)r ∗ + δ 1 + r∗ ] [ (1− α)Y ∗ K∗(1 + g) − C ∗ K∗(1 + g) ϕCA ] Aˆt − [ ϕCK + (1− α)r ∗ + δ 1 + r∗ ] [ G∗ K∗(1 + g) + C∗ K∗(1 + g) ϕCG ] Gˆt + ( ϕCA − (1− α)r ∗ + δ 1 + r∗ ) φAAˆt − ϕCGφGGˆt (30) As this equation must hold for all values of Kˆt, Aˆt and Gˆt, we find the following system of three equations and three unknowns: ϕCK = [ ϕCK + (1− α)r ∗ + δ 1 + r∗ ] [ 1 + r∗ 1 + g − C ∗ K∗(1 + g) ϕCK ] (31) ϕCA = [ ϕCK + (1− α)r ∗ + δ 1 + r∗ ] [ (1− α)Y ∗ K∗(1 + g) − C ∗ K∗(1 + g) ϕCA ] + ( ϕCA − (1− α)r ∗ + δ 1 + r∗ ) φA (32) ϕCG = − [ ϕCK + (1− α)r ∗ + δ 1 + r∗ ] [ G∗ K∗(1 + g) + C∗ K∗(1 + g) ϕCG ] − ϕCGφG (33) Solution of the linearized model Download free ebooks at bookboon.com Pl ea se c lic k th e ad ve rt The Stochastic Growth Model 11 Now note that equation (31) is quadratic in ϕCK : Q0 + Q1ϕCK + Q2ϕ2CK = 0 (34) where Q0 = −(1− α) r∗+δ1+g , Q1 = (1− α) r ∗+δ 1+r∗ C∗t K∗t (1+g) − r∗−g1+g and Q2 = C∗t K∗t (1+g) This quadratic equation has two solutions: ϕCK1,2 = −Q1 ± √ Q21 − 4Q0Q2 2Q2 (35) It turns out that one of these two solutions yields a stable dynamic system, while the other one yields an unstable dynamic system. This can be recognized as follows. Recall that there are three state variables in this economy: K, A and G. A and G may undergo shocks that pull them away from their steady states, but as |φA| and |φG| are less than one, equations (3) and (6) imply that they are always expected to converge back to their steady state values. Let us now look at the expected time path for K, which is described by equation (29). If K is not at its steady state value (i.e. if Kˆ �= 0), K is expected to converge back to its steady state value if the absolute value of the coefficient of Kˆt in equation (29), 1+r∗ 1+g − C ∗ K∗(1+g)ϕCK , is less than one; if |1+r ∗ 1+g − C ∗ K∗(1+g)ϕCK | > 1, Kˆ is expected to increase - which means that K is expected to run away along an explosive path, ever further away from its steady state. Solution of the linearized model © Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers Download free ebooks at bookboon.com The Stochastic Growth Model 12 ϕCG = − [ ϕCK + (1− α) r∗+δ1+r∗ ] G∗ K∗(1+g) 1 + [ ϕCK + (1− α) r∗+δ1+r∗ ] C∗ K∗(1+g) − φG (38) We now have found all the ϕ-coefficients of equation (26), so we can compute Cˆt from period t’s state variables Kˆt, Aˆt and Gˆt. Once we know Cˆt, the other endogenous variables can easily be found from equations (18), (19), (20), (21) and (23). The values of the state variables in period t+1 can be computed from equation (29), and equations (3) and (6) (moved one period forward). Now note that equation (31) is quadratic in ϕCK : Q0 + Q1ϕCK + Q2ϕ2CK = 0 (34) where Q0 = −(1− α) r∗+δ1+g , Q1 = (1− α) r ∗+δ 1+r∗ C∗t K∗t (1+g) − r∗−g1+g and Q2 = C∗t K∗t (1+g) This quadratic equation has two solutions: ϕCK1,2 = −Q1 ± √ Q21 − 4Q0Q2 2Q2 (35) It turns out that one of these two solutions yields a stable dynamic system, while the other one yields an unstable dynamic system. This can be recognized as follows. Recall that there are three state variables in this economy: K, A and G. A and G may undergo shocks that pull them away from their steady states, but as |φA| and |φG| are less than one, equations (3) and (6) imply that they are always expected to converge back to their steady state values. Let us now look at the expected time path for K, which is described by equation (29). If K is not at its steady state value (i.e. if Kˆ �= 0), K is expected to converge back to its steady state value if the absolute value of the coefficient of Kˆt in equation (29), 1+r∗ 1+g − C ∗ K∗(1+g)ϕCK , is less than one; if |1+r ∗ 1+g − C ∗ K∗(1+g)ϕCK | > 1, Kˆ is expected to increase - which means that K is expected to run away along an explosive path, ever further away from its steady state. Solution of the linearized model Download free ebooks at bookboon.com The Stochastic Growth Model 13 We now calibrate the model by assigning appropriate values to α, δ, ρ, A∗t , G∗t , φA, φG, g and L. Let us assume, for instance, that every period corresponds to a quarter, and let us choose parameter values that mimic the U.S. economy: α = 1/3, δ = 2.5%, φA = 0.5, φG = 0.5, and g = 0.5%; A∗t and L are normalized to 1; G∗t is chosen such that G∗t /Y ∗t = 20%; and ρ is chosen such that r∗ = 1.5%.11 It is then straightforward to compute the balanced growth path: Y ∗t = 2.9, K∗t = 24.1, I∗t = 0.7, C∗t = 1.6 and w∗t = 1.9 (while r∗ = 1.5% per construction). Y ∗, K∗, I∗, C∗ and w∗ all grow at rate 0.5% per quarter, while r∗ remains constant over time. Note that this parameterization yields an annual capital- output-ratio of about 2, while C and I are about 55% and 25% of Y , respectively - which seem reasonable numbers. Once we have computed the steady state, we can use equations (36), (37) and (38) to compute the ϕ-coefficients. We are then ready to trace out the economy’s reaction to shocks in A and G. Consider first the effect of a technology shock in quarter 1. Suppose the economy is initially moving along its balanced growth path (such that Kˆs = Aˆs = Gˆs = 0 ∀s < 1), when in quarter 1 it is suddenly hit by a technology shock εA,1 = 1. From equation (3) follows then that Aˆ1 = 1 as well, while equations (29) and (6) imply that Kˆ1 = Gˆ1 = 0. Given these values for quarter 1’s state variables and given the ϕ-coefficients, Cˆ1 can be computed from equation (26); the other endogenous variables in quarter 1 follow from equations (18), (19), (20), (21) and (23). Quarter 2’s state variables can then be computed from equations (28), (3) and (6) - which leads to the values for quarter 2’s endogenous variables, and quarter 3’s state variables. In this way, we can trace out the effect of the technology shock into the infinite future. 6. Impulse response functions Impulse response functions Download free ebooks at bookboon.com The Stochastic Growth Model 14 Figure 1: Effect of a 1% shock in A ... 0 0.05 0.10 0.15 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Kˆ 0 0.2 0.4 0.6 0.8 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Yˆ 0 0.05 0.10 0.15 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Cˆ 0 1 2 3 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Iˆ 0 0.2 0.4 0.6 0.8 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on wˆ −0.5 0 0.5 1.0 1.5 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on E(r)− r∗ Impulse response functions Download free ebooks at bookboon.com The Stochastic Growth Model 15 Figure 2: Effect of a 1% shock in G ... −0.04 −0.03 −0.02 −0.01 0 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Kˆ −0.015 −0.010 −0.005 0 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Yˆ −0.04 −0.03 −0.02 −0.01 0 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Cˆ −0.8 −0.6 −0.4 −0.2 0 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on Iˆ −0.015 −0.010 −0.005 0 0 4 8 12 16 20 24 28 32 36 40 in % quarter ... on wˆ 0 0.