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Koen Vermeylen

The Stochastic Growth Model

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The Stochastic Growth Model © 2014 Koen Vermeylen & bookboon.com ISBN 978-87-7681-284-3

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Contents

The Stochastic Growth Model

Contents 1. Introduction

5

2.

The stochastic growth model

6

3.

The steady state

9

4.

Linearization around the balanced growth path

10

5.

Solution of the linearized model

11

6.

Impulse response functions

15

7.

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

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Koen Vermeylen The Stochastic Growth Model

Introduction

1 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.

2

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 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 depreciation rate δ: Kt+1 = (1 − δ)Kt + It

with 0 ≤ δ ≤ 1

1

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(2)

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.

The Stochastic Growth Model

The stochastic growth model

2 The The stochastic growth modelmodel 2. 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 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 depreciation rate δ: Kt+1 = (1 − δ)Kt + It

with 0 ≤ δ ≤ 1

(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 | < 1

(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

1 = Et

(4) 





Yt+1 1 1−δ α + Et 1 + rt+1 Kt+1 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 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

+ g)

with |φG | < 1

(6)

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 Download free eBooks at bookboon.com 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 | < 1

(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

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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 1+ρ

s−t

ln Cs



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 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 + 1. 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: lim Et

s→∞

 s 

1 1 + rs s =t



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 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.

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(11)



The Stochastic Growth Model

The steady state

3 The steady state 3. 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∗

=

It∗

=

Ct∗ = wt∗ = r∗ =

4

 

α ∗ r +δ α ∗ r +δ

 

α 1−α

1 1−α



A∗t L

(12)

A∗t L

(13)



1

1−α α (g + δ) ∗ A∗t L r +δ    α 1−α α α 1 − (g + δ) ∗ A∗t L − G∗t ∗ r +δ r +δ   α 1−α α (1 − α) ∗ A∗t r +δ (1 + ρ)(1 + g) − 1

(14) (15) (16) (17)

Linearization around the balanced growth path

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 ˆ = ln X − ln X ∗ ). are denoted by aˆ(so that X

PREPARE FOR A LEADING ROLE.

Below are the loglinearized versions of the production function (1), the law of programmes motion of the capital stock (2), the first-order conditions (4) MSc and (5), the Euler in Strategy and management in equation (10) and the equilibrium condition (11):9

Et



ˆ t + (1 − α)Aˆt Yˆt = αK ˆ t + g + δ Iˆt ˆ t+1 = 1 − δ K K 1+g 1+g ˆ ˆt Yt = w rt+1 − r ∗ 1 + r∗



=

international organisations, and Industrial engineering and (18) management. No tuition fees. (19) (20)

 r∗ + δ  ˆ t+1 ) Et (Yˆt+1 ) − Et (K ∗ 1+r

(21)

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α

1−α α A∗t ∗ r +δ = (1 + ρ)(1 + g) − 1

wt∗ = (1 − α) r∗

(16) (17)



The Stochastic Growth Model

Linearization around the balanced growth path

4 Linearization Linearization around the balanced growth path 4. around the balanced growth path 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 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

Et



ˆ t + (1 − α)Aˆt Yˆt = αK ˆ t + g + δ Iˆt ˆ t+1 = 1 − δ K K 1+g 1+g ˆt Yˆt = w  r∗

rt+1 − 1 + r∗

=

r∗

(18) (19) (20)

+δ 

1 + r∗



ˆ t+1 ) Et (Yˆt+1 ) − Et (K 

  rt+1 − r ∗ Cˆt = Et Cˆt+1 − Et 1 + 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 + εG,t+1 ˆ t+1 = φG G G

5

(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: ˆ t + ϕCA Aˆt + ϕCG G ˆt Cˆt = ϕCK K 10

The challenge now is to determine the ϕ-coefficients. Download free eBooks at bookboon.com

First substitute equation (26) in the Euler equation (22):

(26)

Aˆt+1 = φA Aˆt + εA,t+1 ˆ t + εG,t+1 ˆ t+1 = φG G G

(25)



The Stochastic Growth Model

5

(24) Solution of the linearized model

Solution of the linearized model

5. 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 ∗ 1 + 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

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The Stochastic Growth Model

Solution of the linearized 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 = ϕCK + (1 − α) + ϕ − (1 − α) φA Aˆt + ϕCG φG G K t+1 CA 1 + r∗ 1 + 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∗ ˆt ˆ t+1 = 1 + r − ϕCK K K 1+g K ∗ (1 + g)     C∗ C∗ (1 − α)Y ∗ G∗ ˆ ˆ − ϕ + ϕ + − A CA t CG Gt 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    ∗ C∗ r + δ 1 + r∗ ˆt − ∗ ϕCK K = ϕCK + (1 − α) 1 + r∗ 1+g K (1 + g)    C∗ r ∗ + δ (1 − α)Y ∗ − ϕCA Aˆt + ϕCK + (1 − α) 1 + r ∗ K ∗ (1 + g) K ∗ (1 + g)    C∗ r∗ + δ G∗ ˆ + ϕ − ϕCK + (1 − α) CG Gt 1 + r ∗ K ∗ (1 + g) K ∗ (1 + g)   r∗ + δ ˆt + ϕCA − (1 − α) φA Aˆt − ϕCG φG G (30) 1 + r∗ 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







C∗ r∗ + δ 1 + r∗ − ϕCK = ϕCK + (1 − α) (31) 1 + r∗ 1+g K ∗ (1 + g)    C∗ r ∗ + δ (1 − α)Y ∗ − ϕ = ϕCK + (1 − α) CA 1 + r ∗ K ∗ (1 + g) K ∗ (1 + g)   r∗ + δ + ϕCA − (1 − α) φA (32) 1 + r∗     C∗ r∗ + δ G∗ + ϕCG − ϕCG φG = − ϕCK + (1 − α) 1 + r ∗ K ∗ (1 + g) K ∗ (1 + g) (33)

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The Stochastic Growth Model

Solution of the linearized model

Now note that equation (31) is quadratic in ϕCK : Q0 + Q1 ϕCK + Q2 ϕ2CK = 0 where Q0 = −(1 −

∗ +δ α) r1+g ,

(34)

Q1 = (1 −

Ct∗ r ∗ +δ α) 1+r ∗ K ∗ (1+g) t

−

r ∗ −g 1+g

and Q2 =

Ct∗ ∗ Kt (1+g)

This quadratic equation has two solutions: −Q1 ±

ϕCK1,2 =



Q21 − 4Q0 Q2

(35)

2Q2

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 ˆ �= 0), K is expected to converge back to its at its steady state value (i.e. if K ˆ t in equation (29), steady state value if the absolute value of the coefficient of K 1+r ∗ C∗ 1+r ∗ C∗ ˆ 1+g − K ∗ (1+g) ϕCK , is less than one; if | 1+g − K ∗ (1+g) ϕCK | > 1, K is expected to increase - which means that K is expected to run away along an explosive path, ever further away from its steady state. Let us therefore evaluate the coefficient K ∗ (1+g)



1+r ∗ 1+g

∗

∗

− K ∗C(1+g) ϕCK , which we call ϕKK . 

1+r Rewriting yields ϕCK = t C ∗ 1+g − ϕKK . Substituting in the quadratic t equation (34) leads to a quadratic equation in ϕKK . Denote this quadratic equa2 KK ) > 0. This tion as f (ϕKK ) = 0, and note that f (0) > 0, f (1) < 0 and ∂f∂(ϕ 2ϕ KK implies that the quadratic equation f (ϕKK ) = 0 has one solution between 0 and 1, and another solution which is greater than 1. To ensure stable dynamics,10 we retain the solution for ϕKK that is between 0 and 1; which means that of the two solutions for ϕCK (given in equation (35)), we need to retain the largest one:

INNOVATIVE LIKE YOU. ϕCK

=

−Q1 +



Q21 − 4Q0 Q2

(36)

2Q2

If you’re hoping for a truly modern education, where Substituting in equations (32) and (33) yieldsone then theyou’re solutions for ϕCA and ϕCG :encouraged to speak your mind and to think long-term, both when  it comes to your own future and the future of the planet. Then the ∗ ∗ r ∗ +δ r +δ ϕ + (1 − α) (1you. − α) K ∗Y(1+g) − (1 − α) 1+r ∗ φA CK University of Gothenburg is the place 1+r ∗ for

ϕCA =



∗

r +δ 1 + ϕCK + (1 − α) 1+r ∗



C∗ K ∗ (1+g)

− φA

(37)

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ˆ �= 0), K is expected to converge back to its at its steady state value (i.e. if K ˆ t in equation (29), steady state value if the absolute value of the coefficient of K 1+r ∗ C∗ 1+r ∗ C∗ ˆ 1+g − K ∗ (1+g) ϕCK , is less than one; if | 1+g − K ∗ (1+g) ϕCK | > 1, K is expected to The Stochastic Growth Model increase - which means that K is expected to run away along an explosiveSolution path, of the linearized model ever further away from its steady state. Let us therefore evaluate the coefficient K ∗ (1+g)



1+r ∗ 1+g

∗

− K ∗C(1+g) ϕCK , which we call ϕKK . 

∗

1+r Rewriting yields ϕCK = t C ∗ 1+g − ϕKK . Substituting in the quadratic t equation (34) leads to a quadratic equation in ϕKK . Denote this quadratic equa2 KK ) > 0. This tion as f (ϕKK ) = 0, and note that f (0) > 0, f (1) < 0 and ∂f∂(ϕ 2ϕ KK implies that the quadratic equation f (ϕKK ) = 0 has one solution between 0 and 1, and another solution which is greater than 1. To ensure stable dynamics,10 we retain the solution for ϕKK that is between 0 and 1; which means that of the two solutions for ϕCK (given in equation (35)), we need to retain the largest one:

ϕCK

=

−Q1 +



Q21 − 4Q0 Q2

(36)

2Q2

Substituting in equations (32) and (33) yields then the solutions for ϕCA and ϕCG : ϕCA =



∗



∗

∗

r +δ Y r +δ ϕCK + (1 − α) 1+r ∗ (1 − α) K ∗ (1+g) − (1 − α) 1+r ∗ φA

ϕCG = −



∗

r +δ 1 + ϕCK + (1 − α) 1+r ∗





∗

r +δ ϕCK + (1 − α) 1+r ∗ ∗

r +δ 1 + ϕCK + (1 − α) 1+r ∗







C∗ K ∗ (1+g)

G∗ K ∗ (1+g)

C∗ K ∗ (1+g)

− φG

− φA

(37)

(38)

We now have found all the ϕ-coefficients of equation (26), so we can compute ˆ t , Aˆt and G ˆ t . Once we know Cˆt , the other Cˆt from period t’s state variables K 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).

6

Impulse response functions

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 /Yt∗ = 20%; and ρ is chosen such that r ∗ = 1.5%.11 It is then straightforward to compute the balanced growth path: Yt∗ = 2.9, Kt∗ = 24.1, It∗ = 0.7, Ct∗ = 1.6 and wt∗ = 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 capitaloutput-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 ˆs = 0 ˆ s = Aˆs = G is initially moving along its balanced growth path 14 (such that K ∀s < 1), when in quarter 1 it is suddenly hit by a technology shock εA,1 = 1. Download free eBooks at bookboon.com From equation (3) follows then that Aˆ1 = 1 as well, while equations (29) and ˆ 1 = 0. Given these values for quarter 1’s state variables ˆ1 = G (6) imply that K ˆ

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).

The Stochastic Growth Model



Impulse response functions

6 Impulse Impulse response functions 6. response functions 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 /Yt∗ = 20%; and ρ is chosen such that r ∗ = 1.5%.11 It is then straightforward to compute the balanced growth path: Yt∗ = 2.9, Kt∗ = 24.1, It∗ = 0.7, Ct∗ = 1.6 and wt∗ = 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 capitaloutput-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 ˆs = 0 ˆ s = Aˆs = G is initially moving along its balanced growth path (such that K ∀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 ˆ 1 = 0. Given these values for quarter 1’s state variables ˆ1 = G (6) imply that K 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.

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The Stochastic Growth Model

Impulse response functions

Figure 1: Effect of a 1% shock in A ...

ˆ ... on K

in %

... on Yˆ

in %

0.15

0.8 0.6

0.10

0.4 0.05

0.2

0

0 0

4

8 12 16 20 24 28 32 36 40

0

4

quarter

quarter

... on Cˆ

in %

8 12 16 20 24 28 32 36 40

... on Iˆ

in %

0.15

3

0.10

2 1

0.05

0 0 0

4

8 12 16 20 24 28 32 36 40

0

4

quarter

quarter

... on w ˆ

in %

1.5

0.6

1.0

0.4

0.5

0.2

0

0

−0.5 4

... on E(r) − r ∗

in %

0.8

0

8 12 16 20 24 28 32 36 40

8 12 16 20 24 28 32 36 40

0

4

quarter

8 12 16 20 24 28 32 36 40 quarter

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The Stochastic Growth Model

Impulse response functions

Figure 2: Effect of a 1% shock in G ...

ˆ ... on K

in %

... on Yˆ

in %

0

0

−0.01

−0.005

−0.02

−0.010

−0.03 −0.04

−0.015 0

4

8 12 16 20 24 28 32 36 40

0

4

quarter

quarter

... on Cˆ

in %

8 12 16 20 24 28 32 36 40

... on Iˆ

in %

0

0

−0.01

−0.2

−0.03

−0.6

−0.02

−0.4

−0.04

−0.8 0

4

8 12 16 20 24 28 32 36 40

0

4

quarter

quarter

... on w ˆ

in %

0.12

−0.005

0.08

−0.010

0.04

−0.015

0 4

... on E(r) − r ∗

in %

0

0

8 12 16 20 24 28 32 36 40

8 12 16 20 24 28 32 36 40

0

4

8 12 16 20 24 28 32 36 40 quarter

quarter

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The Stochastic Growth Model



Impulse response functions

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 stock K gradually increases as well 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 state values. Consider now the effect of a shock in government expenditures in quarter 1. Assume again that the economy is on a balanced growth path in quarter 0. In quarter 1, however, the economy is hit by a shock in government expenditures εG,1 = 1. From equation (3) follows then that Aˆ1 = 1 as well, while equations ˆ 1 = 0. Once we know the state variables in ˆ1 = G (29) and (6) imply that K quarter 1, 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. Figure 2 shows the economy’s reaction to a shock in government expenditures during the first 40 quarters. As G increases, E(r) increases as well such that C and I fall (to make sure that C +I +G remains equal to Y , which does not change ˆ 1 = 0). As I falls, the capital stock K gradually decreases after in quarter 1 as K period 1, such that Y starts decreasing after period 1 as well. In the meantime, however, the shock in G is dying out, so after a while E(r) decreases again. As a 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&KDOPHUV8QLYHUVLW\RI7HFKQRORJ\FRQGXFWVUHVHDUFKDQGHGXFDWLRQLQHQJLQHHU their steady state values.



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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

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ˆ1 = G ˆ 1 = 0. Once we know the state variables in (29) and (6) imply that K quarter 1, 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 leadsGrowth to theModel values for quarter 2’s endogenous variables and quarter Impulse 3’s response functions The Stochastic state variables, and so on until the infinite future. Figure 2 shows the economy’s reaction to a shock in government expenditures during the first 40 quarters. As G increases, E(r) increases as well such that C and I fall (to make sure that C +I +G remains equal to Y , which does not change ˆ 1 = 0). As I falls, the capital stock K gradually decreases after in quarter 1 as K period 1, such that Y starts decreasing after period 1 as well. In the meantime, however, the shock in G is dying out, so after a while E(r) decreases again. As a 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.

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 undetermined coefficients. Even though the stochastic growth model itself might bear little resemblance to

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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. Conclusions

The Stochastic Growth Model

7

Conclusions

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 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.

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 rate of aggregate production, the interest rate and the capital-output-ratio.

3 4

5

Join the Vestas Graduate Programme 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.

Experience the Forces of Wind Lump-sum taxes do not affect the first-order conditions of the firms and the households, and kick-start therefore doyour not career affect their behavior either. and

6

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 As one of the world leaders in wind power soluor by raising taxes. The reason for this is that given the time path of government expentions with wind turbine installations in over 65 ditures,and every in public debt must sooner or later be matched by an increase in countries moreincrease than 20,000 employees taxes, such that the present discounted value of the taxes which a representative houseglobally, Vestas looks to accelerate innovation hold has to pay is not affected by the way through the development of our employees’ skills how the government finances its expenditures which implies that her current wealth and her consumption path are not affected either. and talents. Our goal is to reduce CO emissions

7

dramatically and ensure sustainable world for See appendix A for athe derivation. future generations.

2

8 9 10

See appendix B for the derivation.

Read about the Vestas See more appendix C for theGraduate derivations. Programme on vestas.com/jobs. The solution with unstable dynamics not only does not make sense from an economic Application period will open March 1 2012.

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.

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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 Conclusions The Stochastic Growth Model and saving in a general equilibrium setting.

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 rate of aggregate production, the interest rate and the capital-output-ratio.

3

See appendix A for derivations.

4

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.

5

Lump-sum taxes do not affect the first-order conditions of the firms and the households, and therefore do not affect their behavior either.

6

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.

7

See appendix A for the derivation.

8

See appendix B for the derivation.

9

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.

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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    1 max Yt − wt Lt − It + Et Vt+1 (Kt+1 ) Vt (Kt ) = 1 + rt+1 {Lt ,It } The maximization problem of the firm can be rewritten as:    1 s.t. Yt = Ktα (At Lt )1−α max Yt − wt Lt − It + Et Vt+1 (Kt+1 ) Vt (Kt ) = 1 + rt+1 {L t ,It }= (1 − δ)Kt + It Kt+1

(A.1) (A.1)

1−α The first-order conditions respectively It , are: s.t. Ytfor =L Kttα, (A t Lt ) α 1−α −α − wt = 0 (A.2) (1(1 −− α)K Kt+1 = δ)K t+ t A t It Lt   ∂Vt+1 1 t+1 ) The first-order conditions Ltt , respectively It ,(K are: = 0 (A.3) −1for +E 1 + rt+1 ∂K t+1 α 1−α −α (A.2) (1 − α)Kt At Lt − wt = 0  implies that  In addition, the envelope theorem ∂Vt+1 (Kt+1 ) 1 = (K0t+1 )  (A.3) −1 + Et ∂Vt (Kt ) ∂Vt+1 1 α−1 1−α ∂K 1 + r t+1 (1 − δ) (A.4) = αKt (At Lt ) t+1 + Et ∂K 1 + 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 (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 ) = αK α−1 (A L )1−α + (1 − δ)  t t   1 α−1 t ∂K t αKt+1 (At+1 Lt+1 )1−α + (1 − δ) = 0 −1 + Et 1 + 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 equa  1 α−1 1−α tion (5): αKt+1 (At+1 Lt+1 ) + (1 − δ) = 0 −1 + Et 1 + rt+1     Yt+1 1 1−δ +E 1 = function Et α equation Substituting the production in the above and reshuffling leads to equat 1 + rt+1 Kt+1 1 + rt+1 tion (5):     Yt+1 1 1−δ + Et 1 = Et α 1 + rt+1 K 1 + rt+1 A2. The maximization problem oft+1the representative household

The maximization problem of the household can be rewritten as:

A2. The The maximization maximization problem ofof the representative household A2. problem the representative household   1 Et [Ut+1 (Xt+1 )] Ut (Xt ) = max ln Ct + 1+ρ {Ct } The maximization problem of the household can be rewritten as:   s.t. Xt+1 = Xt (1 + r1t ) + wt L − Tt − Ct Et [Ut+1 (Xt+1 )] Ut (Xt ) = max ln Ct + 1+ρ {Ct } s.t. Xt+1 = Xt (1 + rt ) + wt L − Tt − Ct

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(A.5) (A.5)



The Stochastic Growth Model

Appendix A

The first-order condition for Ct is:   ∂Ut+1 (Xt+1 ) 1 1 Et = − Ct 1+ρ ∂Xt+1

0

(A.6)

In addition, the envelope theorem implies that   ∂Ut (Xt ) ∂Ut+1 (Xt+1 ) 1 Et = (1 + rt ) ∂Xt 1+ρ ∂Xt+1

(A.7)

Substituting (A.6) in (A.7) yields: ∂Ut (Xt ) ∂Xt

= (1 + rt )

1 Ct

Moving one period forward, and substituting again in (A.6) gives the Euler equation (10):   1 1 + rt+1 1 = 0 − Et Ct 1 + ρ Ct+1

Appendix B If C grows at rate g, the Euler equation (10) implies that Cs∗ (1 + g) =

1 + r∗ ∗ C 1+ρ s

Rearranging gives then the gross real rate of return 1 + r∗ : 1 + r∗ = (1 + g)(1 + ρ) which immediately leads to Scholarships 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:

Open your mind to (A L) αK newgives opportunities Rearranging then the value of K : ∗α−1 t+1

t+1

1−α

=

r∗ + δ

∗ t+1

With 31,000 students, Linnaeus University 1  is  1−α α one of the larger universities in Sweden. ∗ = We At+1 L Kt+1 are a modern university, known for our strong r∗ + δ international profile. Every year more than which is equivalent to students equation (13). 1,600 international from all over the world choose to enjoy the friendly atmosphere and active student life at Linnaeus University. Welcome to join us!

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Moving one period forward, and substituting again in (A.6) gives the Euler equation (10):   1 1 + rt+1 1 = 0 − Et 1 + ρ Ct+1 The Stochastic Growth Model Ct

Appendix Appendix BB If C grows at rate g, the Euler equation (10) implies that Cs∗ (1 + g) =

1 + r∗ ∗ C 1+ρ s

Rearranging gives then the gross real rate of return 1 + r∗ : 1 + 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  1−α

At+1 L

which is equivalent to equation (13).

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Appendix B



The Stochastic Growth Model

Appendix B

Substituting in the production function (1) gives then equation (12): α  1−α  α ∗ Yt = At L r∗ + δ

Substituting (12) in the first-order condition (4) gives equation (16): α  1−α  α ∗ wt = (1 − α) At r∗ + δ

Substituting (13) in the law of motion (2) yields: 1 1   1−α  1−α  α α A L = (1 − δ) At L + It∗ t+1 r∗ + δ r∗ + δ such that It∗ is given by:  It∗ = = =

α r∗ + δ

1  1−α

At+1 L − (1 − δ)



α r∗ + δ

1  1−α α [(1 + g) − (1 − δ)] At L r∗ + δ 1  1−α  α At L (g + δ) ∗ r +δ



1  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 1  1−α  1−α   α α At L − (g + δ) At L − G∗t r∗ + δ r∗ + δ α   1−α  α g+δ 1−α ∗ At L − G∗t r +δ r∗ + δ

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

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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 ϕ2

= =

 

∂ ln {(1 − δ)Kt + It } ∂ ln Kt ∂ ln {(1 − δ)Kt + It } ∂ ln It

∗ ∗

ϕ1 and ϕ2 can be worked out as follows: ∗  ∂ ln {(1 − δ)Kt + It } ∂Kt ϕ1 = ∂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 ϕ2 = ∂It ∂ ln It 26

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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

=

ϕ2

=

 

∂ ln {(1 − δ)Kt + It } ∂ ln Kt ∂ ln {(1 − δ)Kt + It } ∂ ln It

∗ ∗

ϕ1 and ϕ2 can be worked out as follows: ∗  ∂ ln {(1 − δ)Kt + It } ∂Kt ϕ1 = ∂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 ϕ2 = ∂It ∂ ln It

= = =

∗ 1 It (1 − δ)Kt + It ∗  1 It Kt+1 g+δ ...as It∗ /Kt∗ = g + δ and Kt grows at rate g on the balanced growth path 1+g 

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 The first-order condition for the firm’s labor demand27 is given by equation (4): Yt Download = wat bookboon.com (1 − α)free eBooks Lt

t

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 = 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): 1 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 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 = 1 + 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 = 1 + ϕ11 (ln ˆ t+1 + ϕ3 (rt+1 − r∗ ) = 1 + ϕ1 Yˆt+1 + ϕ2 K (C.4)

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The Stochastic Growth Model

Appendix C

where ϕ1

ϕ2

ϕ3

 � Yt+1 ∂ 1+r1t+1 α K + t+1  ∂ ln Yt+1 �  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 1 = α Yt+1 1 + rt+1 Kt+1 r∗ + δ ∗ ∗ = ...using the fact that αYt+1 = (r∗ + δ)Kt+1 1 + r∗ �  � ∗ Yt+1 1−δ ∂ 1+r1t+1 α K + 1+rt+1 ∂Kt+1  t+1 ϕ2 =  ∂Kt+1 ∂ ln Kt+1 �∗ � 1 Yt+1 = − α 2 Kt+1 1 + rt+1 Kt+1 r∗ + δ ∗ ∗ = − ...using the fact that αYt+1 = (r∗ + δ)Kt+1 1 + r∗ � � ��∗ Yt+1 1 α ϕ3 = − + 1 − δ (1 + rt+1 )2 Kt+1 1 = − 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 1 + 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 1 + r∗ 1 + r∗

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(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: 1

= 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

∗ 1 + ϕ1 (ln Ct+1 − ln Ct+1 ) + ϕ2 (ln Ct − ln Ct∗ ) + ϕ3 (rt+1 − r∗ ) 1 + 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 �∗ � 1 + rt+1 Ct = − 2 Ct+1 1 + ρ Ct+1 = −1 ϕ2

ϕ3

�  � 1+r ∗ t+1 Ct ∂ 1+ρ Ct+1 ∂Ct  =  ∂Ct ∂ ln Ct �∗ � 1 + rt+1 1 = Ct 1 + ρ Ct+1 = 1 = =

�

Ct 1 1 + ρ Ct+1

� 1+rt+1 1+ρ

�∗

Ct Ct+1

1 + rt+1

�∗

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(C.8)



The Stochastic Growth Model

Appendix C

1 1 + r∗

=

Substituting in equation (C.8) gives then: Zt+1

=

rt+1 − r∗ 1 − Cˆt+1 + Cˆt + 1 + r∗

Substituting in equation (C.6) and rearranging, gives then equation (22):     rt+1 − r∗ Cˆt = Et Cˆt+1 − Et 1 + 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

=

ϕ2

=

ϕ3

=

  

∂ ln {Ct + It + Gt } ∂ ln Ct ∂ ln {Ct + It + Gt } ∂ ln It ∂ ln {Ct + It + Gt } ∂ ln Gt

∗

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ϕ1 , ϕ2 and ϕ3 can be worked out as follows: ∗  ∂ ln {Ct + It + Gt } ∂Ct ϕ1 = ∂Ct ∂ ln Ct ∗  1 = Ct Ct + It + Gt Ct∗ = Yt∗

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Substituting in equation (C.6) and rearranging, gives then equation (22):     rt+1 − r∗ ˆ ˆ Ct = Et Ct+1 − Et 1 + r∗ The Stochastic Growth Model

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

=

ϕ2

=

ϕ3

=



∂ ln {Ct + It + Gt } ∂ ln Ct



∂ ln {Ct + It + Gt } ∂ ln Gt



∂ ln {Ct + It + Gt } ∂ ln It

∗ ∗ ∗

ϕ1 , ϕ2 and ϕ3 can be worked out as follows: ∗  ∂ ln {Ct + It + Gt } ∂Ct ϕ1 = ∂Ct ∂ ln Ct ∗  1 = Ct Ct + It + Gt Ct∗ = Yt∗

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Appendix C



The Stochastic Growth Model

ϕ2

= = =

ϕ3

= = =

Appendix C

∗ ∂ ln {Ct + It + Gt } ∂It ∂It ∂ ln It ∗  1 It Ct + It + Gt It∗ Yt∗ ∗  ∂ ln {Ct + It + Gt } ∂Gt ∂Gt ∂ ln Gt ∗  1 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∗

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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, 3 (April), 837-846.

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