advanced modern macroeconomics · advanced modern macroeconomics exogenous growth max gillman...

Advanced Modern MacroeconomicsExogenous Growth

Max Gillman

University of Missouri-St. Louis

10 October 2016

Gillman (University of Missouri-St. Louis) Chapter 11 10 October 2016 1 / 49

Chapter 11: Exogenous GrowthChapter Summary

Exogenous technological progress in goods sectorOutput growth rate a simple function of productivity growth.Account for basic facts about the growth process.

Modi�cation to consumption demand and AS � AD.Interest rate no longer simple sum

of time preference rate and depreciation rate;interest rate still function of exogenous parametersbecause of exogeneous growth rate.

Calibration two percent targeted growth rate.Same interest rate and wage rate as beforeif time preference rate is lowered.

Goods, Labor growth shown over 4 periods.Trend downwards in time endowment

captures trend upwards in education time.and trend downwards in labor hours per week.


Building on the Last Chapters

Parts 2, 3, 4 allow technology progress

in terms of increase in output productivity parameter.Dynamic Part 4 assumes zero exogenous growth.

Now exogenous growth rate is positive instead of zero

through continuous increase in output productivity.

Calibration is shown with same interest rate

by having lower rate of time preference.

Goods productivity factor increases at constant rate over time;

standard neoclassical growth model.

Adds time endowment decrease at small rate over time.

Two comparative statics explain trends in growth facts, employment.

Still consistent with �uctuations around trends for business cycle.Growth theory with trends in parameters consistentwith business cycle theory of changes in parameters.


Learning Objective

Baseline dynamic model can have positive exogenous growth.

Focus on intertemporal marginal rate of substitution.

How interest rate is determined once growth rate is given.

Then revise AS � AD framework.

Graphical changes of AS � AD as in growth facts.

Use same comparative static tools:

goods and time endowment changes,as used for business cycles.


Who Made It Happen

Solow 1956 extended Ramsey model of growth.by introducing technological change.simultaneously done by Swan 1956 in Australia.

Subsequent articles by Solow on "growth accounting"compute contribution of labor, capital, technological change.1957 "Technical Change & Aggregate Production Function"1960 "Investment and Technical Progress".

Ramsey�s 1928 set out dynamic optimization modelSolow worked on �rm side with constant savings rate.R.G.D.Allen�s 1968 Macro-Economic Theory :focus on growth but with no utility maximization.

1965 David Cass, T.C Koopmans independentlycombined Solow growth with Ramsey optimization:Ramsey-Cass-Koopmans neoclassical growth model.Brock and Mirman 1972 included uncertaintywith random shock in productivity, as in RBC.


Causes of Sustained Growth

Shift up in output productivity parameter causes sustained growth.

Principle of sustained growth:

each factor of production must grow continuously.Physical capital factor, kt , can already grow:

kt+1 = kt (1� δk ) + it .

If it � δkkt , then capital is growing.

Need labor to grow as well, or "labor augmentation".

Productivity shift acts as labor augmentation.

Raw labor time cannot grow inde�nitely:

limited amount of time endowment.Growth models in essence must augment labor time

Total Factor Productivity (TFP) growth: common approach.


Exogenous Technological Progress

TFP increase same as AG increaseLet AG depend on time: denoted AGt , where

yt = AGt (lt )γ (kt )

1�γ . (1)

Assume AGt grows at rate µ :

AGt+1 = AGt (1+ µ) , (2)

Re-write production function with AGt factoring labor time:

yt =hlt (AGt )

1γ

iγ(kt )

1�γ . (3)

De�ne AGt � (AGt )1γ , then

yt =�lt AGt

�γ(kt )

1�γ .

Result: if both factors lt AGt and kt grow at same ratethen output yt grows at that same rate.


Stylized Growth Facts

Real wage rises over time

Real interest rate remains constant.

Output to capital ratio remains constant

Per capita income rises.


Growth Facts by Assuming Trend in TFP

ytkt

=

�lt AGtkt

�γ

,

rt = MPkt = (1� γ)ytkt= (1� γ)

�lt AGtkt

�γ

=) ∆�lt AGtkt

�= 0; ∆ (lt ) = 0,=) ∆

�AGt

�= ∆kt

1+ g =yt+1yt

=

hlt+1 (AGt+1)

1γ

iγ(kt+1)

1�γhlt (AGt )

1γ

iγ(kt )

1�γ= (1+ µ)

�kt+1kt

�1�γ

,

kt+1kt

=

�1+ g1+ µ

� 11�γ

;

�AGt+1AGt

� 1γ

= (1+ µ)1γ .

=)�1+ g1+ µ

� 11�γ

= (1+ µ)1γ ; 1+ g = (1+ µ)

1γ

=) g =yt+1yt

=kt+1kt

' µ

γ.


Growth Accounting

Baseline calibration: γ = 13 .

also assume growth rate of g = 0.03,

=) g = 0.03 = 3µ =µ

γ,

=) µ = 0.01.

g = µ+ (1� γ)kt+1kt

= µ+ (1� γ)µ

γ.

capital accounts for (1� γ)µγ = 2µ = 0.02,

two-thirds of total g = 0.03.Technology change: one-third of growth

Can be reinterpreted with endogenous growth, human capital.


Balanced Growth Path Equilibrium

All variables that grow do so at common rate g .yt , ct , kt and it grow at rate g , for any t :

yt+1yt

=ct+1ct

=kt+1kt

=it+1it= 1+ g .

Time allocations xt , lt stationary along BGP : i.e. constant.Consumer Demand along BGP : kt+1kt = 1+ g ,

cdt = Twt lt + rtkt � kt+1 + kt (1� δk ) ,

= Twt lt + rtkt � kt (1+ g) + kt (1� δk ) ,

= Twt lt + kt (rt � δk � g) .

lt = 1� αctwt,

cdt =Twt + kt (rt � δk � g)

1+ α.


Intertemporal Marginal Rate of Substitution along BGP

1+ g =(ct+1)

d

(ct )d =

1+ rt � δk1+ ρ

,

rt � δk = (1+ g) (1+ ρ)� 1,rt � δk � g = (1+ g) (1+ ρ)� 1� g = ρ (1+ g) .

cdt =1

1+ α[Twt + ktρ (1+ g)] ,

ypt � Twt + ktρ (1+ g) ,

cdt =ypt1+ α

.

Same as cdt =11+α (Twt + ktρ) when g = 0 in Part 4.


AS-AD with Growth

ydt = cdt + st ; st = kst+1 � kst (1� δk ) , k

dt = k

st = kt ;

st = kst

�kst+1kst

� (1� δk )

�= kst [(1+ g)� (1� δk )] = k

st (g + δk ) .

ydt =Twt + ktρ (1+ g)

1+ α+ kt (g + δk ) ,

AD :1wt=

Tydt (1+ α)� kt [ρ (1+ g) + (g + δk ) (1+ α)]

y st = AG

�γAGwt

� γ1�γ

(kt )γ (kt )

1�γ = A11�γ

G

�γ

wt

� γ1�γ

kt ,

AS :1wt=

(y st )1�γ

γ

γA1γ

G (kt )1�γ

γ

.


AS-AD Analysis: Example 11.1

γ = 13 , α = 0.5, ρ = 0.03, T = 1, δk = 0.03, AGt = 0.15;

g = 0.02; .kt = 1.0161.

1.02 = 1+ g = (1+ µ)1γ = (1+ µ)3 ,

µ = (1.02)13 � 1 = 0.0066227.

1wt

=T

ydt (1+ α)� kt [ρ (1+ g) + (g + δk ) (1+ α)]1wt

=1

ydt (1.5)� kt [0.03 (1.02) + (0.02+ 0.03) (1.5)], (4)

1wt

=(y st )

1�γγ

γA1γ

Gt (kt )1�γ

γ

1wt

=3 (y st )

2

(0.15)3 (1.0161)2. (5)


Shift Back in AS-AD with 2% Growth

0.0 0.1 0.2 0.3 0.40

5

10

15

20

Aggregate Output y

1/w

Figure 11.1. AS � AD Equilibrium With 2% Exogenous Growth and ZeroGrowth in Example 11.1.


Recalibration: Example 11.2

With rt = (1+ g) (1+ ρ) + δk � 1,then g > 0 implies rt decreases; kt decreases.With r = 0.06 and g = 0.02, then change ρ :

rt = (1+ g) (1+ ρ) + δk � 1,rt = (1+ 0.02) (1+ ρ) + 0.03� 1 = 0.06;

ρ =1.03(1.02)

� 1 = 0.0098; =) kt = 2.7778.

1wt

=1

ydt (1.5)� 2.7778 [0.0098039 (1.02) + (0.02+ 0.03) (1.5)],

1wt

=3 (y st )

2

(0.15)3 (2.7778)2.


AS-AD Shift Out, Wage the Same

0.0 0.1 0.2 0.3 0.40

5

10

15

20

Aggregate Output y

1/w

Figure 11.2. AS � AD Equilibrium With 2% Exogenous Growth and aLower Rate of Time Discount


Consumption and Output

cdt =1

1+ α[Twt + ktρ (1+ g)]

=23(0.13889+ (2.7778) (0.0098) (1.02)) = 0.1111.

ydt =1

1+ α(Twt + kt [ρ (1+ g) + (g + δk ) (1+ α)])

=23(0.1389+ (2.7778) ((0.0098) (1.02) + (0.05) (1.5))) = 0.25.

(δk + g) kt = (0.03+ 0.02) (2.7778) = 0.13889.

c falls by 20%, k, y rise by 20%, i rises by 100%.Consumption to output ratio falls from two-thirds to

cd

yd=0.11110.25

= 0.444.


Labor Market

T � l st = xt =αctwt; cdt =

11+ α

[Twt + ktρ (1+ g)] ,

l st = T � α

1+ α

�T +

ktρ (1+ g)wt

�;

ldt =

�γAGwt

� 11�γ

kt .

wt =αktρ (1+ g)T � (1+ α) l st

=(0.5) (2.7778) (0.0098) (1.02)

1� (1.5) l st,

wt = γAG

�ktlt

�1�γ

=13(0.15)

�(2.7778)ldt

� 23

.


Shift out in Labor Supply and Demand

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

Labor

w

Figure 11.3. Higher Employment, Same Wage, g = 0.02 of Example 11.2


The Exogenous Growth Process Over TimeExample 11.3. Continuous Shifts in Productivity

g = 0.02, µ = 0.00662.

AGt = 0.15, AGt+1 = 0.15 (1+ µ) .

AGt+1 = (0.15) (1.0066227) = 0.151,

AGt+2 = (0.150 99) (1.0066227) = 0.152,

AGt+3 = (0.15199) (1.0066227) = 0.153.

=) kt = 2.7778, kt+1 = 2.8331,

kt+2 = 2.89, kt+3 = 2.9478.


AS-AD Over Time

1wt+1

=1

ydt+1 (1.5)� 2.8331 [(0.0098039) (1.02) + (0.05) (1.5)],

1wt+1

=3�y st+1

�2(0.15099)3 (2.8331)2

;

1wt+2

=1

ydt+2 (1.5)� 2.8898 [(0.0098039) (1.02) + (0.05) (1.5)],

1wt+2

=3�y st+2

�2(0.15199)3 (2.89)2

;

1wt+3

=1

ydt+3 (1.5)� 2.9478 [(0.0098039) (1.02) + (0.05) (1.5)],

1wt+3

=3�y st+3

�2(0.153)3 (2.9478)2

.


Output and Real Wage Rises

0.23 0.24 0.25 0.26 0.27 0.285

6

7

8

9

10

Aggregate Output y

1/w

Figure 11.4. AS � AD Equilibria Over Time With 2% Exogenous Growthin Example 11.3.


Labor Market

wt+1 =(0.5) (2.8331) (0.0098039) (1.02)

1� (1.5) l st+1,

wt+1 =13(0.15099)

�(2.8331) /

�ldt+1

�� 23;

wt+2 =(0.5) (2.8898) (0.0098039) (1.02)

1� (1.5) l st+2,

wt+2 =13(0.15199)

�(2.89) /

�ldt+2

�� 23;

wt+3 =(0.5) (2.9478) (0.0098039) (1.02)

1� (1.5) l st+3,

wt+3 =13(0.153)

�(2.9478) /

�ldt+3

�� 23.


Labor Supply, Demand Shift Up, Employment Unchanged

0.590 0.595 0.600 0.605 0.610

0.135

0.140

0.145

0.150

0.155

0.160

Labor

Wage Rate w

Figure 11.5. Labor Market with 2% Exogenous Growth in Example 11.3.


Consumption and Output

Consumption and Output grow at 2% rate:

cdt+1 =23(0.14166+ (2.8331) (0.0098) (1.02)) = 0.11333;

cdt+2 =23(0.144 49+ (2.89) (0.0098) (1.02)) = 0.11559;

cdt+3 =23(0.14739+ (2.9478) (0.0098) (1.02)) = 0.11791.

ydt+1 =23(0.142+ (2.83) ((0.0098) (1.02) + (0.05) (1.5))) = 0.255,

ydt+2 =23(0.144+ (2.89) ((0.0098) (1.02) + (0.05) (1.5))) = 0.260,

ydt+3 =23(0.147+ (2.95) ((0.0098) (1.02) + (0.05) (1.5))) = 0.265.


Growth in Input Market SpaceIsocost Lines Over Time

ISOCOST : yt = wt lt + rtkt ,

0.25 = (0.13889) lt + (0.06) kt ,

kt =0.250.06

� (0.13889) lt0.06

;

kt+1 =0.254980.06

� (0.14166)0.06

lt+1;

kt+2 =0.260080.06

� (0.14449)0.06

lt+2;

kt+3 =0.26530.06

� (0.14739)0.06

lt+3


Growth in Input Market SpaceIsoquant Curves Over Time

ISOQUANT : 0.25 = yt = AG�ldt�γ(kt )

1�γ = 0.15�ldt� 13 γ(kt )

23 ,

kt+1 =

�(0.25498)0.15099

� 32

�ldt+1

� 12

;

kt+2 =

�(0.26008)0.15199

� 32

�ldt+2

� 12

;

kt+2 =

�(0.2653)0.153

� 32

�ldt+2

� 12.


Growth in Input Market SpaceInput Ratios Over Time

ktlt

=2.77780.60

= 4.6297;

kt+1lt+1

=2.83310.60

= 4.7218;

kt+2lt+2

=2.890.60

= 4.816;

kt+3lt+3

=2.94780.60

= 4.913.


Isocosts, Isoquants Shift up, Labor Unchanged

0.4 0.5 0.6 0.7 0.8 0.9

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

Labor

k Capital

Figure 11.6. Factor Market Equilibrium with 2% Exogenous Growth inExample 11.3.


Growth in Output SpaceProduction Possibility Curves Over Time

cdt = y st � it = AG�ldt�γ(kt )

1�γ � (g + δk ) kt ,

cdt = (0.15)�ldt� 13(2.7778)

23 � (0.02+ 0.03) (2.7778) ;

cdt+1 = (0.15099)�ldt� 13(2.8331)

23 � (0.02+ 0.03) (2.8331) ;

cdt+2 = (0.15199)�ldt� 13(2.89)

23 � (0.02+ 0.03) (2.89) ;

cdt+3 = (0.153)�ldt� 13(2.9478)

23 � (0.02+ 0.03) (2.9478) ;


Growth in Output SpaceUtility Level Curves Over Time

u = ln ct + α ln xt = ln ct + α ln (1� lt ) ,�2.452 7 = ln 0.1111+ 0.5 ln 0.4,

�2.452 7 = ln ct + 0.5 ln (1� lt ) ,

ct =e ln 0.1111+0.5 ln 0.4

(1� lt )0.5; (6)

ct+1 =e ln 0.11333+0.5 ln 0.4

(1� lt+1)0.5; (7)

ct+2 =e ln 0.11559+0.5 ln 0.4

(1� lt+2)0.5; (8)

ct+3 =e ln 0.11791+0.5 ln 0.4

(1� lt+3)0.5. (9)


Growth in Output SpaceBudget Lines Over Time

cdt = wt l st + ρ (1+ g) kst ,

cdt = (0.13889) l st + (0.0098039) (1+ 0.02) (2.7778) ;

cdt+1 = (0.14166) l st+1 + (0.0098039) (1+ 0.02) (2.8331) ;

cdt+2 = (0.144 49) l st+2 + (0.0098039) (1+ 0.02) (2.8898) ;

cdt+3 = (0.14739) l st+3 + (0.0098039) (1+ 0.02) (2.9478) .


Production, Utility Shift up, Labor Unchanged

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.06

0.08

0.10

0.12

0.14

0.16

0.18

Labor

c

Figure 11.7. General Equilibrium Consumption and Utility Levels withExogenous Growth


Trend in Time Endowment

Aguiar, Hurst 2009: US 1965-2005

labor hours down 12%, leisure up 5%T = x + l , net decrease of 7% over 40 yearsT down by 0.00182 or 0.182% per yearover 40 years, since (1� 0.00182)40 = 0.937% decrease from 1.

New Experiment: Let T trend down, AG trend up

changing in opposite directions simultaneously,related to business cycle explanation:Chapters 3, 9 : T and AG increased in expansion.Now T trends down slightly by 0.182%while AG trends up more signi�cantly.by 0.72%.


Example 11.4: Opposite Trends in Time, Productivity

AGt = 0.15, Tt = 1;

AGt+1 = 0.15108, Tt+1 = 1� 0.00182 = 0.99818;AGt+2 = 0.15216, Tt+2 = (0.99818) (1� 0.00182) = 0.9964;AGt+3 = 0.15327, Tt+3 = (0.99636) (1� 0.00182) = 0.9946.

kt = 2.7778, kt+1 = 2.8280, kt+2 = 2.8793., kt+3 = 2.9317,

g = 0.02.


Example 11.4: AS-AD

1wt+1

=(0.99818)

ydt+1 (1.5)� (2.8331) [(0.0098) (1.02) + (0.05) (1.5)],

1wt+1

=3�y st+1

�2(0.15108)3 (2.8331)2

;

1wt+2

=(0.99636)

ydt+2 (1.5)� (2.89) [(0.0098) (1.02) + (0.05) (1.5)],

1wt+2

=3�y st+2

�2(0.15216)3 (2.89)2

;

1wt+3

=(0.99455)

ydt+3 (1.5)� (2.948) [(0.0098) (1.02) + (0.05) (1.5)],

1wt+3

=3�y st+3

�2(0.15327)3 (2.948)2

.


AS-AD Shift Out, Wage Rises Over Time

0.23 0.24 0.25 0.26 0.27 0.285

6

7

8

9

10

Aggregate Output y

1/w

Figure 11.8. AS � AD Equilibria With AG Trending Up and T TrendingDown in Example 11.4.


Labor Market Supply, Demand Over Time

wt+1 =(0.5) (2.8337) (0.0098039) (1.02)

(0.99818)� (1.5) l st+1,

wt+1 =13(0.15108)

h(2.8331) /

�ldt+1

�i 23;

wt+2 =(0.5) (2.89) (0.0098039) (1.02)

(0.99636)� (1.5) l st+2,

wt+2 =13(0.15216)

h(2.8902) /

�ldt+2

�i 23;

wt+3 =(0.5) (2.948) (0.0098039) (1.02)

(0.99455)� (1.5) l st+3,

wt+3 =13(0.15327)

h(2.948) /

�ldt+3

�i 23.


Labor Supply Shifts Back with Hours Falling

0.590 0.595 0.600 0.605 0.610

0.135

0.140

0.145

0.150

0.155

0.160

Labor

Wage Rate w

Figure 11.9. Labor Market with AG Trending Up and T Trending Down inExample 11.4.


Employment Falls Slightly Over Time

lt = 0.6,

lt+1 = (0.6) (1� 0.00182) = 0.598 91,lt+2 = (0.598 91) (1� 0.00182) = 0.597 82,lt+3 = (0.597 82) (1� 0.00182) = 0.596 73.


TFP, Japan�s Lost Decade, Minnesota School

AGt :"total factor productivity", or TFP, the "Solow residual".Much research computing TFP, including over time

Hayashi and Prescott 2002: Japan�s 1990s stagnation

"lost decade" explained by a low TFP.TFP accounting used both for growth, business cyclesgiven the cycle is more of a longer term,like occasional change in the trend growth rate.

Kehoe, Prescott 2002 "Great Depressions of the Twentieth Century",

known as "Minnesota approach", including1930s depression in Germany, France, Italy, US and UK,1980s contractions in Argentina, Mexico, Chile1990s Japan .

Extended in Chari et al 2007 "Business Cycle Accounting".


Appendix A.11: Solution Methodology for ExogenousGrowth

1+ g =ct+1ct

=1+ rt � δk1+ ρ

,

=) g (1+ ρ) + ρ+ δk = rt = (1� γ)AG

�ltkt

�γ

,

=) ltkt=

�g (1+ ρ) + ρ+ δk

(1� γ)AG

� 1γ

;wt = γAG

�ltkt

�γ�1

ydt =wtT + kt [ρ (1+ g) + (1+ α) (g + δk )]

1+ α= AG

�ltkt

�γ

kt = y st .


Solving for the Capital Stock from AS-AD

Substitute into ydt = yst equation for w as a function of

ltkt

from marginal product of labor,

and substitute in for ltkt,

solved from intertemporal marginand the marginal product of capital.

=) kt =

Tγ (AG )1γ

�(1�γ)

g (1+ρ)+ρ+δk

� 1�γγ

(1+ α) δk

�γ

(1�γ)

�+ (1+ α)

�g (1+ρ)+ρ(1�γ)

�� [ρ (1+ g) + (1+ α) g ]


Example 11.1: Capital Stock Solution

γ = 13 , α = 0.5, ρ = 0.03, T = 1, δk = 0.03, AGt = 0.15, g = 0.02.

=) kt = 1.0161 =

(0.15)3�

23(0.02(1.03)+0.03+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.03)+0.03)

2

�� (0.03 (1.02) + (1.5) (0.02))

�g = 0.04 instead of g = 0.02, then the capital stock falls almost by half to0.53357 :

g = 0.04 =) kt = 0.53357 =

(0.15)3�

23(0.04(1.03)+0.03+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.04(1.03)+0.03)

2

�� (0.03 (1.04) + (1.5) (0.04))

� ,Gillman (University of Missouri-St. Louis) Chapter 11 10 October 2016 45 / 49

Examples 11.2: Capital Stock Solution

r = 0.06, g = 0.02, ρ = 0.0098039

=) kt = 2.7778 =

(0.15)3�

23(0.02(1.0098)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

�


Examples 11.3: Continuous Productivity Increase

=) kt+1 = 2.8331 =

(0.15099)3�

23(0.02(1.0098039)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

� .=) kt+2 = 2.89 =

(0.15199)3�

23(0.02(1.0098)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

�=) kt+3 = 2.9478 =

(0.153)3�

23(0.02(1.0098)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

� .Gillman (University of Missouri-St. Louis) Chapter 11 10 October 2016 47 / 49

Examples 11.4: Trends in Time, Goods

=) kt+1 = 2.8337 =

0.99818 (0.15108)3�

23(0.02(1.0098)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

�=) kt+2 = 2.8902 =

0.99636 (0.15216)3�

23(0.02(1.0098)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

�=) kt+3 = 2.9480 =

0.99455 (0.15327)3�

23(0.02(1.0098)+0.0098+0.03)

�23�(1.5) (0.03) (0.5) + 1.5

�3(0.02(1.0098)+0.0098)

2

�� (0.0098 (1.02) + 0.03)

�Gillman (University of Missouri-St. Louis) Chapter 11 10 October 2016 48 / 49

Examples 11.4: Growth Rate

kt+1kt

=

Tt+1γ(AGt+1)1γ�

(1�γ)g (1+ρ)+ρ+δk

� 1�γγ

(1+α)δk�

γ(1�γ)

�+(1+α)

�g (1+ρ)+ρ(1�γ)

��[ρ(1+g )+(1+α)g ]

Ttγ(AGt )1γ�

(1�γ)g (1+ρ)+ρ+δk

� 1�γγ

(1+α)δk�

γ(1�γ)

�+(1+α)

�g (1+ρ)+ρ(1�γ)

��[ρ(1+g )+(1+α)g ]

,

kt+1kt

=Tt+1 (AGt+1)

1γ

Tt (AGt )1γ

=Tt (1� 0.00182) [AGt (1.0072)]

1γ

Tt (AGt )1γ

= (1� 0.00182) (1.0072)3 = 1.02.


advanced modern macroeconomics · advanced modern macroeconomics exogenous growth max gillman...

Documents