the solow growth model - · pdf filereview and goals the solow model: toward the law of motion...

Review and GoalsThe Solow Model: Toward the Law of Motion

The Solow Model: simple case n = 0 and g = 0The Solow Model: general case n 6= 0 and g 6= 0

The Solow Growth ModelLectures 5, 6 & 7

Topics in Macroeconomics

Topic 2

October 20, 21 & 27, 2008

Lectures 5, 6 & 7 1/37 Topics in Macroeconomics



From Growth Accounting to the Solow ModelGoal 1: Stylized facts of economic growthGoal 2: Understanding differences over time and across countriesOutline

From Growth Accounting to the Solow Model 2

◮ In growth accounting

→ link of inputs in period t to output in period t→ no link of inputs or output across periods (t versus t + 1)

◮ Solow model links

→ population/labor force, productivity and, in particular,capital stock in year t

to→ labor force, productivity and capital stock in year t + 1

◮ Solow (1956), Solow (1957) and Solow (1960)





From Growth Accounting to the Solow Model 3

◮ Solow’s story about how the capital stock evolves over time

◮ Households save → investment◮ Households save a (constant) fraction s ∈ [0, 1] of their

income every period/year◮ Households consume the rest, i.e., fraction (1 − s) of

income◮ Aggregate income : Yt◮ Aggregate investment = It = sYt

◮ Law of motion of aggregate capital (δ ∈ [0, 1])

Kt+1 = (1 − δ)Kt + It





Kaldor facts: Stylized facts of economic growth 4

1. The labor share and the capital share are almost constantover time.

2. The ratio of aggregate capital to output is almost constantover time.

3. The return to capital is almost constant over time.

4. Output per capita and capital per worker grow at a roughlyconstant and positive rate.

5. Different countries and regions within a country that startout with a different level of income per capita tend toconverge over time.





Understanding growth differencesover time and across countries 5

◮ Why do (developed) countries grow?

◮ Will developing countries catch up to developed countries?

◮ Solow model:a first attempt to explain the mechanics of growth

◮ Implications of Solow’s theory:differences in initial condition, effectiveness of labor andpopulation growth matter





Outline 6

◮ Assumptions◮ Inputs◮ Production function◮ Depreciation◮ Evolution of technology◮ Evolution of population/labor force◮ Consumption and savings

◮ Results◮ Evolution of the capital stock◮ Steady state◮ Balanced Growth





Further steps 7

◮ Comparative statics◮ Savings rate◮ Population growth◮ Technological change

◮ The Golden Rule◮ Implications for

◮ Cross-country differences in GDP levels and growth rates◮ Convergence across countries




AssumptionsAggregationFirm’s problemLaw of motion of aggregate capital stock

Assumptions of the Solow model 8

◮ Assumptions

◮ Inputs: capital, Kt and labor Lt

◮ Production function: neo-classical production function

◮ Depreciation:capital depreciates at rate δ ∈ [0, 1] from t to t + 1

◮ Evolution of technology:At+1 = (1 + g)At ,

◮ Evolution of population (labor force*):Lt+1 = (1 + n)Lt

◮ where δ, g and n are exogenously given parameters





Assumptions of the Solow model 9

◮ Last Assumption

◮ Consumption and savings:

consumers save a constant fraction s of their income, yt ,consume fraction (1 − s) (s parameter)

◮ Per person income is: yt = rtkt + wtℓt

◮ Labor is supplied inelastically & normalized to ℓt = 1

◮ Savings per person are: syt





Aggregating consumers 10

◮ Savings per person are: syt = s(rt kt + wt)

◮ Multiplying by the number of people in period t

pause Aggregate Savings/Investment

= It = Ltsyt = Lts(rtkt + wt) = s(rt Kt + wtLt)





Firm’s problem (lecture 2) 11

max Π(Kt , AtLt) = max[

F (Kt , ALt) − rtKt − wtLt

]

◮ Firms take prices as given and choose inputs K and L

◮ First order conditions

◮∂Πt∂Kt

= FK (Kt , AtLt ) − rt = 0

◮∂Πt∂Lt

= FL(Kt , At Lt) − wt = 0

◮ Firm picks Kt and Lt such that

◮ FK (Kt , At Lt) = rt

◮ FL(Kt , At Lt ) = wt





Law of motion of aggregate capital stock 12

◮ Using the solution to the firm’s problem, we showed that

rtKt + wtLt = F (Kt , AtLt) = Yt (lecture 2)

◮ Using the aggregation over consumers, we saw earlier

It = s(rtKt + wtLt)

◮ Therefore, It = sYt = sF (Kt , AtLt)

◮ Law of motion of aggregate capital

Kt+1 = (1 − δ)Kt + It

◮ Consider Kt+1 as a function of Kt




Law of motion (simple case)Steady state (simple case)Comparative statics (simple case)Comparative dynamics (for n = 0 and g = 0)

The Solow Model (for n = 0 and g = 0)





Law of motion: simple case n = 0 and g = 0 14

◮ Consider Kt+1 as a function of Kt :

Kt+1 = (1 − δ)Kt + It

Kt+1 = (1 − δ)Kt + sYt

Kt+1 = (1 − δ)Kt + sF (Kt , AL)

◮ Since marginal product of K positive,→ law of motion: increasing function

◮ Since marginal product of K diminishing→ law of motion: concave function





Solow’s law of motion 15

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50K_t

K_

t+1

Kt+1 = Kt (45 degree line)

Kt+1 = (1-delta) Kt + s F(Kt,AL)





Solow’s law of motion 16

0

5

10

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20

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40

45

50

0 10 20 30 40 50K_t

K_t+

1







Steady state 17

The state variable of this economy is capital Kt

◮ We say that the economy is at a steady state if the statevariable remains constant.

◮ That is capital is constant at K ∗,K ∗ = Kt = Kt+1

◮ Using the C-D production function, we getKt+1 = (1 − δ)Kt + sK α

t (AL)1−α

K ∗ = (1 − δ)K ∗ + s(K ∗)α(AL)1−α

◮ Solving this equation for K ∗ yields*

K ∗ = (sδ)

11−α AL





Comparative statics 18

K ∗ = (sδ)

11−α AL

◮ If s increases, → K ∗ increases *

◮ If δ increases, → K ∗ decreases*

◮ If A increases, → K ∗ increases*

◮ If L increases, → K ∗ increases*





Comparative dynamics 19

◮ Suppose the level of the capital stock in some economy(country) in year t is at its steady state level

Kt = K ∗ = (sδ)

11−α AL

◮ That is, there is no more growth, i.e. Kt+1 = Kt .

◮ In t + 1, s suddenly increases to s′ > s,

→ sF (Kt , AL) increases to s′F (Kt , AL)

→ K ∗ increases to K ∗′> K ∗

◮ On the graph, we can see that now, the economy startsgrowing again, i.e. Kt+2 > Kt+1 (drawn in class)*

◮ ...until the capital stock reaches the new steady state...K ∗′





Homework 20

◮ Derive the same reasoning for *

◮ If δ decreases or increases*

◮ If A decreases or increases*

◮ If L decreases or increases*





Steady state comparative statics: savings rate s 21

0

5

10

15

20

25

30

35

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45

50

0 10 20 30 40 50K_t

K_

t+1



Please complete as drawn in class.Lectures 5, 6 & 7 21/37 Topics in Macroeconomics




Steady state comparative statics: deprec. rate δ 22

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50K_t

K_

t+1







Steady state comparative statics: productivity A 23

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50K_t

K_

t+1







Next steps 24

◮ What happens if there is exogenous technologicalprogress?

◮ What if there is population growth?

Ramsey Model

◮ What if people explicitly choose how much to save? Doesthe savings rate depend on the rate of technologicalprogress, the rate of depreciation, preferences, labor’sshare in output, taxes..., and if so, how?

Endogenous Growth

◮ What if there is no steady state? can there be endogenousgrowth forever?




Law of motion (general case)Balanced Growth (general case)Goals?

The Solow Model

with population growth (n 6= 0)

and technological progress (g 6= 0)





Outline 26

◮ Introduce n 6= 0 and g 6= 0◮ Derive growth rate of capital stock per worker on balanced

growth path (BGP)◮ Derive growth rate of GDP (output) per capita on BGP◮ Derive growth rate of wages◮ Compare results to Kaldor stylized facts of growth◮ Comparative statics of growth rates on BGP





Balanced growth: n 6= 0 and g 6= 0 27

◮ Evolution of technology:At+1 = (1 + g)At ,

◮ Evolution of population (labour force*):Lt+1 = (1 + n)Lt

◮ Law of motion of aggregate capitalKt+1 = (1 − δ)Kt + sF (Kt , AtLt)

◮ Want to find growth rate of capital per worker, kt = KtLt

and

GDP per capita yt = YtLt





Redefine variables per unit of effective labour 28

◮ With technological progress and population growth(g 6= 0, n 6= 0)Law of motion: Kt+1 is not a stable function of Kt

→ transform law of motion to get stable function

◮ Letyt = Yt

At Ltoutput per unit of effective labour

kt = KtAt Lt

capital per unit of effective labour

◮ Then we can writeytAtLt = Yt

ktAtLt = Kt





→ Law of Motion 29

◮ Law of motion becomes

Kt+1 = (1 − δ)Kt + sYt or,

kt+1At+1Lt+1 = (1 − δ)kt AtLt + sytAtLt or,

kt+1(1 + g)At(1 + n)Lt = (1 − δ)kt AtLt + sytAtLt

kt+1(1 + g)(1 + n) = (1 − δ)kt + syt





→ Law of Motion 30

◮ Law of motion for capital per unit of effective labour

kt+1 = 1(1+g)(1+n)

[

(1 − δ)kt + syt

]

◮ Note that yt = YtAt Lt

= F (Kt ,At Lt )At Lt

= kαt





Steady state in units of effective labour (=BGP) 31

◮ Using yt = kαt , law of motion for kt

kt+1 = 1(1+g)(1+n)

[

(1 − δ)kt + skαt

]

◮ Law of motion: kt+1 is now a stable function of kt

(none of the parameters depends on t)

◮ Show that kt+1 is an increasing and concave function of kt

if α, δ ∈ [0, 1], g, n ∈ [−1, 1]





Solow’s law of motion (capital per u. of eff. labour) 32

0

5

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20

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0 10 20 30 40 50k(hat)_t

k(h

at)

_t+

1

k(hat)_t+1 = k(hat)_t (45 degree line)

k(hat)_t+1 = (1-delta) k(hat)_t + s k(hat)t^alpha





Per capita/worker variables along BGP 33

◮ Steady state ito capital per u. of eff. labour

kt+1 = 1(1+g)(1+n)

[

(1 − δ)kt + skαt

]

◮ This again can be solved for k∗, the value for which capitalper unit of effective labour does not change anymore, i.e.kt = kt+1 = k∗

* k∗ =(

sg+n+ng+δ

)1

1−α





Capital per worker and GDP per capita 34

◮ When the capital stock per unit of effective labour, kt ,reaches its steady state level k∗, we get:

◮ Growth rate of capital per worker:

kt+1

kt=

Kt+1Lt+1

KtLt

=

At+1Kt+1At+1Lt+1

At KtAt Lt

=At+1k∗

At k∗

= (1 + g)

◮ Growth rate of output per capita:

yt+1

yt=

kα

t+1A1−α

t+1

kαt A1−α

t

=(kt+1

kt

)α(At+1

At

)1−α

= (1+g)α(1+g)1−α

yt+1

yt= (1 + g)





The wage rate and rental rate 35

◮ Growth rate of wages

wt+1

wt=

FL(t + 1)

FL(t)=

(1 − α)K αt+1(At+1Lt+1)

−αAt+1

(1 − α)K αt (AtLt)−αAt

=( k∗

k∗

)α(At+1

At

)

= (1 + g)

◮ Show that the rental rate on capital, rt ,is constant along the BGP*





Kaldor facts: Stylized facts of economic growth 36

1. The labor share and the capital share are almost constantover time.

2. The ratio of aggregate capital to output is almost constantover time.

3. The return to capital is almost constant over time, whilewages grow at a roughly constant rate.

4. Output per capita and capital per worker grow at a roughlyconstant and positive rate.

5. Different countries and regions within a country that startout with a different level of income per capita tend toconverge over time.





Predictions of the Solow model 37

1. What does the Solow model have to say about the growthexperience in Chad versus the UK, for example?

2. What is the Solow model missing relative to theoriescoming up later in the course? (keep this question in mindfor later)

3. What is the Solow model missing according to Romer,Barro and Sala-i-Martin (textbooks)?

4. What is the Solow model missing according to Easterly(book)?

5. What is the Solow model missing according to Lucas(article)?


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