the solow growth model - southamptonlecture 5 & 6 2/23 topics in macroeconomics review and goals...

Review and GoalsToday: The Solow Model

The Solow Model: Toward the Law of MotionThe Solow Model: simple case n = 0 and g = 0

The Solow Growth ModelLecture 5 & 6

Topics in Macroeconomics

October 22 & 23, 2007

Lecture 5 & 6 1/23 Topics in Macroeconomics



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

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 differences across time andacross 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




Today’s stepsFurther steps

Today’s steps 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




Today’s stepsFurther steps

Further steps 7

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

◮ The Golden Rule◮ Dynamic Inefficiency◮ 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)

Law of motion: simple case n = 0 and g = 0 13

◮ 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 14

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K_

t+1

Kt+1 = Kt (45 degree line)

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





Solow’s law of motion 15

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Steady state 16

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 17

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 18

◮ 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 19

◮ 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 20

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Please complete as drawn in class.Lecture 5 & 6 20/23 Topics in Macroeconomics




Steady state comparative statics: deprec. rate δ 21

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Steady state comparative statics: productivity A 22

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Next steps 23

◮ 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?


the solow growth model - southamptonlecture 5 & 6 2/23 topics in macroeconomics review and goals...

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