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CAPITAL ACCUMULATION

AND GROWTH: THE BASIC SOLOW

MODEL

Chapter 3 – first lecture

IntroducingAdvanced Macroeconomics:Growth and business cycles

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The basic Solow model • How can a nation become rich, i.e., initiate a growth

process leading to higher GDP/consumption per capita in the long run.

• The basic Solow model provides some first answers: It predicts how the evolution and the long run levels of GDP and consumption per capita depend on structural parameters such as the rate of investment and the growth rate of the labour force.

• Key elements of the Solow model:– In each period output is determined by the supplies of

capital and labour through the production function– Exogenous savings/investment rate, s, exogenous growth

rate of labour force, n, and exogenous depreciation rate, δ.– Explicit description of capital accumulation: – Accumulation of capital is the main driving force for wealth.

“Basic” model: No technological progress.1t t t tK K I K

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The ”micro world” of the Solow model

• Object: Closed economy.• Time: A sequence of periods/years,• Agents: Households and firms (and

government).• Commodities and markets: Output, capital

services and labour services (one asset = physical capital).

• The market for output: Supply = firms’ output, . Demand from households for consumption and investment = . Relative price = 1.– One-sector model: Output can be used either for

consumption or for investment.

0 1 2t , , ,...

tY

t tC I

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• The market for capital services: Consumers own the capital stock, , and rent its services to firms.Supply of capital services = . Firms’ demand =– Relative price (in units of output) for renting one unit of

capital for one period: = real rental rate for capital.– Real interest rate: , where is the rate of

depreciation. Or: .(Alternative interpretation: The firms own the capital, borrow for the purchase of capital at an interest rate of and bear the cost of depreciation themselfes).

– User cost .

• Labour market: Households supply = (= the labour force). Demand from firms = . Relative price: = the real wage rate.

• Competitive markets: and adjust to equate supply and demand in all markets full (or natural) utilization of ressources.

tKstK

tr

t tr t tr

t

t tr

tLdtL

tw

dtK

tr tw

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The production side • … is modelled as if all production (all of GDP)

comes from one profit maximizing firm that produces value added, , from of capital services, (machine-years), and labour services, (man-years), according to the production function:

1. Constant returns to scale: . The replication argument!

2. Positive marginal products:

tYdtK

dtL

d dt t tY F K ,L

d d d dt t t tF K , L F K ,L

0 0d d d dK LF' K ,L , F ' K ,L .

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3. Marginal products are decreasing in the amount of the factor used:

(diminishing returns) and growing in the amount of the other factor used:

• Profit maximization: Given and , the firm chooses and to:

The ususal necessary conditions for an optimum:

(These two equations do not determine and from given and , they only determine )

0 0KK LLF , F '' ''

tr twdtL

dt tY ,K

max s.td d d dt t t t t t t tY rK w L , . Y F K ,L

d d d dK t t t L t t tF K ,L r , F K ,L w ' '

dtK

dtL

d dt tK / L

0 0LK LKF , F '' ''

tr tw

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• Competitive market clearing and , where and are the supplies in period t:

Since and are predetermined in any given period, and are determined this way.

and predetermined: what does that mean?

dt tK K d

t tL LtK tL

K t t t L t t tF K ,L r , F K ,L w ' '

tK tL

tr tw

tK tL

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The income distribution

1. No pure profits. Euler’s rule

2. The functional income distribution

The income share of each factor is the elasticity of the production function with respect to the factor in question.

t K t t t t K t t t

t L t t t t L t t t

r F K ,L rK F K ,L K

w F K ,L w L F K ,L L

' '

' '

0t t K t t t L t t tF K ,L F K ,L K F K ,L L ' '

and

K t t t L t t tt t t t

t tt t t t

F K ,L K F K ,L LrK w K

Y YF K ,L F K ,L

' '

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According to empirics, labour’s share is relatively constant around 2/3 over long periods except for short run fluctuations:

Labour’s share of domestic factor incomes

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Is there a production function that fulfills all of our assumptions and has fixed output elasticities independently of and ? Yes, the Cobb-Douglas production function:

where is total-factor-productivity (TFP).Check: It follows that

The Cobb-Douglas function seems as a realistic long run assumption. We even have reason to believe that .

tK tL

1 0 0 1t t t t t tF K ,L B K L , B , ,

tB

1

and 1

and 1

t tt K t t L t

t t

t t t t

t t

K Kr F B w F B

L L

rK w L

Y Y

' '

1 3/

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Households• The number of households in period is , which is

predetermined. Household behaviour:1. Each supplies one unit of labour inelastically. Total supply

.2. Own the capital stock, , which is predetermined in period

. Supply = (as long as ).3. The representative household decides given , and

hence . The intertemporal budget constraint:

We assume that the result of the consumer’s considerations is:

4. ”Biology”:

t tL

tL

tK t

tK 0tr

tC tY

t t tS Y C

1 0 1t t t tK K S K ,

t tS sY .

1 1 1t tL n L , n

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The complete model

• Parameters: and . NB: No subscript on : ”Basic” Solow model.

• Endogenous variables: andof which and are state variables:

1t t tY BK L

1

tt

t

Kr B

L

1 tt

t

Kw B

L

t tS sY

1 1t tL n L ,

B, ,s, n t B

t t t t tY , K , L , r , w tS tK tL

1t t t tK K S K

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• Given and the model determines

• Government in the model? Yes, simply interpret as private plus government savings.

• We viewed the capital accumulation equation:

as the household’s budget constraint.

0K 0L t t tY , K , L ...

1t t t tK K S K

tS

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Alternatively: By definition we have that

The condition for equilibrium in the output market, or the national accounting identity, is: , and by definition: . Hence

Combining and gives the capital accumulation equation again: is the savings rate and the investment rate.

t t tY C I t t tS Y C

( )t tI S

s

1 ( )t t t tK K I K

( ) ( )

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Analysing the basic Solow model

1. Define: and . 2. From we get the per capita

production function:

Note:

t t ty Y / L t t tk K / L1

t t tY BK L

0 1t ty Bk ,

1 1y k

t t t t t tln y ln y ln k ln k g g

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3. Insert into to get:

4. Divide by on both sides to find that:

5. Insert to get the transition equation:

6. Subtracting from both sides of the transition equation gives the Solow equation:

1t t t tK K S K t tS sY

1 1t t t tK K sY K

1 1t tL n L

1

11

1t t tk sy kn

t ty Bk

1

11

1t t tk sBk kn

tk

1

1

1t t t tk k sBk n kn

savings per capita

Replacement investment to compensate for depreciation and growth of labour force

tsy”technical term” appearing because of discrete time

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The transition diagram 1

1

111

1 1t tt

t t tt

sBk kdkk sBk k .

n dk n

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• About the slope, :– It is everywhere strictly positive– It goes to infinity as goes to zero – It goes to as , and

. Indeed we assume that (realistic!)

• The figure shows that in the long run. Hence ,

and etc. • The values etc. define the ”steady state”.

1 1/ n

1d dt tk / k

1 1 1/ n

*tk k

* *ty y B k

1* *

tc c s y 1* *tr r B k

tk

tk 0n

* *k , y

n

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The Solow diagram

Why does growth in and have to stop? Diminishing returns!

1

1

1t t t tk k sBk n kn

k t

sBk t

k*

(n + )k t

tk ty

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Steady state• The long run levels , etc. depend on

parameters. How? What makes a nation rich? • Look at the Solow equation:

In steady state . Insert this to find

*y*k

1

1

1t t t tk k sBk n kn

1 0t tk k

11

11

11

1

* *

*

* *

sB k n k

sk B

n

sy B k B

n

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• Some sharp predictions of the Solow model:

The elasticity of wrt. is (since we believe that ): an increase in of 10%, e.g. from 20 to 22%, should give an increase in of 5%!

The elasticity of wrt. and wrt. are and , respectively. Why is the

latter not one? Capital accumulation!

• We have reached empirically testable hypotheses! Empirics:

1ln ln ln

1 1*y B s n .

*y s

1

1 2

1 3

1 2

n B

s*y

1

1 2

1

3

*y

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Real GDP per worker against the average investment share across 85 countries

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Real GDP per worker against the average annual labour force growth rate across 85 countries