Long run economic growth, part 2. The Solow growth model

Macroeconomics IIJoanna Siwińska-Gorzelak

WNE UW

The Solow growth model

• The seminal Solow growth model dates back to 1950’s and belongs to the fundamentals of growth theory

• The Solow model is remarkable for its simplicity

• The Solow model is a good starting point and a springboard for further models

• We will analyze it in continuous time, we make the time units (the difference between t and t+1) as small as possible

• At the centre of the model is the neoclassical production function

The neoclassical production function

Production function is neoclassical when is characterised by :

1. Constant returns to scale in capital and labour

aY = A aK; aN

2. Positive & diminishing marginal product of capital and labour:

𝐹𝐾 =𝑑𝑌

𝑑𝐾> 0;

𝑑𝐹𝐾𝑑𝐾

< 0

𝐹𝑁 =𝑑𝑌

𝑑𝑁> 0;

𝑑𝐹𝑁𝑑𝑁

< 0

3. Inada conditions

lim𝐾→0

𝐹𝐾 = ∞ lim𝐾→∞

𝐹𝐾 = 0

lim𝑁→0

𝐹𝑁 = ∞ lim𝑁→∞

𝐹𝑁 = 0

Other important assumptions of the Solow

model• One sector economy that produces a homogenous good that can

be consumed or invested

– The economy is closed, there is no government sector:𝐺𝑡 = 0; 𝑁𝑋𝑡 = 0; 𝑌𝑡 = 𝐶𝑡 + 𝐼𝑡

• The saving rate ‘s’ is constant: 𝑆𝑡 = 𝑠𝑌𝑡; 𝑠 > 0

𝑆𝑡 = 𝑠𝑌𝑡 = 𝐼𝑡

• The population growth ‘n’ rate is constant

𝑛 =ሶ𝑁𝑡𝑁

• The depreciation rate of capital ‘d’ is constant

A temporary assumption (just for now)

• To make our analysis easier, we will assume that

technology is constant and equal to 1

• We will change this assumption very soon

• For now, the (neoclassical) production function is:

Y=A F (K, N) = F (K, N)

Per capita production function

• Since the production function is characterised by

constant returns to scale, we can write :

•𝑌𝑡

𝑁𝑡= 𝐹

1

𝑁𝑡𝐾𝑡 ,

1

𝑁𝑡𝑁𝑡 = 𝐹

𝐾𝑡

𝑁𝑡, 1

• We will use the following notation:

• Hence, the production function can be written in

intensive form as

• 𝑦𝑡 = 𝑓(𝑘𝑡)

N

Kk

N

Yy == ,

Per capital production function…

• …is still characterised by:

• 𝑓𝑘 =𝜕𝑓𝑡

𝜕𝑘𝑡> 0,

𝜕𝑓𝑘

𝜕𝑘𝑡< 0

• lim𝑘→0

𝑓𝑘 = ∞, lim𝑘→∞

𝑓𝑘 = 0

Qunatities „per worker”

• Since 𝑌𝑡 = 𝐶𝑡 + 𝐼𝑡, then if we express this equation per worker, we will get:

𝑌𝑡𝑁𝑡

=𝐶𝑡𝑁𝑡

+𝐼𝑡𝑁𝑡

𝑦𝑡 = 𝑐𝑡 + 𝑖𝑡,

• All „per worker” qunatities are denoted by lower case (small) letters .

• An excpetion is the saving reate (s), or the saved fraction of income.

• Consumption per worker is given by:𝑐𝑡 = 1 − 𝑠 𝑦𝑡

8

A neoclassical per capita production

function

Notice that

• The per capita production function depends only on

per capita capital stock

• If we understand the dynamics of capital per capita

we can understand economic growth!

The accumulation of capital per person

• We know that capital in period t+1 is given by:𝐾𝑡+1= 𝐾𝑡 + 𝐼𝑡 − 𝑑𝐾𝑡

𝐾𝑡+1−𝐾𝑡 = ሶ𝐾𝑡 = 𝑠𝑌𝑡 − 𝑑𝐾𝑡

• Notice that we are interested in the dynamics of capital per worker (k)

𝜕𝑘𝑡𝜕𝑡

=𝜕𝐾𝑡𝑁𝑡𝜕𝑡

=ሶ𝐾𝑡𝑁𝑡 − ሶ𝑁𝑡𝐾𝑡

𝑁𝑡2 =

ሶ𝐾𝑡

𝑁𝑡

𝑁𝑡𝑁𝑡

−ሶ𝑁𝑡𝐾𝑡

𝑁𝑡2 =

=𝑠𝑌𝑡 − 𝑑𝐾𝑡

𝑁𝑡−

ሶ𝑁𝑡

𝑁𝑡

𝐾𝑡

𝑁𝑡= 𝑠𝑦𝑡 − 𝑑𝑘𝑡 − 𝑛𝑘𝑡

11

The accumulation of capital per person

Fundamental equation of the (simple) Solow growth model:

ሶ𝒌 = 𝒔𝒚𝒕 − (𝒅 + 𝒏)𝒌𝒕

Saving and investment

y

k

f(k)

sf(k)

k1

y1

i1

c1

The „effective deprecation” or „break-even

investment”

y

k

(d+n) k

Investment needed to keep ‘k’at a constant level

Moving toward the steady-state

y

k

sf(k)

(d+n)k

k*k1

investment

depreciation

k

The steady state

• Recall: ሶ𝑘 = 𝑠𝑦𝑡 − (𝑠 + 𝑛)𝑘𝑡

•If investment is just enough to finance effective depreciation:

𝑠𝑦𝑡 = (𝑑 + 𝑛)𝑘𝑡

•then capital per worker will remain constant: ሶ𝑘 = 0

This constant value, denoted k*, is called the steady state value of capital stock.

The steady-state

The steady-state

y

k

y*

y=f(k)

s f(k)

k*

(+n) k

The steady-state

• The only possible steady-state capital–labor ratio is k*

• Output at that point is y* = f(k*); consumption is c* = f(k*) – s f(k*) = f(k*)-(n + d)k*

• If k begins at some level other than k*, it will move toward k*

– For k below k*, saving > the amount of investment needed to keep k constant, so k rises

– For k above k*, saving < the amount of investment needed to keep k constant, so k falls

The steady-state

• We have established that in the steady-state, capital per person is constant.

• That, of course implies that output per person and consumption per person are also constant:

0===•••

cyk

The Solow’s suprise

• Investment in new capital cannot lead to continued growth in per capita income.

• What can lead to sustained growth of output per person?

• As we will see next week – technology!

The steady-state

• What about the growth rates of K (capital) and Y (output)in the steady-state?

• Recall that K=kN; Y=yN

• Therefore:

•ሶ𝐾

𝐾=

ሶ(𝑘𝑁)

𝑘𝑁=

ሶ𝑘𝑁+𝑘 ሶ𝑁

𝑘𝑁=

ሶ𝑘

𝑘+

ሶ𝑁

𝑁

• In the steady-state:

•ሶ𝐾

𝐾= 0 + 𝑛 = 0

• Exactly the same reasoning applies to ሶ𝑌

𝑌

The steady-state - summary

• In the steady-state, the per worker values are stable:

• In the steady-state, Y & K & C exhibit are characterized by steady growth rate equal to the rate of growth of population:

0===

•••

c

c

y

y

k

k

nC

C

Y

Y

K

K===

•••

Steady-state: a Cobb-Douglas production function

• Assume a Cobb-Douglas function:

• 𝑌𝑡 = 𝐾𝑡∝𝑁𝑡

∝

• Intensive form (per worker):

• 𝑦𝑡 = 𝑘𝑡∝

• The steady-state:

• ሶ𝑘 = 0

• ሶ𝑘 = 𝑠𝑘∝ − 𝑑 + 𝑛 𝑘 = 0

• 𝑠𝑘∝ = 𝑑 + 𝑛 𝑘

•𝑠

(𝑑+𝑛)= 𝑘1−∝

−

+=

1

1

)(*

nd

sk

The Solow Model

• To summarize:– With no productivity growth, the economy reaches a

steady state, with constant capital per person (or capital–labor ratio), output per person, and consumption per person

– In the steady-state, the amount of capital per person output per person and consumption per person depend, among others on:

• the saving rate (s)

• the rate of population growth (n)

• and on the depreciation rate (d)

The effect of an increased saving rate on the steady-state capital and output per person

An increase in the saving rate

An increase of s increases k*, and y*.

y

k

y1*

y=f(k)

s1 f(k)

k1*

(+n) k

k2*

y2*

s2 f(k)

Prediction:

• Higher s higher k*.

• And since y = f(k) , higher k* higher y* .

• Thus, the Solow model predicts that countries with

higher rates of saving and investment

will have higher levels of capital and income per

worker in the long run.

The effect of a higher population growth rate on the steady-state capital–labor ratio

Population growth rate and GDP per capita

30

The Solow Model

• Should a policy goal be to reduce population growth?– Doing so will raise consumption per worker– Note however that the Solow model also assumes that

the proportion of the population of working age is fixed (exactly: population = workers)

• But when population growth changes, this may change the % of the working-age population

• Changes in cohort sizes may cause problems for social security systems and areas like health care

The Golden Rule: Introduction

• Different values of s lead to different steady states. How do we know which is the “best” steady state?

• The “best” steady state has the highest possible consumption per person: c* = (1–s) f(k*).

• An increase in s

– leads to higher k* and y*, which raises c*

– reduces consumption’s share of income (1–s), which lowers c*.

• So, how do we find the s and k* that maximize c* (in the steady-state)?

The Golden Rule capital stock

the Golden Rule level of capital per worker:the steady state value of k that maximizes steady-state consumption per person.

*

goldk =

To find it, first express c* in terms of k*:

c* = y* − i*

= f (k*) − sy*

= f (k*) − (d+n)k*

In the steady state:

sy* = (d+n)k*

• Then, differentiate with respect to k, to find the value of k* that maximises c*:

The Golden Rule capital stock

)(*)('

0)(*)('*

*

0*

*

*)(*)(*

ndkf

ndkfdk

dc

dk

dc

kndkfc

+=

=+−=

=

+−=

Graph

f(k*) and (d+n)k*;

look for the

point where

the gap between

them is biggest.

The Golden Rule capital stocksteady state output and

depreciation

steady-state capital per

worker, k*

f(k*)

(d+n)k*

*

goldk

*

goldc

* *

gold goldi k=* *( )gold goldy f k=

The Golden Rule capital stock

c* = f(k*) − (d+n)k*

is biggest where the

slope of the

production function

equals

the slope of the

depreciation line:

steady-state capital per

worker, k*

f(k*)

k*

*

goldk

*

goldc

MPK = (d+n)

The golden rule: an example using C-D p.f.• Let’s assume a Cobb-Douglas production function:

• 𝑌𝑡 = 𝐾𝑡∝𝑁𝑡

∝

• In intensive form:

• 𝑦 = 𝑘∝

• Consumption in the steady-state is:

• ሶ𝑐∗ = 𝑓 𝑘 − 𝑑 + 𝑛 𝑘

• Maximise it with respect to capital

•𝑑𝑐∗

𝑑𝑘=∝ 𝑘𝐺𝑂𝐿𝐷

∝−1 − d + n = 0

• 𝑘𝐺𝑂𝐿𝐷∝−1 =

𝑑+𝑛

∝

• 𝑘𝐺𝑂𝐿𝐷∗ = (

∝

𝑑+𝑛)

1

1−∝

• This is the steady-state level of capital per worker that maximises consumption per worker

How much do we need to save, to achieve „k GOLD”?

• Recall that in the steady-state:

• 𝑘∗ =𝑠

𝑑+𝑛

1

1−∝

• Since the „golden-rule” level of capital per person is also the steady-state capital per person, then we know that

• 𝑘∗ = 𝑘𝐺𝑂𝐿𝐷∗

•𝑠

𝑑+𝑛

1

1−∝= (∝

𝑑+𝑛)

1

1−∝

• This implies that the saving rate s that is needed to reach maximum consumption in the steady state is equal to α

• 𝑠𝐺𝑂𝐿𝐷 =∝

The transition to the „Golden Rule” steady-state

• The economy does NOT have a tendency to move toward the „Golden Rule” steady state.

• Achieving the Golden Rule requires thatpolicymakers adjust s to reach sgold

• This adjustment leads to a new steady state with higher consumption.

• But what happens to consumption during the transition to the Golden Rule?

Starting with too much capital

then increasing c*

requires a fall in s.

In the transition to

the Golden Rule,

consumption is

higher at all points

in time.

If goldk k* *

timet0

c

i

y

Starting with too little capital

then increasing c*

requires an increase in s.

Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption.

If goldk k* *

timet0

c

i

y

Summary

• The economy will reach a steady-state, where the values of capital & output & consumption per worker will be constant

• Investment in new capital (and the growth of population) cannot lead to continued growth in per capita income.

• GDP per worker in the steady-state depends on (among others): the saving rate and the population growth rate

What’s ahead?

• Discussing convergence (next week)

• Adding technology to the Solow’s model (next week)

• A very quick glimpse at different growth models