1

8. Economic growth 8.1. Production function: Capital as production factor

Y Nα=

( ),Y F K N=

Diminishing marginal productivity of capital and labor:

( ),

0F K L

K∂

>∂

( )2

2

,0

F K LK

∂<

∂

( ),

0F K L

L∂

>∂

( )2

2

,0

F K LL

∂<

∂

Constant return to scale:

( ),xY F xK xN=

,Y K NFN N N

=

( )y f k=

where output per worker is denoted by YyN

= , and capital per worker is

denoted by KkN

= , the latter of which is also called a labor equipment ratio.

2

Profit-maximization:

( ) ( ),

max ,K L

F K L r K wLδΠ = − + −

( ),F K L

rK

δ∂

= +∂

( ),F K L

wL

∂=

∂

( ) ( )max 1k

f k r k wπ δ= − + − ×

( )f k r δ′ = +

or ( )r f k δ′= −

3

Cobb-Douglas production function: 1Y AK Nα α−=

y Akα=

where A is called total factor productivity. An increase in A implies the improvement of productivity or technological progress. 1 1r AK Lα αδ α − −+ =

capital income share: ( )r K

Yδ

α+

=

Note that depreciation costs are included in capital income.

labor income share: 1wLY

α= −

1r Ak αδ α −+ =

capital income share: ( )r kyδ

α+

=

4

Economic growth as a consequence of capital accumulation Economic growth as a consequence of technological progress

5

8.2. A case without any technological progress

1t t ti k kδ+= ∆ +

ti : gross fixed investment

1 1t t tk k k+ +∆ = − : net fixed investment

( )1

t t t

t t t

y c ic k kδ+

= +

= + ∆ +

( )1 1t t t t t tk k k f k c kδ+ +∆ = − = − −

( )t tc cf k=

( ) ( )( )

1t t t

t

t

s y cc f k

sf k

= −

= −

=

( )1t t tk sf k kδ+∆ = −

6

a steady state:

( ) 0k sf k kδ∆ = − =

sAk kα δ=

1

1* sAkα

δ− =

7

8.3. Golden rule A fundamental question: An increase in capital unambiguously leads to an increase in output. Is it a case for consumption? At a steady state or 0k∆ = , consumption is maximized under the golden rule.

( )maxk

c f k kδ= −

( )f k δ′ = ( ) 0r f k δ′= − =

1Akαα δ− =

1

1Akαα

δ− =

Compare between 1

1Akαα

δ− =

and

11Akαα

δ− =

8

The attached excel file includes data on nominal GDP, and the shares by expenditures (household consumption, government consumption, fixed investment, and net export) for 35 countries (11 in Asia, 3 in North America, 2 in South America, 16 in Europe, 2 in Africa, and 1 in Oceania). The sample period is between 2009 and 2011. Now, we define saving rates as follows:

No min al GDP-Nominal Household Consumption-Nominal Government ConsumptionNo min al GDP

Compute the three-year average of the above-defined saving rate for

each country, and discuss the state of capital accumulation in terms of dynamic efficiency under the assumption 0.4α = .

Note: Rigorously, a saving rate should be computed from real terms.

Given limited data availability, nominal data are employed as a first approximation purpose.

9

From 2009 to 2011

10

8.4. A case with technological progress and population growth

( ) ( )1 01 1 tt N t NN g N g N−= + = +

( ) ( )1 01 1 tt A t AA g A g A−= + = +

Output per effective worker, and capital per effective worker

tt

t t

YyA N

=

tt

t t

KkA N

=

11

1 1 1

ln lnt tt t

t

t t t

t t t

t t tA N

t

k k k kkK A NK A N

Y C K g gK

δ

++

+ + +

−≈ −

∆ ∆ ∆≈ − −

− −= − −

( )

( )( ) ( )

1t t t t

t A N tt t t

t t A N t

t t A N t

K Y C Kk g g kA N K

y c g g k

f k c g g k

δ

δ

δ

+

− −∆ = − +

= − − + +

= − − + +

Golden rule:

( ) A Nf k g gδ′ = + +

( ) A Nr f k g gδ′= − = +

11

8.5. Growth account

1t t t tY A K Nα α−=

( )ln ln ln 1t t t tY A K Nα α= + + −

( ) ( ) ( )( )

( )

11

1 1 1

1 1

ln ln

ln ln ln ln 1 ln ln

1

tt t

t

t t t t t t

t t t

t t t

Y Y YY

A A K K N NA K NA K N

α α

α α

++

+ + +

+ +

∆≈ −

= − + − + − −

∆ ∆ ∆≈ + + −

( )1 1 11t t t t

t t t t

A Y K NA Y K N

α α+ + +∆ ∆ ∆ ∆= − − −

1 1t t t

t t t

y A ky A k

α+ +∆ ∆ ∆= +

1 1t t t

t t t

A y kA y k

α+ +∆ ∆ ∆= −

12

Alwyn Young reports the following growth account results of the four Asian countries in his article entitled "The tyranny of numbers: Confronting the statistical realities of the East Asian growth experience" in The Quarterly Journal of Economics, 1995. Comment on his results.

Annual GDP growth

( YY∆ )

Annual TFP growth

( AA∆ )

Hong Kong (1966-1991) 7.3% 2.3%

Singapore (1966-1990) 8.7% 0.2%

Korea (1966-1990) 10.3% 1.7%

Taiwan (1966-1990) 9.4% 2.6%

13

8.6. Issues on technological progress 8.6.1. Productivity and output 8.6.2. Productivity and unemployment

8.6.3. Technological progress and unemployment

14

8.7. Revisit of consumption and investment by Ramsey model 8.7.1. Ramsey model

Let us start with a simple economy with y c i= + in per-worker terms. In the Keynesian framework, given an increase in fixed investment, consumption and output increases together thanks to the multiplier effects. This multiplier effect is so impressive among students of introductory macroeconomics that for such students, capital accumulation by intensive investment is always good news. As the argument of the golden rule implies, however, aggressive capital accumulation results in an increase in output, but not necessarily in an increase in consumption. Capital accumulation may enhance output at the sacrifice of consumption. According to Solow’s growth model with the Cobb-Douglas production function, if the saving rate s is beyond the capital income share α , then the steady-state level of capital exceeds the golden-rule capital; that is, an economy with a high saving rate may result in too much capital with too little consumption. While Solow’s model delivers several important implications for capital accumulation, its serious problem is the assumption that the saving rate is fixed over time; the total output is always divided between consumption and saving/investment at a fixed proportion. In Solow’s growth model, whether the steady state capital exceeds the golden-rule level depends on how the saving rate is picked up. Hence, we cannot discuss how capital is accumulated efficiently or how over-accumulation emerges. Then, let us explore how output is divided between consumption and investment over time using Ramsey model. Pick up the problem of the consumption allocation between time t and t+1 as follows:

( ) ( )1

1,

1max1t t

t tc cu c u c

ρ++

+ +

subject to ( )( )1 1 1 1t t t t t tc s w r w c+ + ++ = + + − . A time preference is introduced that

time t+1 utility may be compared with time t utility as of time t. From a set of the first-order conditions for the above utility maximization problem, we obtain

( ) ( )111

tt t

ru c u cρ +

+′ ′=+

15

Employing the mathematical tools with a particular form of utility function

( )11

11

cu cσ

σ

−

=−

, we can derive the following approximated relationship.

( )1 1t t tt

t t

c c c rc c

σ ρ+ +∆ −= = −

The above equation is often called the Euler equation. The introduction of the Euler equation immediately makes the capital accumulation model

extremely rich. Instead of replacing consumption by ( ) ( )1t tc s f k= − , we still

make consumption explicit.

( )1 1t t t t t tk k k f k c kδ+ +∆ = − = − −

From the profit maximization in competitive markets, we have

( )t tr f k δ′= − .

Suppose below that a technological progress is absent. At the golden rule,

( )gf k δ′ = . Thus, if gk k< , then r is positive, otherwise it is negative.

0k∆ = locus:

( )c f k kδ= −

0c∆ = locus:

( )mgf k ρ δ′ = +

Note that ( ) ( )mg gf k f k′ ′> or mg gk k< .

16

8.7.2. Asset pricing

1tt

t

S rS

ρ+∆+ =

( )1tt

t

S f kS

δ ρ+∆ ′+ − =

( )( ) ( )

1 1

1lim

1 1t t t

t

S f k SS τ τ ττ τττ

δ

ρ ρ

∞− + − + +

→∞=

′ − = ++ +

∑

When sk k= ,

( )

lim 01

tS τττ ρ

+

→∞=

+

( )

( )1 1

1 1t t

t

S f kS τ τ

ττ

δ

ρ

∞− + − +

=

′ − =+

∑

When gk k> ,

( )

lim1

tS τττ ρ

+

→∞= ∞

+

( )1tt

t

S f kS

ρ δ ρ+∆ ′= + − >

( )

( )1 1

1 1t t

t

S f kS τ τ

ττ

δ

ρ

∞− + − +

=

′ − >+

∑

17

When k converges to zero,

( )1tt

t

S f kS

ρ δ ρ+∆ ′= + − <

18

8.7.3. Tobin’s q theory Keynesian investment function:

( ) ( )( )t tI I r I f k δ′= = − , where ( ) 0tI r′ < .

( )( )

1 1

1 1t t

t

S f kS τ τ

ττ

δ

ρ

∞− + − +

=

′ − =+

∑ : the present value of future returns yielded by one

unit of capital as stock prices

tp : a unit price of investment goods

Tobin’s q theory:

Tobin’s q: tt

t

Sqp

=

( )t t ti i q= , where ( ) 0ti q′ > .

19

8.7.4. Consumption

at＋1＝at(1＋rt)＋wt－ct

( )1 11 1

1 1t t t tt t

a a c wr r+ += + −

+ +

a1＝a0(1＋r0)＋w0－c0

a2＝a1(1＋r1)＋w1－c1

a3＝a2(1＋r2)＋w2－c2,

⋮

a0＝a1－w0＋c0

1＋r0

a1＝a2－w1＋c1

1＋r1

a2＝a3－w2＋c2

1＋r2

⋮

a0 + �wτ

∏ (1＋ri)τ

i=0

=∞

τ=0

�cτ

∏ (1＋ri)τ

i=0

+ limτ→∞

𝑎𝑎τ

∏ (1＋ri)τ

i=0

∞

τ=0

(16-14)

limτ→∞

aτ/��1＋ri� <

τ

i＝0

0

limτ→∞

aτ/�(1＋ri) ≥

τ

i＝0

0

a0 + �wτ

∏ (1＋ri)τ

i=0

=∞

τ=0

�cτ

∏ (1＋ri)τ

i=0

∞

τ=0

20

( )1 1t t tt

t t

c c c rc c

σ ρ+ +∆ −= = − with tr ρ= ,

0tc c=

a0＋ �wτ

(1＋ρ)τ＋1

∞

τ＝0

＝c0 �1

(1＋ρ)τ＋1

∞

τ＝0

c0=ρ �a0＋ �wτ

(1＋ρ)τ＋1

∞

τ＝0

�

21

8.7.5. Consumption and net investment

( )NDPt t t t ty y k f k kδ δ= − = − (16-23)

( )NDTt

tt

dy f kdk

δ′= −

( )( )( )

( )1 1

1

NDP NDPt t t t t

t t t

y y f k k k

r k k

δ+ +

+

′− = − −

= −

( )1NDPt t t ty k k c+= − +

( )1NDP NDP NDPt t t t ty y r y c+ − = −

1

1

NDPNDP t t tt

t

r c yyr++

=+

(16-24-1)

1 1 21

11

NDPNDP t t tt

t

r c yyr

+ + ++

+

+=

+ (16-24-2)

2 2 32

21

NDPNDP t t tt

t

r c yyr

+ + ++

+

+=

+ (16-24-3)

( ) ( )

1 1

1

1

lim11

NDPNDP t tt

ii

r c yyrr

τ τ ττ τττ τ

∞+ − + +

→∞=

=

= +++

∑Π

( )

1

1

11

NDP tt

ii

r cyr

τ ττ

τ

∞+ −

=

=

=+

∑Π

(16-25)

22

1NDP

t t t tk k y c+ − = −

( )

11

1

11

tt t t

ii

r c c k kr

τ ττ

τ

∞+ −

+=

=

− = −+

∑Π

( )1

1111 1

1

c rr c crr

r

ττ

∞

=

= =++ −

+

∑

1t t t

t t

c c k kc c

+− −=

(16-26)

1980s 1990s 2000s 2010 to2013

(i) Ten-year average of net fixedinvestment/real household consumption(%, 2010 to 2013 for 2010s)

23.0% 17.9% 4.1% -0.5%

(ii) Ten-year average of real householdconsumption (billion yen, 2010 to 2013for 2010s)

206,832.0 271,115.7 305,494.2 321,190.2

(iii) Consumption growth rate from thecurrent decade to the next decade (%,2010 to 2013 for 2010s)

31.1% 12.7% 5.1% ???

Table 1: Net fixed investment and future consumption