@K.R.Bhattarai, Business School, University of Hull, UK. 1 Economic Modelling Lecture 3 Steady State and Golden Rule of Saving in Solow Model.

@K.R.Bhattarai, Business School, University of Hull, UK.

1

Economic Modelling

Lecture 3

Steady State and Golden Rule of Saving in Solow Model

@K.R.Bhattarai, Business School, University of Hull, UK.

2

Solow Growth ModelProduction function with capital and labour as its inputs:

(Close Economy without Government)

tLtKtY

tItCtY

Market clearing:

tsYtS

tKntI

tItS tIt

KtK 1

1

Households’ Saving:

Investment requirement:

Closure rule:

Dynamics: Capital accumulation:

Firms’ Production Function

@K.R.Bhattarai, Business School, University of Hull, UK.

3

Per Capita Output and Per Capita Capital Stock in the Steady State

ky

L

Yy

L

Kk

sksyS

kni

0.5ks ks

SST

@K.R.Bhattarai, Business School, University of Hull, UK.

4

Capital Stock and output in the Steady State in the Solow Model

Differentiate it with respect to time to get growth rate of k :

Fundamental equation of economic growth: nk

dksk 1

nskk

dk

1

L

Kk Per Capita Capital Stock

Take log both sides LKk lnlnln

LdL

KdK

kdk n

K

KsY

k

dk

@K.R.Bhattarai, Business School, University of Hull, UK.

5

Capital Stock and output in the Steady State in the Solow Model

Fundamental equation of economic growth: nk

dksk 1

In steady state 0k

dk nsk 1

1

1

nsssk

1

nsssy

Per Capita Capital Stock in the Steady State:

Per Capita Output in the Steady State:

@K.R.Bhattarai, Business School, University of Hull, UK.

6

Solow Growth ModelProduction function with Labour augmenting technology

(Close Economy without Government)

tALtKtY

tItCtY

Market clearing:

tsYtS

tKagntI

tItS tIt

KtK 1

1

Households’ Saving:

Investment requirement:

Closure rule:

Dynamics: Capital accumulation:

Firms’ Production Function

@K.R.Bhattarai, Business School, University of Hull, UK.

7

Capital Stock and output in the Steady State in the Solow Model with technical progress

Differentiate it with respect to time to get growth rate of k :

Fundamental equation of economic growth: agnk

kdks 1~

~

~agnksk

kd

1~~

~

AL

Kk ~

Per Capita Effective Capital Stock

Take log both sides ALKk lnlnln~

ln

AdA

LdL

KdK

kkd ~~

agnK

KsY

k

kd

~

~

AL

Yy ~

@K.R.Bhattarai, Business School, University of Hull, UK.

8

Capital Stock and output in the Steady State in the Solow Model with technical progress

Fundamental equation of economic growth: agn

k

kdks 1~

~

~

In steady state 0~

~

k

kd agnks 1~

11

~

agnsssk

1~

agnsssy

Per Capita Effective Capital Stock in the Steady State:

Per Capita Effective Output in the Steady State:

@K.R.Bhattarai, Business School, University of Hull, UK.

9

Results from the steady state:

1. Countries with higher saving rate have higher steady state level of output and countries with lower saving rate have lower level of output in the steady state.

2. Countries with higher level of technology have higher level of output and countries with lower level of technology have lower level of output in the steady state.

3. Countries with higher rate of population growth rate have lower level of output in the steady state.

4. Countries with higher capital share have higher output in the steady state.

5. Countries which differ in the initial capital stock eventually reach to the same output level in the steady state.

6. Growth of per capita income is zero in the steady state

@K.R.Bhattarai, Business School, University of Hull, UK.

10

Calculation of Steady State: A Numerical Example

Let s = 0.32 and = 0.08, and n =0 3

1

1

nssskssyOutput in the steady state

22

1

08.0

32.0

3

11

3

1

008.0

32.01

n

sssy

SSSSss sYYC = 2-0.32(2) = 1.36.Consumption in the steady state:

41.12

1

08.0

16.0

3

11

3

1

008.0

16.01

n

sssy

Let s = 0.16 and = 0.08, and n =0 3

1

@K.R.Bhattarai, Business School, University of Hull, UK.

11

How does a higher saving rate affect the level of output in the steady state?

11

sksyS

ky

1y

L

Kk

222

sksyS 22

kni

2y

1k 2k

Low saving country

High saving country

Note: Saving rate affects level of income but not the growth rates.

@K.R.Bhattarai, Business School, University of Hull, UK.

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How does a higher rate of population growth affect the level of output in the steady state?

sksyS

ky

1y

L

Kk

kni

11

2y

1k 2k

Low saving country

High saving country

Higher population growth rate

means lower output and

capital stock in the steady state

kni

22

@K.R.Bhattarai, Business School, University of Hull, UK.

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Golden Rule for Saving and Capital Accumulation

ky

L

Yy

L

Kk

sksyS

kni

KssKg

C-max

nMPK

nk 1

Golden rule Steady State

@K.R.Bhattarai, Business School, University of Hull, UK.

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Saving rate

C-max = 1.25

Con

sum

ptio

n

s*=0.5

y=2.5

k = 25

y = 0.5*k0.5

s1 s2 s4 s5

How High Should be the Saving Rate? Saving Rate that Maximises Consumption

C

@K.R.Bhattarai, Business School, University of Hull, UK.

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Golden Rule of Savingiyc

knyc

knycMax

01 nk

dk

c

nk 1 nMPK

1

1

nk

1

1

nssskGolden Rule

Steady State

@K.R.Bhattarai, Business School, University of Hull, UK.

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Reading and ReferencesText:Blanchard Chapters 10, 11

Mankiw 7

Burda Wypslosz 3

Miles and Scot 5-6Jones, Charles (CJ) Introduction to economic growth, 2002, 2nd Edition, Norton.

Articles:• Freeman Richard • Kaldor N. (1961) “ Capital Accumulation and Economic Growth” in F.A. Lutz and D.C. Hague ed. The

Theory of Capital, New York, St. Martin.• Lucas R.E. (1988) "On the Mechanics of Economic Development", Journal of Monetary Economics, 22,

3-42.• Mankiw N.G., D. Romer and D. N. Weil (1992) “Contribution to the Empirics of Economic Growth”

Quarterly Journal of Economics, 107 407-437.• Parente S.L. and Prescott E. C. (1993) Changes in the Wealth of Nations, Federal Reserve Bank of

Minneapolis, Quarterly Review, Spring, pp. 3-16.• Romer, Paul (1989) Endogenous Technological Change, Journal of Political Economy, vol. 98, no. 5.

Pt. 2, pp. S71-S102.• Temple, Jonathan R. W. and Voth, Hans-Joachim (1998).

Human capital, equipment investment, and industrialization. European Economic Review, 42(7), July, 1343-1362.