@k.r.bhattarai, business school, university of hull, uk. 1 economic modelling lecture 3 steady state...
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@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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.