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ECONOMIC GROWTH (Continued…):The Golden Rule Capital Stock

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Macroeconomics ECO 403
VU
LESSON 21
ECONOMIC GROWTH (Continued...)
The Golden Rule: introduction
�
Different values of s lead to different steady states.
How do we know which is the "best" steady state?
�
Economic well-being depends on consumption, so the "best" steady state has the highest
possible value of consumption per person:
c* = (1­s) f(k*)
�
An increase in s
� leads to higher k and y , which may raise c
*
*
*
� reduces consumption's share of income (1­s),
which may lower c*
�
So, how do we find the s and k* that maximize c*?
The Golden Rule Capital Stock
K*gold = the Golden Rule level of capital, the steady state value of k that maximizes
consumption.
To find it, first express c* in terms of k*:
=  y* - i*
C*
- i*
= f (k*)
- δk*
= f (k*)
In general:
i = Δk + δk
In the steady state:  i* = δk* because Δk = 0.
Then, graph f(k*) and δk*, and look for the point where the gap between them is biggest.
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Macroeconomics ECO 403
VU
steady state
δ k*
output and
depreciation
f(k*)
C*gold
i*gold = δk*gold
k*gold
Steady-state
Y*gold = f(k*gold)
capital per
worker, k*
c* = f(k*) - δk*is biggest where the slope of the production function equals the slope of the
depreciation line: MPK = δ
steady state
δ k*
output and
depreciation
f(k*)
*
C gold
k*gold
Steady-state
capital per
worker, k*
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Macroeconomics ECO 403
VU
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 that Policymakers adjust s.
�
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
If k  * > k  gold
*
then increasing c* requires a fall in s.
In the transition to the Golden Rule, consumption is higher at all points in time.
y
c
i
t0
Time
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Macroeconomics ECO 403
VU
Starting with too little capital
If k  * < k  gold
*
then increasing c* requires an increase in s.
Future generations enjoy higher consumption, but the current one experiences an initial drop
in consumption.
y
c
i
Time
t0
�
The basic Solow model cannot explain sustained economic growth. It simply says that high
rates of saving lead to high growth temporarily, but the economy eventually approaches a
steady state.
�
We need to incorporate two sources of growth to explain sustained economic growth:
population and technological progress.
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Macroeconomics ECO 403
VU
Population Growth
�
Assume that the population--and labor force-- grow at rate n. (n is exogenous)
ΔL
= n
L
EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02).
�
Then ΔL = n L = 0.02 � 1000 = 20,
so L = 1020 in year 2.
Break-even investment
(δ + n)k = break-even investment, the amount of investment necessary to keep k constant.
Break-even investment includes:
�
δ k to replace capital as it wears out
�
n k to equip new workers with capital
(otherwise, k would fall as the existing capital stock would be spread more thinly over a
larger population of workers)
The equation of motion for k
�
With population growth, the equation of motion for k is
Δk = s f(k) - (δ + n) k
Where
S f(k)= actual investment
(δ + n) k = breakeven investment
The impact of population growth
(δ +n2) k
Investment,
(δ +n1) k
break-even
investment
sf(k)
k2*
k1*
Capital per
worker, k
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Macroeconomics ECO 403
VU
Prediction:
Higher n ⇒ lower k*.
�
And since y = f(k) ,
�
lower k* ⇒ lower y* .
Thus, the Solow model predicts that countries with higher population growth rates will
�
have lower levels of capital and income per worker in the long run.
The Golden Rule with Population Growth
To find the Golden Rule capital stock, we again express c* in terms of k*:
=  y* - i*
c*
= f (k* ) - (δ + n) k*
c* is maximized when
MPK = δ + n
or equivalently,
MPK - δ = n
In the Golden Rule Steady State, the marginal product of capital net of depreciation equals the
population growth rate.
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Table of Contents:
  1. INTRODUCTION:COURSE DESCRIPTION, TEN PRINCIPLES OF ECONOMICS
  2. PRINCIPLE OF MACROECONOMICS:People Face Tradeoffs
  3. IMPORTANCE OF MACROECONOMICS:Interest rates and rental payments
  4. THE DATA OF MACROECONOMICS:Rules for computing GDP
  5. THE DATA OF MACROECONOMICS (Continued…):Components of Expenditures
  6. THE DATA OF MACROECONOMICS (Continued…):How to construct the CPI
  7. NATIONAL INCOME: WHERE IT COMES FROM AND WHERE IT GOES
  8. NATIONAL INCOME: WHERE IT COMES FROM AND WHERE IT GOES (Continued…)
  9. NATIONAL INCOME: WHERE IT COMES FROM AND WHERE IT GOES (Continued…)
  10. NATIONAL INCOME: WHERE IT COMES FROM AND WHERE IT GOES (Continued…)
  11. MONEY AND INFLATION:The Quantity Equation, Inflation and interest rates
  12. MONEY AND INFLATION (Continued…):Money demand and the nominal interest rate
  13. MONEY AND INFLATION (Continued…):Costs of expected inflation:
  14. MONEY AND INFLATION (Continued…):The Classical Dichotomy
  15. OPEN ECONOMY:Three experiments, The nominal exchange rate
  16. OPEN ECONOMY (Continued…):The Determinants of the Nominal Exchange Rate
  17. OPEN ECONOMY (Continued…):A first model of the natural rate
  18. ISSUES IN UNEMPLOYMENT:Public Policy and Job Search
  19. ECONOMIC GROWTH:THE SOLOW MODEL, Saving and investment
  20. ECONOMIC GROWTH (Continued…):The Steady State
  21. ECONOMIC GROWTH (Continued…):The Golden Rule Capital Stock
  22. ECONOMIC GROWTH (Continued…):The Golden Rule, Policies to promote growth
  23. ECONOMIC GROWTH (Continued…):Possible problems with industrial policy
  24. AGGREGATE DEMAND AND AGGREGATE SUPPLY:When prices are sticky
  25. AGGREGATE DEMAND AND AGGREGATE SUPPLY (Continued…):
  26. AGGREGATE DEMAND AND AGGREGATE SUPPLY (Continued…):
  27. AGGREGATE DEMAND AND AGGREGATE SUPPLY (Continued…)
  28. AGGREGATE DEMAND AND AGGREGATE SUPPLY (Continued…)
  29. AGGREGATE DEMAND AND AGGREGATE SUPPLY (Continued…)
  30. AGGREGATE DEMAND IN THE OPEN ECONOMY:Lessons about fiscal policy
  31. AGGREGATE DEMAND IN THE OPEN ECONOMY(Continued…):Fixed exchange rates
  32. AGGREGATE DEMAND IN THE OPEN ECONOMY (Continued…):Why income might not rise
  33. AGGREGATE SUPPLY:The sticky-price model
  34. AGGREGATE SUPPLY (Continued…):Deriving the Phillips Curve from SRAS
  35. GOVERNMENT DEBT:Permanent Debt, Floating Debt, Unfunded Debts
  36. GOVERNMENT DEBT (Continued…):Starting with too little capital,
  37. CONSUMPTION:Secular Stagnation and Simon Kuznets
  38. CONSUMPTION (Continued…):Consumer Preferences, Constraints on Borrowings
  39. CONSUMPTION (Continued…):The Life-cycle Consumption Function
  40. INVESTMENT:The Rental Price of Capital, The Cost of Capital
  41. INVESTMENT (Continued…):The Determinants of Investment
  42. INVESTMENT (Continued…):Financing Constraints, Residential Investment
  43. INVESTMENT (Continued…):Inventories and the Real Interest Rate
  44. MONEY:Money Supply, Fractional Reserve Banking,
  45. MONEY (Continued…):Three Instruments of Money Supply, Money Demand