Growth Accounting From GA to Solow to Ramsey Endogenous long-run growth

Growth Theory: Review Lecture 1

Economic Policy in Development 2, Part 2

March 9, 2009

Lecture1,EndogenousGrowth 1/33 in Development 2, Part 2 Growth Accounting From GA to Solow to Ramsey Endogenous long-run growth Outline

Growth Accounting Growth Accounting: Objective and Technical Framework Results

From GA to Solow to Ramsey Solow Model Ramsey Model

Endogenous long-run growth Ak model Akh model

Lecture1,EndogenousGrowth 2/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Growth Accounting: Objective 3

◮ Many factors play role to determine output in a country

◮ Certainly, size of the labour force and stock do ◮ But also, education, government regulation, weather,...

◮ Any theory of chooses which of these factors to emphasize as

◮ sources of GDP growth within countries ◮ explanation for differences in levels/growth rates across countries

◮ Growth accounting: tool to evaluate relative importance of such factors → Theory & Policy Implications

Lecture1,EndogenousGrowth 3/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Technical framework 4

◮ Ignore the demand side for now ◮ Carefully specify the supply side

◮ Inputs: capital, K , and labour, L ◮ Output, Y ◮ State of technology, A

Lecture1,EndogenousGrowth 4/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Technology, F 5

Y = F(K , AL) where Y = output K = capital (input / factor) L = labour (input / factor) A = state of technology H = AL = effective labour Assumptions ◮ Marginal products positive and diminishing ◮ Constant

Lecture1,EndogenousGrowth 5/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Marginal products 6

◮ Marginal product of labour

◮ ∂F ∂L = FL > 0 positive

2 ◮ ∂ F ∂L2 = FLL < 0 and diminishing

◮ Marginal product of capital

◮ ∂F ∂K = FK > 0 positive

2 ◮ ∂ F ∂K 2 = FKK < 0 and diminishing

Lecture1,EndogenousGrowth 6/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Constant returns to scale (CRS) 7

F(λK , AλL)= λF(K , AL) for λ> 0

◮ Implications of CRS

◮ Size (of firms) does not matter → representative firm ◮ Euler’s theorem: Factor payments exhaust the output ◮ See example

Lecture1,EndogenousGrowth 7/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Various functions 8

◮ Capital and labour: Y = F(K , AL)= K α(AL)1−α ◮ Physical and Human Capital and labour: Y = F(K , H, AL)= K αHβ(AL)1−α−β ◮ Capital and Land: Y = F(K , L) ◮ Capital and population/labor: Y = F(K , N)

For some questions it is not important which factor is accumulable and which is not, for other questions it is (see later).

Lecture1,EndogenousGrowth 8/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth A particular exercise: Steps for growth accounting 9

◮ TFP residual, At , for K only

◮ TFP residual, At , across countries: K only

◮ TFP residual, At , including human capital

◮ TFP residual, At , across countries: K and H ◮ Decomposing growth in GDP per worker: K only

◮ Decomposing growth in GDP per worker: K and H

◮ Summary of results

◮ Critique

Lecture1,EndogenousGrowth 9/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth

TFP residual, At , across countries: K only 10

UK, South Korea and India

TFP (Accounting for Physical Capital Only) 1400.00

1200.00

1000.00

800.00 U.K. Korea 600.00 India

400.00

200.00

0.00 1960 1965 1970 1975 1980 1985 1990 1995

Lecture1,EndogenousGrowth 10/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth

TFP residual, At , across countries: K and H 11

UK, South Korea and India

TFP (Accounting for Physical and Human Capital) 1400.00

1200.00

1000.00

800.00 U.K. Korea 600.00 India

400.00

200.00

0.00 1960 1965 1970 1975 1980 1985 1990 1995

Lecture1,EndogenousGrowth 11/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Summary of Results 12

◮ TFP in the UK > TFP Korea AND India → For given inputs output in the UK is higher than in Korea and India

◮ When accounting for higher educational attainment, differences in TFP are smaller → Adding H as input shows that India is not that much less productive than the UK; educational attainment (on average) is lower

Lecture1,EndogenousGrowth 12/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth UK, Korea and India 13

Growth Accounting with physical and human capital

Lecture1,EndogenousGrowth 13/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Summary of Results 14

◮ GDP growth accounting: increase in human capital (average years of education) accounts for a major part of growth in India

◮ Hence, omitting human capital in growth accounting can lead to erroneous conclusions

Lecture1,EndogenousGrowth 14/33 Economic Policy in Development 2, Part 2 Growth Accounting Growth Accounting: Objective and Technical Framework From GA to Solow to Ramsey Results Endogenous long-run growth Critique 15

◮ Interpretation of TFP? ◮ Technological change? ◮ Deregulation? ◮ Regulation?? ◮ Why did trend change?

◮ Other factors ◮ Human capital? → Done ◮ Capital-skill complementarities? ◮ Quality of capital?

Lecture1,EndogenousGrowth 15/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth From Growth Accounting to the Solow Model 16

◮ In growth accounting → link of inputs in period t to output in period t → no link of inputs or output across periods (t versus t + 1)

◮ Solow model links → population/labor force, and, in particular, capital stock in year t to → labor force, productivity and capital stock in year t + 1

◮ Solow (1956), Solow (1957) and Solow (1960)

Lecture1,EndogenousGrowth 16/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Solow Model 17

Solow Model ◮ Production function: same as before

◮ Exogenously given rate s

◮ Law of motion for physical capital

Lecture1,EndogenousGrowth 17/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Solow: Law of motion: n = 0 and g = 0 18

◮ Consider Kt+1 as a function of Kt :

Kt+1 = (1 − δ)Kt + It

Kt+1 = (1 − δ)Kt + sYt ¯¯ Kt+1 = (1 − δ)Kt + sF(Kt , AL) ◮ Since marginal product of K positive, → law of motion: increasing function

◮ Since marginal product of K diminishing → law of motion: concave function

Lecture1,EndogenousGrowth 18/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Solow’s law of motion 19

50 Kt+1 = Kt (45 degree line) 45 Kt+1 = (1-delta) Kt + s F(Kt,AL) 40

35

30

25 K_t+1 20

15

10

5

0 0 10 20K_t 30 40 50

Lecture1,EndogenousGrowth 19/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Solow: Balanced growth: n 6= 0 and g 6= 0 20

◮ Evolution of technology: At+1 = (1 + g)At , ◮ Evolution of population (labour force*): Lt+1 = (1 + n)Lt ◮ Law of motion of aggregate capital Kt+1 = (1 − δ)Kt + sF(Kt , At Lt ) α 1−α Kt+1 = (1 − δ)Kt + sKt (At Lt ) ˆ ˆ ˆα (1 + g)(1 + n)kt+1 = (1 − δ)kt + skt

Lecture1,EndogenousGrowth 20/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Solow: Balanced Growth 21

◮ Law of motion ito capital per u. of eff. labour

ˆ 1 ˆ ˆα kt+1 = (1+g)(1+n) (1 − δ)kt + skt Short-run growthh rate decreases wheni n increases.

◮ This can be solved for kˆ∗, the for which capital per unit of effective labour does not change anymore, i.e. ˆ ˆ ˆ∗ kt = kt+1 = k

1 ˆ∗ s 1−α * k = g+n+ng+δ   ◮ Higher population growth implies lower level of capital stock per unit of effective labor in the long run, but long-run growth rate of per capita variables unaffected

Lecture1,EndogenousGrowth 21/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Solow Model → Results 22

Solow Model ◮ Unless there is exogenous technological change (At+1 = (1 + g)At , g > 0), the economy converges to a steady state in per capita variables.

◮ No long-run growth except due to technological change.

◮ Golden Rule steady state level of consumption requires s = α Investments result from choices: the Solow model has nothing to say about savings/investment decisions → Ramsey model

Lecture1,EndogenousGrowth 22/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Ramsey Model: Savings Choice 23

Ramsey Model ◮ Households choose how much to save and how much to consume → dynamics result from this choice.

∞ t max∞ β u(ct ) (a ,ct ) t+1 t=0 Xt=0 s.t. ct + at+1 = wt + (1 + r)at , for all t = 0, 1, 2, ...

◮ Euler equation: In general,

′ ′ u (ct )= β(1 + rt+1)u (ct+1)

Lecture1,EndogenousGrowth 23/33 Economic Policy in Development 2, Part 2 c1−σ Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Ramsey Model and Population Growth 24

Ramsey Model ◮ Production function/firms: same as Solow model

◮ Still, no fertility choice, i.e. population growth exogenous.

In terms of population growth rates, Ramsey model gives same conclusions as Solow model.

Lecture1,EndogenousGrowth 24/33 Economic Policy in Development 2, Part 2 Growth Accounting Solow Model From GA to Solow to Ramsey Ramsey Model Endogenous long-run growth Ramsey Model → Results 25

Ramsey Model ◮ Unless there is exogenous technological change (At+1 = (1 + g)At , g > 0), the economy converges to a steady state in per capita variables (g = 0). ◮ No long-run growth except due to technological change. ◮ Modified Golden Rule steady state level of consumption and capital are smaller than Golden Rule. This is due to impatience of households. ◮ If government consumption is financed with taxes on ∗ capital gains, HH save less and the steady state (c τk ) and ∗ (k τk ) are lower than with LS taxes. How does long-run growth occur endogenously?

Lecture1,EndogenousGrowth 25/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth A simple model of endogenous long-run growth 26

Ak Model ◮ Take Ramsey Model but change 1 assumption: α = 1

◮ Thus the production function becomes: f (k)= Ak (or F(K , L)= AK , i.e. labor does not enter into the production function)

◮ That is, f ′(k)= A > 0 → used in Euler equation ◮ That is, f ′′(k)= 0. There are no more diminishing marginal returns to capital. It is constant returns to capital alone!

How does long-run growth occur endogenously?

Lecture1,EndogenousGrowth 26/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Ak law of motion 27

Lecture1,EndogenousGrowth 27/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Ak Model: Theorem 28

Ak Model: Theorem Consider the social planner’s problem with linear technology f (k)= Ak and CEIS preferences. Suppose (β, σ, A, δ) satisfy

1 1 − β(A + 1 − δ) > 1 > β σ (A + 1 − δ) σ 1

Then the economy exhibits a balanced growth path where capital, output and consumption all grow at a constant rate given by

k y c t+1 = t+1 = t+1 = 1 + γ = [β(A + 1 − δ)]1/σ kt yt ct The growth rate is increasing in A and β and it is decreasing in δ and σ.

Lecture1,EndogenousGrowth 28/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Setup: Production function 29

Production function with human capital

Yt = F(Kt , Ht )= F(Kt , ht Lt ) ◮ F: Neoclassical production function (Lecture 3): → A1: Constant returns to scale (CRS) F(λK , λH)= λF(K , H) → A2: Marginal products positive and diminishing FK > 0, FH > 0, FKK < 0, FHH < 0 F(K ,H) → Use CRS , write F in per capita terms L = F(k, h) → For example, f (k, h)= Ak αh1−α with α< 1

◮ Ht = ht Lt is effective labor.

Lecture1,EndogenousGrowth 29/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Social Planner’s problem 30

With the usual function, Social planner’s problem can be written as

∞ t max∞ β u(ct ) (k ,ct ) t+1 t=0 Xt=0 s.t.

ct + kt+1 + ht+1 = F(kt , ht ) + (1 − δ)kt + (1 − δ)ht , for all t = 0, 1, 2, ...

k0, h0 > 0 given

Lecture1,EndogenousGrowth 30/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Balanced growth path theorem for Akh model 31

Akh Model: Theorem Consider the social planner’s problem with linear technology f (kˆ)= Aˆkˆ and CEIS preferences. Suppose (β, σ, A, δ, α) satisfy

1 1 − β(Aˆ + 1 − δ) > 1 > β σ (Aˆ + 1 − δ) σ 1

− 1 − 1 − β(A(1 − α)1 ααα + 1 − δ) > 1 > β σ (A(1 − α)1 ααα + 1 − δ) σ 1 1−α and suppose h0 = α k0.Then the economy exhibits a balanced growth path where capital, output and consumption all grow at a constant rate given by

kˆ k h y c t+1 = t+1 = t+1 = t+1 = t+1 = 1 + γ = [β(Aˆ + 1 − δ)]1/σ kˆt kt ht yt ct

Lecture1,EndogenousGrowth 31/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Concluding remarks 32

◮ This model exhibits long run growth, even though some form of labor is taken into account.

◮ This is because the QUALITY of labor is taken into account.

◮ This quality, human capital, is accumulable.

◮ Therefore the diminishing marginal returns don’t kick in. Both factors grow simultaneously and therefore allow the economy to grow FOREVER.

Lecture1,EndogenousGrowth 32/33 Economic Policy in Development 2, Part 2 Growth Accounting Ak model From GA to Solow to Ramsey Akh model Endogenous long-run growth Think about other Interpretations 33

1. Capital and land enter the production function, Y = F(K , L) → land is not accumulable, it is fixed → decreasing marginal products kick in → steady state → land is accumulable, e.g. US expansion East to West → economy expands 2. Capital and population/labor enter the production function, Y = F(K , L) → labor is not accumulable, it is fixed → decreasing marginal products kick in → steady state Y = F(K , N) → population is accumulable → endogenous fertility models → economy expands

Lecture1,EndogenousGrowth 33/33 Economic Policy in Development 2, Part 2