Part IIB. Paper 2 Michaelmas Term 2009 Economic Growth Lecture 2: Neo-Classical Growth Model Dr. Tiago Cavalcanti Readings and Refs Texts: (*)Jones ch.2; BX chs.1,10; Romer ch.1. Original Articles: Solow R. (1956) A contribution to the theory of economic growth Quarterly Journal of Economics, 70, 65-94. Solow R. (1957) Technical change and the aggregate production function Review of Economics and Statistics, 39, 312-320. Swan T. (1956) Economic growth and capital accumulation
Economic Record, 32, 334-361. The Neoclassical Growth model Solow (1956) and Swan (1956) simple dynamic general equilibrium model of growth Neoclassical Production Function Output produced using aggregate production function Y = F (K , L ), satisfying: A1. positive, but diminishing returns FK >0, FKK<0 and FL>0, FLL<0 A2. constant returns to scale (CRS) F (K , L) F ( K , L), for all 0 replication argument Production Function in Intensive Form
Under CRS, can write production function K Y F ( K , L) Y L.F ( ,1) L Alternatively, can write in intensive form: y = f(k) - where per capita y = Y/L and k = K/L Exercise: Given that Y=L f(k), show: FK = f(k) and FKK= f(k)/L . Competitive Economy representative firms maximise profits and take price as given (perfect competition) can show: inputs paid their marginal products: r = FK and w = FL inputs (factor payments) exhaust all output: wL + rK = Y general property of CRS functions (Eulers THM)
A3: The Production Function F(K,L) satisfies the Inada Conditions lim K 0 FK ( K , L) and lim K FK ( K , L) 0 lim L 0 FL ( K , L) and lim L FL ( K , L) 0 Note: As f(k)=FK have that limk 0 f ' (k ) and limk f ' (k ) 0 Production Functions satisfying A1, A2 and A3 often called Neo-Classical Production Functions Technological Progress = change in the production function Ft Yt Ft ( K , L) 1. Ft ( K , L) Bt F ( K , L) Hicks-Neutral T.P.
2. Ft ( K , L) F ( K , ( At L)) Labour augmenting (Harrod-Neutral) T.P. 3. Ft ( K , L) F ((Ct K ), L) Capital augmenting (Solow-Neutral) T.P. A4: Technical progress is labour augmenting Ft ( K , L) F ( K , ( At L)) and At A0 e gt Note: For Cobb-Douglas case three forms of technical progress equivalent:
Ft ( K , L) Bt K L(1 ) K ( At L) (1 ) ( Dt K ) L(1 ) when Bt At (1 ) Dt Under CRS, can rewrite production function in intensive form in terms of effective labour units ~ ~ y f (k ) ~ K Y ~
where y and k AL AL -note: drop time subscript to for notational ease - Exercise: Show that ~ ~ f ' (k ) FK and f ' ' (k ) AL FKK Model Dynamics A5: Labour force grows at a constant rate n Lt L0e nt A6: Dynamics of capital stock:
dK K I K dt net investment = gross investment - depreciation capital depreciates at constant rate closing the model National Income Identity Y = C + I + G + NX Assume no government (G = 0) and closed economy (NX = 0) Simplifying assumption: households save constant fraction of income with savings rate 0 s 1 I = S = sY Substitute in equation of motion of capital: K sY K sF ( K , AL) K Fundamental Equation of
Solow-Swan model ~ ~ dk ~ ~ k sy (n g )k dt ~ K ~ Proof : k ln k ln K ln A ln L AL ~ d ln k d ln K d ln A d ln L dt
dt dt dt ~ k K sY s~ y ~ g n (n g ) ~ (n g ) K k K k Steady State Definition: Variables of interest grow at constant rate (balanced growth path or BGP) ~ k 0 ~ y c~ 0
at steady state: ~* ~* sf k (n g )k 0 Solow Diagram _ = (~ ~ ) ( + + )~ ~= (~ ) ~
( + + )~ ~= (~) 0 0 ~ (0) ~ ~ 4 x 10 Existence of Steady State
From previous diagram, existence of a (nonzero) steady state can only be guaranteed for all values of n,g and if limk~ 0 ~ ~ f ' (k ) and limk~ f ' (k ) 0 - satisfied from Inada Conditions (A3). Transitional Dynamics ~ ~* If k k ,
then savings/investment exceeds ~ depreciation, thus k~ 0 g k~ 0. k ~ ~* If k k , then savings/investment lower than ~ depreciation, thus k~ 0 g k~ 0. k By continuity, concavity, and given that f(k) satisfies the INADA conditions, there ~* ~* ~* k such that f (k ) (n g )k must exists an unique ~ k
~ k Transitional Dynamics _ ~ ~ = ( ~) ~ ( + + ) ~
~ = ~ 0 ~ 0 ~ ~ 0 Properties of Steady State 1. In steady state, per capita variables grow at the rate g, and aggregate variables grow at rate (g + n) ~ K
K and k Proof: as k AL L d log K d log L d log k gK n g k dt dt dt ~ d log k d log A d log k gk g g k~
dt dt dt g in Steady State 2. Changes in s, n, or will affect the levels of y* and k*, but not the growth rates of these variables. - Specifically, y* and k* will increase as s increases, and decrease as either n or increase Prediction: In Steady State, GDP per worker will be higher in countries where the rate of investment is high and where the population growth rate is low - but neither factor should explain differences in the growth rate of GDP per worker. Golden Rule and Dynamic Inefficiency Definition: (Golden Rule) It is the saving rate
that maximises consumption in the steady-state. ~* ~* ~* * ~ max c (1 s) f (k ) f (k ) (n g )k s ~* ~* ~* * ~ ~* c f (k ) k k ~* (n g ) 0 f ' (kGR ) (n g )
s s s k ~* Given kGR ,we can use to find sGR . ~* ~* sf (kGR ) (n g )kGR Golden Rule and Dynamic Inefficiency ~ )
= (1 ) (~ ~ 0 0 1 Changes in the savings rate Suppose that initially the economy is in the ~*
~* steady state: sf (k1 ) (n g )k1 ~* ~* ~ If s increases, then sf (k1 ) (n g )k1 k 0 Capital stock per efficiency unit of labour grows until it reaches a new steady-state Along the transition growth in output per capita is higher than g. Linear versus log scales 4 x 10 Log-Scale L inear-Scale
( ()) ) ( ) = (0) = _ () ( ()) ( ( )) 0
0 0 0 = Changes in the savings rate ( ()) Log of capital per capita ( ())
per capita Log of output ( ()) Log of consumption per capita Next lecture Testing the neo-classical model: 1. Convergence 2. Growth Regressions 3. Evidence from factor prices
