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Unformatted text preview: AK Model Prof. Lutz Hendricks August 7, 2009 L. Hendricks () AK Model August 7, 2009 1 / 38 AK Model We study the simplest model with endogenous growth . Endogenous growth means: The balanced growth rate is a/ected by choices. Endogenous growth models are used to study the determinants of longrun growth. A key empirical question: Do countries&longrun growth rates di/er? L. Hendricks () AK Model August 7, 2009 2 ¡ 38 Necessary Conditions for Sustained Growth How can growth be sustained without exogenous productivity growth? A necessary condition: constant returns to the reproducible factors . Vaguely: The production function must be linear in factors that can be produced. This motivates a simple class of models in which 1 only K can be produced and 2 the production function is AK . This can be thought of as a reduced form for more complex models (we& ll see examples). L. Hendricks () AK Model August 7, 2009 3 / 38 Necessary Conditions for Sustained Growth Solow AK model To see what is required for endogenous growth, consider the Solow model: g ( k ) = s f ( k ) / k & ( n + δ ) (1) Positive longrun growth requires: As k ! ∞ it is the case that f ( k ) / k > n + δ (2) L&Hopital&s rule implies (if f has a limit): lim f ( k ) / k = lim f ( k ) (3) Sustained growth therefore requires: lim k ! ∞ f ( k ) > n + δ (4) L. Hendricks () AK Model August 7, 2009 4 / 38 Necessary Conditions for Sustained Growth This argument is more general than the Solow model. It does not matter how s is determined. If lim k ! ∞ f ( k ) exists, the production function has asymptotic constant returns to scale . f ( k ) ! Ak (5) It is &ne to have diminishing returns for &nite k . Examples: 1 f ( k ) = Ak + Bk α with < α < 1 2 CES production function with high elasticity of substitution: F ( K , L ) = h μ K θ + ( 1 & μ ) L θ i 1/ θ (6) L. Hendricks () AK Model August 7, 2009 5 / 38 AK Solow Model In the Solow model, assume f ( k ) = A k . Law of motion: g ( k ) = s A & n & δ (7) Changes in parameters alter the growth rate of k . The model does not have any transitional dynamics: k always grows at rate sA & n & δ . L. Hendricks () AK Model August 7, 2009 6 / 38 AK Solow Model It is not necessary to have constant returns in all sectors of the economy. Imagine that c is produced from k with diminishing returns to scale: c = [( 1 & s ) Ak ] ϕ with ϕ < 1 . The law of motion for k is unchanged (so is the balanced growth rate of k ). This model still has a balanced growth path with a strictly positive growth rate, but not c and k grow at di/erent constant rates: g ( c ) = ϕ g ( k ) (8) L. Hendricks () AK Model August 7, 2009 7 ¡ 38 AK Neoclassical Growth Model L. Hendricks () AK Model August 7, 2009 8 / 38 AK neoclassical growth model This model adds optimizing consumers to the Ak model....
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This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.
 '09
 LUTZHENDRICKS

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