Likewise if the initial capital level k was to the right of k there will be a

# Likewise if the initial capital level k was to the

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Euler condition would no longer be satisfied. Likewise, if the initial capital level 0 k was to the right of * k , there will be a unique stable manifold, satisfying the Euler and Transversality conditions and the capital transition constraint, that will converge towards the steady state * * ( , ) k c . If this was not to be the case, the economy would have ended up with zero capital or zero consumption violating the physical or the Transversality constraint. Summarizing results, we have established the following: Proposition 1 (Neoclassical Growth Theory): The Neoclasical Growth Model economy, introduced in Section A, has a unique competitive equilibrium that coincides with the solution to the Social Planner’s problem, (I.19).Along this equilibrium, capital and consumption per efficient household satisfy conditions (I.35)-(I.38), such that the following are true: (a) There exists a unique (interior) steady state * * ( , ) k c , characterized by (I.41) and (I.42). (b) If 0 k < * k , { { , } 1 1 0 k c t t t increases monotonically as it converges towards * * ( , ) k c And, if 0 k > * k , { , } 1 1 0 k c t t t decreases monotonically as it converges towards * * ( , ) k c Exercise 3 : Show that the competitive equilibrium of the Neoclassical Growth Model satisfies the Kaldor Stylized Facts (i.e., output capital per capita increases, output per 17
capita increases, the capital – output ratio is constant, the return to capital is constant, the capital and labor income shares are constant), along the steady state. Exercise 4 : Show that the competitive equilibrium of the Neoclassical Growth Model satisfies the Convergence hypothesis (i.e., the growth rate of less developed countries will converge to the growth rate of developed countries monotonically). E. Local Stability Analysis We can also characterize quantitatively the behavior of the competitive equilibrium around the steady state. This is called local stability analysis. Here, it will be convenient to work with the second order difference equation (I.34), directly. In fact, it is more convenient to begin with the more general case of the Euler condition of the Basic Problem, (I.27). If a steady state, x , exists, as in the case of the Neoclassical Growth Model, this steady state must satisfy the following: ( ) ( ) 2 1 , , 0 x x x x b F + F = (I.43) Moreover, if F is twice continuously differentiable, it follows by Taylor’s Theorem that the LHS of (I.27) may be approximated, for 1 2 ( , , ) t t t x x x + + sufficiently close to ( , , ) x x x , as the LHS of the following: ( ) ( ) ( ) ( ) ( ) ( ) 2 1 21 22 1 11 1 12 2 , , , ( ) , ( ) , ( ) , ( ) 0 t t t t x x x x x x x x x x x x x x x x x x x x b b + + + F + F +F - +F - + F - +F - = which, in view of (I.43) simplifies to the homogeneous second order linear difference equation with constant coefficients: ( ) ( ) ( ) ( ) 12 2 11 22 1 21 , ( ) [ , , ]( ) , ( ) 0 t t t x x x x x x x x x x x x x x b b + + F - + F +F - +F - = (I.44) For the Neoclassical Growth Model, omitting function arguments, the coefficients of (I.44) are given by: 1 2 n z 2 11 12 n z 2 22 n z Φ =u [f +(1- δ)] >0 Φ =u [(-1)(1+ g )(1+ g )]<0 Φ =u [f +(1- δ)] +u f <0 Φ =u [f +(1- δ)] [(-1)(1+ g )(1+ g )]> 0 Φ =u [(1+ g )(1+ g )] <0     18
To solve (I.44) in a way that satisfies the initial and Transversality conditions, or