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Ramsey Growth Model Dynamics

Ramsey Growth Model Dynamics. ECGA 7020 Macro Theory II Fall 2005 Fordham University Professor Darryl McLeod Thanks to Rosendo Ramirez for the dynamic action slides. Consumption Laws of Motion.

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Ramsey Growth Model Dynamics

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  1. Ramsey Growth Model Dynamics ECGA 7020 Macro Theory II Fall 2005 Fordham University Professor Darryl McLeod Thanks to Rosendo Ramirez for the dynamic action slides

  2. Consumption Laws of Motion • The key Euler equation determines the trajectory of consumption, the marginal rate of substitution between present and future is always equal to MRT between periods as determined by the production function, (the Keynes-Ramsey rule) • Note that when f ’(k) is greater than  + θg, c rises, and vice versa,

  3. Law of Motion for capital • The main innovation of the Ramsey model is to make savings s = y - c = f(k) – c endogenous, though we need a constant intemporal elasticity of substitution such as that provided by the CRRA utility function to do so.

  4. Utility Maximization Utility ( c ) falls when f’(k) < ρ+θg (n + g)k y Utility maximized whenf ’(k) = ρ+θg y = f(k) Utility U(ct ) increases as long as f ’(k) > ρ+ θg > n + g k k*, f ’(k) = ρ+θg

  5. when, f ‘(k*)=ρ+θg c(t) falls when f ‘(k) < ρ + θg ρ + θg c(t) increases when f ‘(k) > ρ + θg f ’(k) k k*

  6. f ’(k) = n+g c f ’(k)< ρ+θg f ’(k) > ρ+θg k k*

  7. Dynamics of k: recall the golden rule: consumption reaches maximum when f’(k) = n +g y When f ’(k) = n+g, c reaches a maximum as c = f(k) – (n + g)k(golden rule max c ) (n+g)k y = f(k) When f ’(k) < n + g, c falls. When f ’(k) > n+g, c increases. k kgr (golden rule)

  8. Dynamics of k (the change in k depends on s = y-c ) c when k falls in the green region because savings, s < (n+g)k . when, Savings just covers investment per capita, s = y – c = (n+g)k Savings, s > (n+g)k when, k

  9. Note that k* < kgr (golden rule – see the next slide) c k kgr (golden rule) k*

  10. Note that golden rule kgr > k* • The boundedness condition: we assume the discount rate ρ > 0 is large enough to assure that the present discounted value of utility is finite, that is, β = ρ – n – (1-θ)g > 0. (see eq. 2.12 on page 52 of Romer (2001). • And this β > 0 condition implies ρ+θg > n + g, so that k* < kgr.

  11. c k k*

  12. c k k*

  13. c k k*

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