204A-noteOG - Supplementary Note on the OG Model Econ 204A...

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1 Supplementary Note on the OG Model Econ 204A - Prof. Bohn Here are some comments on Romer’s exposition and on general OG dynamics. Read Romer’s Section 2.9 carefully as it lays out the individual problem. In Romer’s Section 2.10, the key equation for the dynamics is (2.59), or equivalently (2.67). Log-utility/Cobb-Douglas is a special case. The ‘Speed of Convergence’ section examines convergence in this special case. Romer then discusses the general case, but without examining convergence. This note examines convergence in general, as in class, but provides more specialized cases as example. Slight change to notation: Let me follow Romer and use w t for the wage per efficiency unit . Let’s denote per-worker wage income by W t = w t A t = wage per A t units of work. The individual problem with general utility function Preferences: u ( C 1 t ) + ! " u ( C 2 t + 1 ) Budget equation for workers: C 1 t + a t = W t Budget equation for retirees: C 2 t + 1 = (1 + r t + 1 ) ! a t For a graphical analysis, combine the budget equations to obtain the Intertemporal Budget Constraint W t = C 1 t + 1 1 + r t + 1 ! C 2 t + 1 . For the algebraic solution, there are several approaches: set up a Lagrangian with preferences subject to IBC; solve IBC for C 1 and insert into preferences, then maximize with respect to C 2 ; or solve the budget equations for consumption, insert into preferences, and maximize with respect to assets. The latter yields u ( W t ! a t ) + " # u [(1 + r t + 1 ) # a t ] as the objective function. The first order condition is ! u '( W t ! a t ) + " # (1 + r t + 1 ) # u '((1 + r t + 1 ) # a t ) = 0 The solution is a function a t = a ( W t , r t + 1 ) of the exogenous current wage and next period’s interest rate. Its derivatives can be determined by taking total differentials (or equivalently, apply the Implicit Function Theorem): [ ! u "( C 1 t )]( dW t ! da t ) + " u ' # dr t + 1 + (1 + r t + 1 ) # u "( C 2 t + 1 ) # [ a t dr t + 1 + (1 + r t + 1 ) # da t ] = 0 => da t = [ ! u "( C 1 t )] dW t + [ " u ' # dr t + 1 ! a t (1 + r t + 1 ) # u "( C 2 t + 1 )] # dr t + 1 [ ! u "( C 1 t ) ! (1 + r t + 1 ) 2 # u "( C 2 t + 1 )] Because u”<0, the denominator is positive; also 0 < da t + 1 / dW t < 1 . The sign of da t + 1 / dr t + 1 is ambiguous because substitution (positive) and income effects (negative) conflict. The properties of the savings rate s=a/W follow from the properties of a. Note that if utility is homothetic, optimal consumption and asset are proportional to W, so the savings rate depends only on r and not on W.
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2 Aggregate dynamics with general production function
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