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Unformatted text preview: Review Questions: In&nite Horizon Models in Continuous Time Prof. Lutz Hendricks. August 7, 2009 1 Household Behavior 1.1 Wealth tax [Romer 2.3] Suppose it is known in advance that the government will con&scate some wealth at a future date t . Does consumption change disontinuously at t ? Why or why not? What is the equation governing consumption around t ? Consider two cases: 1. The government con&scates half of each household¡s wealth. 2. The government con&scates half of the average wealth held by all households. 2 Equilibrium 2.1 Exponential utility [Barro and Sala-i-Martin 2.3] Consider a version of the Ramsey model with exponential utility function u ( c ) = & (1 =& ) e & &c (1) where & > . 1. Compute the intertemporal elasticity of substitution. How does it change with c ? 2. Find the household¡s Euler equation. 3. Draw a phase diagram and characterize how the economy converges to a steady state. Assume g ( A ) = 0 . 4. Does the economy have a balanced growth path if g ( A ) > ? 2.2 CES Production function Consider a Ramsey model with land & . Output is produced according to Y = h a & K ¡ L 1 & ¡ ¡ + (1 & a ) & i 1 = (2) 1. Show that this production function has constant returns to scale. 2. Calculate the elasticity of substitution between & and Q = K ¡ L 1 & ¡ . Show that it is simply a function of . This is why the production function is called CES or Constant Elasticity of Substitution. 3. Under what conditions on is Y=L constant in steady state? Under what conditions does Y=L decline in the long run? Can Y=L grow in the long run? 2.2.1 Answer: CES production function 1. Easy. 2. The answer is in any micro text book. The elasticity is 1 = (1 & ) . 1 3. The easiest method is probably to look at Y=K = " a + (1 & a ) & & K & L 1 & & ¡ # 1 = (3) If Y=K is constant, then from g ( K ) = sY=K & & we know that K and Y both grow at constant rates. Case 1: > . The substitution elasticity is greater than 1. Land is "not essential" in production. As L and K grow, X = & & K & L 1 & & ¡ ! and Y=K ! a . There is a steady state (asymptotically) which looks exactly like the one of an economy without land. Case 2: < . Land is essential. Over time X ! 1 and ( a + [1 & a ] X ) ! because < . Y=K ! and g ( K ) ! & & . Y=L ! . [Is there an easier way of doing this?] Persistent growth is not possible. 2.3 Capital adjustment costs Consider a version of the standard one sector, neoclassical growth model with no technological progress, inelastic labor supply and and zero population growth. We will examine the planner&s problem in an economy in which capital is costly to adjust. The planner maximizes the lifetime utility of the representative household: Z 1 u ( c t ) e & ¡t dt: (4) where u ( c ) = c 1 & & & 1 1 & ¢ : There is one ¡nal good which is produced using labor and capital using a production technology which can be written in intensive form as: y t = f ( k t ) ; (5) where f ( ¡ ) > , f 00 ( ¡ ) < , and the usual Inada conditions apply....
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This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.