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Unformatted text preview: ECON 714: MACROECONOMIC THEORY II TA: TIM LEE JANUARY 29, 2010 1. The Roadmap Before you jump into all this math, a few things to note: 1. Econ 6 = Math. 2. Unfortunately, you’re going to have to know all this stuff at least for this course and the prelims. Following that, If you don’t do macro, no one’s really going to care whether you remember all this. Even if you do macro, no one’s really going to care whether you remember all this. BUT... (a) If you do quantitative macro, it’s nice to know whatever you tell your computer to do actually has some theory behind it. And you might want to cite some theorems just to let people know you’re not stupid. (b) If you do macro theory, it’s nice to know there’s a bunch of theorems you can use even if you don’t really remember why they’re true. To be a bit serious, it’s terribly boring if you just go through the math for the sake of math. There’s a reason we’re doing this: consider the following dynamic programming problem V ( x ) = sup y [ F ( x , y ) + β V ( y )] s . t . y is feasible given x , where x is the beginning of period state variable, y is the control variable, and F ( x , y ) is the current period return function. As we said in class, it’s the value function V and the solution to the sup problem, y * = g ( x ) , that we’re going after. To do this, we proceed in the following steps. 1. Find conditions such that an iterative procedure will admit V . What we’re going to do is create a sequence of guesses, and wish that it converges to the true function. (a) The Contraction Mapping Theorem (CMT) tells us conditions under which we get convergence. 1 (b) To use the theorem, we need to define the space in which V lives, and a notion of convergence for this space. (c) Blackwell’s Sufficiency Conditions characterize when the CMT applies to our space. 2. So we know how to get V and g . The Theorem of the Maximum characterizes what these guys will look like. In short, that’s the whole point about Stokey and Lucas ( 1989 ) Ch.3. Proving whether this value function is actually identical to the sequence problem is another issue, which is the subject of Ch.4. 2. Spaces and Sequences DEFINITION 1 A metric space is a set S , together with a metric ρ : S × S → R , such that for all x , y , z ∈ S : 1. ρ ( x , y ) ≥ 0, with equality if and only if x = y 2. ρ ( x , y ) = ρ ( y , x ) and 3. ρ ( x , z ) ≤ ρ ( x , y ) + ρ ( y , z ) EXERCISE 1 STOKEY AND LUCAS ( 1989 ), 3.3C, P.45 The set S of all continuous, strictly increasing functions on [ a , b ] with ρ ( x , y ) = max a ≤ t ≤ b  x ( t ) y ( t )  is a metric space. Proof: ( i ) ρ ( x , y ) = max a ≤ t ≤ b  x ( t ) y ( t )  =  x ( t * ) y ( t * )  ≥ 0, where t * is the maximizer, ρ ( x , y ) = max a ≤ t ≤ b  x ( t ) y ( t )  = iff x ( t ) = y ( t ) ∀ t ∈ [ a , b ] , i.e. x = y ( ii ) ρ ( x , y ) = max a ≤ t ≤ b  x ( t ) y ( t )  = max a ≤ t ≤ b  y ( t ) x ( t )  =...
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 Spring '08
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 Topology, Metric space, ρ, Cauchy

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