Matthias Doepke - Marcroeconomics - Chapter 2 docx

... Cobb-Douglas function exhibits constant returns to scale, since (1 )+( ) = 1. Figure 2. 1 is a three-dimensional rendering of this function for particular values of and . 0 0 .2 0.4 0.6 0.8 1 0 0 .2 0.4 0.6 0.8 1 0 0 .2 0.4 0.6 0.8 1 product ... consumption . Formally, Crusoe’s problem is: max ( ) such that: (2. 1) and: (2. 2) = ( ) (2. 3) There are two elements of equation (2. 1). First,...

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... countries at that date? Exercises 5 0 2 4 6 8 0 5 10 15 20 25 Year (n) V(n) Figure 1.1: Exponential Growth 0 1 2 0 5 10 15 20 25 Year (n) ln[V(n)] Figure 1 .2: Log of Exp Growth The two ln($130 ) ... be: (2) = (1) + (1) = (1 + ) +(1+ ) =(1+ ) 2 Similarly, after years, the value will be: ( )=(1+ )(1 .2) Of course, this formula works only an integral numbers of years. For non-inte...

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... Labor 139 0 0.1 0 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0 .2 0 .25 Tax revenue when γ=0.5 τ T(τ) Figure 13.1: A government revenue func- tion that exhibits a Laffer curve. A Lump-sum Tax Now ... )=max 2 The first-order condition for optimality is: 1=0 Solving for produces: ( )= + 2 The notation is meant to imply lump-sum tax: there is a surfeit of notation involving and in t...

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... )=0.Wehad: 1 ( )= 1 [ 2 + 1 (1 + )] (1 + )(1 + ) so we set the left-hand side to zero and solve for : 1 = [ 2 + 1 (1 + )] (1 + )(1 + ) (3.11) 3 .2 A Two-Period Model 23 3 .2 A Two-Period Model We begin ... budget equation is: 1 = 1 + 1 (3 .2) and in period = 2 it is: 2 + 1 (1 + )= 2 (3.3) The household’s problem is to choose consumptions 1 and 2 and first-period bond hold- ing...

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... budget constraint. Since the consumer chooses 1 2 even though ¯ 1 ¯ 2 ¯ yields higher utility, it has to be the case that ¯ 1 ¯ 2 ¯ violates the consumer’s budget con- straint: =1 ¯ =1 (5.4) Otherwise, the ... allocation ¯ 1 ¯ 2 ¯ =1 satisfied market-clearing. Therefore the terms inside the brackets are all zero. This implies 0 0, which is a contradiction. There- fore, no such allocati...

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... 1 ]+ 2 [ ( 2 )+ 1 (1 + ) 2 ] There are seven first-order conditions: 1 ( 1 1 )+ 1 [ ]=0(FOC 1 ) 1 ( 2 2 )+ 2 [ ]=0(FOC 2 ) 2 ( 1 1 )+ 1 [ ( 1 )] = 0(FOC 1 ) 2 ( 2 2 )+ 2 [ ( 2 )] = 0 and:(FOC 2 ) 1 [ ... labor, 1 and 2 .Theincome takes the place of the endowment in the model from Chapter 3. The household’s maximization problem is: max 1 2 1 2 1 ( 1 1 )+ ( 2 2...

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... monetary pol- icy. While the prime emphasis of the cash-in-advance model is the inefficiency of holding cash instead of interest-bearing assets, Chapter 19 turns to the issue of expected versus un- expected ... inflation that we will pick up later in this book. Chapter 18 is concerned with the coordination of monetary 8.3 A Cash-in-Advance Economy 61 8.3 A Cash-in-Advance Economy In this se...

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... single shock has long-run effects, and investment goes through a full cycle in response to this shock. Response to a Single Shock -0 .01 -0 .005 0 0.005 0.01 0.015 0. 02 0. 025 123 45678 Periods Deviation ... respect to is: = 1 1 2 1 1 1 1 2 1 1 +(1 ) 1 1 and for investment we get: = 1 2 1 1 1 2 1 1 (1 ) 1 1 It turns out that the relative change in investment is larger. A one...

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... yields: = ( ) 1 1 = 1 1 1 = 1 1 (11 .20 ) Equation (11 .20 ) relates output per worker to capital per worker 1 .Ifweuse lower case letters to denote per-worker values ( = , 1 = 1 ), we can write equation (11 .20 ) as: = 1 1 (11 .21 ) 11.4 ... production function equa- tion (11.1) exhibits constant returns to scale, which means that if we double both inputs, output also doubles. Our choice of...

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... 15 20 25 30 1.5 1.6 1.7 1.8 1.9 2 2.1 2. 2 Capital Stock time t K(t) Figure 12. 4: Time path of the capital stock before, during and after a war. The sol- id line gives assuming that govern- ment ... cross-multiply to find: (1 + ) (1 ) = + [ 2 (1 + )][(1 ) ]= + [ 2 (1 + )](1 ) =[1+ 2 (1 + )] and: 2 (1 + ) [1 + 2 (1 + )] = [1 + 2 (1 + )] Dividing both sides by 1 + 2 (1 + )pr...

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