Matthias Doepke - Marcroeconomics - Chapter 2 docx
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,...
Matthias Doepke - Marcroeconomics - Chapter 1 docx
... 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...
Matthias Doepke - Marcroeconomics - Chapter 13 docx
... 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...
Matthias Doepke - Marcroeconomics - Chapter 3 ppsx
... )=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...
Matthias Doepke - Marcroeconomics - Chapter 5 pptx
... 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...
Matthias Doepke - Marcroeconomics - Chapter 6,7 doc
... 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...
Matthias Doepke - Marcroeconomics - Chapter 8 pdf
... 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...
Matthias Doepke - Marcroeconomics - Chapter 9 doc
... 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...
Matthias Doepke - Marcroeconomics - Chapter 11 pot
... 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...
Matthias Doepke - Marcroeconomics - Chapter 12 pdf
... 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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