Econ 712: 2010 Final Exam Answer Key
Prepared by Scott Swisher
1/23/2011
1
Question 1: Pure Endowment Economy (60 points)
Consider an innite horizon economy where times ows t = 0, 1, 2, . . .; there are three households, indexed by
j cfw_1, 2, 3. There is
Econ 712: Macroeconomic Theory I 1
Assignment 6: Answer Key
1
Dynamic GE Model: Pricing a Lucas Tree
1 Week
13, 12/2/2011, UWMadison; TA Scott Swisher.
1
2
2
Dynamic GE Model: Pricing a Share in the Firm
3
4
5
6
7
Econ 712: Macroeconomic Theory I 1
Assignment 7: Answer Key
1
Neoclassical Growth Model: Steady State
(a)
Parameters: = 1/3, g = 0.02, = 0.025, = 0.95, L = L = 1 (the model assumes inelastic labor supply), Z
set so model output in steady state matches ini
Econ 712: Macroeconomic Theory I 1
Assignment 8: Answer Key
1
Optimality of Lumpsum Taxes
This one is fairly easy to show and intuitive. Linear taxes are distortionary, and enter agents rstorder conditions
and Euler equations; lumpsum taxes do not, exc
Econ 712: Macroeconomic Theory I 1
Discussion Section 2: Answer Key
1
ArrowDebreu equilibrium in a pure endowment economy
(AD Equilibrium with Switching Endowments) An economy consists of two types of consumers indexed by
i = 1, 2. There is one nonstor
Econ 712: Macroeconomic Theory I 1
Discussion Section 3: Answer Key
1
Neoclassical growth model with a representative agent
(Government Spending and Labor Supply) Consider an economy populated by a large number of identical
agents. The instantaneous utili
Econ 712: Macroeconomic Theory I 1
Discussion Section 4: Answer Key
1
Skillbiased technical change in general equilibrium
(Fall 2007 Problem Set 1, Noah Williams, Question 3) This problem considers a twosector static model,
where the two sectors corresp
Econ 712: Macroeconomic Theory I 1
Discussion Section 5: Answer Key
1
Endogenous fertility
(Rody Manuelli) Consider an economy populated by a large number of identical dynasties with utility functions
given by
t Nt u(ct ) , (0, 1)
U
t=0
where u(c) = c1 /
Econ 712: Macroeconomic Theory I 1
Discussion Section 6: Answer Key
1
Ramsey Problem
(Linear Taxes) Consider the Neoclassical growth model with linear taxation, but instead of there being taxes on
capital and labor income, there are linear taxes tc on con
2
Questions
Problem 1 (Productivity and Drrrable Consumption)
Consi'd"er an econorng poputith uti'h'ty functi'ons giaen by
ulated, by a large rrumber of id,arti,u,l d,ynosties
u:Lp'lu(c,)+u(il,
o<0
<r,
t:0
where u(c) and. u(z) are stri,ctly incrensing an
8. OVERLAPPING GENERATION MODELS
110
Exercise 8.1.
each date I > 1, an economy consists of overlapping generations of a constant
number 1[ of twoperiodlived agents. Young agents born in f have preferences
over consumption streams of a single good that a
Problem Sel ff6 Answer Key
Economics 808: Macroeconomic Theory
Fall 2004
1
a)
Overlapping generations with CobbDouglas production
The Lagrangean is:
L:lrrq,t*
So the
Blnc2,1*1f )1 (a,1 
ct,t
kt+t

br+t)
0t
(rt+tkt+t
*
Rt+tbt+t

cz,t+t)
first order c
Problem 2
Consid,er an economA uith twa period,s. There are tuo states of
nature: The gaod stI,te and, the bad, state. The good, state has probabilita 0.9
the bad state has probdbilitU 0.1. There i,s only one dgerLt in the econom!the agent has ane unit of
Econ 712: Macroeconomic Theory I 1
Assignment 5: Answer Key
1
Necessary and Sucient Conditions with Exogenous Wealth
1 Week
10, 11/11/2011, UWMadison; TA Scott Swisher.
1
2
2
Necessary and Sucient Conditions with Exogenous Income
3
4
5
6
3
Aggregate Impl
Econ 712: Macroeconomic Theory I 1
Assignment 3: Answer Key
1
Data facts
1 Week
7, 10/21/2011, UWMadison; TA Scott Swisher.
1
2
3
4
2
Growth accounting (percapita terms, actual data)
5
3
Approximation error in the Taylor expansion
6
4
Calibration of Sol
Econ 712: Macroeconomic Theory I 1
Assignment 2: Answer Key
1
Equivalence of equilibrium in Solow and Neoclassical growth models
1 Week
5, 10/7/2011, UWMadison; TA Scott Swisher.
1
2
2
Existence and uniqueness of steady state in the Solow model
3
4
3
Opt
Econ 712: Macroeconomic Theory I 1
Assignment 1: Answer Key
1
Aggregate production function and Eulers theorem
1 Week
4, 9/30/2011, UWMadison; TA Scott Swisher.
1
2
Feasibility of equilibrium allocation and rms objective
2
3
3
Noarbitrage condition for
Econ 712: Macroeconomic Theory I 1
Assignment 2: Answer Key
1
Equivalence of equilibrium in Solow and Neoclassical growth models
1 Week
5, 10/7/2011, UWMadison; TA Scott Swisher.
1
2
2
Existence and uniqueness of steady state in the Solow model
3
4
3
Opt
Econ 712: Macroeconomic Theory I 1
Assignment 3: Answer Key
1
Data facts
1 Week
7, 10/21/2011, UWMadison; TA Scott Swisher.
1
2
3
4
2
Growth accounting (percapita terms, actual data)
5
3
Approximation error in the Taylor expansion
6
4
Calibration of Sol
Econ 712: Macroeconomic Theory I 1
Assignment 4: Answer Key
1
Dynamic general equilibrium model with exogenous income
1 Week
7, 10/21/2011, UWMadison; TA Scott Swisher.
1
2
Determining equilibrium loan prices
2
3
Proving the sequential markets, ArrowDeb
Results behind the Negishi algorithm and their proofs.
Proposition 1
(Equivalence of SM and AD equilibria).
1. Suppose (cj , bj )J=1 , qt ) is a SM equilibrium. Let:
t
tj
t=0
p = 1
0
p = q0 qt1 ,
t
t 1.
Then (cj )J=1 , p ) is an AD equilibrium.
tj
t t=0
2
Econ 712: Macroeconomic Theory I 1
Assignment 5: Answer Key
1
Necessary and Sucient Conditions with Exogenous Wealth
1 Week
10, 11/11/2011, UWMadison; TA Scott Swisher.
1
2
2
Necessary and Sucient Conditions with Exogenous Income
3
4
5
6
3
Aggregate Impl
Econ 712: Macroeconomic Theory I 1
Assignment 6: Answer Key
1
Dynamic GE Model: Pricing a Lucas Tree
1 Week
13, 12/2/2011, UWMadison; TA Scott Swisher.
1
2
2
Dynamic GE Model: Pricing a Share in the Firm
3
4
5
6
7
Econ 712: Macroeconomic Theory I 1
Assignment 7: Answer Key
1
Neoclassical Growth Model: Steady State
(a)
Parameters: = 1/3, g = 0.02, = 0.025, = 0.95, L = L = 1 (the model assumes inelastic labor supply), Z
set so model output in steady state matches ini
Econ 712: Macroeconomic Theory I 1
Assignment 8: Answer Key
1
Optimality of Lumpsum Taxes
This one is fairly easy to show and intuitive. Linear taxes are distortionary, and enter agents rstorder conditions
and Euler equations; lumpsum taxes do not, exc
(p , c , s+1 )T=0
t
t
t
t
max
(ct ,st+1 )T=0
t
ct +
T
(c , s+1 )T=0
t
t
t
t u( c t ) ,
t=0
p st+1
t
( p + y t ) st ,
t
ct , st 0,
t
t
s0 = 1
c = yt ,
t
s = 1 ,
t
p =
t
t
t
u (yt+1 )
(p + yt+1 )
u (yt ) t+1
(p , c , s+1 )T=0
t
t
t
t
(c , s+1 )T=0
t
t
t
1 t
max
(kt , kt+1 ),
(k t )
1+r
t=1
t=0
k0 = k0
V :K R
1 t
V (k ) = max
(kt , kt+1 ),
(kt )
1+r
t=1
t=0
k0 = k
kK
V (k ) = max (k, k ) +
k K
1
V (k ) ,
1+r
k K.
g : K K
1
g (k ) arg max (k, k ) +
V (k )
K
k
1+r
kK
V
k0 = k 0
kt+1 = g (kt )
( kt )
Econ 712: Macroeconomic Theory I 1
Assignment 1: Answer Key
1
Aggregate production function and Eulers theorem
1 Week
4, 9/30/2011, UWMadison; TA Scott Swisher.
1
2
Feasibility of equilibrium allocation and rms objective
2
3
3
Noarbitrage condition for
Econ 712: Macroeconomic Theory I 1
Discussion Section 11: Answer Key
1
RBC Model with a Fixed Factor of Production
(Ken West) This question asks you to consider the role of a xed factor, say land, in a simplied version of a
standard RBC model, using the a