Review Questions: AK Models
Econ720. Fall 2009. Prof. Lutz Hendricks
1
Growth model with two capital goods
Consider the following endogenous growth model with two capital goods. Households: There is a single, representativeP household who lives forever. P
Review Questions: Search and Matching
Econ720. Fall 2015. Prof. Lutz Hendricks
1
McCall Model
Romer, Advanced Macro, exercises 9.10, 9.11.
Ljunqvist & Sargent, Recursive Macroeconomic Theory, 2nd ed., exercises 26.2, 26.3.
Work out the McCall model wit
Money in OLG
Prof. Lutz Hendricks
September 2, 2009
L. Hendricks ()
Money in OLG
September 2, 2009
1 / 32
Money in OLG Models
We study the value of money in OLG models. We develop an important model of money. The model can be used to price other assets.
L
Review Problems: Money in OLG Models
Econ720. Fall 2009. Lutz Hendricks
1
Government Purchases
Consider a two-period OLG model with money. At each date t a cohort of size Nt = (1 + n)t is born. Each household lives for two periods and derives utility from
Problem Set 2: OLG Models With Money
Econ720. Fall 2009. Lutz Hendricks
1
Money and Storage
Consider a two-period OLG model with at money and a storage technology. The timing is a follows: The old enter period t holding aggregate capital Kt = Nt 1 kt and
Problem Set 2: OLG Models With Money
Econ720. Fall 2009. Lutz Hendricks
1
Money and Storage
Consider a two-period OLG model with at money and a storage technology. The timing is a follows: The old enter period t holding aggregate capital Kt = Nt 1 kt and
Bequests and Dynamic E ciency
Prof. Lutz Hendricks
August 5, 2009
L. Hendricks ()
Bequests and Dynamic E ciency
August 5, 2009
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Bequests
If parents leave bequests to their children, does that solve the dynamic e ciency problem? The answer is no - be
Money in the utility function
Prof. Lutz Hendricks
August 7, 2009
L. Hendricks ()
Sidrauski model
August 7, 2009
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Money in the utility function
A shortcut for getting money valued in equilibrium: assume that households gain utility from holding mone
Review Questions: Money in Continuous Time
Prof. Lutz Hendricks. August 7, 2009
1
Money in the production function
R1 Consider a standard Sidrauski model with a single representative household who maximizes 0 e t u(ct ; mt ) dt subject to the budget const
Midterm Exam. Econ720. Spring 2009
Professor Lutz Hendricks Answer all questions. Write legibly! Write legibly! Write legibly! Write on only one side of each sheet. The total time is 1:30 hours. A good answer should explain what you are doing. For example
The Growth Model In Continuous Time
Prof. Lutz Hendricks
August 6, 2009
L. Hendricks ()
Continuous time
August 6, 2009
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Topics
1
Tools:
1 2
Solving models in continuous time (optimal control). Phase diagrams.
2 3
Solow model. Cass-Koopmans / neoclas
Review Questions: Innite Horizon Models in Continuous Time
Prof. Lutz Hendricks. August 7, 2009
1
1.1
Household Behavior
Wealth tax
[Romer 2.3] Suppose it is known in advance that the government will conscate some wealth at a future date t0 . Does consump
Problem Set 5: Growth Model in Continuous Time
Econ720. Fall 2009. Lutz Hendricks
1
Capital income tax
[Romer 2.9] Consider a Ramsey economy on its balanced growth path. At time 0 the government starts to tax capital income, so that the interest rate faci
Problem Set 5: Growth Model in Continuous Time
Econ720. Fall 2009. Lutz Hendricks
1
Capital income tax
[Romer 2.9] Consider a Ramsey economy on its balanced growth path. At time 0 the government starts to tax capital income, so that the interest rate faci
Comparative Dynamics
Prof. Lutz Hendricks
October 7, 2009
L. Hendricks ()
Comparative Dynamics
October 7, 2009
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Comparative Dynamics
We use phase diagrams to uncover the dynamic response to shocks. We study tax changes in a growth model.
L. Hendrick
Aggregation
Prof. Lutz Hendricks
August 7, 2009
L. Hendricks ()
Aggregation
August 7, 2009
1 / 20
Notes on Aggregation
We have assumed a representative household. How restrictive is this assumption? If households are not identical, do they "aggregate" int
Econ 720 - Macro I - Fall 2008
Prof. Neville Francis
T.A. Fernando Chague
Recitation #3
A Dynamic Optimization Problem as an Optimal Control Problem
We want to solve:
max
RT
0
F (t; y; u) dt subject to y = f (t; y; u) and y (0) = y0
_
This problem can be
Econ 720 - Macro I - Fall 2008
Prof. Neville Francis
T.A. Fernando Chague
Homework 1 - Solutions
1. Solve the following innite-time horizon consumption model for the optimal consumption path. Assume that
r < 0.
max
Z1
e
t
c
ac2 dt
0
s.t. x = rx
_
c
x0 giv
Problem Set 6: Innovation and Growth
Econ720. Fall 2015. Prof. Lutz Hendricks
1
Stochastic patent duration
[Due to Matt Doyle] Consider a version of the Expanding Variety of Goods model in which innovators monopoly
power diminishes over time. Otherwise th
Problem Set 1: OLG Models
Econ720. Fall 2015. Prof. Lutz Hendricks. August 11, 2015
1
OLG with Arrow-Debreu
Consider the standard OLG production economy with Arrow-Debreu trading.
Demographics: Nt young are born in t. Each lives for 2 periods.
Endowments:
Problem Set 2: OLG Models with Money
Econ720. Fall 2015. Prof. Lutz Hendricks. August 11, 2015
1
Money and Heterogeneity
Consider a two-period OLG model with at money.
Demographics: In each period Nt = (1 + n)t persons are born. Each lives for 2 periods.
Problem Set 5: Growth Model in Continuous Time
Econ720. Fall 2015. Prof. Lutz Hendricks
1
Relative Wealth Preferences
Consider the following version of the growth model in continuous time.
Notation: c: consumption, k: capital, k: average capital in the ec
Problem Set 4: Cash-in-Advance Model
Econ720. Fall 2015. Prof. Lutz Hendricks
1
Shopping time
Demographics:
Preferences:
There is a single representative household who lives forever.
The household values consumption (c) and leisure (l) according to
1
X
t
Problem Set 3: Ramsey Model in Discrete Time
Econ720. Fall 2014. Prof. Lutz Hendricks. August 18, 2014
1
Land Prices with Capital Accumulation
Consider the following economy with land and capital.
Demographics: There is a representative household of unit
Problem Set 7: Asset Pricing
Econ720. Fall 2015. Prof. Lutz Hendricks. October 29, 2015
1
Lucas Fruit Trees With Crashes
Demographics: There is a single, representative household who lives forever.
P
Preferences: U = E 1 t u (ct ) where u (c) = c1 / (1
).
Endogenous Growth:
AK Model
Prof. Lutz Hendricks
Econ720
October 21, 2015
1 / 35
Endogenous Growth
Why do countries grow?
A question with large welfare consequences.
We need models where growth is endogenous.
The simplest model is a variation of the Ramse
Econ 720 - Macro I - Fall 2008
Prof. Neville Francis
T.A. Fernando Chague
Recitation 11: Log-Linearization - Solution
1. Consider the CRRA utility function
c1 a
t
1
and assume a real interest rate rt and a discount rate . Recall the consumption Euler equa
Econ 720 - Macro I - Fall 2008
Prof. Neville Francis
T.A. Fernando Chague
Recitation 10: Money in Utility Function Model
1. Consider the following MIU (Sidrauski (1967) model. The representative household solves the dynamic
programming problem
V (! t ) =
Econ 720 - Macro I - Fall 2008
Prof. Neville Francis
T.A. Fernando Chague
Recitation 3: Dynamic Optimization in Continuous Time
Solutions
1. Denote the following maximization problem:
Z
Max V =
1
u2 dt
0
s.t. y = y + u
_
(a) Write down the Hamiltonian and
Econ 720 - Macro I - Fall 2008
Prof. Neville Francis
T.A. Fernando Chague
Recitation 2: IS-LM Model
Solutions
Consider the following IS-LM model:
C = a + bY
(Consumption)
lR
I=I
(Investment)
G=G
(Government)
L = kY
(Money Demand)
hR
(Money Supply)
M =M
Th