Econ 714: Macroeconomic Theory II1
Assignment 5: Answer Key2
1
Capacity choice
Here is a problem facing a monopolist. He faces a demand curve each period given by q = 1 p; that is, if the
price is p, he can sell the quantity q . Production is costless, bu
Lecture 6: The Equity Premium Puzzle
Economics 714, Spring 2016
1
Pricing General Claims
Price of any contingent claim g(s0 ) one-period ahead:
Z
g
p (s) =
0
u0 (s0 ) 0
g(s )Q(s, ds0 )
u0 (s)
0
Component m = uu0(s(s) is called stochastic discount factor
F
Lecture 16: Incomplete Markets Models
Limited Commitment
Economics 714, Spring 2014
1
Stationary Distributions
1.1
Background
We want to establish under what conditions a stationary distribution of assets exists
For general Markov transition P , measure o
Lecture 12: Asset Pricing and the Equity Premium
Economics 714, Spring 2014
1
Asset Pricing
1.1
Multi-Period Claims
For multi-period claims, can chain together one-step claims:
q j (s, sj ) =
q(s, s )q j1 (s , sj )ds
Ownership of tree is claim to entire c
Lecture 15: Incomplete Markets Models
Stationary Distributions
Economics 714, Spring 2014
1
Incomplete Markets Models
1.1
Consumption-Savings Model
Put consumption-savings model with borrowing constraint into general equilibrium
Only asset is risk free bo
Lecture 14: The Stochastic Growth Model and Real
Business Cycles
Economics 714, Spring 2014
1
Review: Optimal Growth
Planners problem: Neoclassical, Ramsey-Cass-Koopmans optimal growth model
t u(Ct )
max
cfw_Ct ,Kt+1
t=0
subject to:
Ct + Kt+1 (1 )Kt = f
Lecture 19: Asymmetric Information
Economics 714, Spring 2014
1
Decentralizing Allocations
Alvarez and Jermann (2000) show how to decentralize ecient allocations in more general setting with limited commitment as a competetive equilibrium with (endogenous
Lecture 17: Incomplete Markets Models
Limited Commitment
Economics 714, Spring 2014
1
Krusell-Smith (1998): Incomplete Markets with
Aggregate Risk
Aggregate production function now has Markov productivity shock zt Qz :
Yt = zt F (Kt , Nt )
Gives usual mar
Noah Williams
Department of Economics
University of Wisconsin
Economics 714
Macroeconomic Theory
Spring 2014
Midterm Examination
Instructions: This is a 75 minute exam with three questions worth a total of 100 points.
Allocate your time wisely. In order t
Lecture 21
Optimal Unemployment Insurance
Optimal Taxation with Private Information
Noah Williams
University of Wisconsin - Madison
Economics 714
Williams
Economics 714
Optimal Unemployment Insurance
Preferences over consumption c and job search eort a:
More on the New Keynesian Model
Noah Williams
University of Wisconsin-Madison
Noah Williams (UW Madison)
New Keynesian model
1 / 49
Households
The households decision problem can be dealt with in two stages.
1
2
Regardless of the level of Ct , it will alw
Welfare
Dene welfare as the average utility:
W = M V1 + (1 M) V0
Then:
rW = (1 M ) [(1 M) y + M ] (u c)
Note that welfare is increasing in .
16
Welfare = 1
Note:
rW = ( 1 M) [ ( 1 M)y + M] ( uc)
2
Maximize W with respect to M :
1
1 2y
if y <
2 2y
2
Lecture 20
More on Asymmetric Information:
More on Thomas and Worrall (1990)
Optimal Unemployment Insurance
Noah Williams
University of Wisconsin - Madison
Economics 714
Williams
Economics 714
Claim: The household and moneylender prefer high endowments.
Lecture 11: Asset Pricing
Economics 714, Spring 2014
1
General Equilibrium In an Endowment Economy
1.1
Lucas (1978) Asset Pricing Model
Large number of identical agents, single nonstorable consumption good (fruit), given o
by productive units (trees) with
Lecture 8: More on Dynamic Programming
Economics 714, Spring 2014
1
Dynamic Programming
1.1
Constructing Solutions to the Bellman Equation
Theorem:
Under (1), (F1), T : C(X) C(X) is a contraction, and hence has a
unique xed point v C(X) and for all v0 C(X
Lecture 5: Lucas Model and Asset Pricing
Economics 714, Spring 2017
1
Asset Pricing
1.1
Lucas (1978) Asset Pricing Model
Large number of identical agents, single nonstorable consumption good (fruit), given off
by productive units (trees) with net supply o
Lecture 5: Lucas Model and Asset Pricing
Economics 714, Spring 2016
1
GE: Arrow-Debreu Complete Markets Model
Each household solves single optimization problem. Representative first order condition:
(i )t uic (cit (st )P (st |s0 ) = i qt0 (st )
Cross sect
Lecture 6: Asset Pricing and the Equity Premium
Puzzle
Economics 714, Spring 2017
1
Pricing Long-lived Assets
For multi-period claims, can chain together one-step claims:
j
j
Z
q (s, s ) =
q(s, s0 )q j1 (s0 , sj )ds0
Ownership of tree is claim to entire c
Lecture 2: Search Equilibrium
Economics 714, Spring 2017
1
Equilibrium Search Model
Pissarides (1985) model, later modified by Mortensen-Pissarides (1994)
Continuous time, constant interest rate r.
Continuum L of identical workers, risk neutral preference
Lecture 2: Expectations, Search Model
Economics 714, Spring 2014
1
Expectations
1.1
Indeterminacy
What if |a| > 1? Cant solve forward
Say mt 1
1
Et pt+1 = pt c
a
so
1
pt+1 = pt c + et+1
a
Solve backward:
pt+1
1.2
c
=
+
1a
1
a
t+1
t+1
p0 +
j=0
1
a
Multivar
Lecture 1: The Role of Expectations
Economics 714, Spring 2014
1
The Muth Model
Demand:
xd = pt
t
Supply:
xs = + pe + ut
t
t
Equilibrium:
pt =
e 1
pt ut
What determines pe ?
t
1.1
Adaptive expectations
pe = pe + (pt1 pe )
t
t1
t1
Solve backward:
pe =
t
Lecture 3: Labor Market Search
Economics 714, Spring 2014
1
Search Labor Model
1.1
McCall (1970) Model
Risk-neutral agent searches for job:
t xt
E0
t=0
xt = w if employed, xt = z if unemployed
Job oers i.i.d. draw from F (w).
Recursive formulation: state
Lecture 6: Consumption-Savings Problem,
Finite Horizon
Economics 714, Spring 2014
1
Basic Consumption-Savings Problem
1.1
Setup
Agent preferences:
T
t u(ct )
t=0
Where u is C 2 , u > 0, u < 0.
Sometimes well also assume u is bounded, satises Inada: limc0
Lecture 7: Dynamic Programming
Economics 714, Spring 2014
1
Finite Horizon Consumption-Savings
1.1
Permanent Income Theory
Suppose R = 1, then ct = ct. Then intertemporal constraint gives:
c=
T
t=0
1 t
yt +
R
T
1 t
t=0 R
x0
Even with large variation in cf
Lecture 10: More on the Stochastic
Consumption-Savings Problem,
General Equilibrium
Economics 714, Spring 2014
1
Consumption-Savings Problem
1.1
Euler Equations
If the constraints dont bind, we can proceed as before:
v(a, y) =
max
u(c) +
v(a , y )Q(y, dy
Lecture 9: Consumption-Savings Problem
Under Uncertainty
Economics 714, Spring 2014
1
Consumption Savings-Problem: Innite Horizon
1.1
Basic Problem
Write choice variable as savings s = x c + y:
V (x) = max cfw_u(x + y s) + V (Rs)
s
First-order condition,
Lecture 5: Equilibrium Search
Economics 714, Spring 2013
1
Pissarides (1985) Equilibrium Search Model
1.1
Wage Determination
Solve for wages by Nash bargaining solution:
w = arg max (W (w) U ) (J(w) V )1
w
Optimality condition:
W (w)
J (w)
= (1 )
W U
J V
Lecture 4: Consumption-Savings Problem
Under Uncertainty
Economics 714, Spring 2016
1
Consumption-Savings Problem under Uncertainty
1.1
Basic Problem
X
max E0
cfw_ct ,at+1
s.t.
t u(ct )
t=0
ct + at+1 = Rat + yt
a0 , y0 given.
Constraints: ct 0, at a. De
Lecture 1
Labor Market from a Flow Perspective
Noah Williams
University of Wisconsin - Madison
Economics 714
Williams
Economics 714
Labor Market Flows and Search
We will introduce dynamic models of the labor market
with equilibrium unemployment
Labor mark
Lecture 4: Consumption-Savings Problem
Under Uncertainty
Economics 714, Spring 2017
1
Consumption-Savings Problem under Uncertainty
1.1
Basic Problem
X
max E0
cfw_ct ,at+1
s.t.
t u(ct )
t=0
ct + at+1 = Rat + yt
a0 , y0 given.
Utility function: u0 > 0, u
Lecture 2: Search Equilibrium
Economics 714, Spring 2016
1
Search Labor Model
1.1
Adding Separations and Imperfect Job Finding
Now employed worker value:
W (w) = w + [sU + (1 s)W (w)]
w + sU
1 (1 s)
=
Unemployed worker finds job with probability p:
Z
max
Lecture 3: Search Equilibrium
Economics 714, Spring 2016
1
More on Pissarides Model
1.1
Firm Decision
Wage w, hours fixed at 1, either party can freely break contract.
J value of filled job, V value of vacant job.
rV
= pc + q()(J V )
rJ = p w sJ
So we hav