Solutions to Problem Set 10
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
For Extra Practice - Not Collected or Graded
1. Linear-Quadratic Dynamic Programming
The problem is to choose sequences cfw_zt and
Solutions to Problem Set 4
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 8
The two welfare theorems of economics tell us that optimal and equilibrium resource allocations coincide but o
Problem Set 3
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 1
Many famous results from microeconomic theory are now understood to be special cases
of the envelope theorem. This problem
Problem Set 1
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Thursday, September 12
1. Prot Maximization
Consider a rm that produces output y with capital k and labor l according to the technology
descri
Problem Set 4
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 8
The two welfare theorems of economics tell us that optimal and equilibrium resource allocations coincide but only under cer
Solutions to Final Exam
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, December 17 at 12 noon
1. Dynamic Pricing by an Industry Leader
The industry leader chooses continuously dierentiable funct
Solutions to Midterm Exam
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Thursday, October 31, 1:30 - 2:45pm
1. The Kuhn-Tucker Theorem
The problem is nding values for two variables, x and y, to maximize the
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
October 12, 2011
Problem Set 3 Solutions
4. Hotellings Lemma
(a) The Lagrangian for the rms problem is
L(y, k, l, ) = py rk wk + [f (k, l) y].
(b) According to the Kuhn-Tucke
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
December 9, 2011
Problem Set 9 Solutions
1. Stochastic Linear-Quadratic Dynamic Programming
(a) Using the guess that the value function for this problem depends only on yt an
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
December 3, 2011
Problem Set 8 Solutions
1. Human Capital Accumulation and Economic Growth
(a) The Bellman equation for the problem can be written as
v(kt , ht ; t) = max ln(
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
November 11, 2011
Problem Set 6 Solutions
1. Investment with Adjustment Costs
(a) The current-value Hamiltonian for this problem can be dened as
H(K, I, ) = K I I 2 + (I K).
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
October 13, 2011
Midterm Exam - Solutions
1. Quasilinear Preferences
(a) There are a number of ways to dene the Lagrangian for this problem, depending
on how one chooses to t
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
November 22, 2011
Problem Set 7 Solutions
1. Linear-Quadratic Dynamic Programming
(a) The Bellman equation for this problem is
2
2
v(yt ; t) = max Ryt + Qzt + v(Ayt + Bzt ; t
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
December 13, 2011
Final Exam - Solutions
1. Job Search
(a) The agents utility function has no explicit dependence on time, except through
the exponential discount factor, and
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
October 24, 2011
Problem Set 4 Solutions
1. Optimal Lending
The lenders dynamic optimization problem can be stated formally as:
max ln(cL ) + ln(cL ) subject to cL + sL 1 and
EC720 - Problem Set 1 Solutions
Samson Alva
Department of Economics, Boston College
September 22, 2011
Exercise 2.1
1a. See notes.
1b. () Let x f 1 (B B ). Then there exists y f (x) B B , since x has a
unique image. Then, y B and so x f 1 (B) but y B and
EC 720 - Math for Economists
Samson Alva
Department of Economics, Boston College
November 8, 2011
Problem Set 5 Solutions
1. Life Cycle Consumption and Saving in Discrete Time
(a) The Hamiltonian is
H(t, ct , kt+1 , t+1 )
1
t ct
1
+ t+1 (wt + rt kt ct ).
Problem Set 2
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, September 24
1. Utility Maximization - Second-Order Conditions
The following result specializes Theorem 19.8 from Simon and Blumes bo
Final Exam
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, December 17 at 12 noon
This exam has two questions on ve pages; please check to make sure that your copy has
both questions and all ve p
Solutions to Problem Set 5
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 15
1. Optimal Lending
The lenders dynamic optimization problem can be stated formally as:
max ln(cL ) + ln(cL )
Solutions to Problem Set 3
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 1
Many famous results from microeconomic theory are now understood to be special cases
of the envelope theorem.
Solutions to Problem Set 2
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, September 24
1. Utility Maximization - Second-Order Conditions
The following result specializes Theorem 19.8 from Simon
Solutions to Problem Set 1
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Thursday, September 12
1. Prot Maximization
Consider a rm that produces output y with capital k and labor l according to the tech
Solutions to Problem Set 6
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 22
1. The Permanent Income Hypothesis
The consumer chooses c0 , c1 , and s to maximize the utility function
ln(c
Solutions to Problem Set 7
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, October 29
Consider an economy populated by a large number of identical consumers, each of whom
takes s0 as given, and c
Solutions to Problem Set 11
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, December 3
Human Capital Accumulation and Economic Growth
In this version of the Uzawa-Lucas model, the representative
Solutions to Problem Set 9
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, November 19
1. Natural Resource Depletion
A social planner chooses continuously dierentiable functions c(t) and s(t) for
Solutions to Problem Set 8
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Due Tuesday, November 12
1. Life Cycle Saving
+1
The consumer chooses sequences cfw_ct T=0 and cfw_kt T=1 to maximize
t
t
T
t
t=0
1
c
Solutions to Problem Set 12
EC720.01 - Math for Economists
Boston College, Department of Economics
Peter Ireland
Fall 2013
Practice for Final - Not Collected or Graded
1. Stochastic Linear-Quadratic Dynamic Programming
The problem is to choose contingency
c) We need to introduce constraints that describe the evolution of stock variables over time:
e.g., larger ows of savings or investment today will lead to larger stocks of wealth or
capital tomorrow.
2
2.1
The Maximum Principle: Discrete Time
A Dynamic Op