L. Karp Notes for Dynamics
VI. Two Stochastic Control Problems 1) Stochastic Control of Jump Process (i) Derivation of DPE (ii) Application to pollution control problem in section 4 2) Finite Markov chanins i) Discuss range management application. i

L. Karp Notes for Dynamics
VIII. "Nonconvex" Control Problems 1) 2) 3) 4) 5) 6) 6) Describe growth model that leads to non-convex control problem Sketch phase portrait of solutions to FOCs Identify optimal candidate. Economic interpretation. A pollu

Environmental & Resource Economics (2005) 32: 445493 DOI 10.1007/s10640-005-4681-y
Springer 2005
Declining Discount Rates: The Long and the Short of it
BEN GROOM1; , CAMERON HEPBURN2 , PHOEBE KOUNDOURI3 , and DAVID PEARCE4
Department of Economics,

Climate Policy when the Distant Future Matters: Catastrophic events with hyperbolic discounting
Larry Karp Yacov Tsur
November 10, 2006
Abstract We study climate change policy with hyperbolic discounting under the risk of catastrophic events. Usin

Larry Karp Notes for Dynamics September 2001
III. The Maximum Principle
1) Necessary conditions using Maximum Principle. 2) Relation between COV and Maximum Principle approaches. 3) Various terminal conditions, sufficiency. 4) An example with two s

July 28, 2000
Taxes and Quotas for a Stock Pollutant with Multiplicative Uncertainty
Michael Hoel and Larry Karp
Abstract We compare taxes and quotas when firms and the regulator have asymmetric information about the slope of firms' abatement cost

L. Karp Notes.1 for Dynamics, September 10, 2001
I. Basic Ideas of ODE's 1) Basic terms of ordinary differential equations (ODE's). 2) Basics of phase plane analysis. 3) Solutions and stability of linear ODE's. 4) Linear approximations of nonlinear

Numerical Analysis of Non-constant Discounting with an Application to Renewable Resource Management
Tomoki Fujii Larry Karp
May 31, 2006
Abstract The possibility of non-constant discounting is important in environmental and resource management prob

Estimating Intertemporal Preferences for Natural Resource Allocation
Richard E. Howitta, Siwa Msangia, Arnaud Reynaudb, and Keith C. Knappc Abstract
In this paper we show how the degree of risk aversion, discounting, and preference for intertemporal

Solutions to Problem Set 8
ARE 261 December 15, 2004
Question 1
1. The dynamic programming equation (DPE) for the control problem is 1 -rt 2 2 -Jt (xt , t) = max - e (ut + xt ) + Jx (xt , t)(xt + ut ) (1) ut 2 where J(xt , t) is the value function

Problem set 9 (Inspired by a paper by Craig Bond and Hossein Farzin) Consider the following linear-quadratic control problem: N is a n-dimensional vector of stocks of nutrients and F is the flow of fertilizer application. The flow of nutrients availa

Solutions to Problem Set 2
ARE 261 September 15, 2000
The following two .les contain MATLAB code to solve the problem set: (i) predator_prey2.m; and (ii) prey2.m. Ftp these .les from the are ftp site /pub/classes/are261/.
1

Problem Set 4
ARE 261 September 30, 2002
Question 1
Consider the following optimization problem: max S(y, u) = where y0 is given and where yi+1 - yi = ui x i = 0, ., n - 1 (2)
n-1 X i=0
F (xi , yi , ui )x
(1)
Note that x can be interpreted as the

Solutions to Problem Set 5
ARE 261 November 4, 2001
Question 1
The Euler equation, using Calculus of Variations, is x=0 (1)
If we integrate this equation twice and use the boundary conditions we get the following solution x(t) = t + 2 (2) The nece

Solutions to Problem Set 4
ARE 261 September 30, 2002
Question 1
The control is ui , the state is yi and the co-state variable is i . The first order conditions with respect to each of these variables are given by F L = + i x = 0, i = 0, 1, .n - 1

Exam ARE 263 Fall 2004 Answer all three questions. Each question is worth the same number of points. (If you find yourself drowning in algebra and running out of time, step back and explain in words how you would proceed if there was no time pressure

Larry Karp Notes for Dynamics September 19, 2001 II. General Dynamic Problem in Resources, Calculus of Variations
1) Description of dynamic optimization problem 2) Statement of necessary condition (Euler equation) 3) Interpretation of Euler Equation

L. Karp Notes for Dynamics
V. Dynamic Programming 1) 2) 3) 4) 5) 6) 7) 8) The basic idea of dynamic programming (discrete time). The linear quadratic (LQ) discrete time control problem with additive errors. Two problems related to the LQ problem. De

Final exam, AREP 263 Fall 2006 You have three hours to work on this exam - closed book, closed notes. Please return the exam to me or to my mailbox in 207 Giannini by 5 PM on Friday. Answer all questions. You cannot avoid using mathematics to answer

Why We Should Be Willing to Devote More Resources to Avoid Climate Change
Larry Karp
An important body of empirical models recommends modest efforts to reduce greenhouse gases. Although ostensibly scientific, these conclusions are actually largely dr

Final Exam ARE 261 (2001) Answer all parts of all three questions. The value of the question is given in parentheses next to the question number. Question 1) (45 points) S is an index of environmental quality and x is the flow of pollution. The evol

Partial solution to Problem set 9 Answer to part a) y = F + N
The DPE for the autonomous problem is: b 2 0 rJ(N) = max ay - y + J (N) (AN + BF ) F 2
I am going to "guess" that the value function is quadratic in the state: N2 2
J = + N +
Subst

L. Karp Notes for Dynamics
VII. Limit Cycles in Intertemporal Adjustment Models 1. Describe basic models of convex adjustment, with one state variable. 2. Review basic fish problem with no costs of adjusting control. 3. Add adjustment costs to basic

L Karp Course Outline ARE 263, Fall 2006 Methods of Dynamic Analysis and Control
I am going to wait to see who is enrolled in this class before finalizing the syllabus. You should treat this document as a provisional syllabus. It will at least give

Reading List Are 261, last two lectures I will discuss two types of dynamic games. In the first type ("Nash"), strategic agents make decisions simultaneously in each period (each instant of time). For example, agents extract a common property resourc

Exam ARE 263 Fall 2003 Answer both questions 1 and 2 and either question 3 or 4. Each question is worth the same number of points.
1) When an agent harvests ht fish in a period and the current stock is St , the stock the next period is St+1 = f (St

Problem Set 7 The purpose of this problem set is to be sure that you can solve a linearquadratic control problem, and to be sure that you know what the Principle of Certainty Equivalence says, and when it holds. 1. (Hoel and Karp 2001, Taxes versus Q