Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims

Size: px
Start display at page:

Download "Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims"

Transcription

1 Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable, please prin ou figures and codes. This problem se is due in class on Thursday, March 30, (1) Preference Shocks and Variable Capial Uilizaion: Suppose we have a model wih variable capial uilizaion and a ime-varying disuiliy of labor. A represenaive household owns he capial sock, K, and makes capial accumulaion decisions. This agen also ges o pick how inensively o uilize capial, u, leasing capial services (he produc of uilizaion and physical capial), K u K, o a represenaive firm on a period-by-period basis. The cos of more inensive uilizaion of capial is faser depreciaion. The problem of he household can be wrien as follows: { } E 0 β ln C ν θ N 1+χ C,K +1,B +1,N,u =0 s.. C + K +1 (1 δ(u ))K + B +1 w N + R u K + Π + (1 + r 1 )B δ(u ) = δ 0 + φ 1 (u 1) + φ 2 2 (u 1) 2 ν is a ime-varying shock o preferences over labor, obeying a saionary AR(1) process ha is mean zero in he log: ln ν = ρ ν ln ν 1 + ε ν, (a) Derive he firs order condiions for he household problem. The firm produces oupu using labor and capial services, K u K, and akes facor prices, w and R, as given. Is problem is: N, K Π = A Kα N 1 α (b) Derive he firs order condiions for he firm problem. Produciviy obeys an AR(1) process in he log: w N R K ln A = ρ a ln A 1 + ε a, (c) Derive he aggregae resource consrain using he he definiion of profi and he bond marke-clearing condiion. (d) Provide a condiion on he parameer φ 1 o ensure ha seady sae capial uilizaion is 1. 1

2 (e) Derive a condiion giving he value of θ necessary o arge seady sae labor hours of N = 1/3. (f) Wrie a Dynare code o solve he model and compue impulse responses o boh produciviy and preference shocks using a firs order log-linear approximaion. Use he following parameer values: β = 0.99, δ = 0.02, α = 1/3, χ = 1, ρ a = 0.97, ρ ν = 0.95, s a = 0.01 (sandard deviaion of produciviy shock), and s ν = 0.02 (sandard deviaion of he preference shock). Use he value of θ consisen wih seady sae hours of 1/3 found in (e) above. Consider hree differen values of φ 2 : 100, 0.1, Graphically show impulse responses of oupu, hours, consumpion, invesmen, uilizaion, he real wage, and he real ineres rae o boh shocks. Commen on how he impulse responses o boh shocks vary wih he parameer φ 2, and provide some inuiion for your resuls. (g) Suppose ha you measure oal facor produciviy no aking ino accoun variable uilizaion. Tha is, le  = ln Y α ln K (1 α) ln N. Show impulse responses of measured TFP o boh shocks for he hree differen values of φ 2 from above. How does φ 2 affec he response of measured TFP o boh shocks? (h) How does he inclusion of he preference shock affec he relaive volailiy of HP filered hours (relaive o GDP) in he model? To see his, compue he relaive volailiy of hours o oupu wih boh shocks urned on and again wih he preference shock urned off (e.g. se he sandard deviaion of ha shock o zero). Use a value of φ 2 = 0.1 in doing his par. (2) GHH vs. Tradiional Preferences and he Effecs of Governmen Spending and Produciviy Shocks: Suppose you have a RBC model wih wo sochasic shocks: a shock o produciviy and a shock o governmen spending. We will consider wo differen preference specificaions: sandard separable preferences and GHH preferences. The household problem can be wrien: C,K +1,N,B +1 E 0 s.. β U(C, N ) C + K +1 (1 δ)k + B +1 w N + R K + Π T + (1 + r 1 )B (a) For he arbirary specificaion of household preferences, U(C, N ), find he firs order condiions for a soluion o he problem. The firm problem is he same in boh seups: N,K =0 Π = A K α N 1 α w N R K (b) Find he firs order condiions necessary for a soluion o he firm problem. (c) The governmen chooses is spending exogenously and balances is budge each period, G = T. Wha hen mus be rue abou bond-holding by households in equilibrium? Wrie down he aggregae resource consrain. Suppose ha preferences are given by he sandard separable form: U(C, N ) = ln C θ N 1+χ 2

3 (d) Suppose ha he non-sochasic seady sae value of A is A = 1, while he seady sae value of governmen spending is G = ωy, where Y is he seady sae value of oupu and 0 < ω < 1. Using hese preferences, derive an expression for he value of θ consisen wih seady sae hours of N = 1/3, and provide expressions for he seady sae values of Y, C, I, K, w, and R as a funcion of parameers. Suppose ha G and A follows saionary AR(1) processes in he log: ln A = ρ A ln A 1 + ε A, ln G = (1 ρ G ) ln G + ρ G ln G 1 + ε G, (e) Solve he model using a firs order log-linear approximaion in Dynare using he following parameer values: β = 0.99, δ = 0.02, α = 1/3, χ = 1, ρ A = 0.97, ρ G = 0.95, s A = 0.01 (sandard deviaion of produciviy shock), s G = 0.01 (sandard deviaion of governmen spending shock), ω = 0.20, and he value of θ consisen wih N = 1/3. Produce impulse response graphs of Y, C, I, N, w, and r o each shock over a 20 period horizon. Calculae he governmen spending muliplier, defined as he raio of he impac response of he level of oupu o he impac response of he level of governmen spending following a governmen spending shock. Now insead suppose ha preferences are given by he GHH variey: U(C, N ) = ln ( ) C θ N 1+χ (f) Using hese preferences, derive an expression for he value of θ consisen wih seady sae hours of N = 1/3, and provide expressions for he seady sae values of Y, C, I, K, w, and R as a funcion of parameers. (g) Solve he model using a firs order log-linear approximaion using he same parameer values as above. Produce impulse response graphs of Y, C, I, N, w, and r o each shock over a 20 period horizon. Calculae he governmen spending muliplier, defined as he raio of he impac response of he level of oupu o he impac response of he level of governmen spending following a governmen spending shock. (h) Compare and conras your impulse responses o he wo shocks wih he wo differen preference specificaions. Provide some inuiion for your findings. (3) News Shocks: This problem asks you o work ou a business cycle model where here are anicipaed shocks o produciviy, which he lieraure has called news shocks. In paricular, suppose ha he process for exogenous produciviy, A, obeys he following sochasic process: ln A = ρ ln A 1 + ε 4 This is he sandard AR(1) in he log, bu he shock is observed 4 periods in advance of when produciviy changes. In his way, a posiive (or negaive) realizaion of ε provides news abou he level of A in 4 periods. The model ha we will consider is a varian of Jaimovich and Rebelo (2009, American Economic Review): Can News Abou he Fuure Drive he Business Cycle? I is a real model (nohing nominal) wih hree main wiss relaive o a sandard RBC model: invesmen adjusmen coss, variable capial uilizaion, and GHH ype preferences. For now, le s assume a generic preference specificaion, where flow uiliy 3

4 is is U = U(C, N ) and is increasing in consumpion and decreasing in labor hours. A represenaive household owns he capial sock, makes invesmen decisions, and also makes uilizaion decisions. I leases capial services (he produc of uilizaion and he physical capial sock) o a represenaive firm. The household problem can be wrien: C,K +1,I,N,u E 0 s.. β U(C, N ) =0 C + I w N + R u K + Π [ K +1 = 1 κ ( ) ] 2 I 1 I + (1 δ(u ))K 2 I 1 δ(u ) = δ 0 + δ 1 (u 1) + δ 2 2 (u 1) 2 (a) Derive he firs order condiions for he household problem. I is bes o wrie a Lagrangian wih wo consrains; e.g. le λ be he muliplier on he flow budge consrain and µ he muliplier on he capial accumulaion equaion. The firm problem is sandard, where i picks capial services, K individually: = u K, no capial or uilizaion N,K Π = A Kα N 1 α (b) Derive he firs order condiions for he firm problem. w N R K (c) Wrie down he definiion of a compeiive equilibrium and derive he aggregae resource consrain. (d) Derive a resricion on he parameer δ 2 necessary o ensure ha seady sae uilizaion equals 1 (you do no need o know he specific form of preferences o do his). Suppose ha preferences are given by he sandard addiively separable form: U(C, N ) = ln C θ N 1+χ (e) Wrie a Dynare file o solve his model using a firs order log-linear approximaion. Use parameer values α = 1/3, β = 0.99, δ 0 = 0.025, χ = 1, he value of δ 1 you derived above, and a value of θ consisen wih seady sae labor hours of 1/3. Use values of ρ = 0.95 and he sandard deviaion of he shock of I is sraighforward o include he news shock in Dynare jus wrie e(-4) where you would normally wrie e in he shock process. For now, assume ha δ 2 = 1000 (effecively, no variable uilizaion) and κ = 0 (no invesmen adjusmen coss). Produce and prin ou impulse responses of oupu, consumpion, hours, and invesmen o he news shock. Wha happens o hese variables in he period beween he news his (on impac ) and when he shock ranslaes ino higher produciviy four periods laer? Can you provide some inuiion for his? (f) Re-do he exercise in (e), bu his ime se δ 2 = 0.01, effecively urning on variable uilizaion (bu keep κ = 0). Commen on how he impulse responses are differen, and ry o provide some inuiion. (g) Repea he exercise in (e), bu insead urn on invesmen adjusmen coss, seing κ = 3 (bu se δ 2 = 1000). Commen on how he impulse responses differ from he baseline, and ry o provide some 4

5 inuiion. (h) Now urn boh of hese feaures on simulaneously, seing δ 2 = 0.01 and κ = 3. Produce he impulse responses o he news shock. Knowing wha you know abou co-movemens among aggregae variables in he acual ime series daa, can news shocks be an imporan driving force of he business cycle in his model? Now insead consider he GHH preference specificaion, where: ( ) U(C, N ) = ln C θ N 1+χ (i) Use he same parameer values you did in par (e) (e.g. urn off variable uilizaion and he invesmen adjusmen cos), compue impulse responses o he news shock when he household has hese preferences. Use a value of θ consisen wih seady sae labor hours of 1/3 (noe, his value of θ will be differen han wha you found above). Commen on how he impulse responses look differen from wha you found in par (e), and ry o provide some inuiion. (j) Coninue o use GHH preferences, bu now urn on boh variable uilizaion and he invesmen adjusmen cos (δ 2 = 0.01 and κ = 3). Produce impulse response o he news shock, and compare hem wha you found for he base RBC model wih separable preferences, no uilizaion, and no adjusmen coss. Commen on he differences, and ry o provide some inuiion for how each of he hree differen changes relaive o he baseline (preferences, uilizaion, and adjusmen coss) help accoun for he differences. 5

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3 Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has

More information

Online Appendix to Solution Methods for Models with Rare Disasters

Online Appendix to Solution Methods for Models with Rare Disasters Online Appendix o Soluion Mehods for Models wih Rare Disasers Jesús Fernández-Villaverde and Oren Levinal In his Online Appendix, we presen he Euler condiions of he model, we develop he pricing Calvo block,

More information

( ) (, ) F K L = F, Y K N N N N. 8. Economic growth 8.1. Production function: Capital as production factor

( ) (, ) F K L = F, Y K N N N N. 8. Economic growth 8.1. Production function: Capital as production factor 8. Economic growh 8.. Producion funcion: Capial as producion facor Y = α N Y (, ) = F K N Diminishing marginal produciviy of capial and labor: (, ) F K L F K 2 ( K, L) K 2 (, ) F K L F L 2 ( K, L) L 2

More information

Cooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m.

Cooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m. Cooperaive Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS Augus 8, 213 8:45 a.m. o 1: p.m. THERE ARE FIVE QUESTIONS ANSWER ANY FOUR OUT OF FIVE PROBLEMS.

More information

1 Answers to Final Exam, ECN 200E, Spring

1 Answers to Final Exam, ECN 200E, Spring 1 Answers o Final Exam, ECN 200E, Spring 2004 1. A good answer would include he following elemens: The equiy premium puzzle demonsraed ha wih sandard (i.e ime separable and consan relaive risk aversion)

More information

Suggested Solutions to Assignment 4 (REQUIRED) Submisson Deadline and Location: March 27 in Class

Suggested Solutions to Assignment 4 (REQUIRED) Submisson Deadline and Location: March 27 in Class EC 450 Advanced Macroeconomics Insrucor: Sharif F Khan Deparmen of Economics Wilfrid Laurier Universiy Winer 2008 Suggesed Soluions o Assignmen 4 (REQUIRED) Submisson Deadline and Locaion: March 27 in

More information

Graduate Macro Theory II: Extensions of Basic RBC Framework

Graduate Macro Theory II: Extensions of Basic RBC Framework Graduae Macro Theory II: Exensions of Basic RBC Framework Eric Sims Universiy of Nore Dame Spring 25 Inroducion The basic RBC model which is jus a sochasic neoclassical growh model wih variable labor is

More information

Lecture Notes 3: Quantitative Analysis in DSGE Models: New Keynesian Model

Lecture Notes 3: Quantitative Analysis in DSGE Models: New Keynesian Model Lecure Noes 3: Quaniaive Analysis in DSGE Models: New Keynesian Model Zhiwei Xu, Email: xuzhiwei@sju.edu.cn The moneary policy plays lile role in he basic moneary model wihou price sickiness. We now urn

More information

Economics 8105 Macroeconomic Theory Recitation 6

Economics 8105 Macroeconomic Theory Recitation 6 Economics 8105 Macroeconomic Theory Reciaion 6 Conor Ryan Ocober 11h, 2016 Ouline: Opimal Taxaion wih Governmen Invesmen 1 Governmen Expendiure in Producion In hese noes we will examine a model in which

More information

The Brock-Mirman Stochastic Growth Model

The Brock-Mirman Stochastic Growth Model c December 3, 208, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social planner

More information

E β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.

E β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem. Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke

More information

Graduate Macroeconomics 2 Problem set 4. - Solutions

Graduate Macroeconomics 2 Problem set 4. - Solutions Graduae Macroeconomics Problem se. - Soluions In his problem, we calibrae he Roemberg and Woodford (995) model of imperfec compeiion. Since he model and is equilibrium condiions are discussed a lengh in

More information

BOKDSGE: A DSGE Model for the Korean Economy

BOKDSGE: A DSGE Model for the Korean Economy BOKDSGE: A DSGE Model for he Korean Economy June 4, 2008 Joong Shik Lee, Head Macroeconomeric Model Secion Research Deparmen The Bank of Korea Ouline 1. Background 2. Model srucure & parameer values 3.

More information

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u (

More information

Lecture 19. RBC and Sunspot Equilibria

Lecture 19. RBC and Sunspot Equilibria Lecure 9. RBC and Sunspo Equilibria In radiional RBC models, business cycles are propagaed by real echnological shocks. Thus he main sory comes from he supply side. In 994, a collecion of papers were published

More information

Explaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015

Explaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015 Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become

More information

Lecture Notes 5: Investment

Lecture Notes 5: Investment Lecure Noes 5: Invesmen Zhiwei Xu (xuzhiwei@sju.edu.cn) Invesmen decisions made by rms are one of he mos imporan behaviors in he economy. As he invesmen deermines how he capials accumulae along he ime,

More information

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given.

More information

T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 2011 EXAMINATION

T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 2011 EXAMINATION ECON 841 T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 211 EXAMINATION This exam has wo pars. Each par has wo quesions. Please answer one of he wo quesions in each par for a

More information

MONOPOLISTIC COMPETITION IN A DSGE MODEL: PART II OCTOBER 4, 2011 BUILDING THE EQUILIBRIUM. p = 1. Dixit-Stiglitz Model

MONOPOLISTIC COMPETITION IN A DSGE MODEL: PART II OCTOBER 4, 2011 BUILDING THE EQUILIBRIUM. p = 1. Dixit-Stiglitz Model MONOPOLISTIC COMPETITION IN A DSGE MODEL: PART II OCTOBER 4, 211 Dixi-Sigliz Model BUILDING THE EQUILIBRIUM DS MODEL I or II Puing hings ogeher impose symmery across all i 1 pzf k( k, n) = r & 1 pzf n(

More information

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, 65-70 ISSN: 2241-0384 (prin), 2241-0376 (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models

More information

Final Exam. Tuesday, December hours

Final Exam. Tuesday, December hours San Francisco Sae Universiy Michael Bar ECON 560 Fall 03 Final Exam Tuesday, December 7 hours Name: Insrucions. This is closed book, closed noes exam.. No calculaors of any kind are allowed. 3. Show all

More information

Final Exam Advanced Macroeconomics I

Final Exam Advanced Macroeconomics I Advanced Macroeconomics I WS 00/ Final Exam Advanced Macroeconomics I February 8, 0 Quesion (5%) An economy produces oupu according o α α Y = K (AL) of which a fracion s is invesed. echnology A is exogenous

More information

Different assumptions in the literature: Wages/prices set one period in advance and last for one period

Different assumptions in the literature: Wages/prices set one period in advance and last for one period Øisein Røisland, 5.3.7 Lecure 8: Moneary policy in New Keynesian models: Inroducing nominal rigidiies Differen assumpions in he lieraure: Wages/prices se one period in advance and las for one period Saggering

More information

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate. Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since

More information

Rational Bubbles in Non-Linear Business Cycle Models. Robert Kollmann Université Libre de Bruxelles & CEPR

Rational Bubbles in Non-Linear Business Cycle Models. Robert Kollmann Université Libre de Bruxelles & CEPR Raional Bubbles in Non-Linear Business Cycle Models Rober Kollmann Universié Libre de Bruxelles & CEPR April 9, 209 Main resul: non-linear DSGE models have more saionary equilibria han you hink! Blanchard

More information

Problem Set #3: AK models

Problem Set #3: AK models Universiy of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Se #3: AK models Jorge F. Chavez December 3, 2012 Problem 1 Consider a compeiive economy, in which he level of echnology, which is exernal

More information

A Dynamic Model of Economic Fluctuations

A Dynamic Model of Economic Fluctuations CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model

More information

A New-Keynesian Model

A New-Keynesian Model Deparmen of Economics Universiy of Minnesoa Macroeconomic Theory Varadarajan V. Chari Spring 215 A New-Keynesian Model Prepared by Keyvan Eslami A New-Keynesian Model You were inroduced o a monopolisic

More information

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon 3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of

More information

Problem Set on Differential Equations

Problem Set on Differential Equations Problem Se on Differenial Equaions 1. Solve he following differenial equaions (a) x () = e x (), x () = 3/ 4. (b) x () = e x (), x (1) =. (c) xe () = + (1 x ()) e, x () =.. (An asse marke model). Le p()

More information

Solutions Problem Set 3 Macro II (14.452)

Solutions Problem Set 3 Macro II (14.452) Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.

More information

Robert Kollmann. 6 September 2017

Robert Kollmann. 6 September 2017 Appendix: Supplemenary maerial for Tracable Likelihood-Based Esimaion of Non- Linear DSGE Models Economics Leers (available online 6 Sepember 207) hp://dx.doi.org/0.06/j.econle.207.08.027 Rober Kollmann

More information

13.3 Term structure models

13.3 Term structure models 13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)

More information

The general Solow model

The general Solow model The general Solow model Back o a closed economy In he basic Solow model: no growh in GDP per worker in seady sae This conradics he empirics for he Wesern world (sylized fac #5) In he general Solow model:

More information

Appendix to The Macroeconomics of Trend Inflation Journal of Economic Literature, September 2014

Appendix to The Macroeconomics of Trend Inflation Journal of Economic Literature, September 2014 Appendix o The Macroeconomics of Trend Inflaion Journal of Economic Lieraure, Sepember 204 Guido Ascari Universiy of Oxford and Universiy of Pavia Argia M. Sbordone Federal Reserve Bank of New York Sepember

More information

A User s Guide to Solving Real Business Cycle Models. by a single representative agent. It is assumed that both output and factor markets are

A User s Guide to Solving Real Business Cycle Models. by a single representative agent. It is assumed that both output and factor markets are page, Harley, Hoover, Salyer, RBC Models: A User s Guide A User s Guide o Solving Real Business Cycle Models The ypical real business cycle model is based upon an economy populaed by idenical infiniely-lived

More information

Optimal Monetary Policy with the Cost Channel: Appendix (not for publication)

Optimal Monetary Policy with the Cost Channel: Appendix (not for publication) Opimal Moneary Policy wih he Cos Channel: Appendix (no for publicaion) Federico Ravenna andcarlewalsh Nov 24 Derivaions for secion 2 The flexible-price equilibrium oupu (eq 9) When price are flexible,

More information

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures. HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2015 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Chapter 15 A Model with Periodic Wage Contracts

Chapter 15 A Model with Periodic Wage Contracts George Alogoskoufis, Dynamic Macroeconomics, 2016 Chaper 15 A Model wih Periodic Wage Conracs In his chaper we analyze an alernaive model of aggregae flucuaions, which is based on periodic nominal wage

More information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

Introduction to choice over time

Introduction to choice over time Microeconomic Theory -- Choice over ime Inroducion o choice over ime Individual choice Income and subsiuion effecs 7 Walrasian equilibrium ineres rae 9 pages John Riley Ocober 9, 08 Microeconomic Theory

More information

Graduate Macro Theory II: A New Keynesian Model with Price Stickiness

Graduate Macro Theory II: A New Keynesian Model with Price Stickiness Graduae Macro Theory II: A New Keynesian Model wih Price Sickiness Eric Sims Universiy of Nore Dame Spring 215 1 Inroducion This se of noes lays and ou and analyzes he canonical New Keynesian NK model.

More information

( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:

( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively: XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

LABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012

LABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012 LABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem ma 0 ( ) f Ξ v, n + 1 = 0 ( f y wn h g v ) Discoun facor beween ime 0

More information

COMPETITIVE GROWTH MODEL

COMPETITIVE GROWTH MODEL COMPETITIVE GROWTH MODEL I Assumpions We are going o now solve he compeiive version of he opimal growh moel. Alhough he allocaions are he same as in he social planning problem, i will be useful o compare

More information

= ( ) ) or a system of differential equations with continuous parametrization (T = R

= ( ) ) or a system of differential equations with continuous parametrization (T = R XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3. Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies

More information

Final Spring 2007

Final Spring 2007 .615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o

More information

Chapter 13 A New Keynesian Model with Periodic Wage Contracts

Chapter 13 A New Keynesian Model with Periodic Wage Contracts George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 13 A New Keynesian Model wih Periodic Wage Conracs The realizaion of he insabiliy of he original Phillips curve has gradually led o a paradigm

More information

Simulating models with heterogeneous agents

Simulating models with heterogeneous agents Simulaing models wih heerogeneous agens Wouer J. Den Haan London School of Economics c by Wouer J. Den Haan Individual agen Subjec o employmen shocks (ε i, {0, 1}) Incomplee markes only way o save is hrough

More information

Economics 6130 Cornell University Fall 2016 Macroeconomics, I - Part 2

Economics 6130 Cornell University Fall 2016 Macroeconomics, I - Part 2 Economics 6130 Cornell Universiy Fall 016 Macroeconomics, I - Par Problem Se # Soluions 1 Overlapping Generaions Consider he following OLG economy: -period lives. 1 commodiy per period, l = 1. Saionary

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check

More information

Macroeconomics Qualifying Examination

Macroeconomics Qualifying Examination Macroeconomics Qualifying Examinaion January 205 Deparmen of Economics UNC Chapel Hill Insrucions: This examinaion consiss of four quesions. Answer all quesions. If you believe a quesion is ambiguously

More information

Problem 1 / 25 Problem 2 / 20 Problem 3 / 10 Problem 4 / 15 Problem 5 / 30 TOTAL / 100

Problem 1 / 25 Problem 2 / 20 Problem 3 / 10 Problem 4 / 15 Problem 5 / 30 TOTAL / 100 eparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and Policy Miderm Exam Suggesed Soluions Professor Sanjay hugh Fall 2008 NAME: The Exam has a oal of five (5) problems

More information

Dynamics of Firms and Trade in General Equilibrium. Robert Dekle, Hyeok Jeong and Nobuhiro Kiyotaki USC, KDI School and Princeton

Dynamics of Firms and Trade in General Equilibrium. Robert Dekle, Hyeok Jeong and Nobuhiro Kiyotaki USC, KDI School and Princeton Dynamics of Firms and Trade in General Equilibrium Rober Dekle, Hyeok Jeong and Nobuhiro Kiyoaki USC, KDI School and Princeon real exchange rae.5 2 Figure. Aggregae Exchange Rae Disconnec in Japan 98 99

More information

Intermediate Macro In-Class Problems

Intermediate Macro In-Class Problems Inermediae Macro In-Class Problems Exploring Romer Model June 14, 016 Today we will explore he mechanisms of he simply Romer model by exploring how economies described by his model would reac o exogenous

More information

Lecture 3: Solow Model II Handout

Lecture 3: Solow Model II Handout Economics 202a, Fall 1998 Lecure 3: Solow Model II Handou Basics: Y = F(K,A ) da d d d dk d = ga = n = sy K The model soluion, for he general producion funcion y =ƒ(k ): dk d = sƒ(k ) (n + g + )k y* =

More information

Econ107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)

Econ107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8) I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression

More information

Problem 1 / 25 Problem 2 / 25 Problem 3 / 35 Problem 4 / 20 TOTAL / 100

Problem 1 / 25 Problem 2 / 25 Problem 3 / 35 Problem 4 / 20 TOTAL / 100 Deparmen of Applied Economics Johns Hopkins Universiy Economics 60 acroeconomic Theory and Policy Final Exam Suggesed Soluions Professor Sanjay Chugh Spring 009 ay 4, 009 NAE: The Exam has a oal of four

More information

This document was generated at 7:34 PM, 07/27/09 Copyright 2009 Richard T. Woodward

This document was generated at 7:34 PM, 07/27/09 Copyright 2009 Richard T. Woodward his documen was generaed a 7:34 PM, 07/27/09 Copyrigh 2009 Richard. Woodward 15. Bang-bang and mos rapid approach problems AGEC 637 - Summer 2009 here are some problems for which he opimal pah does no

More information

Graduate Macro Theory II: A New Keynesian Model with Both Price and Wage Stickiness

Graduate Macro Theory II: A New Keynesian Model with Both Price and Wage Stickiness Graduae Macro Theory II: A New Keynesian Model wih Boh Price and Wage Sickiness Eric Sims Universiy of Nore Dame Spring 27 Inroducion This se of noes augmens he basic NK model o include nominal wage rigidiy.

More information

Chapter 14 A Model of Imperfect Competition and Staggered Pricing

Chapter 14 A Model of Imperfect Competition and Staggered Pricing George Alogoskoufis, Dynamic Macroeconomic Theory, 205 Chaper 4 A Model of Imperfec Compeiion and Saggered Pricing In his chaper we presen he srucure of an alernaive new Keynesian model of aggregae flucuaions.

More information

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion

More information

The Brock-Mirman Stochastic Growth Model

The Brock-Mirman Stochastic Growth Model c November 20, 207, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social

More information

Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 1

Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 1 Macroeconomics I, UPF Professor Anonio Ciccone SOUTIONS PROBEM SET. (from Romer Advanced Macroeconomics Chaper ) Basic properies of growh raes which will be used over and over again. Use he fac ha he growh

More information

On the Desirability of Nominal GDP Targeting *

On the Desirability of Nominal GDP Targeting * On he Desirabiliy of Nominal GDP Targeing * Julio Garín Universiy of Georgia Rober Leser Colby College July 21, 2015 Eric Sims Universiy of Nore Dame & NBER Absrac This paper evaluaes he welfare properies

More information

1 Price Indexation and In ation Inertia

1 Price Indexation and In ation Inertia Lecures on Moneary Policy, In aion and he Business Cycle Moneary Policy Design: Exensions [0/05 Preliminary and Incomplee/Do No Circulae] Jordi Galí Price Indexaion and In aion Ineria. In aion Dynamics

More information

Unit Root Time Series. Univariate random walk

Unit Root Time Series. Univariate random walk Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he

More information

Essential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems

Essential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems Essenial Microeconomics -- 6.5: OPIMAL CONROL Consider he following class of opimizaion problems Max{ U( k, x) + U+ ( k+ ) k+ k F( k, x)}. { x, k+ } = In he language of conrol heory, he vecor k is he vecor

More information

Linear Dynamic Models

Linear Dynamic Models Linear Dnamic Models and Forecasing Reference aricle: Ineracions beween he muliplier analsis and he principle of acceleraion Ouline. The sae space ssem as an approach o working wih ssems of difference

More information

Let us start with a two dimensional case. We consider a vector ( x,

Let us start with a two dimensional case. We consider a vector ( x, Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our

More information

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070

More information

A Test of Identification for Government Spending Shocks.

A Test of Identification for Government Spending Shocks. A Tes of Idenificaion for Governmen Spending Shocks. Anna Kormilisina December 14, 2015 Absrac The response of consumpion o an increase in governmen spending in SVAR models may be influenced by he shock

More information

Financial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS

Financial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS Name SOLUTIONS Financial Economerics Jeffrey R. Russell Miderm Winer 009 SOLUTIONS You have 80 minues o complee he exam. Use can use a calculaor and noes. Try o fi all your work in he space provided. If

More information

This document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC

This document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,

More information

Midterm Exam. Macroeconomic Theory (ECON 8105) Larry Jones. Fall September 27th, Question 1: (55 points)

Midterm Exam. Macroeconomic Theory (ECON 8105) Larry Jones. Fall September 27th, Question 1: (55 points) Quesion 1: (55 poins) Macroeconomic Theory (ECON 8105) Larry Jones Fall 2016 Miderm Exam Sepember 27h, 2016 Consider an economy in which he represenaive consumer lives forever. There is a good in each

More information

Unemployment and Mismatch in the UK

Unemployment and Mismatch in the UK Unemploymen and Mismach in he UK Jennifer C. Smih Universiy of Warwick, UK CAGE (Cenre for Compeiive Advanage in he Global Economy) BoE/LSE Conference on Macroeconomics and Moneary Policy: Unemploymen,

More information

Lecture 5. Time series: ECM. Bernardina Algieri Department Economics, Statistics and Finance

Lecture 5. Time series: ECM. Bernardina Algieri Department Economics, Statistics and Finance Lecure 5 Time series: ECM Bernardina Algieri Deparmen Economics, Saisics and Finance Conens Time Series Modelling Coinegraion Error Correcion Model Two Seps, Engle-Granger procedure Error Correcion Model

More information

1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC

1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC This documen was generaed a :45 PM 8/8/04 Copyrigh 04 Richard T. Woodward. An inroducion o dynamic opimizaion -- Opimal Conrol and Dynamic Programming AGEC 637-04 I. Overview of opimizaion Opimizaion is

More information

Seminar 4: Hotelling 2

Seminar 4: Hotelling 2 Seminar 4: Hoelling 2 November 3, 211 1 Exercise Par 1 Iso-elasic demand A non renewable resource of a known sock S can be exraced a zero cos. Demand for he resource is of he form: D(p ) = p ε ε > A a

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON4325 Moneary Policy Dae of exam: Tuesday, May 24, 206 Grades are given: June 4, 206 Time for exam: 2.30 p.m. 5.30 p.m. The problem se covers 5 pages

More information

Dynamic firm profit-maximization problem. max ( ( )) Total output sold in perfectlycompetitive

Dynamic firm profit-maximization problem. max ( ( )) Total output sold in perfectlycompetitive LABOR SEARCH MODELS: BASIC DSGE IMPLEMENTATION NOVEMBER 2, 200 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem f f Ξ 0 y wn g v v, n + = 0 ma ( ( Discoun facor beween ime 0 and because

More information

Two Coupled Oscillators / Normal Modes

Two Coupled Oscillators / Normal Modes Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own

More information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

FINM 6900 Finance Theory

FINM 6900 Finance Theory FINM 6900 Finance Theory Universiy of Queensland Lecure Noe 4 The Lucas Model 1. Inroducion In his lecure we consider a simple endowmen economy in which an unspecified number of raional invesors rade asses

More information

Currency Misalignments and Optimal Monetary Policy: A Reexamination

Currency Misalignments and Optimal Monetary Policy: A Reexamination Appendix: No for Publicaion Currency Misalignmens and Opimal Moneary Policy: A eexaminaion Charles Engel Universiy of isconsin July 8, Appendix A Model Equaions Aa Households The represenaive household

More information

SOLUTIONS TO ECE 3084

SOLUTIONS TO ECE 3084 SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no

More information

Linear Response Theory: The connection between QFT and experiments

Linear Response Theory: The connection between QFT and experiments Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and

More information

On the Desirability of Nominal GDP Targeting

On the Desirability of Nominal GDP Targeting On he Desirabiliy of Nominal GDP Targeing Julio Garín Universiy of Georgia Rober Leser Colby College Eric Sims Universiy of Nore Dame & NBER This Version: May 4, 2016 Absrac This paper evaluaes he welfare

More information

15.023J / J / ESD.128J Global Climate Change: Economics, Science, and Policy Spring 2008

15.023J / J / ESD.128J Global Climate Change: Economics, Science, and Policy Spring 2008 MIT OpenCourseWare hp://ocw.mi.edu 15.023J / 12.848J / ESD.128J Global Climae Change: Economics, Science, and Policy Spring 2008 For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.

More information

The Goals of his Research To undersand financial crises wih a model of muliple seady sae equilibria To undersand he role of fiscal policy in resoring

The Goals of his Research To undersand financial crises wih a model of muliple seady sae equilibria To undersand he role of fiscal policy in resoring Fiscal Policy Can Reduce Unemploymen: Bu There is a Beer Alernaive Federal Reserve Bank of Alana January 9 h 2010 Roger E. A. Farmer Deparmen of Economics UCLA 1 The Goals of his Research To undersand

More information

Fiscal and Monetary Policy in a New Keynesian Model with Tobin s Q Investment Theory Features

Fiscal and Monetary Policy in a New Keynesian Model with Tobin s Q Investment Theory Features MPRA Munich Personal RePEc Archive Fiscal and Moneary Policy in a New Keynesian Model wih Tobin s Q Invesmen Theory Feaures Sylianos Giannoulakis Ahens Universiy of Economics and Business 4 May 2017 Online

More information

Problem set 3: Endogenous Innovation - Solutions

Problem set 3: Endogenous Innovation - Solutions Problem se 3: Endogenous Innovaion - Soluions Loïc Baé Ocober 25, 22 Opimaliy in he R & D based endogenous growh model Imporan feaure of his model: he monopoly markup is exogenous, so ha here is no need

More information

BU Macro BU Macro Fall 2008, Lecture 4

BU Macro BU Macro Fall 2008, Lecture 4 Dynamic Programming BU Macro 2008 Lecure 4 1 Ouline 1. Cerainy opimizaion problem used o illusrae: a. Resricions on exogenous variables b. Value funcion c. Policy funcion d. The Bellman equaion and an

More information

Problem Set 1 "Working with the Solow model"

Problem Set 1 Working with the Solow model Problem Se "Working wih he Solow model" Le's define he following exogenous variables: s δ n savings rae depreciaion rae of physical capial populaion growh rae L labor supply e n (Normalizing labor supply

More information

Fall 2015 Final Examination (200 pts)

Fall 2015 Final Examination (200 pts) Econ 501 Fall 2015 Final Examinaion (200 ps) S.L. Parene Neoclassical Growh Model (50 ps) 1. Derive he relaion beween he real ineres rae and he renal price of capial using a no-arbirage argumen under he

More information

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.

More information