# Solutions Problem Set 2 Macro II (14.452)

Save this PDF as:

Size: px
Start display at page:

Download "Solutions Problem Set 2 Macro II (14.452)"

## Transcription

1 Solutions Problem Set 2 Macro II (4.452) Francisco A. Gallego 4/22 We encourage you to work together, as long as you write your own solutions. Intertemporal Labor Supply Consider the following problem. The consumer problem is: Max fc tg;fn tg E s:t:! t=t X t f C t \$ + 2 N t \$2 g t=o A t+ = R(A t + N t W t C t ) Where C is consumption, N is labor supply, A is initial wealth, R = + r, and the greek letters are parameters. Only W is stochastic ). Derive and interpret the rst-order conditions for C t and N t. What are the intertemporal and intratemporal optimal conditions There are at least two (equivalent ways of setting up and solving this problem). I will implement both of them. Lagrangian method I: In this case the Lagrangian is: Max fc tg;fn tg;fa t+g \$ = E F.O.C.! t=t X t f C t \$ + 2 N t \$2 t (A t+ R(A t + N t W t C t )g t=o s.t. A fc t t =, \$ C \$ t = t R ()

2 @\$ fn t g : 2 \$ 2 N t \$2 = t W t R t The interpretation is more or less natural. The rst condition tells you that the marginal utility of C t has to be equal to the marginal utility of wealth (do not worry too much about R it is an implication of the fact that C in this case is realized at the beginning of the period t so the value of saving one unit of consumptions is times R). The second condition tells you that marginal disutility of work has to be equal to the marginal utility of wealth times how much consumption you can get from one our of work (the wage rate). Why no expectation? This is important, at time t, the only source of uncertainty is realized (W t ) and, roughly speaking, t is a function of expected future wages (see below). Combining both equations we get the intratemporal condition: 2 \$ 2 N t \$2 = W \$ C \$ t To get the intertemporal condition, we need t+ =, t t = t+ t+ R Using the FOC for consumption, we get our intertemporal condition: 2. Lagrangian method II: t \$ C \$ t = t+ RE( \$ C \$ = RE " Ct+ C t \$ # t+ ) Notice that you can iterate the constraint and write the intertemporal budget constraint. TX R t C t = A + t= TX R t N t W t Two comments: () Why no expectation? This has to hold exactly. (2) what about A T +? Max fc tg;fn tg \$ = E F.O.C. t=o t= t=t X t f C t \$ + 2 N t \$2 g ( TX T! X R t C t A R t N t W t ) t= t= 2

3 fc t t =, t \$ C \$ t = R t (2) fn t t =, t 2 \$ 2 N \$2 t = W R t (3) If you combine both equations, you get the same intratemporal condition as before. To get the intertemporal condition write the FOC for t + taking expectations at h fc t+ g : =, E t \$ C \$ t+ R t i = t+ Using the FOC for C t : h E t t+ \$ C \$ t+ t \$ C \$ t R t i = and you get the same as before: = RE " Ct+ C t \$ # What is more interesting about this method is that can be interpreted as the su cient statistic you need to solve the problem. Notice that () and (2) imply that: t R = (R) t How can you interpret this condition? We can plug () and (3) into the budget constraint and we will get a decreasing implicit function of as a function of fw i g with i t and, of course, some constant terms (among them R). Thus, the only thing you need to compute is to know something about fw i g. Therefore, you will keep your constant in the future if the realizations of fw i g were more or less what you expected in the past. 2). What is the link between C t and N t in this model? What assumption(s) is (are) producing this result The only link between both variables is (or t ). This implies the intratemporal condition and the negative correlation between both variables if W is kept constant. And, the lack of correlation if W increases without a ecting. Assumption: () both terms are additively separable in the utility function and (2) perfect capital markets (what may happen if we introduce frictions as A t > ) 3

4 3). How can you analyze changes to wages that do not a ect the expected wealth of the consumers? (Hint: what is the Lagrange multiplier in this model?). Let s de ne the wealth-constant ln Nt of labor supply as ln W t when wealth is constant. What is in this model? Is it positive or negative? Why? Given our previous discussion, these are changes that do not a ect, therefore we can take logs in (3) and get: Then log( 2 \$ 2 (R) t ) + (\$ 2 ) log(n t ) = log W t = log(n t) = log W t (\$ 2 ) Notice that >. There are several ways of understanding this. Here is one: First, notice that obviously > ; 2 < ; \$ <. Why? People like C, dislike N, and marginal utility of consumption is decreasing in general. Therefore if the instantaneous utility function is to be concave (and, therefore, the solution we just proposed has sense), it has to be true 2 2 = 2\$ 2 (\$ 2 ) N \$ 2 2 t < ) (\$ 2 ) > 4). Take the FOCs and discuss the e ect of the following situations on fc t g and fn t g :. Cross-sectional di erences associated with permanent differences in human capital (assume W varies in the crosssection) Higher human capital will probably imply higher W. Therefore, #, C ". What about N? Assume all other parameters do not vary in the cross-section and take two individuals i and j log( N j N i ) = (\$ 2 ) log W j W i + log j i So e ect is ambiguous. One of them is richer, but at the same time the alternative cost of leisure is higher. Life-cycle changes in wages (i.e. if Mincerian equations are correct, wages follow an inverted-u behavior over the lifecycle) Notice that these evolutionary changes were predictable at the beginning of the life-cycle and, therefore, as W changes, is constant. Thus C is also constant see (2). But given that > and is constant, N moves together with W. 4

5 Unexpected temporary shocks to wages If change is temporary, it shouldn t a ect by much, then C is constant, and N moves together with W. Unexpected permanent shocks to wages In this case should be a ected. Therefore, C moves in the same direction as W and the e ect on N is unclear. 2 Log-Linear RBC Consider the following problem. Consumers maximize 2 3 X max E t 4 b j U( C e t+j ; N t+j ) 5 j= Subject to ek t+ = R tkt e + W f t N t Ct e U( C e t ; N t ) = log ect + log( N t ) while rms solve nn t ; e K t o and we assume = arg max Z t F (A t N; K) e (R t + ) K e W f t N N; K e F (A t N; e K) = Y = Z e K A t+ = A t : (A t N) This is the standard RBC model. trending variables are those with a hat. ) De-trend the model De ne V = e V A.. Utility function: U(C t ; N t ) = log (C t A t )+ log( N t ) = log (A t ) {z } +log (C t )+ log( N t ) Does not matter 2. Consumers B.C.K t+ A t+ = R t K t A t + W t A t N t C t A t K t+ A t+ = R t K t A t + W t A t N t C t A t A t+, K t+ = R t K t + W t N t A {z t } C t, K t+ = R t K t + W t N t C t 5

6 3. Pro ts fn t ; K t g = arg max N;K Z t (K t A t ) (A t N) (R t + ) A t K t A t W t N t, fn t ; K t g = arg max N;K Z t (K t ) (N) (R t + ) K t W t N t 2)Solve for the F:O:C of the consumer problem, either using a Lagrangian or a Bellman Equation. You should nd two nal equations, the Euler Equation and the labor supply.. Bellman = V (K) = Max C;N flog (C) + log( N) + be [V (K )]g s:t: K = RK + W [V (K C + [V (K N [V (K E = (4) N + W [V (K E = (5) [V = = [V E (a) Euler Equation: Using (5) and (4), we get: Then, = R C C = b R E C = (6) (a) Labor supply (5) and (4) imply: N = C W (7) 6

7 2. Lagrangean: Max \$ = E fc tg;fn tg! t=t X b t flog (C) + log( N) t (K t+ (R t K t + W t N t C t ))g t=o FOC From (5) and (4) we get: Take additional FOC: fc t g : fc t g : C = t (8) N = tw t (9) N = C W fk t+ g : t t = E t+ t+ R t+ Then, combining (6) and (), we get our Euler equation: t = E t+ R t+, = t C t = E R t+ C t C t+ C t C t+ () () 3)Solve the rm problem to nd the labor and capital demands. Give the law of motion for capital (re-write K e t+ = R tkt e + W f t N t Ct e to get the usual equation) This should be easy., fn t ; K t g = arg max = arg max Z t (K t ) (N) (R t + ) K t W t N t N;K N;K FOC: fn t g t =, Z t Kt N t = W t fk t g t =, ( ) Z t Nt K t = (R t + ) 4) We have a total of 5 equations. Discuss how would you proceed to log-linearize our system (you don t need to do all the math, just apply the method to one or two equations). There is no unique way of log-linearize things. I will use here my way (which I think is simpler and does not use any math trick you didn t know). Let go one by one: 7

8 . Consumers BC: K t+ = R t K t + W t N t C t, K t+ = ( )K t + Y t C t It is also true that (where X denotes the steady state of X): Then: K = ( )K + Y C K t+ K = ( ) K t K + (Y t Y ) C t C K t+ K = ( ) K t K + (Y t Y ) Y C t C C K K Y K C K De ne x t = log (X t ) log X (Xt X) X, k t+ = ( ) Y K k t + y t C K c t (2) But, take the production function (I ll skip some steps):, y t = z t + n t + ( ) k t Then (2) becomes: k t+ + + ( ) Y k t K 2. The wage equation: C K c t + Y K n t + Y K z t = It is also true that Kt Z t N t = W t Then Z t Z Z Kt K N K N N t = W = W t W Take logs and use small letters to denote log deviations from the steady state and you get: z t + ( )(k t n t ) = w t 8

9 3. Labor supply Then, Take logs N t = C t W t N = C W N N t = CW t C t W log( N ) = w t c t (3) N t Let s de ne L = N and l = log(l) log(l): Then, the LHS can be written as: log( N ) = N t l N N N N nn = n N N Then, (3) becomes: n N N = w t c t So there is a mistake in the log-linearization that appears in the program. 4. Price of capital As with the wage equation: Nt ( ) Z t = (R t + ) K t Then (after some steps) Take logs: Z t Z Nt N K = (R t + ) R + K t z t +n t k t (R t + ) R + R + = R t R R + R R + r t 9

10 5. Euler equation b C t = E R t+! C t+ b C = C R Then, Ct C R t+ = E = E [( + c t ) ( C C t+ R c t+ ) ( + r t+ )] E [ + c t c t+ + r t+ ] 5) Our log-linearized system is the following (small variables denote logs): The law of motion and the market clearing conditions: k t+ + + ( ) Y C k t K K c t + Y K n t + Y K z t = (k t n t ) w t + z t = N N c t + n t N N w t = R k t + R + r t n t z t = The Euler Equation (or any other asset pricing equation): We assume E t [r t+ c t+ ] + c t = : z t = z t + " t : We want to write the system of six equations in order to solve it numerically using the process in Uhlig(997). Let s t be the vector of endogenous state variables (capital stock), x t = [c t ; r t ; n t ; w t ] the endogenous control variables and z t the stochastic processes. To avoid confusion, matrices are named using two capitalized letters. There are two conceptually di erent sets of equations. The rst set describes the law of motion of the economy, together with the market equilibrium conditions: I use M_ for these matrices (Motion and Market). In the second set, the equations are forward looking and they involve the expectations of the agents: I use F _ for these equations (Forward). Finally, we have the equation for the stochastic process z t. Write the system in the following way: MS s t+ + MS s t + MX x t + MZ z t = E t [F S 2 s t+2 + F S s t+ + F Ss t + F X x t+ + F Xx t + F Z z t+ + F Zz t ] = z t+ ZZ z t " t+ = :

11 In other words, what does each matrix contain? Just by inspecting the log-linearize system we get: MS = MX = ; MS = ( ) Y K ( ) 3 C K C K ( ) N N N N R R ; 7 5 ; MZ = Y K F S 2 = ;F S = ;F S = ;F X = ; F X = ; F Z = ;F Z = ) We want to nd numerically a solution of the type s t+ = b S s t + c SZ z t x t = b X s t + d XZ z t ; because we are looking at small deviations from steady state. You have to solve that using matlab. I give you a program, linear, containing the function linear which does the solution of the system, so basically what you have to do is create a program containing the following: a) Values of the parameters of the model. We will assume = :25; = :4, = 2 ; = ; = :979; 3 R = :63; Y=K = =2=4; K = ; C = Y 68=; N = :2; e = :72 Before solving the problem, notice that we need a few clari cations here: There are some conditions that the steady state values have to satisfy: ( ) Z N Y = ( ) K K Y ) = R + K ( ) = R + = 3 (:63 + :25) 8:7

12 Y So is not a parameter that you can change. K K = ( )K + Y C, C K = ( ) + Y K ) C Y = ( )K Y + 8:7( :25 :4) + :7629 So the value that is in the PS is inconsistent with the budget constraint. Sorry about that. Moreover, N = C W = C = Y N C Y N, N = + C Y :38 So, if we want to have N = :2, we need 6=! Fortunately, this does not a ect the simulations Olivier showed in class because it is the sum of two mistakes that cancel out each other. In any case, keep in mind that if you change N you have to change : This is completely intuitive because is a measure of how much you value leisure. Notice that if =, then N = : b) The matrices derived in 5). c)then you have to call the function linear. 7) Now we want to do impulse-response functions. For that, attach to your program (copy and paste after part c)) the program linear 2. This programs calls two other functions, impulse and impulse-plot. The rst nds the reduced form model and the second hits it with a shock. That should be the end of your program. Explain the results. How do they depend on the values assumed in a)? (run the program for di erent values of at least two parameters and compare results). Interpret. Several exercises can be implemented: Base case (Figure ) Minor di erences with the gure Olivier showed in class, same intuition. 2

13 Baseline case Change to.999 and.3 (Figures 2 and 3) = :999 3

14 = :9 Change N = :8 (Figure 4). N = :8 R = : (Figure 5) 4

15 R = : = : (Figure 6) = : 3 (Optional) Discrete Time Dynamic Program- 5

16 ming Use the discrete time setup of Philippon and Segura-Cayuela (23) (up to section 5) and the MATLAB programs contained in the folder DynamicP rogramming. Read tutorial:doc and the comments in DP:m. ) Derive the Euler equation in discrete time in the model with exogenous labor, rst using dynamic programming and then using the Lagrange multiplier method. Compare it to the continuous time Euler equation. The problem is: t= Max \$ = X fc tg t=o s:t: b t ek t+ = R t e Kt + f W t e Ct A t+ = A t ect Divide all the trending variables by A t :De ne V = e V A and notice that A t = A t Utility function: t= X MaxV = Max t=o b t = Max A t=t X = Max t=o t C ta t t= X t=o The budget constraint (same as before): b {z } (C t) (C t) t W t K t+ = R t K t + W t t=t X Thus, the problem becomes Max C t Bellman equation C t t (C t) t=o s.t. K t+ = R t K t + This exercise is optional in the sense that you could skip it if you feel comfortable with stochastic DP, and do it if you want to get more training. 6

17 t V (K) = Max C s:t: (C) K = RK + W N C + E [V (K = C C [V (K + = C C [V E = (4) [V = @C Using (5) and (4), we get: Then, = RC = R 2 3 = e C E 4R C 5 C e [V (K (5) But notice that: Then = b ec 2 3 e C = be 4R C 5 C Lagrangean: t= Max \$ = X t fc tg (C t) t (K t+ (R t K t + W t N t C t )) FOC t=o fc t g : C t = t (6) fk t+ g : t t = E t+ t+ R t+ (7) Then, combining (6) and (), we get our Euler equation: t C t = E " i # h t+ C t+ R Ct t+, = E Rt+ C t+ 7

18 Continuous time Max fc tg Z ect exp ( s:t: ek t = (R t ) K e t + W f t Ct e t) dt A t+ = A t Detrending: Max Z = MaxA ect exp ( Z Notice that () + : Then MaxA = Max = Max Z t) dt = Max (C t) () t exp ( t) dt = exp log ( Z ) = exp (C t) () t exp ( t) dt (C t) exp with The budget constraint becomes Z (C t) exp ( t) dt Z (C ta t ) exp ( t) dt log () exp, K t = (R t )K t + W t C t t dt Then the Euler equation will be: C t = (R t ) = (R t C t = (R t ) ) 8

19 2) Numerical analysis using DP:m. Describe the dynamics of the economy when it starts far away from its steady state (say 3% below). How does the recovery depend on the evolution of productivity? What does it tell you about the rapid growth in Europe after the second world war? (Hint: for the question on the evolution of productivity you may want to replicate the model, say times, and see what happens with the recovery path) See gures Evolution of output after the recovery (Figure 7): Recovery Mean and median and max and min in simulations (Figures 8 and 9) 9

20 Mean and median ( recoveries) Fastest and Lowest Recovery ( simulations) 3)DP:m produces 5 plots. Describe brie y each of them (what they represent and how they are computed. I am not asking for the details of the computations. 2

21 Simply show me that you understand the architecture of the program). Figure : 3D graph showing optimal consumption as a function of realizations of Z and initial capital stock. By-product of the search for the policy rule. Figure 2: Policy rule. K t+ as a function of K t. Capital in the steady state goes from 4.6 to around 5.4 Figure 3: Evolution of consumption in a stochastic simulation after starting from some initial capital that is below the steady state value. Figure 4: Evolution of consumption in the same simulation. Figure 5: stationary distribution of capital in the steady state. 5) Numerical analysis using DP:m. We have seen in the continuous time setup that the reaction of consumption to a positive technology shock depends on the elasticity. Use the policy function estimated by DP:m to illustrate this nding. (note: in the program, they de ne gamma, the CRRA instead of. How are they related? Why?) Hint: DP:m produces 5 plots. The rst one is called consumption, and it is 3 dimensional. Consumption is measured on the vertical axis. The two horizontal axis are for K and Z. If you do not see well, you can rotate the picture with the mouse. Notice that CRRA = Di erent = f:; ; g Figures to 2. 2

22 = : = 22

23 = 6) Show numerically how the behavior of consumption is a ected by the persistence of the shocks. Interpret. Di erent = f:5; :99g = :5 23

24 = :99 24

### Handout 7: Business Cycles

University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout 7: Business Cycles We now use the methods that we have introduced to study modern business

More information

### Great Depressions from a Neoclassical Perspective. Advanced Macroeconomic Theory

Great Depressions from a Neoclassical Perspective Advanced Macroeconomic Theory 1 Review of Last Class Model with indivisible labor, either working for xed hours or not. allow social planner to choose

More information

### The Real Business Cycle Model

The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the co-movements in the uctuations

More information

### Economics 702 Macroeconomic Theory Practice Examination #1 October 2008

Economics 702 Macroeconomic Theory Practice Examination #1 October 2008 Instructions: There are a total of 60 points on this examination and you will have 60 minutes to complete it. The rst 10 minutes

More information

### Real Business Cycle Models

Real Business Cycle Models Lecture 2 Nicola Viegi April 2015 Basic RBC Model Claim: Stochastic General Equlibrium Model Is Enough to Explain The Business cycle Behaviour of the Economy Money is of little

More information

### UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON4310 Intertemporal macroeconomics Date of exam: Thursday, November 27, 2008 Grades are given: December 19, 2008 Time for exam: 09:00 a.m. 12:00 noon

More information

### Real Business Cycle Theory. Marco Di Pietro Advanced () Monetary Economics and Policy 1 / 35

Real Business Cycle Theory Marco Di Pietro Advanced () Monetary Economics and Policy 1 / 35 Introduction to DSGE models Dynamic Stochastic General Equilibrium (DSGE) models have become the main tool for

More information

### Topic 5: Stochastic Growth and Real Business Cycles

Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar

More information

### In ation Tax and In ation Subsidies: Working Capital in a Cash-in-advance model

In ation Tax and In ation Subsidies: Working Capital in a Cash-in-advance model George T. McCandless March 3, 006 Abstract This paper studies the nature of monetary policy with nancial intermediaries that

More information

### Chapter 4. Simulated Method of Moments and its siblings

Chapter 4. Simulated Method of Moments and its siblings Contents 1 Two motivating examples 1 1.1 Du e and Singleton (1993)......................... 1 1.2 Model for nancial returns with stochastic volatility...........

More information

### 2. Real Business Cycle Theory (June 25, 2013)

Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 13 2. Real Business Cycle Theory (June 25, 2013) Introduction Simplistic RBC Model Simple stochastic growth model Baseline RBC model Introduction

More information

### Discrete Dynamic Optimization: Six Examples

Discrete Dynamic Optimization: Six Examples Dr. Tai-kuang Ho Associate Professor. Department of Quantitative Finance, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan 30013,

More information

### ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

More information

### A Comparison of 2 popular models of monetary policy

A Comparison of 2 popular models of monetary policy Petros Varthalitis Athens University of Economics & Business June 2011 Varthalitis (AUEB) Calvo vs Rotemberg June 2011 1 / 45 Aim of this work Obviously,

More information

### Lecture 14 More on Real Business Cycles. Noah Williams

Lecture 14 More on Real Business Cycles Noah Williams University of Wisconsin - Madison Economics 312 Optimality Conditions Euler equation under uncertainty: u C (C t, 1 N t) = βe t [u C (C t+1, 1 N t+1)

More information

### The RBC methodology also comes down to two principles:

Chapter 5 Real business cycles 5.1 Real business cycles The most well known paper in the Real Business Cycles (RBC) literature is Kydland and Prescott (1982). That paper introduces both a specific theory

More information

### Dynamics of current account in a small open economy

Dynamics of current account in a small open economy Ester Faia, Johann Wolfgang Goethe Universität Frankfurt a.m. March 2009 ster Faia, (Johann Wolfgang Goethe Universität Dynamics Frankfurt of current

More information

### Advanced Macroeconomics (ECON 402) Lecture 8 Real Business Cycle Theory

Advanced Macroeconomics (ECON 402) Lecture 8 Real Business Cycle Theory Teng Wah Leo Some Stylized Facts Regarding Economic Fluctuations Having now understood various growth models, we will now delve into

More information

### Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

More information

### ECON 5010 Class Notes Business Cycles and the Environment

ECON 5010 Class Notes Business Cycles and the Environment Here, I outline a forthcoming paper by Garth Heutel in the Review of Economic Dynamics. The title is "How Should Environmental Policy Respond to

More information

### Why Does Consumption Lead the Business Cycle?

Why Does Consumption Lead the Business Cycle? Yi Wen Department of Economics Cornell University, Ithaca, N.Y. yw57@cornell.edu Abstract Consumption in the US leads output at the business cycle frequency.

More information

### MA Advanced Macroeconomics: 7. The Real Business Cycle Model

MA Advanced Macroeconomics: 7. The Real Business Cycle Model Karl Whelan School of Economics, UCD Spring 2015 Karl Whelan (UCD) Real Business Cycles Spring 2015 1 / 38 Working Through A DSGE Model We have

More information

### 4. Only one asset that can be used for production, and is available in xed supply in the aggregate (call it land).

Chapter 3 Credit and Business Cycles Here I present a model of the interaction between credit and business cycles. In representative agent models, remember, no lending takes place! The literature on the

More information

### Graduate Introductory Macroeconomics: Exam 1

Graduate Introductory Macroeconomics: Exam Name: Student ID: You have 60 minutes. Answer in English or Japanese.. Consumption tax in a one-period model 0 points) The household determines the consumption

More information

### Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan

Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes

More information

### Lecture 2 Dynamic Equilibrium Models : Finite Periods

Lecture 2 Dynamic Equilibrium Models : Finite Periods 1. Introduction In macroeconomics, we study the behavior of economy-wide aggregates e.g. GDP, savings, investment, employment and so on - and their

More information

### Real Business Cycle Theory

Real Business Cycle Theory Barbara Annicchiarico Università degli Studi di Roma "Tor Vergata" April 202 General Features I Theory of uctuations (persistence, output does not show a strong tendency to return

More information

### 3 The Standard Real Business Cycle (RBC) Model. Optimal growth model + Labor decisions

Franck Portier TSE Macro II 29-21 Chapter 3 Real Business Cycles 36 3 The Standard Real Business Cycle (RBC) Model Perfectly competitive economy Optimal growth model + Labor decisions 2 types of agents

More information

### Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2

Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2 1 Consumption with many periods 1.1 Finite horizon of T Optimization problem maximize U t = u (c t ) + β (c t+1 ) + β 2 u (c t+2 ) +...

More information

### Economic Growth: Lecture 9, Neoclassical Endogenous Growth

14.452 Economic Growth: Lecture 9, Neoclassical Endogenous Growth Daron Acemoglu MIT November 29, 2011. Daron Acemoglu (MIT) Economic Growth Lecture 9 November 29, 2011. 1 / 41 First-Generation Models

More information

### Intertemporal approach to current account: small open economy

Intertemporal approach to current account: small open economy Ester Faia Johann Wolfgang Goethe Universität Frankfurt a.m. March 2009 ster Faia (Johann Wolfgang Goethe Universität Intertemporal Frankfurt

More information

### Dynamics of Small Open Economies

Dynamics of Small Open Economies Lecture 2, ECON 4330 Tord Krogh January 22, 2013 Tord Krogh () ECON 4330 January 22, 2013 1 / 68 Last lecture The models we have looked at so far are characterized by:

More information

### A Two-Period Model of the Current Account Obstfeld and Rogo, Chapter 1

A Two-Period Model of the Current Account Obstfeld and Rogo, Chapter 1 1 Small Open Endowment Economy 1.1 Consumption Optimization problem maximize U i 1 = u c i 1 + u c i 2 < 1 subject to the budget constraint

More information

### 14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

More information

### Graduate Macroeconomics 2

Graduate Macroeconomics 2 Lecture 1 - Introduction to Real Business Cycles Zsófia L. Bárány Sciences Po 2014 January About the course I. 2-hour lecture every week, Tuesdays from 10:15-12:15 2 big topics

More information

### Slides 2: Extending the RBC Model Endogenous Labour Supply and Real Rigidities

Slides 2: Extending the RBC Model Endogenous Labour Supply and Real Rigidities Bianca De Paoli November 29 The social planner's maximisation problem In previous lectures presented the following problem:

More information

### Business Cycle Accounting for South Africa

Business Cycle Accounting for South Africa May 20 Albert Touna Mama University of Cape Town albert.tounamama@uct.ac.za Nicola Viegi University of Pretoria and ERSA nicola.viegi@up.ac.za Abstract Quantitative

More information

### The Basic New Keynesian Model

The Basic New Keynesian Model January 11 th 2012 Lecture notes by Drago Bergholt, Norwegian Business School Drago.Bergholt@bi.no I Contents 1. Introduction... 1 1.1 Prologue... 1 1.2 The New Keynesian

More information

### 14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model

14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model Daron Acemoglu MIT November 1 and 3, 2011. Daron Acemoglu (MIT) Economic Growth Lectures 2 and 3 November 1 and 3, 2011. 1 / 96 Solow Growth

More information

### Noah Williams Economics 312. University of Wisconsin Spring 2013. Midterm Examination Solutions

Noah Williams Economics 31 Department of Economics Macroeconomics University of Wisconsin Spring 013 Midterm Examination Solutions Instructions: This is a 75 minute examination worth 100 total points.

More information

### Long-Term Debt Pricing and Monetary Policy Transmission under Imperfect Knowledge

Long-Term Debt Pricing and Monetary Policy Transmission under Imperfect Knowledge Stefano Eusepi, Marc Giannoni and Bruce Preston The views expressed are those of the authors and are not necessarily re

More information

### Maximum Likelihood Estimation of an ARMA(p,q) Model

Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates

More information

### Real Business Cycle Models

Phd Macro, 2007 (Karl Whelan) 1 Real Business Cycle Models The Real Business Cycle (RBC) model introduced in a famous 1982 paper by Finn Kydland and Edward Prescott is the original DSGE model. 1 The early

More information

### Preparation course MSc Business&Econonomics: Economic Growth

Preparation course MSc Business&Econonomics: Economic Growth Tom-Reiel Heggedal Economics Department 2014 TRH (Institute) Solow model 2014 1 / 27 Theory and models Objective of this lecture: learn Solow

More information

### Interest Rates and Real Business Cycles in Emerging Markets

CENTRAL BANK OF THE REPUBLIC OF TURKEY WORKING PAPER NO: 0/08 Interest Rates and Real Business Cycles in Emerging Markets May 200 S. Tolga TİRYAKİ Central Bank of the Republic of Turkey 200 Address: Central

More information

### Prep. Course Macroeconomics

Prep. Course Macroeconomics Intertemporal consumption and saving decision; Ramsey model Tom-Reiel Heggedal tom-reiel.heggedal@bi.no BI 2014 Heggedal (BI) Savings & Ramsey 2014 1 / 30 Overview this lecture

More information

### New Keynesian model. Marcin Kolasa. Warsaw School of Economics Department of Quantitative Economics. Marcin Kolasa (WSE) NK model 1 / 36

New Keynesian model Marcin Kolasa Warsaw School of Economics Department of Quantitative Economics Marcin Kolasa (WSE) NK model 1 / 36 Flexible vs. sticky prices Central assumption in the (neo)classical

More information

### Real Business Cycle Theory

Real Business Cycle Theory Guido Ascari University of Pavia () Real Business Cycle Theory 1 / 50 Outline Introduction: Lucas methodological proposal The application to the analysis of business cycle uctuations:

More information

### Teaching modern general equilibrium macroeconomics to undergraduates: using the same t. advanced research. Gillman (Cardi Business School)

Teaching modern general equilibrium macroeconomics to undergraduates: using the same theory required for advanced research Max Gillman Cardi Business School pments in Economics Education (DEE) Conference

More information

### 14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth

14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth Daron Acemoglu MIT November 15 and 17, 211. Daron Acemoglu (MIT) Economic Growth Lectures 6 and 7 November 15 and 17, 211. 1 / 71 Introduction

More information

### Macroeconomic Effects of Financial Shocks Online Appendix

Macroeconomic Effects of Financial Shocks Online Appendix By Urban Jermann and Vincenzo Quadrini Data sources Financial data is from the Flow of Funds Accounts of the Federal Reserve Board. We report the

More information

### 6. Budget Deficits and Fiscal Policy

Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 2012 6. Budget Deficits and Fiscal Policy Introduction Ricardian equivalence Distorting taxes Debt crises Introduction (1) Ricardian equivalence

More information

### Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

More information

### VI. Real Business Cycles Models

VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized

More information

### Sample Midterm Solutions

Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few

More information

### Graduate Macro Theory II: The Real Business Cycle Model

Graduate Macro Theory II: The Real Business Cycle Model Eric Sims University of Notre Dame Spring 2011 1 Introduction This note describes the canonical real business cycle model. A couple of classic references

More information

### Notes 10: An Equation Based Model of the Macroeconomy

Notes 10: An Equation Based Model of the Macroeconomy In this note, I am going to provide a simple mathematical framework for 8 of the 9 major curves in our class (excluding only the labor supply curve).

More information

### Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem

Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become

More information

### Optimal Investment. Government policy is typically targeted heavily on investment; most tax codes favor it.

Douglas Hibbs L, L3: AAU Macro Theory 0-0-9/4 Optimal Investment Why Care About Investment? Investment drives capital formation, and the stock of capital is a key determinant of output and consequently

More information

### The Real Business Cycle model

The Real Business Cycle model Spring 2013 1 Historical introduction Modern business cycle theory really got started with Great Depression Keynes: The General Theory of Employment, Interest and Money Keynesian

More information

### Business Cycles, Theory and Empirical Applications

Business Cycles, Theory and Empirical Applications Seminar Presentation Country of interest France Jan Krzyzanowski June 9, 2012 Table of Contents Business Cycle Analysis Data Quantitative Analysis Stochastic

More information

### Econ 8105 MACROECONOMIC THEORY DYNAMIC PROGRAMMING FOR MACRO. Prof. L. Jones. Fall 2010

Econ 8105 MACROECONOMIC THEORY DYNAMIC PROGRAMMING FOR MACRO Prof. L. Jones Fall 2010 These notes are a condensed treatment of the chapters in SLP that deal with Deterministic Dynamic Programming used

More information

### Universidad de Montevideo Macroeconomia II. The Ramsey-Cass-Koopmans Model

Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The Ramsey-Cass-Koopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous

More information

### Chapter 1 continued. Wouter J. Den Haan. University of Amsterdam. November 20, 2010

Chapter 1 continued Wouter J. Den Haan University of Amsterdam November 20, 2010 Wouter (University of Amsterdam) Chapter 1 continued November 20, 2010 1 / 18 Environment of the competitive equilibrium

More information

### Time Preference and the Distributions of Wealth and. Income

Time Preference and the Distributions of Wealth and Income Richard M. H. Suen This Version: February 2010 Abstract This paper presents a dynamic competitive equilibrium model with heterogeneous time preferences

More information

### A Classical Monetary Model - Money in the Utility Function

A Classical Monetary Model - Money in the Utility Function Jarek Hurnik Department of Economics Lecture III Jarek Hurnik (Department of Economics) Monetary Economics 2012 1 / 24 Basic Facts So far, the

More information

### ECON 20310 Elements of Economic Analysis IV. Problem Set 1

ECON 20310 Elements of Economic Analysis IV Problem Set 1 Due Thursday, October 11, 2012, in class 1 A Robinson Crusoe Economy Robinson Crusoe lives on an island by himself. He generates utility from leisure

More information

### Constrained optimization.

ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

More information

### Directed search and job rotation

MPRA Munich Personal RePEc Archive Directed search and job rotation Fei Li and Can Tian University of Pennsylvania 4. October 2011 Online at http://mpra.ub.uni-muenchen.de/33875/ MPRA Paper No. 33875,

More information

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

More information

### Economics 326: Budget Constraints and Utility Maximization. Ethan Kaplan

Economics 326: Budget Constraints and Utility Maximization Ethan Kaplan September 12, 2012 Outline 1. Budget Constraint 2. Utility Maximization 1 Budget Constraint Two standard assumptions on utility:

More information

### Common sense, and the model that we have used, suggest that an increase in p means a decrease in demand, but this is not the only possibility.

Lecture 6: Income and Substitution E ects c 2009 Je rey A. Miron Outline 1. Introduction 2. The Substitution E ect 3. The Income E ect 4. The Sign of the Substitution E ect 5. The Total Change in Demand

More information

### Oligopoly. Chapter 10. 10.1 Overview

Chapter 10 Oligopoly 10.1 Overview Oligopoly is the study of interactions between multiple rms. Because the actions of any one rm may depend on the actions of others, oligopoly is the rst topic which requires

More information

### We refer to player 1 as the sender and to player 2 as the receiver.

7 Signaling Games 7.1 What is Signaling? The concept of signaling refers to strategic models where one or more informed agents take some observable actions before one or more uninformed agents make their

More information

### University of Saskatchewan Department of Economics Economics 414.3 Homework #1

Homework #1 1. In 1900 GDP per capita in Japan (measured in 2000 dollars) was \$1,433. In 2000 it was \$26,375. (a) Calculate the growth rate of income per capita in Japan over this century. (b) Now suppose

More information

### Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems

Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 E-Mail: r.j.rossana@wayne.edu

More information

### Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey

Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey MURAT ÜNGÖR Central Bank of the Republic of Turkey http://www.muratungor.com/ April 2012 We live in the age of

More information

### Summing up ECON4310 Lecture. Asbjørn Rødseth. University of Oslo 27/

Summing up 4310 ECON4310 Lecture Asbjørn Rødseth University of Oslo 27/11 2013 Asbjørn Rødseth (University of Oslo) Summing up 4310 27/11 2013 1 / 20 Agenda Solow, Ramsey and Diamond Real Business Cycle

More information

### Chapter 3: The Multiple Linear Regression Model

Chapter 3: The Multiple Linear Regression Model Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics

More information

### Math 21A Brian Osserman Practice Exam 1 Solutions

Math 2A Brian Osserman Practice Exam Solutions These solutions are intended to indicate roughly how much you would be expected to write. Comments in [square brackets] are additional and would not be required.

More information

### Financial Development and Macroeconomic Stability

Financial Development and Macroeconomic Stability Vincenzo Quadrini University of Southern California Urban Jermann Wharton School of the University of Pennsylvania January 31, 2005 VERY PRELIMINARY AND

More information

### Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

More information

### Graduate Macro Theory II: Notes on Investment

Graduate Macro Theory II: Notes on Investment Eric Sims University of Notre Dame Spring 2011 1 Introduction These notes introduce and discuss modern theories of firm investment. While much of this is done

More information

### Our development of economic theory has two main parts, consumers and producers. We will start with the consumers.

Lecture 1: Budget Constraints c 2008 Je rey A. Miron Outline 1. Introduction 2. Two Goods are Often Enough 3. Properties of the Budget Set 4. How the Budget Line Changes 5. The Numeraire 6. Taxes, Subsidies,

More information

### CDMA07/04 Investment Frictions and the Relative Price of Investment Goods in an Open Economy Model * OCTOBER 2008

CENTRE FOR DYNAMIC MACROECONOMIC ANALYSIS WORKING PAPER SERIES CDMA07/04 Investment Frictions and the Relative Price of Investment Goods in an Open Economy Model * Parantap Basu University of Durham OCTOBER

More information

### Labour markets. Spring 2013. Suppose workers can freely choose the amount of hours they work, wages are given, no unemployment.

Labour markets Spring 2013 1 A Static Model of Labour Supply Suppose workers can freely choose the amount of hours they work, wages are given, no unemployment. Interpretations: Literally choosing hours

More information

### Equilibrium with Complete Markets

Equilibrium with Complete Markets Jesús Fernández-Villaverde University of Pennsylvania February 12, 2016 Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, 2016 1 / 24 Arrow-Debreu

More information

### This is a simple guide to optimal control theory. In what follows, I borrow freely from Kamien and Shwartz (1981) and King (1986).

ECON72: MACROECONOMIC THEORY I Martin Boileau A CHILD'S GUIDE TO OPTIMAL CONTROL THEORY 1. Introduction This is a simple guide to optimal control theory. In what follows, I borrow freely from Kamien and

More information

### Collusion: Exercises Part 1

Collusion: Exercises Part 1 Sotiris Georganas Royal Holloway University of London January 010 Problem 1 (Collusion in a nitely repeated game) There are two players, i 1;. There are also two time periods,

More information

### Endogenous Growth Models

Endogenous Growth Models Lorenza Rossi Goethe University 2011-2012 Endogenous Growth Theory Neoclassical Exogenous Growth Models technological progress is the engine of growth technological improvements

More information

### Inflation. Chapter 8. 8.1 Money Supply and Demand

Chapter 8 Inflation This chapter examines the causes and consequences of inflation. Sections 8.1 and 8.2 relate inflation to money supply and demand. Although the presentation differs somewhat from that

More information

### Dynamic programming. Martin Ellison

Dynamic programming Martin Ellison 1 Motivation Dynamic programming is one of the most fundamental building blocks of modern macroeconomics. It gives us the tools and techniques to analyse (usually numerically

More information

### INTERMEDIATE MICROECONOMICS MATH REVIEW

INTERMEDIATE MICROECONOMICS MATH REVIEW August 31, 2008 OUTLINE 1. Functions Definition Inverse functions Convex and Concave functions 2. Derivative of Functions of One variable Definition Rules for finding

More information

### EconS Bargaining Games and an Introduction to Repeated Games

EconS 424 - Bargaining Games and an Introduction to Repeated Games Félix Muñoz-García Washington State University fmunoz@wsu.edu March 25, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March 25,

More information

### Chapter 6: Economic Growth: Malthus to Solow

Chapter 6: Economic Growth: Malthus to Solow El-hadj Bah EC 313-Intermediate Macroeconomics Second Summer Session Arizona State University July, 2007 1 Economic Growth Facts 1. Pre-1800 (Industrial Revolution):

More information

### Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan

Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian

More information

### ECON 5110 Class Notes Overview of New Keynesian Economics

ECON 5110 Class Notes Overview of New Keynesian Economics 1 Introduction The primary distinction between Keynesian and classical macroeconomics is the flexibility of prices and wages. In classical models

More information

### Quality differentiation and entry choice between online and offline markets

Quality differentiation and entry choice between online and offline markets Yijuan Chen Australian National University Xiangting u Renmin University of China Sanxi Li Renmin University of China ANU Working

More information

### 1 A simple two period model

A QUICK REVIEW TO INTERTEMPORAL MAXIMIZATION PROBLEMS 1 A simple two period model 1.1 The intertemporal problem The intertemporal consumption decision can be analyzed in a way very similar to an atemporal

More information

### Estimating baseline real business cycle models of the Australian economy

Estimating baseline real business cycle models of the Australian economy Don Harding and Siwage Negara y University of Melbourne February 26, 2008 1 Introduction This paper is concerned with the issues

More information