Lectures 1 and 2 Adam Gulan · Lectures 1 and 2 Adam Gulan Bank of Finland January 2019. Course...

Macroeconomic TheoryLectures 1 and 2

Adam Gulan

Bank of Finland

January 2019

Course outline

I The lectures are given by Adam Gulan, e-mail [email protected]

I Lectures 1, 2 and 3: Introduction to business cycle fluctuations,stochastic growth model

I Lectures 4 and 5: Methods of solving and analyzing DSGE models(log-linearization, numerical solution methods, impulse responses,computation of moments, stochastic simulations)

I Lectures 6, 7 and 8: The basic real business cycle model + someextensions

I Lectures 9 and 10: Open economy RBC models

I Lectures 11 and 12: Introduction to financial frictions

Course material

Lectures, lecture notes (slides), readings, exercisesReadings for lectures 1-2:

I DeJong and Dave (2011), Ch.6

I Cooley (1995), Ch.1

I King and Rebelo (1999), ”Resuscitating Real Business Cycles”(Handbook of Macroeconomics, Ch.14)

I Wickens (2011), Ch.2.1-2.5 and 3.1-3.4

U.S. (log of) GDP, 1947-2014, constant 2009 pricesexp(7.5) = 1808 billion $, exp(9.7) = 16318 billion $

Motivation

I Business cycles a central feature in modern market economies.

I Data on business cycles since the 19th century

I Pre-WWII NBER research program: Burns & Mitchell, Kuznets,Mills

I The study of business cycles is one of classic themes inmacroeconomics (Keynes, Friedman, Lucas, Prescott etc.)

I The events of the past few years show that the business cycle is verymuch alive.

Cooley and Prescott (1995), p.27:

Every researcher who has studied growth and/or businesscycle fluctuations has faced the problem of how to representthose features of economic data that are associated withlong-term growth and those that are associated with thebusiness cycle - the deviation from the growth path. Kuznets,Mitchell and Burns and Mitchell employed techniques (movingaverages, piecewise trends etc.) that define the growthcomponent of the data in order to study the fluctuations ofvariables around the long-run growth path defined by thegrowth component. Whatever choice one makes about this issomewhat arbitrary. There is no single correct way to representthese components. They are simply different features of thesame observed data.

Methods of isolating business cyclesNote: the most common practice among macroeconomists interested inBCs to work with natural logs of variables (usually real) at quarterlyfrequency.

I linear trend

I (log-) differencing / growth rate dynamics

I (log-) business cycle dating (Bry, Boschan, NBER 1971, Harding,Pagan, JME 2002, Canova, EER 1994)

I Baxter-King (BK) filter (REStat, 1999)- type of band-pass filter- motivated by Burns & Mitchell definition of business cycles- isolates fluctuations between 6 and 32 quarters- discards longer-term (i.e. lower) frequencies and seasonaloscillations- optimal MA parameter K ≥ 12 for quarterly data

I Unobserved component models + Kalman Filter, etc...

I Beveridge-Nelson decomposition (JME, 1981) - ARIMA(p,q) can bedecomposed into sum of A) deterministic trend, B) stochastic trend(pure random walk), C) cyclical component

Hodrick Prescott filter,1981, JMCB 1997

I Let ln(Yt) = yt = yTt + yCt

I Minimize the loss function

min{yCt ,yTt }

T

∑t=1

(yCt )2 + λT

∑t=1

[(yTt+1 − yTt

)−(yTt − yTt−1

)]2

I λ is pre-specified parameterIt controls the smoothness of the trend yTtAs λ→ ∞, yTt → linear trendAs λ→ 0, yTt → yt and yCt = 0

I At quarterly frequency, it is customary to use λ = 1600.

I At annual data λA = 6.25 (Ravn and Uhlig, REStat, 2002)consistent with λQ = 1600.

I

Critique of Hodrick Prescott filter

I Two-sided filter (moving average)- trend observation at any time t based on whole sample 1 . . .T- i.e. uses FUTURE into to obtain PAST data

I One-sided filter - trend observation at any time t based on data1 . . . t- To execute 1SHP, run repetitively 2SHP, cutting at time t andtaking the last observation (because the last observation of 2SHP isactually 1S...)

I Critique: Cogley and Nason (JEDC, 1995), Canova (JME, 1998),Hamilton (REStat, 2018)- HP filter introduces artificial cycles where there is none!

I Hamilton’s (REStat, 2018) proposal

yt+h = β0 + β1yt + · · ·+ βp+1yt−p + µt+h

where p is the degree of integration (i.e., y ∼ I (p)) and h is thecycle horizon (e.g. 8 quarters or 2 years)

Blue: U.S. (log of) GDP, 1947-2014, constant 2009 pricesRed: linear trendexp(7.5) = 1808 billion $, exp(9.7) = 16318 billion $

Blue: U.S. (log of) GDP, 1947-2014, constant 2009 pricesRed: HP trendexp(7.5) = 1808 billion $, exp(9.7) = 16318 billion $

US GDP cycle

US GDP cycle

US GDP cycle

Filter gains (source: DeJong and Dave)

Filter gains, cont. (source: Stock and Watson, 1999)

US GDP cycle - HP vs BK filters

I HP and BK yield similar results if one works with deseasonalized data

I BK allows to work with NSA data but involves losses of data points

To summarize...

I Output movements exhibit some, but far from perfect regularity

I Fluctuations are quite large and persistent.

Business Cycle Statistics of Real GDP, US (1947-2014)

Filter Mean S.E AR(1)QoQ growth rate 3.1% 1.0% 0.83Hodrick Prescott 0 1.7% 0.84

I Using words of Long and Plosser (1983): the term ”business cycles”refers to the joint time-series behavior of a wide range of economicvariables (output, prices, employment, consumption, investment).The main aim must then be to build models which explain such jointbehavior. We will learn a lot more of this joint behavior later on.

Consumption (non-durables + services)

Investment (gross fixed capital formation)

Hours (total hours worked)

Stylized Business Cycle Facts

Source: King and Rebelo (1999)Note: Y is output, C is consumption, I is investment, N is hours worked,w is real wage, r is real interest rate, A is total factor productivity (Solowresidual).

Stylized Business Cycle Facts, cont.

I Investment is much more volatile than output

I Non-durable consumption is considerably less volatile than output

I Investment and consumption are strongly correlated with output

I Employment (unemployment) is procyclical (countercyclical) andmuch more strongly correlated with output than labor productivity(Y /N) or real wages.

I Real wages are roughly acyclical.

I These characteristics are similar for many developed countries andfor many macroeconomic time series.See Sorensen and Whitta-Jacobsen (2005, Ch. 13) for business cyclefacts for a number of European countries.

General Motivation

I Develop a framework for studying and understanding business cycles

I Build a bridge with business cycles and growth theories

I Account for basic empirical business cycle and growth facts

I Use standard growth model as point of departure (e.g. RCK)

I BC models are explicitly dynamic (variables evolve over time)

I Stochastic. Economic fluctuations are due to shocks, e.g.:- technology and productivity shocks- exogenous policy changes- shifts in preferences

I General equilibrium. Many markets which clear via price mechanism.

I Dynamic stochastic general equilibrium (DSGE) theory has becomea major paradigm in macroeconomics. It has found its way to policyinstitutions who make policy analysis and forecasts.

Lucas Critique

I Structural Econometric Models (SEM) in 1950s and 1960s:- very large (100s of eqs.) and detailed description of the economy- equations guided by economic theory but not strictly- adaptive expectations (based on past data only)- successful until late 1960s, then failed miserably

I Lucas (CRCSoPP, 1976):

Given that the structure of an econometric modelconsists of optimal decision rules of economic agents, andthat optimal decision rules vary systematically withchanges in the structure of series relevant to the decisionmaker, it follows that any change in policy willsystematically alter the structure of econometric models.

- SEM are OK for short-term forecasting only- any policy change will invalidate past reduced-from relationships

I Solution:- specifying micro fundamentals (they are microfounded)- adopting the rational expectations paradigm (Muth, 1961)

General Principles of Model Specification

In practice that means the need to:

I Define the agents who make optimal decisions. These decisions aretypically dynamic (inter-temporal).

I Profit-maximizing firms: We specify the technology available forfirms. Technology describes how certain inputs can be transformedinto outputs.

I Utility-Maximizing Households: We specify their preferences overconsumption bundles and leisure/labor (other extensions include e.g.money).

I Government: In the second part of the course (given by AnttiRipatti) we touch upon models where the government (central bank)maximizes a well-defined objective function.

General Principles of Model Specification, cont.

I We need also to decide what information is available for the agents(information set). We discuss for instance how uncertainty affectson consumption decisions.

I We need an equilibrium concept. We focus on competitive (inRBC models) and monopolistically competitive (in New Keynesianmodels) equilibria.

I Working with such theoretical framework is elegant and convenient.But models that we study can easily become impossible to solveanalytically.

I We need numerical solution methods to solve these models. Severalsoftware packages have been developed to implement these methods.

General Principles of Model Specification, cont.

I We need a way to assess empirically these models.

I We use very simple methods (calibration, first and second moments,impulse responses)

I Yet Bayesian estimation theory of DSGE models has developed andis available for rigorous model validation.

I Ultimately, our aim is to build quantitatively realistic models. Wewill not quite achieve this goal during this course, but we will learnthe basis on which one can build such models.

Discussion on Lucas Critique

I crazy-smart agents vs. modesty postulate

I rationality as outcome of learning (Evans, Honkapohja, 2001)

I is Lucas Critique empirically relevant?- is it as (more/less) important as model misspecification?- invariance to policy changes only when no misspecification (Cogley,Yagihashi, BEJM 2010)- Linde (AER 2001) vs Rudebusch (JMCB 2005) and Hendry

I Relevance of LC when expectations are not fully rational?- Gabaix (2016): perhaps not so much

I How often are policy RULES really changed?- Modest Policy Interventions (Leeper, Zha, JME 2003)

I Are modern DSGE models really microfounded?- representative agent- aggregation problems

General features of real business cycle (RBC) models

I RBC models focus on the real side of the economy:- Quantities (aggregate production, consumption, employment etc.)- Real prices (real wage, real interest rate)

I Classical dichotomy: nominal variables do not affect real variablesand are usually not explicitly modeled.

I Therefore they are ill-suited for studying inflation, nominal interestrates and monetary policy

I In the simplest possible (or baseline) RBC models there are virtuallyno frictions or market imperfections:- no asymmetric information- no price or wage rigidities- no externalities

I Perfect competition in all markets

I All prices adjust instantaneously

I The competitive equilibrium is Pareto optimal

Adding frictions to RBC models

I Nominal frictions: nominal prices and/or nominal wages adjustsluggishly, e.g. Calvo, JME, 1983 staggered pricing

I Labor market frictions: search&matching models

I Finding a job typically requires time and effort.People may be unemployed while looking for a job

I It also takes time for employers to fill a vacancy

I Financial market frictions

I Asymmetric information: the lender does not know the borrowerI Collateral constraint: In order to borrow, one must post a collateral

(e.g. a house). The value of the collateral determines how much onecan borrow.

I The shape of a firm’s balance sheet affects the size of the externalfinance premium. If the firm has a low equity to assets ratio, it hasto pay a higher price for external funds.

The simplest real business cycle model

I Stochastic extension of the Ramsey-Cass-Koopmans neoclassicalgrowth model:- optimal choice between consumption and saving- no endogenous labor supply

I Firms are identical. They are price takers

I Households are identical and live for ever (representative agents)

I The measures of households and firms are both normalized to 1

I Business cycles driven by total factor productivity shocks

Profit maximizing firms and technology

I Cobb Douglas technology

Yt = AtKαt H

1−αt (1)

where Yt is (real) output, At is (stochastic) total factor productivity(TFP) process, Kt is capital stock, Ht is labor and α is capital shareparameter

I At evolves according to:

ln(At) = (1− ρ) lnA+ ρ ln(At−1) + εt (2)

where εt is iid, 0 < ρ < 1 and A is NSSS value of At .

I The firm hires workers at wage Wt and rents capital from thehouseholds (which is owned by the households).Rental rate of capital isrKt = rt + δwhere rt is the real interest rate and δ is depreciation rate of capital.

I Firms’ profit maximization problem reads:

maxHt ,Kt

(Yt −WtHt − rKt Kt)

s.t. (3)

Yt = AtKαt H

1−αt

Kt ,Ht ≥ 0

I Notice that this is a static problem.

Firm’s first order conditions

Wt = (1− α)Yt

Ht(4)

rKt = αYt

Kt(5)

I Firms hire labor and capital until the (real) wage rate is equal tomarginal product of labor and (real) rental rate of capital is equal tomarginal product of capital

I Price taking firms assumption implies that profits are zero. You canverify this easily by plugging 4 and 5 into Yt −WtHt − rKt Kt . Youcan also easily verify that C-D production function implies that laborshare (WtHt/Yt) and capital share are constant (rKt Kt/Yt)

I What if production function was CES: Yt = At [aK αt + (1− a)Hα

t ]1α

Would profits be still zero?

Households - Preferences

U(C0,C1..., β) = E0

[∞

∑t=0

βtU(Ct)

], β < 1 (6)

I β < 1 is the time discount factor. U ′(C ) > 0, U ′′(C ) < 0β < 1 assumption imply that subjective utility does not ”blow up”.

It is often assumed that U(Ct) =C1−σt

1−σ (CRRA)or U(Ct) = ln(Ct) (CRRA with σ = 1).

I Notice that households have no preference over leisure. Householdswork one unit of time each period, Ht = 1 ∀t, i.e. labor supply isperfectly inelastic (we will introduce leisure choice later).

I Households are born with initial assets B0 > 0. They earn wages Wt

and rents rKt . They also own the firms so they receive all the profitsΠt .

Households - Budget constraint

Ct + Bt+1︸︷︷︸consumption+asset purchases

= Wt × 1 + RtBt + Πt︸︷︷︸labor income+capital income

(7)

I Bt is predetermined endogenous variable. In other words, Bt is takenas given at the beginning of period t, but household at t can affectthe value of Bt+1. Paths for Wt and Rt ≡ (1 + rt) are taken asgiven. They are stochastic and depend on the realizations on thetechnology process At .

I Consumption is numeraire good and its price is normalized to 1.Assets are real and pay out in terms of consumption good. Inequilibrium profits Πt are zero at all times so we ignore it in whatfollows.

I Transversality condition

limT→∞

T

∏s=1

R−1s BT+1 = 0

Households - Optimization problem

I Given {Wt , Rt}, household solves

max{Ct ,Bt+1}∞

t=0

E

[∞

∑t=0

βtU(Ct)

]s.t. (8)

Ct + Bt+1 = Wt + RtBt

Ct ≥ 0

limT→∞

T

∏s=1

R−1s BT+1 = 0

Dynamic programming

I Basic idea:

1. Collapse infinite horizon problem into two period problem.2. Turn the constrained problem into an unconstrained one.

Bellman equation(also known as functional equation)

V (Bt ,Wt ,Rt)︸︷︷︸value function

= maxEt∞

∑j=0

βjU(Ct+j )

= max

(U(Ct) + βEt

∞

∑j=0

βjU(Ct+j+1)

)V (Bt ,Wt ,Rt) = max

{Ct ,Bt+1}{U(Ct) + β× Et [V (Bt+1,Wt+1,Rt+1)]}

s.t.

Bt+1 = RtBt +Wt − Ct

I We have transformed infinite horizon decision problem into recursiveconstrained two period problem.

I V (.) is by definition the maximized value of the discountedconsumption stream. Wt and Rt are exogenous state variables (theyfollow a stochastic process that is exogenous to the household).

First order condition from unconstrained problem

I Plug constraint into the utility function

V (Bt ,Wt ,Rt) = maxBt+1

{U(RtBt +Wt − Bt+1) (9)

+βEtV (Bt+1,Wt+1,Rt+1)}

I Now we have specified the problem such that instead of choosingtoday’s consumption Ct we choose tomorrow’s state Bt+1.

I Assume that V (.) is differentiable. The first order condition (withrespect to Bt+1) becomes:

−U ′(Ct)︸︷︷︸marginal cost

of saving

+ βEtV′(Bt+1,Wt+1,Rt+1)︸︷︷︸

marginal expected discounted

shadow value of future wealth

= 0 (10)

The envelope condition

I Differentiate the Bellman equation (9) with respect to Bt

V ′ (Bt ,Wt ,Rt) = RtU′ (Ct)︸︷︷︸

direct effect

+{−U ′(Ct) + βEt

[V ′ (Bt+1,Wt+1,Rt+1)

]} dBt+1

dBt︸︷︷︸indirect effect

I Indirect effect arises, because Bt affects the optimal choice of Bt+1.

I But the first-order condition (10) tells that−U ′(Ct) + βEtV

′(Bt+1,Wt+1,Rt+1) = 0, so we can ignore theindirect effect

I ThusV ′ (Bt ,Wt ,Rt) = RtU

′ (Ct) (11)

Stochastic Euler equation

I Utilizing the envelope condition V ′(Bt ,Wt ,Rt) = RtU′(Ct) and

iterating it one period forward we get

V ′(Bt+1,Wt+1,Rt+1) = Rt+1U′(Ct+1)

Taking expectations:

Et[V ′(Bt+1,Wt+1,Rt+1)

]= Et

[Rt+1U

′(Ct+1)]

(12)

I Combining (10) and (12) and re-arranging

U ′(Ct) = βEt[Rt+1U

′(Ct+1)]

(13)

1 = Et

[βRt+1

U ′(Ct+1)

U ′(Ct)

]I This is a stochastic version of the Euler equation.

βU ′(Ct+1)U ′(Ct )

is the stochastic discount factor.

Stochastic Euler equation, cont.

I Interpretation: Households equate the cost from saving oneadditional unit of today’s consumption, i.e. the loss of U ′(Ct), tothe benefit of obtaining more consumption tomorrow(saving additional unit of consumption today givesβEt [Rt+1U

′(Ct+1)] more expected utility tomorrow).

I Characteristics of the solution

I Optimized consumption plan depends upon expected future wealth,as opposed to just current income.

I Households prefer intertemporally smooth consumption profile.(at least as long as U ′(C ) > 0,U ′′(C ) < 0)

Stochastic Euler equation, cont.

I Example: The utility function is CRRA

U (C ) =C1−σ

1− σ

where σ = −CU ′′U ′ measures the rate of relative risk aversion, and

1/σ is the intertemporal elasticity of substitution.

I Then the consumption Euler equation

U (Ct) = βEt [Rt+1U (Ct+1)]

takes the formC−σ = βEt

[Rt+1C

−σt+1

]or

1 = Et

[β

(Ct

Ct+1

)σ

Rt+1

]

Aggregate resource constraint

I Recall that we want to describe a competitive equilibrium of thiseconomy. In a competitive equilibrium, all the resources of theeconomy are in use (at equilibrium prices).

I The economy’s total production is Yt = AtKαt H

1−αt . Output can be

used for two purposes: Consumption and investment It .

I Investment is used to replace depreciated capital stock as well as tobuild new capital, so that

It︸︷︷︸Investment

= δKt︸︷︷︸Depreciation

+Kt+1 −Kt︸︷︷︸Net change

= Kt+1 − (1− δ)Kt (14)

I Aggregate resource constraint then reads as:

Ct + It = Yt (15)

Ct +Kt+1 − (1− δ)Kt︸︷︷︸Demand

= AtKαt H

1−αt︸︷︷︸

Supply

Competitive equilibrium

DefinitionGiven initial asset endowment B0, a competitive equilibrium is a set ofallocations of the representative household {Bt+1}∞

t=0, allocations of the

representative firm {Kt ,Ht}∞t=0 and prices {Wt , rKt , rt}∞

t=0 such that:a) households’ allocation solves the households’ problem (8)b) firms’ allocation solves the firms’ problem (3)c) markets clear, i.e.

Ct +Kt+1 − (1− δ)Kt = AtKαt H

1−αt (16)

Bt = Kt (17)

Ht = 1 (18)

∀t = 0, ...∞

Competitive equilibrium, cont.The competitive equilibrium is characterized by the following equations:

I The consumption Euler equation from the household’s problem

U (Ct) = βEt [Rt+1U (Ct+1)] (19)

I The calculation of return, derived from the firm’s maximizationproblem

Rt+1 = αAt+1 (Kt+1)α−1 + 1− δ (20)

I Recall that Rt+1 ≡ 1 + rt+1 = 1 + rKt+1 − δ

I The aggregate resource constraint (goods market equilibrium)

Ct +Kt+1 − (1− δ)Kt = AtKαt H

1−αt (21)

I The labor market equilibrium condition

Ht = 1 (22)

I Law of motion of total factor productivity

ln(At) = (1− ρ) lnA+ ρ ln(At−1) + εt , εt is iid (23)

Competitive equilibrium, cont.

I We can characterize the equilibrium by three equations if we

I Iterate eq. (20) one period forward, and then plug into the Eulerequation (19)

I Insert the labor market equilibrium (22) into the aggregate resourceconstraint (21)

I The three equations are

Et

[U ′(Ct)

βU ′(Ct+1)

]︸︷︷︸

marginal rate of

intertemporal substitution

= Et [αAt+1 (Kt+1)α−1 + 1− δ)]︸︷︷︸

marginal production +

capital left over after production

(24)

I

Ct +Kt+1 − (1− δ)Kt = AtKαt (25)

I

ln(At) = (1− ρ) lnA+ ρ ln(At−1) + εt , εt is iid (26)

Steady state

I We define a non-stochastic steady state (NSSS) as an equilibriumwhere all variables are constant over time, i.e. Ct = Ct+1 = C ,Kt = Kt+1 = K , At = At+1 = A, εt = εt+1 = 0.

I In a steady state, when Ct+1 = Ct = C , the Euler equation

1 = Et

[βRt+1

U ′(Ct+1)

U ′(Ct)

]boils down to

1 = βR

I Thus in steady state the real interest rate r and the gross interestrate R = 1 + r are given by

R =1

βr =

1

β− 1 (27)

Steady state, cont.

I Then, using NSSS versions of equations (24) and (20), we find that

R = αA (K )α−1 + 1− δ

r + δ = αA (K )α−1

K = K =

(αA

r + δ

) 11−α

(28)

I (28) is modified golden rule: Optimal SS capital stock is such thatmarginal product of capital (at K ) equals the depreciation rate plusthe time discount rate.

Golden rule

Golden rule capital stock (K#) maximizes NSSS consumption (C#)

Modified golden rule

I Modified golden rule: People are impatient, therefore steady statecapital stock (K ) is lower than the golden rule capital stock (K#).

I Also: steady state consumption (C ) is lower than C#.

I If β→ 1 (and r → 0), K → K# and C → C#.

Golden rule and modified golden rule, redux

Figure: a

Steady state, cont.I Optimal steady state level of consumption can be solved by using

the resource constraint (16).

Ct +Kt+1 −Kt︸︷︷︸=0

+ δKt = AtKαt

C = AK α − δK

I Steady state consumption equals steady state output minusdepreciation

I Steady state investment equals depreciation

I = δK

I Also notice: steady state consumption can be re-expressed as follows

C = W︸︷︷︸wage income

+ rK︸︷︷︸capital income

I W = (1− α)Y is steady state wage rate, and also, because ofH = 1, wage income.

”Great ratios” and calibration

I ”Great ratios” in steady state

K

Y=

α

r + δ

I

Y=

δK

Y=

δ

r + δα

C

Y= 1− I

Y=

r + δ (1− α)

r + δ

I ”Great ratios” can be used in calibration

I to compute parameter values: we want the model to reproduce thegreat ratios found in the data

I ... or as a cross-check: for example, the parameters values are takenfrom the literature, and we want to check whether of not thesevalues make sense

”Great ratios” and calibration, cont.

I Example: Calibration using quarterly data:

r = 0.01 δ = 0.025 α = 0.36 (capital share)

yieldsKY = 10.3 I

Y = 0.26 CY = 0.74

I Here: great ratios used as a cross-check

Examining the steady state: phase diagram

I Set At+s = 1, s = 0, 1, 2, ...

I Define

gCt+1=

Ct+1 − Ct

Ct

(the growth rate of consumption)

I Assume for concreteness that U (C ) = C1−σ

1−σ

I Then from the Euler equation (24) we get

gCt+1={

β[α (Kt+1)

α−1 − (r + δ)]+ 1}1/σ

− 1

I Clearly

gCt+1=

> 0 if Kt+1 < K= 0 if Kt+1 = K< 0 if Kt+1 > K

Phase diagram: dynamics of consumption

Figure: a

Examining the steady state: phase diagram

I Next define

gKt+1=

Kt+1 −Kt

Kt

(the growth rate of the capital stock)

I From the resource constraint (25) we get

gKt+1=

(K αt − δKt − Ct)

Kt

I Clearly,

gKt+1=

> 0 if Ct < K αt − δKt

= 0 if Ct = K αt − δKt

< 0 if Ct > K αt − δKt

Phase diagram: dynamics of the capital stock

Figure: a

Phase diagram

Figure: a

Phase diagram: the steady state is saddle point stable

Figure: a

Alternative solution strategy: solve the social planner’sproblem

The planner’s problem is of the form

max{Ct ,Kt+1}∞

t=0

E0

[∞

∑t=0

βtU(Ct)

]s.t.

Kt+1 = (1− δ)Kt + AtKαt H

1−αt − Ct

ln(At) = (1− ρ) lnA+ ρ ln(At−1) + εt , εt is iid

Planner’s problem, cont.

I The social planner’s value function V (Kt ,At) satisfies the recursiveBellman equation

V (Kt ,At) = maxKt+1

U (AtKαt + (1− δ)Kt −Kt+1)

+βEt [V (Kt+1,At+1)] (29)

(Note: Ht = 1)

I First order condition

−U ′ (Ct)︸︷︷︸marginal costof investing

+ βEt[V ′ (Kt+1,At+1)

]︸︷︷︸marginal expected discounted

shadow value of capital

= 0 (30)


Envelope condition

I Differentiating the Bellman equation (29) with respect to Kt yields

V ′ (Kt ,At) = U ′ (Ct)(

αAtKα−1t + 1− δ

)︸︷︷︸

direct effect

+{−U ′ (Ct) + βEt

[V ′ (Kt+1,At+1)

]} dKt+1

dKt︸︷︷︸indirect effect

I But the first-order condition tells us that−U ′ (Ct) + βEt [V ′ (Kt+1,At+1)] = 0, and the indirect effect canbe ignored

I ThusV ′ (Kt ,At) = U ′ (Ct)

(αAtK

α−1t + 1− δ

)


I Iterating the envelope condition one period forward yields

V ′ (Kt+1,At+1) = U ′ (Ct+1)(

αAt+1Kα−1t+1 + 1− δ

)and taking expectations gives

Et[V ′ (Kt+1,At+1)

]= Et

[U ′ (Ct+1)

(αAt+1K

α−1t+1 + 1− δ

)](31)


I Combining (31) and (30) gives rise to the familiar intertemporalmaximization condition!

Et

[U ′(Ct)

βU ′(Ct+1)

]︸︷︷︸

marginal rate of

substitution

= Et [αAt+1 (Kt+1)α−1 + 1− δ)]︸︷︷︸

additional production +


I The other two equations characterizing the planner’s solution are

I The resource constraint

Kt+1 = (1− δ)Kt + AtKαt − Ct

and the law of motion of total factor productivity

ln(At ) = (1− ρ) lnA+ ρ ln(At−1) + εt , εt is iid

I These are exactly the same as the equations (24), (25) and (26),characterizing the competitive equilibrium!


I The competitive equilibrium is equivalent to the socially optimaloutcome (the solution to the planner’s problem).

I Perfect competition in all markets, prices adjust immediately, noexternalities, symmetric information ⇒ the welfare theorems can beapplied.

Fluctuations and growth in the model

I The model does not exhibit any growth. But we know that output(even per capita) does grow over time (the trend component).

I Although we will be ultimately interested mainly in business cyclefluctuations, addressing this issue makes the model more realistic.

I Long-term growth also affects the calibration of the model, as weshall see below.

I Basic idea:

I We first introduce growth in the form of technological progressI Then we detrend the modelI Business cycles ≡ fluctuations around the trend

Fluctuations and growth in the model, cont.

I It is (relatively) easy to put growth into the model.

I We assume thatYt = AtK

αt (XtHt)

1−α (32)

where Xt is labor-augmenting technology(technology growth is Harrod-neutral).

I We also assume that labor-augmenting technology grows at some(gross) exogenous rate γ

Xt+1 = γXt = γt+1 (since X0 = 1) (33)

I Deterministic trend, with constant trend growth rate:we consider the simplest possible case

I Now, plugging (33) into (32) allows us to write the productionfunction as

Yt = AtKαt (γ

t)1−α (34)

I How would you introduce population growth?This is left as an exercise!


I The representative household, or the social planner, maximizes

max{Ct ,Kt+1}∞

t=0

E0

[∞

∑t=0

βtU(Ct)

]s.t.

Kt+1 = (1− δ)Kt + AtKαt − Ct


I This optimization problem is expressed in terms of aggregatevariables (Kt , Yt , Ct), which are trending over time.

I Now the basic idea to reformulate the optimization problem in termsof variables that are stationary, and constant along thenon-stochastic balanced growth path.


I The variables Kt , Yt , Ct all grow at the common (gross) rate γ

I Then the detrended variables

Yt =Yt

Xt,Ct =

Ct

Xt,Kt =

Kt

Xt(35)

are stationary (and constant along the balanced growth path)

I Also, using these definition together with the production function(34), we get

Yt = AtKαt

I Yt ,Ct ,Kt are output, consumption and capital per effective unit oflabor.

I The detrended variables measure fluctuations around the trend, orbusiness cycles.


I Rewriting the resource constraint in terms of the detrended variables

Kt+1 + Ct = Yt + (1− δ) Kt

can be rewritten as

Xt+1Kt+1 + XtCt

= XtYt + (1− δ) XtKt

and finallyγKt+1 + Ct = AtK

αt + (1− δ)Kt (36)


I Rewriting the objective function in terms of detrended variables

E0

∞

∑t=0

βt C1−σt

1− σ= E0

∞

∑t=0

βt(Xt)1−σ C1−σ

t

1− σ

and finally we get

E0

∞

∑t=0

βt C1−σt

1− σ=

∞

∑t=0

βtU (Ct) (37)

whereβ = βγ(1−σ) (38)


I Rewriting the maximization problem in terms of the detrendedvariables

max{Ct ,Kt+1}∞

t=0

E0

[∞

∑t=0

βtU(Ct)

]s.t.

γKt+1 = (1− δ)Kt + AtKαt − Ct



I The social planner’s value function V (Kt ,At) satisfies the recursiveBellman equation

V (Kt ,At) = maxKt+1

U (AtKαt + (1− δ)Kt − γKt+1)

+βEt [V (Kt+1,At+1)] (39)

I First order condition

−γU ′ (Ct)︸︷︷︸marginal costof investing

+ βEt[V ′ (Kt+1,At+1)

]︸︷︷︸marginal expected discounted

shadow value of capital

= 0 (40)


Envelope condition

I Differentiating the Bellman equation (39) with respect to Kt yields

V ′ (Kt ,At) = U ′ (Ct)(

αAtKα−1t + 1− δ

)︸︷︷︸

direct effect

+{−γU ′ (Ct) + βEt

[V ′ (Kt+1,At+1)

]} dKt+1

dKt︸︷︷︸indirect effect

I But the first-order condition tells us that−γU ′ (Ct) + βEt [V ′ (Kt+1,At+1)] = 0, and the indirect effect canbe ignored

I ThusV ′ (Kt ,At) = U ′ (Ct)

(αAtK

α−1t + 1− δ

)


I Iterating the envelope condition one period forward yields

V ′ (Kt+1,At+1) = U ′ (Ct+1)(

αAt+1Kα−1t+1 + 1− δ

)and taking expectations gives

Et[V ′ (Kt+1,At+1)

]= Et

[U ′ (Ct+1)

(αAt+1K

α−1t+1 + 1− δ

)](41)


I Combining (41) and (40) gives rise to the intertemporalmaximization condition

Et

[γU ′(Ct)

βU ′(Ct+1)

]︸︷︷︸

marginal rate of

intertemporal substitution

= Et [αAt+1 (Kt+1)α−1 + 1− δ)]︸︷︷︸

additional production +


(42)


I Plugging in the explicit utility function U(Ct) =C1−σt

1−σ and

β = βγ(1−σ) and defining the real (gross) interest rate (”calculus ofreturn”)

Rt+1 = αAt+1 (Kt+1)α−1 + 1− δ (43)

allows us to rewrite (42) as

1 = Et

[βγ−1Rt+1

(Ct

Ct+1

)σ](44)


I The dynamic equilibrium (or equivalently the planner’s solution) ischaracterized by:i) the Euler equation (42)ii) the detrended resource constraint (36)iii) the law of motion of the (stationary part) of total factorproductivity (2)

I The equations (42) and (36) characterize the evolution of thedetrended variables (Ct and Kt), given the (exogenous) dynamics ofTFP (At)

I Also:production Yt = AtK

αt

investment It = Yt − Ct

real wage Wt = (1− α)Yt


I Moving to the original (trending) variables is easy:

Ct = γtCt , Kt = γtKt ,

It = γt It , Wt = γtWt

I Notice: the real interest rate

Rt = αAt (Kt)α−1 + 1− δ

does not have a trend

Balanced growth pathI Along the balanced growth path, the detrended variables are

constant over timeI When Ct+1 = Ct = C the Euler equation

1 = Et

[βγ−1Rt+1

(Ct

Ct+1

)σ]yields

1 = βγ−1R = βγ−σR

I Thus over the balanced growth path, the real interest rate r and thegross interest rate R = 1 + r are given by

R =γσ

β; r =

γσ

β− 1 (45)

and define R0 = 1β

to be the gross real interest rate with no

long-term growth. Hence R = R0γσ. Therefore

lnR = lnR0 + σ ln γ

sor ≈ r0 + σ (γ− 1)

Balanced growth path, cont.

I Next, along the balanced growth path, the calculus of return

Rt = αAt (Kt)α−1 + 1− δ

yields (letting A = 1)

R = αK α−1 + 1− δ

and we get

K =

(α

r + δ

) 11−α

≈(

α

r0 + σ (γ− 1) + δ

) 11−α

I This is modified gold rule in a model with long-term growth.

Balanced growth path, cont.

I Along the balanced growth path, the resource constraint

γKt+1 + Ct = AtKαt + (1− δ)Kt

yieldsC = K α + (1− δ− γ)K

andI = Y − C = K α − C = (γ− 1 + δ)K

”Great ratios” along the balanced growth path

K

Y=

α

r + δ≈ α

r0 + σ (γ− 1) + δ

I

Y= (γ− 1 + δ)

K

Y=

(γ− 1 + δ

r + δ

)α ≈

(γ− 1 + δ

r0 + σ (γ− 1) + δ

)α

C

Y= 1− I

Y=

r + δ (1− α)− α (γ− 1)

r + δ

≈ r0 + δ (1− α) + (σ− α) (γ− 1)

r0 + σ (γ− 1) + δ

”Great ratios” along the balanced growth path, cont.

I How do the ”great ratios” depend on the growth rate of theeconomy γ?

I It is easy to see that KY ↓ when γ ↑: the faster is growth, the less

capital there will be compared to output.

I For the ratios IY (the GDP share of investment) and C

Y (the GDPshare of consumption) the relationship is in general ambiguous, andit depends on σ

I If σ is low, we have IY ↑ and C

Y ↓ as γ ↑I If σ is high, we have I

Y ↓ and CY ↑ as γ ↑

I If σ = 1 (logarithmic utility) we have IY ↑ and C

Y ↓ as γ ↑I the precise threshold is above 1, and depends on r0 and δ

This figure suggests that σ should not be too high.

”Great ratios” and calibration along the balanced growthpath

I Long-term growth also affects calibration

I As a simple example, assume that σ = 1 (logarithmic utility)

I ”Great ratios” along the balanced growth path take the form

K

Y≈ α

r0 + δ,

I

Y≈(

δ

r0 + δ

)α,

C

Y≈ r0 + (1− α) δ

r0 + δ

whereδ = δ + (γ− 1) (46)

”Great ratios” and calibration, cont.

I Hence, in calibrating the model, we have to use a modified (andhigher) rate of decay (δ) for the capital stock. This modified decayrate takes into account long-run (trend) growth γ

I Essentially, along the balanced growth path, you need to invest notonly to offset the physical decay (δ), but also to keep pace with(labor augmenting) technical change.

I In this simple example, expression (46) is the only modification wehave to make to the calibration, compared to the basic model withno growth.

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Lectures 1 and 2 Adam Gulan · Lectures 1 and 2 Adam Gulan Bank of Finland January 2019. Course...

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