Macroeconomic Theory

Lecture 1

Macroeconomic Theory Lecture 1 Markus Haavio Bank of Finland

January 2011

Lecture 1

Course outline I

The lectures are given by Markus Haavio, e-mail markus.haavio@bof.…

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Lecture 1, January 13: Intruduction to business cycle analysis, stochastic growth model, dynamic programming

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Lecture 2, January 20: Methods of solving and analyzing DSGE models (loglinearization, method of undetermined coe¢ cients, impulse responses, computation of moments, stochastic simulations)

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Lecture 3, January 27: The basic real business cycle model, introduction to Dynare

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Lecture 4, February 10: Asset pricing, introduction to open economy RBC models (+ possibly an introduction to …nancial frictions)

Lecture 1

Course material

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Lectures + lecture notes (slides) + exercises

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Core readings: I I

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George McCandless (2008), "The ABCs of RBCs", chapters 1,3-7, 13 Harald Uhlig’s "A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily” King, R. and S. Rebelo (1999), "Resuscitating Real Business Cycles" Schmitt-Grohe, S. and Uribe, M. (2003), "Closing Small Open Economy Models"

More advanced readings: the remaining titles on the reading list

Lecture 1

Motivation

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Business cycles a central feature in modern market economies. I

Data on business cycles since the 19th century

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The study of business cycles one of classic themes in macroeconomics (Keynes, Friedman, Lucas, Prescott etc.)

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The events of the past few years show that the business cycle is very much alive.

Lecture 1

Motivation I

To study business cycles, we need dynamic models.

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Dynamic general equilibrium theory has become a major paradigm in macroeconomics. DGE theory has found its way to policy institutions who make policy analysis and forecasts.

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It builds on intertemporal optimizing behavior of economic agents and perfectly or monopolistically competitive markets

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Market clearing happens through price mechanism. In RBC models prices are perfectly ‡exible. In New Keynesian models, price adjustment is sluggish.

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Fluctuations in macroeconomy are due to di¤erent (real) shocks, such as I

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technology shocks, policy shocks, shifts in preferences.

We aim at improving our understanding of these ‡uctuations by means of DGE models.

Lecture 1

General Principles for Model Speci…cation

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We need to specify the entities who make decision under certain information set. We assume that these decisions are typically based on dynamic optimisation. I

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Pro…t maximizing …rms: We specify the technology available for …rms. Technology just tells us how certain inputs can be transformed into outputs. Utility Maximising Households: We specify their preferences over consumption and leisure (or in more general over commodities). Government: In the second part of the course (given by Antti Ripatti) we touch upon models where the government (Central bank) has well de…ned objective function which it maximises

Lecture 1

General Principles for Model Speci…cation cnt’d I

We need also to decide what information is available for the agents. We discuss for instance how uncertainty a¤ects on consumption decisions.

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We need an equilibrium concept. We focus on competitive (RBC models) and monopolistically competitive (New Keynesian models) equilibria.

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Macro models which share these basic principles are often cited as "microfounded".

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Working with such a theoretical framework is elegant and convenient. But models that we study become easily impossible to solve analytically.

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We need numerical solution methods to solve these models. We do not go very deep into details, but we use software which does.

Lecture 1

General Principles for Model Speci…cation cnt’d

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We need a way to evaluate empirically these models.

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We use very simple methods (calibration, …rst and second moments, impulse responses)

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Yet Bayesian estimation theory of DGE models has developed and is available for rigorous model validation.

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Ultimately, our aim is to build quantitatively realistic models. We do not quite achieve this target during this course, but we construct basis on which we can build such models.

Lecture 1

Fluctuations – Basic Business Cycle Facts I

The main objects of interest in the macroeconomy are GDP (per capita) or production and its components.

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As we are interested in ‡uctuations, we need …lter the GDP such that we get rid of its trend.

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There are many …ltering methods available. The simplest …lter is a linear-trend

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In practise, ‡uctuations are measured by I I I I

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Growth rates Hodrick-Prescott …lter Band pass …lter Unobserved component models

It is very common for macroeconomists interested in BCs to work with natural logarithmic variables at quarterly frequency.

Lecture 1

Fluctuations – Real GDP

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exp(7.4) = 1636 billion $, exp(9.3) = 10938 billion $.

Lecture 1

Lecture 1

Cooley and Prescott (1995): Every researcher who has studied growth and/or business cycle ‡uctuations has faced the problem of how to represent those features of economic data that are associated with long-term growth and those that are associated with the business cycle - the deviation from the growth path. Kuznets, Mitchell and Burns and Mitchell employed techniques (moving averages, piecewise trends etc.) that de…ne the growth componentof the data in order to study the ‡uctuations of variables around the long-run growth path de…ned by the growth component. Whatever choice one makes about this is somewhat arbitrary. There is no single correct way to represent these components. They are simply di¤erent features of the same observed data.

Lecture 1

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HP …lter (likewise Band pass …lter) provides a possibility to extract ‡uctuations at 3-8 year frequency.

Lecture 1

Hodrick Prescott …lter I I

Let log(Yt ) = log(Yttrend ) + log(Ytcycle ) = yttrend + yt How to disentangle business cycle component from the time series of interest? T

min

∑

fyt ,yttrend g t =1

yt2 + λ

T

∑ (∆yttrend +1

∆yttrend )2

t =1

s.t. yt + yttrend I

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= log(Yt ),

∆yttrend = yttrend

yttrend 1

Idea: by choosing λ, we decide how much the trend component tracks the actual data. High λ makes it "optimal" to have a trend component with a relatively constant slope. This is because ∆yttrend ∆yttrend = (yttrend yttrend ) (yttrend yttrend +1 +1 1 ) is change in the growth rate of trend component. At quarterly frequency, it is customary to use λ = 1600. This extracts ‡uctuations of roughly 8 years or longer into the trend component. At yearly data λ = 6.25.

Lecture 1

Lecture 1

Basic Business Cycle Facts - GDP I I I

Output movements are not regular (and neither are the ‡uctuations in investment, consumption, employment etc.) Fluctuations are quite large and persistent. This is typical for many developed countries and for many macroeconomic time series

Business Cycle Statistics of Real GDP, US (1947-2004) Filter Mean S.E AR(1) Growth Rate 3.3% 2.6% 0.83 Hodric 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 econ variables (output, prices, employment, consumption, investment). The main aim must then be to build models which explain such joint behavior. We will learn a lot more of this joint behavior later on.

Lecture 1

Basic 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 (Solow residual).

Lecture 1

Stylized Business Cycle Facts

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Investment is much more volatile than output

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Non-durables consumption is considerably less volatile than ouput

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Investment and consumption are strongly correlated with output

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Employment (unemployment) is procyclical (contracyclical) and much more strongly correlated with output than labour productivity (Y /N) and real wages.

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Real wages are roughly acyclical.

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See Sörensen and Whitta-Jacobsen (2005, Ch. 14) for business cycle facts for a number of European countries (including Finland).

Lecture 1

General features of real business cycle models

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RBC models focus on the real side of the economy I I

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Quantities (aggregate production, consumption, enployment etc.) Relative prices (real wage, real interest rate)

Classical dichotomy: money is a veil; nominal variables do not a¤ect real variables I

These models are not well suited for studying in‡ation, nominal interest rates and monetary policy

Lecture 1

General features of real business cycle models

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In the simplest possible (or baseline) RBC models there are virtually no frictions or imperfections I I I

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Perfect competition in all markets All prices adjust instantaneously No asymmetric information

The competitive equilibrium is Pareto optimal

Lecture 1

Adding frictions to RBC models I

Labour market frictions: search / matching models I

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Finding a job typically requires time and e¤ort. While searching for a job people may be unemployed for quite a while Also for employers it often takes time to …ll a vacancy

Financial market frictions I

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Asymmetric infromation: the lender does not know / trust the borrower Collateral constraint: In order to borrow, one must post a collateral (e.g. a house can serve as a collateral). The value of the collateral determes how much one can borrow. The shape of a …rm’s balance sheet a¤ects the size of the external …nance premium. If the the …rm has a low capital ratio, it has to pay a higher price for external funds.

Lecture 1

The simplest real business cycle model

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Known as Cass-Koopmanns neoclassical growth model: no endogenous labour supply.

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Firms are identical. They are price takers.

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Households are identical and they live for ever

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A measure of households and …rms is normalised to 1.

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Business cycles driven by techology shocks

Lecture 1

Pro…t maximizing …rms and technology I

C-D technology Yt = Zt Ktα L1t

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α

(1)

Yt is (real) production, Zt is (stochastic) technology process, Kt is capital stock, Lt is labour, α is capital share parameter.

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log(Zt ) = (1

ρ) log Z + ρ log(Zt 1 ) + εt , εt is iid

Here 0 < ρ < 1 and Z are parameters. I

The …rm hires workers at wage Wt and rents capital from the households (which is owned by the households). Rental rate of capital is rtK = rrt + δ, whererrt is the real interest rate and δ is capital depreciation rate.

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In fact, the …rms are owned by the households, and the assets to which households save is the physical capital stock of the economy.

Lecture 1

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Pro…t maximisation problem reads: max (Yt

L t ,K t

s.t. Yt Kt , Lt I

Wt Lt

rtK Kt ) (2)

α 1 α

= Zt Kt Lt 0

Notice that this is a static problem.

Lecture 1

Firm’s …rst order conditions

Wt

= (1

α)Zt

rtK

= αZt

Kt Lt

Kt Lt

α

(3)

α 1

(4)

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Firms hire labour and capital until the (real) wage rate is equal to marginal product of labour and (real) rental rate of capital is equal to marginal product of capital

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Price taking …rms assumption implies that pro…ts are zero. You can verify this easily by plugging 3 and 4 into Yt Wt Lt rtK Kt . You can also easily verify that C-D production function implies that labour share (Wt Lt /Yt ) and capital share are constant (rtK Kt /Yt )

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What if production function is CES form. Would pro…ts be still zero?

Lecture 1

Households - Preferences

U (C0 , C1 ..., β) = E I

"

∞

∑β

t =0

t

#

U (Ct ) , β < 1

β < 1 is the time discount factor. U 0 (C ) > 0, U 00 (C ) < 0. β < 1 assumption imply that subjective utility does not "blow up". Often C

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(5)

1 η

we assume that U (Ct ) = log(Ct ), or that U (Ct ) = 1t η . Notice that households have no preference over leisure. Households work one unit of time each period, no matter what. (We will introduce leisure choice in later during this course). Households are born with initial assets B0 > 0. They earn wage Wt . Remember that they also own the …rms, so they receive all the pro…ts (Πt ).

Lecture 1

Households - Budget constraint Ct + Bt +1 | {z }

=

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1 + Rt Bt + Πt {z }

(6)

labour income+capital income

consumption+asset purchases I

Wt |

Bt is predetermined endogenous variable. In other words, it is taken as given at the beginning of period, but household can a¤ect on evolution of B. Paths for Wt and Rt (1 + rrt ) are taken as given. They are stochastic, and depend on the realizations on the techology process Zt . Consumption is numeraire good and we have normalized price of consumption good equal to 1. Assets are real. They pay out in terms of consumption good. Notice that in the competitive equilibria pro…ts Πt are zero at all times. We ignore Πt in what follows. No Ponzi-game condition t

∏ Rs 1 Bt = 0 t !∞ lim

s =1

Lecture 1

Households - Optimisation problem I

Given {Wt , Rt }, household solves max

fC t ,B t +1 gt∞=0

Ct + Bt +1 Ct

s.t. = Wt + Rt Bt 0

t

∏ Rs 1 Bt t !∞ lim

s =1

= 0

E

"

∞

∑ βt U (Ct )

t =0

# (7)

Lecture 1

Dynamic programming

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Basic idea: 1. Collapse in…nite horizon problem into two period problem . 2. Turn the constrained problem into an unconstrained one.

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Bellman equation (also known as functional equation) ∞

V (Bt ; Wt , Rt ) | {z }

= max Et

∑ β j U ( Ct + j )

j =0

value function

∞

= max U (Ct ) + βEt

∑ β U (Ct +j +1 )

j =0

V (Bt ; Wt , Rt )

=

max

fC t ,B t +1 g

f U ( Ct ) + β

j

!

Et [V (Bt +1 ; Wt +1 , Rt +1 )]g

s.t. Bt +1 I I

= Rt Bt + Wt

Ct

We have transformed in…nite horizon decision problem into recursive constrained two period problem. V (.) is by de…nition the maximized value of the discounted consumption stream. Wt and Rt are exogenous state variables (they follow a stochatic process that is exogenous to the household).

Lecture 1

First order condition from unconstrained problem I

Plug constraint into the utility function V (Bt ; Wt , Rt )

= max fU (Rt Bt + Wt B t +1

Bt + 1 )

(8)

+ βEt V (Bt +1 ; Wt +1 , Rt +1 )g I

Now we have speci…ed the problem such that instead of choosing today’s consumption Ct we choose tomorrow’s state Bt +1 .

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Assume that V (.) is di¤erentiable. The …rst order condition (with respect to Bt +1 ) becomes: U 0 (Ct ) | {z }

marginal cost of saving

+ βEt V 0 (Bt +1 ; Wt +1 , Rt +1 ) = 0 | {z } expected discounted shadow value of wealth

(9)

Lecture 1

The envelope condition I

Di¤erentiate the Bellman equation (8) with respect to Bt V 0 ( B t ; W t , Rt ) = Rt U 0 ( C t ) | {z }

direct e¤ect

+

0

0

U (Ct ) + βEt V (Bt +1 ; Wt +1 , Rt +1 ) {z | indirect e¤ect

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dBt +1 dBt }

Indirect e¤ect arises, because Bt a¤ects the otpimal choice of Bt +1 . But the …rst-order condition (9) tells that U 0 (Ct ) βEt V 0 (Bt +1 ; Wt +1 , Rt +1 ) = 0, and we can ignore the indirect e¤ect Thus V 0 ( B t ; W t , Rt ) = Rt U 0 ( C t )

(10)

Lecture 1

Stochastic Euler equation I

Utilising the envelope condition V 0 (Bt ; Wt , Rt ) = Rt U 0 (Ct ) and shifting it one period forward we get V 0 (Bt +1 ; Wt +1 , Rt +1 ) = Rt +1 U 0 (Ct +1 ) Taking expectations: Et V 0 (Bt +1 ; Wt +1 , Rt +1 ) = Et Rt +1 U 0 (Ct +1 )

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(11)

Combining (9) and (11) and re-arranging U 0 ( Ct )

=

1

=

βEt Rt +1 U 0 (Ct +1 ) U 0 ( Ct + 1 ) Et βRt +1 U 0 ( Ct ) βU 0 (C

(12)

)

This is a stochastic version of Euler equation. U 0 (Ct +)1 is stochastic t discount factor. Model is closed by imposing transversality condition and using in…nite horizon budget constraint. (holds in expectation)

Lecture 1

Stochastic Euler equation cont’d

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Intrepretation: Household equates the cost from saving one additional unit of today’s consumption (the loss of U 0 (Ct )) to the bene…t of obtaining more consumption tomorrow (saving additional unit of consumption today gives βEt [Rt +1 U 0 (Ct +1 )] more expected utility tomorrow).

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Characteristics of the solulution I

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Optimised consumption plan depends upon expected future wealth, as opposed to current income (PIH). Households prefer smooth (intertemporal) consumption pro…le. (Well, at least as long as U 0 (C ) > 0, U 00 (C ) < 0.)

Lecture 1

Stochastic Euler equation, cont’d I

Example: The utility function is CRRA U (C ) =

C1 η 1 η

00

where η = UU C measures the rate of relative risk aversions, and 1/η is the intertermporal elasticity of substitution. I

Then the consumption Euler equation U 0 (Ct ) = Et Rt +1 U 0 (Ct +1 ) takes the form C or

η

h i η = Et Rt +1 Ct +1

1 = Et β

Ct Ct +1

η

Rt + 1

Lecture 1

Aggregate resource constraint I

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Recall that we want to describe competitive equilibria to this economy. In competitive equilibria, all the resources of the economy are in use (at equilibrium prices). The economy’s total production is Yt = Zt Ktα Lt1 α . Output can be used for two purposes: Consumption and investment (It ). Investments are used to replace depreciated (broken) capital stock as well as to build new capital, so that It |{z}

Investment I

=

δKt |{z}

Depreciation

+ Kt +1 Kt = Kt +1 | {z }

(1

δ)Kt

(13)

Net change

Aggregate resource constraint then reads as: C t + It Ct + Kt +1 (1 | {z Demand

δ)Kt }

= Yt = Zt Ktα L1t α | {z } Suppy

(14)

Lecture 1

Competitive equilibrium De…nition Given initial asset endowment B0 , a competitive equilibrim is allocations for the representative household {Bt }t∞=0 , allocations for the representative …rm {Kt , Lt }t∞=0 and prices {Wt , rtK , rrt }t∞=0 such that a) household allocation solves the household problem (7), b) the …rm allocation problem (2) and c) markets clear such that Ct + Kt +1

(1

δ)Kt Bt Lt

= Zt Ktα L1t = Kt = 1

α

(15) (16) (17)

for all t=0,...T. I

Notice that there are 3 (T + 1) market clearing conditions, but only 2 (T + 1) prices that can be used to clear the markets. The market’s clear because of Walras Law: If there is excess supply in one market then there must be a excess demand elsewhere.

Lecture 1

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Either all markets are in equilibrium, or more than one is in disequilibrium, but we can’t have a situation where only one market is in disequilibrium.

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So, if in our case 2 markets clear, then 3rd will also clear.

Lecture 1

Competitive equilibrium cont’d The competitive equilibirum is characterized by the following equations: I

The consumption Euler equation from the household’s problem U 0 (Ct ) = βEt Rt +1 U 0 (Ct +1 )

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The calculation of return, derived from the …rm’s maximization problem Rt = αZt +1 (Kt +1 )α 1 + 1 δ (19) I

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Recall that Rt

1 + rrt = 1 + rtK

δ

The aggregate resource constraint (goods market equilibrium) Ct + Kt +1

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(18)

(1

δ)Kt = Zt Ktα Lt1

α

The labour market equilibrium condition Lt = 1

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(20)

(21)

Law of motion of total factor productivity log(Zt ) = (1

ρ) log Z + ρ log(Zt 1 ) + εt , εt is iid

(22)

Lecture 1

Competitive equilibrium cont’d I

We can characterize the equilibrium by three equations if we I I

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Lead eq. (19), and then plug into the Euler equation (18) Insert the labour market equilibrium (21) into the aggregate resource constraint (20)

The three equations are Et

U 0 (Ct ) = Et [αZt +1 (Kt +1 )α 1 + 1 βU 0 (Ct +1 ) | {z | {z }

marginal rate of substitution

δ)] }

(23)

additional production + capital left over after production

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Ct + Kt +1

(1

δ)Kt = Zt Ktα

(24)

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log(Zt ) = (1

ρ) log Z + ρ log(Zt 1 ) + εt , εt is iid

(25)

Lecture 1

Steady state I

We de…ne a (non-stochastic) steady state as an equilibrium where all variables are constant over time (Ct = Ct +1 = C¯ , Kt = Kt +1 = K¯ , Zt = Zt +1 = Z , εt = εt +1 = 0).

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In steady state, when Ct +1 = Ct = C the Euler equation 1 = Et βRt +1

U 0 ( Ct + 1 ) U 0 (Ct )

yields 1 = βR I

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

1 ; β

rr =

1 β

1

(26)

Lecture 1

Steady state I

Then using equations (23) and (19) we …nd that 1 β rr + δ

= [αZ (K )α α = αZ (K¯ )

1 1

+1

δ]

(use 26) 1

K¯ I

=

αZ rr + δ

1 α

(27)

(27) is modi…ed golden rule: Optimal ss capital stock is such that marginal product of capital (at K¯ ) equals the depreciation rate plus the time discount rate.

Lecture 1

Steady state cnt’d I

Optimal steady state level of consumption can be solved by using the resource constraint (15). Ct + Kt +1 Kt + δKt | {z }

= Zt Ktα

=0

C¯

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= Z K¯ α

δK¯

Steady state consumption equals steady state output minus depreciation Steady state investment equals depreciation I = δK¯

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"Great ratios" in steady state K α = ; rr + δ Y

I δK δ C rr + δ (1 α) = = α, = rr + δ rr + δ Y Y Y

Lecture 1

Alternative solution strategy: solve the social planner’s problem

The planner’s problem is of the form max

fC t ,K t +1 gt∞=0

E0

"

∞

∑ β t U ( Ct )

t =0

#

s.t.

= (1 log(Zt ) = (1 Kt +1

δ)Kt + Zt Ktα L1t

α

Ct

ρ) log Z + ρ log(Zt 1 ) + εt , εt is iid

Lecture 1

Planner’s problem cont’d I

The social planner’s value function V (Kt , Zt ) satis…es the recursive Bellman equation V (Kt ; Zt )

= max U (Zt Ktα + (1 K t +1

δ)Kt

Kt +1 )

+ βEt [V (Kt +1 ; Zt +1 )]

(28)

(Note: Lt = 1) I

First order condition U 0 (Ct ) + βEt V 0 (Kt +1 ; Zt +1 ) = 0 | {z } | {z } marginal cost marginal expected of investing shadow value of capital

(29)

Lecture 1

Planner’s problem cont’d Envelope condition I

Di¤erentiating the Bellman equation (28) with respect to Kt yields V 0 (Kt ; Zt )

= U 0 (Ct ) αZt Ktα | {z

1

direct e¤ect

+

+1

δ }

U 0 (Ct ) + βEt V 0 (Kt +1 ; Zt +1 ) | {z indirect e¤ect

dKt +1 dKt }

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But the …rst-order condition tells us that U 0 (Ct ) + βEt [V 0 (Kt +1 ; Zt +1 )] = 0, and the indirect e¤ect can be ignored

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Thus

V 0 (Kt ; Zt ) = U 0 (Ct ) αZt Ktα 1 + 1

δ

Lecture 1

Planner’s problem cont’d

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Leading the envelope condition by one period yields V 0 (Kt +1 ; Zt +1 ) = U 0 (Ct +1 ) αZt +1 Ktα+11 + 1

δ

and taking expectations gives h Et V 0 (Kt +1 ; Zt +1 ) = Et U 0 (Ct +1 ) αZt +1 Ktα+11 + 1

δ

i

(30)

Lecture 1

Planner’s problem cont’d I

Combining (30) and (29) gives rise to the familiar intertemporal maximization condition! Et

U 0 (Ct ) = Et [αZt +1 (Kt +1 )α 1 + 1 βU 0 (Ct +1 ) {z | | {z }

additional production + capital left over after production

marginal rate of substitution I

δ)] }

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

The resource constraint K t +1 = (1

δ)Kt + Zt Ktα

Ct

and the law of motion of total factor productivity log (Zt ) = (1 I

ρ) log Z + ρ log (Zt

1 ) + εt ,

εt is iid

These are exactly the same as the equations (23), (24) and (25), characterizing the competitive equilibrium!

Lecture 1

Planner’s problem cont’d

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The competitive equilibrium is equivalent to the socially optimal outcome (the solution to the planners problem). Perfect competition in all markets, prices adjust immediately, no externalities, a asymmetric information => the welfare theorems can be applied

Lecture 1

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 deterministic trend component). It is (relatively) easy to put growth into the model. The basic idea is to assume that labour force grows at some exogenous rate (n)

Lt

and that

= ( 1 + n ) t L0 , L0 = 1 = (1 + n )t Yt = Zt Ktα (ΛLt Lt )1

where ΛLt is labour augmenting technology.

α

(31)

Lecture 1

Fluctuations and growth in the model cnt’d I

Assuming that ΛLt

= (1 + g )t ΛL0 , ΛL0 = (1 + g )t

we can write production function as Yt = Zt Ktα ((1 + g )t Lt )1 I

α

(32)

Basic idea is then to reformulate the optimisation problems in terms of variables that are stationary (and constant in the non-stochastic balanced growth path). For instance, resource constraint becomes c˜t + (1 + g )(1 + n )k˜ t +1

(1

δ)k˜ t = y˜t = Zt k˜ tα

(33)

where Yt Ct Kt , c˜t = , k˜ t = , (1 + n )t (1 + g )t (1 + n )t (1 + g )t (1 + n )t (1 + g )t (34) are detrended variables. y˜t =

Lecture 1

Fluctuations and growth in the model cnt’d I

Lifetime utility can then also be expressed in terms of "de-trended" consumption . ∞

∑ β˜

t

U (c˜ )

t =0 I

For instance, if U (C ) = C1 I

1 γ

γ

, β˜ = β(1 + g )1

γ

Note: Ct is aggregate consumption, and Ct /(1 + n )t = c˜t (1 + g )t is per capital consumption. U 00 C U0

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γ is a parameter of relative risk aversion γ = 1?.

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Although we are not interested in explaining deterministic growth, having models with "balanced growth path" are more realistic. It also matters for calibration, as we learn later on.

= γ. What if