Macroeconomics 2Lecture 2 - Labor and Leisure Choice

Zsofia L. Barany

Sciences Po

2014 January

Recap of last lecture

Last lecture we:

I went over the business cycle facts

I considered the neoclassical growth model, and different waysof solving it

I looked at the effects of uncertainty

I replicated fairly well the co-movements in output,consumption and investment

I preview of the methods of solving such models

This lecture:

I to really assess the model, add labor/leisure choice → theRBC model, initially due to Prescott

I choose a special case → which we can solve in closed form

Baseline RBC model

Neoclassical growth model with three modifications:

I it’s in discrete time

I it’s stochastic

I it features a labor-leisure choice

competitive markets and no money

Production function

Yt = Kαt (AtLt)

1−α 0 < α < 1

goal: match the datain the data the capital share is constant ⇒ Cobb-Douglas

Technological progress

At = A∗t At

I deterministic component

A∗t = G tA

long-run non-stochastic log-linear trend, G > 1

I shock process

ln(At) = ρ ln(At−1) + εA,t

E (εA,t) = 0 and is iid

Capital accumulation

Kt+1 = (1− δ)Kt + Yt − Ct

Household objective function

U = Et

∞∑i=0

βiU(Ct+i , 1− Lt+i )

Ct is consumption and Lt is the fraction of time spent workingU1,U2 > 0 and U11,U22 < 0

Imagine the following:

I the economy is initially on its non-stochastic BGP

I there is a sudden realization of εA,t 6= 0

I this moves the economy away from its non-stochastic BGP

I over time the economy moves back to its BGP

I interpret the economy’s deviation from the BGP as a businesscycle

I does it look like what we see in the data?

Household behavior I.

inter-temporal FOC

U1(Ct , 1− Lt) = βEt (Rt+1U1(Ct+1, 1− Lt+1)) (1)

interpretation (should be familiar)

the household should not prefer any perturbation to the solution,i.e. should be indifferent:

I decrease consumption by ε, so decrease utility byU1(Ct , 1− Lt)ε

I save and get Rt+1ε next period, so an increase in expectedutility of Et (Rt+1U1(Ct+1, 1− Lt+1)) ε

Household behavior II.

intra-temporal FOC

U1(Ct , 1− Lt)Wt = U2(Ct , 1− Lt) (2)

interpretation

the household should not prefer any perturbation to the solution,i.e. should be indifferent:

I increase work by ε, so decrease in utility by U2(Ct , 1− Lt)ε

I extra wage: Wtε → increase consumption → increase inutility by U1(Ct , 1− Lt)Wtε

U1(Ct , 1− Lt)Wt = U2(Ct , 1− Lt)

if Wt was constant, then if (1− Lt) ↓ ⇒ U2(Ct , 1− Lt) ↑⇒ for intra-temporal FOC to hold we need:U1(Ct , 1− Lt)Wt ↑ ⇒ Ct ↓⇔ work more ⇒ consume less from the intra-temporal FOC

Barro and King (1984) insight: to have both consumption andlabor pro-cyclical (as in the data) need:

* pro-cyclical wage and/or* high substitutability between leisure and consumption

quantitative question: is the observed pro-cyclicality of the wageenough given the large pro-cyclicality of labor?

Balanced growth restrictions

To progress further we need to specify the utility function of thehousehold.

What can the utility function look like?

Imposing balanced growth path restrictions might help.What do they mean? And are they reasonable?

in the steady state:

I L, labor is constant ⇒ production side: we need laboraugmenting technological progress

I C and W are growing at rate G , which is the rate oftechnological progress

can identify a set of utility functions that allow a BGP when technis labor augmenting (King, Plosser, Rebelo (1988, JME))

two cases that are often used (where utility is separable in leisureand consumption):

1. U(C , 1− L) = lnC + b ln (1− L)

→ use this today

2. U(C , 1− L) = lnC + θ (1−L)1−γ

1−γ→ most used New-Keynesian specification, look at it later on

Are balanced growth restrictions reasonable?

Greenwood and Vandenbroucke: Hours worked: Long run trends,Figure 1, NBER working paper 11629

Are balanced growth restrictions reasonable?

Source: Federal Reserve Bank of St. Louis

Using the specification U(C , 1− L) = ln(C ) + v(1− L) we get thefollowing:

I the intra-temporal foc becomes:

Wt

Ct= v ′(1− Lt)

equalize the marginal utility of leisure to the wage times themarginal value of capital

I while the inter-temporal foc becomes:

1 = Et

(βRt+1

Ct

Ct+1

)this is the usual condition for consumption

What are the effects of a positive technological shock? It increasescurrent and future R and W .

I consumptionincome effect: people feel richer ⇒ consumption upsubstitution effect: saving is worth more ⇒ consumption downnet effect is probably consumption up

I leisureincome effect: people feel richer → they want to enjoy moreleisure → leisure upsubstitution effect: higher wage ⇒ leisure downnet effect depends on the relative strength of the two forces

* transitory shock → smaller wealth effect and strongersubstitution effect* permanent shock → it is possible that consumption goes up andemployment goes down

Employment effects another way

combine inter- and intra-temporal conditions and assume thatv(1− L) = b ln(1− L)

I the intra-temporal condition is:

Wt

Ct=

b

1− Lt

I using this in the inter-temporal condition we get:

1 = Et

(βRt+1

Wt

Wt+1

1− Lt1− Lt+1

)* transitory shock → Wt ↑ but not Wt+1 ⇒ (1− Lt)/(1− Lt+1) ↓→ employment increases today* permanent shock → Wt/Wt+1 pretty much constant ⇒(1− Lt)/(1− Lt+1) constant as well → employment does notchange

Back to our very special case

Two additional assumptions:

I δ = 1, full depreciation ⇒ almost like a two-period model

I separable log-utility:

U(Ct , 1− Lt) = ln(Ct) + b ln(1− Lt)

The two FOCs become:

1

Ct= βEt

(Rt+1

Ct+1

)(3)

andWt

Ct=

b

1− Lt(4)

As in the homework, using that Ct = (1− st)Yt , that

Rt = α(AtLtKt

)1−α+ (1− 1), and that Kt+1 = stYt , we can

manipulate (3) to get:

st1− st

= αβEt

(1

1− st+1

)the optimal saving rate has to be constant ⇒ st = αβ ⇒

Ct

Yt= 1− αβ

Using that CtYt

= 1− αβ and that Wt = (1− α)YtLt

in (4) gives

(1− α)YtLt

(1− αβ)Yt=

(1− α)

(1− αβ)Lt=

b

1− Lt

rearranging for Lt we get:

Lt =1− α

b(1− αβ) + 1− α

1. ⇒ constant working hours

2. very pro-cyclical wage

not very good news - far from where we want to be

Intuition:

I re-write (4) as: Lt = 1− bCtWt

I can think of Ct as capturing the income effects(remember: 1/Ct is the marginal value of wealth)Ct moves around a lot in cycles ⇒ the income effect is verylarge ⇒ people feel much richer ⇒ they want to consumemore and enjoy more leisure

I and of Wt as capturing the substitution effectshigher Wt ⇒ people want to work harder to take advantageof the higher wages

I here they exactly cancel each other out

in models where Ct is less variable, maybe substitution effect candominate ⇒ labor up in booms

Summary so far:

I consumption is too pro-cyclicaltoo pro-cyclical, moves one-for-one with Yt

I investment It = stYt = αβYt also pro-cyclicalbut not enoughinvestment in the data is super pro-cyclical, with constantsaving rate it is only pro-cyclicalwe need a pro-cyclical saving rate to match the data:

Var(ln It) = Var(ln st) + Var(lnYt) + 2Cov(ln st , lnYt)

for this a pro-cyclical interest rate is needed

effects of R on saving?r ↑ ⇒ inter-temporal substitution effect, Ct ↓⇒ income effect, Ct ↑

(in this model they cancel out)

I Wt is too pro-cyclical

Output dynamics

I constant saving rate and labor supply ⇒ like the Solow model

in discrete time: y∗t = Y ∗t

A∗t L

∗ =(αβG

) α1−α

, rearranged:

Y ∗t =

(αβ

G

) α1−α

A∗tL∗

I What we are interested in is ln Yt = lnYt − lnY ∗t . Assumethat initially we are in the steady state, ln Y0 = 0. Then wecan express

ln Yt = (1−α)(αt ln A0 + αt−1 ln A1 + ...+ α ln At−1 + ln At

)

One way to show this (normalize L∗ = 1 for simplicity):

Y1 = Kα1 A

1−α1 = (sY0)αA1−α

1 = (sY ∗0 )αA1−α1

=

(s( s

G

) α1−α

A∗0

)αA1−α1

=( s

G

)αGα( s

G

) α2

1−α(A∗0)α A1−α

1

=( s

G

)α(1−α)1−α

+ α2

1−αGα (A∗0)αA1−α

1

=( s

G

) α1−α

(A∗1)αA1−α1 =

( s

G

) α1−α

(A∗1)α(A∗1A1

)1−α=( s

G

) α1−α

A∗1A1−α1 = Y ∗1 A

1−α1

Now move to Y2 and keep going.

The economy’s reaction to a shock

experiment: there is one shock at time 0 and then never again

ln At = ρtεA,0

what is the economy’s log deviation from trend? what is theeconomy’s impulse response function?use formula from before:

ln Yt = (1− α)(αt ln A0 + αt−1 ln A1 + ...+ α ln At−1 + ln At

)= (1− α)

(αt + αt−1ρ+ ...+ αρt−1 + ρt

)εA,0

in the long run output goes back to the BGP level:

limt→∞

ln Yt = 0

plot: ε (green), technology (red), output (blue)

Higher persistence, higher ρ

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.3, α=0.33333

Higher persistence, higher ρ

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.5, α=0.33333

Higher persistence, higher ρ

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.7, α=0.33333

Higher persistence, higher ρ

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.9, α=0.33333

Higher capital share, higher α

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.5, α=0.16667

Higher capital share, higher α

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.5, α=0.33333

Higher capital share, higher α

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.5, α=0.5

Higher capital share, higher α

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.5, α=0.66667

Higher capital share, higher α

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ρ=0.5, α=0.83333

The graphs show

I hump shape in the impulse response of Yt

→ this is due to the increase in total investment, since thesaving rate is constant, and output increasesso there is amplificationthis comes in part from α, but not onlyoutput today is only higher because A is higher, outputtomorrow is higher because K is higher (due to α) and A ishigher (due to ρ)

I output stays above trend longer than the shock (which is justone period) → so there is persistencetwo sources:

1. exogenous persistence, ρ > 0, this is the intrinsic persistence ofthe technology shockthis determines mostly the persistence and it is chosen by us

2. endogenous persistence, α > 0, which works throughinvestment

Conclusions so far

special case does not look good

I not enough persistence: very quick return to the BGP unlessvery high ρ

I not enough amplification

I need a more general model

I and possibly need to look for other sources of shocks

need to:

I get consumption to be less pro-cyclicaland investment to be more pro-cyclical

I get labor effort to respond

I get more endogenous persistence