Master in EconomicsLecture 4: IRBC and Heterogenous Firms

International Business Cycle

Jose Ignacio LopezHEC Paris

October 2015ENSAE

Heterogeneity

• The extensive margin is at the core of trade models withlove-for-variety

• Exporting rms tend to be larger and more productive

• Firm Heterogeneity helps to explain potential gains inproductivity after trade liberalizations. Can they help to explaininternational business cycles?

Basic Facts about Exporting Firms

• Only 21% of manufacturing plants in the US export to othermarkets

• Exporting rms serve both domestic and external markets.Moreover, around 2/3 of exporters sell than 10% of theirproduction in foreign markets

• Exporting rms are larger (measured in assets, labor,production) and more productive.

• These facts suggest that within industry reallocation can be animportant force in trade liberalizations

Heterogenous Firmsand the Aggregate Economy

• Hopenhayn (1992) studies a model with heterogenous rms,entry and exit and establishes the conditions for the existenceof stationary equilibrium under perfect competition

• Melitz (2003)uses a heterogenous rms to account for thereallocation of factors within an industry after a tradeliberalization

Melitz (2003) -Demand• A CES utility function over a continuum of goods index

byz ∈ Zt

Ct =

[∫z ∈Zt

ct (z )θ−1θ dz

] θθ−1

Zt is the set of available varieties and θ is the elasticity ofsubstitution.

• Under perfect competion, the demand for an specif variety

ct (z ) =

[pt (z )

Pt

]−θCt

• Aggregate Price:

Pt =

[∫z ∈Zt

pt (z )1−θdz

] 11−θ

Production

• Continuum of Firms. Labor is the only factor of production• Firms have dierent levels of productivity: ct (z ) = zl (z ) and

have a xed production cost f .• Wages are normalized to one. The total cost of the rm is:TC (z ) = f + c/z

• Optimization problem max d (z ) = p (z ) c (z ) − c (z )z − f .

Using the demand function:

p (z ) =θ

θ − 11z

• If PC = R, revenues of a specic rm:r (z ) =

(θθ−1

)1−θ(Pz )θ−1 R

• For any two rms, z1 and z2, r (z1)r (z2)=

(z1z2

)θ−1

Aggregation• An equilibrium in this economy is fully characterized by a mass

of rms (varieties) M and a distribution of productivities µ (z )• The aggregate price index is:

Pt =θ

θ − 1M

11−θ

[∫0zθ−1µ (z ) dz

] 11−θ

• We can dene the average productivity of rms:

z =

[∫0zθ−1µ (z ) dz

] 11−θ

• Use this as summary statistic:

P =θ

θ − 1M

11−θ z

C =θ − 1θ

Mθ

1−θ z

Endogenous Distribution• There is a large pool of prospective entrants• Firms pay a sunk intial cost of entry fe in order to draw from

the common distribution g (z ) with positive support. G (z )denotes the cummulative distribution

• Conditional on the draw rms decide whether to stay andproduce or to leave the market.

• There is a constant probability of exogenous death δ• The value of a rm is: v (z ) =

∑t=0 (1 − δ )

t d (z ) − fe

• Zero-prot will determine a productivity threshold

µ (z ) =

g (z )1−G (z ) if z ≥ z

0 otherwise

z =

[1

1 − G (z )

∫zzθ−1g (z ) dz

] 11−θ

Zero Prot Condition• Zero prot implies: r (z ) =

(θθ−1

)1−θ(Pz )θ−1 R = f

• Ratio of revenues: r (z )r (z ) =

(zz

)θ−1• Prots average rm:

d (z ) = r (z ) − f = f *,

(z

z

)θ−1− 1+

-• Ex-ante value of rm:

ve = (1 − G (z ))∑t=0

(1 − δ )t d (z ) − fe = (1 − G (z ))d (z )

δ− fe

• Free entry implies:ve = 0

d (z ) =δ fe

1 − G (z )

Steady-State

• The mass of entrants Me

• In steady-state:[1 − G (z )]Me = δM

• Labor-Market Clearing

L = Lp + Le

Le = Me fe = Me[1 − G (z )] d (z )

δ= δM

[1 − G (z )] d (z )

δ=

∏

International Trade

A CES utility function over a continuum of goods index byz ∈ Zt

Ct =

[∫zat (z )

θ−1θ dMt (z ) +

∫zx∗t (z )bt (z )

θ−1θ dM∗t (z )

] θθ−1

at (z ) =

[pat (z )

Pt

]−θCt (1)

bt (z ) =

[pbt (z )

Pt

]−θCt (2)

Pt =

[∫zpat (z )

1−θdMt (z ) +

∫zx∗t (z )pbt (z )

1−θdM∗t (z )

] 11−θ

(3)

Production

• Continuum of Firms. Labor is the only factor of production• Firms have dierent levels of productivity: ct (z ) = zl (z ) and

have a xed production cost f .

d (z ) = maxpat ,p

∗at ,a(z ),a

∗ (z ),x, lpa (z )a(z ) + xp

∗a (z )a

∗ (z ) − l − f ‘ − xfx

a(z ) + xτa∗ (z ) = zl (z )

• Prot maximization problem of each variety producer gives:• p(z ) = θ

θ−11z p∗a (z ) =

1ε

θθ−1

τz =

1ε τp(z )

• d (z ) = 1θ

[p(z )P

]1−θCt − f dX (z ) = Qt

[p∗a (z )P

]1−θC ∗ − fx

Entry into Domestic and External Markets• Zero-prot will determine a productivity threshold

µ (z ) =

g (z )1−G (z ) if z ≥ z

0 otherwise

z =

[1

1 − G (z )

∫zzθ−1g (z ) dz

] 11−θ

• The exporting costs will determine a threshold for exporting

Qt

[p∗a (zx )

P

]1−θ

C ∗ = fx

• The probability of exporting (conditional on succesful entry):

x =1 − G (zx )

1 − G (z )

Aggregation

• As before we can compute the average productivity of all rmsand exporting rms

• The aggregate price index is:

P =θ

θ − 1

[z

1−θ

d + τ 1−θ z∗1−θ

x

] 11−θ

• We can dene the average productivity of rms:

z =

[∫zzd

θ−1Mdµ (z ) dz

] 11−θ

zx =

[∫zx

zxθ−1Mxµ (z ) dz

] 11−θ

M = Md + xMx

Impact of Trade

• What happens when the economy moves from the closedeconomy to the open economy?

• Firms nd a new source of prots. Only more productive rmsexport to the foreign market. Higher prots induce higherentry (the value of the average rms increases)

• There is higher labor demand in the economy. As labor supplyis xed, wages increase forcing the least productive rms toexit.

• Aggregate prices fall (welfare improves) because of the positivereallocation torwards the most productive rms.

Ghironi and Melitz (2005)

• Ghironi and Melitz present a general equilibrium, two-countrymodel with heteregeneous rms, which face sunk entry cost inthe domestic market and both xed and variables export costs(not xed costs of production)

• This paper goes along in the tradition of trade literature relatedto the Harrod-Balassa-Samuelson eect

• The HBS eect can be dened as: the observation thatconsumer price levels in wealthier countries are systematicallyhigher than in poorer ones

• The classical explanation for this phenomenon has been thatproductivity growth-rates vary more by country in the tradedgoods’ sectors than in other sectors (the Balassa-Samuelsonhypothesis).

GM Model

• In this model the division between traded and nontradedsectors is endogeneously determined and evolves over time

• Positive aggregate productivity shocks, expand the tradedsector and translates into higher domestic prices. Therefore, themodel replicates the HBS eect without relying on specicshocks to the traded sector

• The inclusion of per-unit export costs also allow the model toexplain for persistent deviations from PPP, which also show upin cross-country price dierences for tradable goods (Engel1993, 1999).

Households• Households solve (similarly for foreign household):

maxEt

∞∑s=t

βs−t

C1−γs

1 − γ

Bt+1 + vtNH,txt+1 + Ct = (1 + rt )Bt + (dt + vt )ND,txt + wtL∫ω ∈Ωt

pt (ω)ct (ω)dw = PtC

Ct =

[∫ω ∈Ωt

ct (w )θ−1θ dω

] θθ−1

• vt : date t price of claim to future prot stream of the mutualfund.

• dt : average total prot (to be dened later)• NH,t ≡ ND,t + NE,t Survivers: ND,t+1 = (1 − δ )NH,t .

Production Side

• Ex ante identical rms pay xed entry cost of fE,t (f ∗E,t ) eective

labor units.→ Wt fE,tPtZt

(W ∗

t f∗E,t

P∗t Z∗t

)• Upon entry, productivity is drawn from G (z ), z ∈ [zmin,∞)

(identical distribution for foreign rms).• Relative productivity is kept until death, which occurs with

probability δ• Fixed per-period export cost of fX,t (f

∗X,t ) units of eective

labor→ Wt fX,t

PtZt

(W ∗

t f∗X,t

P∗t Z∗t

)• Iceberg Cost: τt ≥ 1 (τ ∗t ≥ 1)

Optimization Problem

• Prot maximization problem of each variety producer gives:• PD,t (z ) =

θθ−1

WtZtz

; PX,t (z ) =1εt

θθ−1

τtWtZtz= 1

εtτtPD,t (z )

• Expressing prices in real terms, relative to Price Index indestination market:

ρD,t =PD,t (z )

Pt=

θ

θ − 1wt

Ztz; ρX,t =

PX,t (z )

P∗t= Q−1t τtρD,t (z )

• where Qt =εtP∗tPt

denotes the real exchange rate.• Similarly for the foreign country

Prots and Exporting Choice• Prots in real terms relative to Price Index of where the rm is

located (similarly for foreign rms):

dD,t (z ) =ΠD (z )

Pt= ρD,t (z )cD,t (z ) −

wt

ZtzcD,t (z )

dD,t (z ) =1θ

[ρD,t (z )]1−θ Ct

dX,t (z ) = QtρX,t (z )c∗X,t (z ) −

wt

Ztzc∗X,t (z ) −

wt

ZtzfX,t

dX,t = Qt [ρX,t (z )]1−θ C ∗t −

wt

ZtzfX,t

• Export Decision (similarly for foreign country):• A rm with productivity z exports ⇐⇒ z ≥ zX,t , wherezX,t = inf

z : dX,t (z ) > 0

• Endogenously determined non-traded sector: ex-ante each

variety is tradeable, but some will not be traded ex-post

Firm Averages

• In every period, a mass ND,t (N∗D,t ) produces in each country

• NX,t = [1 − G (zX,t )]ND , N∗X,t =[1 − G (z∗X,t )

]N∗D

• zD =[∫ ∞

zminzθ−1dG (z )

] 1θ−1 , zX,t =

[∫ ∞zX ,t

11−G (zX ) z

θ−1dG (z )] 1θ−1 (analogoulsy for the foreign

country)• Convenient denition of averages allows for:• dD,t = dD,t (zD ) where dD,t =

∫ ∞zmin

dD,t (z )dG (z )

• dX,t = dX,t (zX,t ) where dX,t =∫ ∞zx,t

11−G (zX,t )

dX,t (z )dG (z )

• Thus dt = dD,t + [1 − G (zX , t )] dX,t represent average totalprots of home producers (similarly for foreigners)

Aggregate Prices• Recall denition of welfare based price index:

P =

ND,t

∫ ∞

zmin

p1−θ

D + N∗X,t

∫ ∞

z∗x,t

11 − G (z∗

X,t)p∗1−θX,t

dG (z )

11−θ

Denition of Averages is also such that;∫ ∞

zmin

pD,t (z )1−θdG (z ) = [pD,t (zD )]

1−θ

∫ ∞

z∗x,t

11 − G (z∗

X,t)p∗X,t (z )

1−θdG (z ) =[p∗X,t (z

∗X,t )

]1−θ

• Thus, Price Index becomes:

Pt =

[ND,t [pD,t (zD )]

1−θ+ N∗X,t

[p∗X,t (z

∗X,t )

]1−θ] 1

1−θ

or

1 =[ND,t [ρD,t (zD )]

1−θ+ N∗X,t

[ρ∗X,t (z

∗X,t )

]1−θ]

Free Entry

• Unbounded mass of prospective forward looking entrants inboth countries

• Entrants at time t start producing at t+1

• Free Entry Condition:wZtfE,t = vt = Et

∑∞s=t+1 [β (1 − δ )]s−t

(CsCt

)−γds

• Law of Motion for mass of rms: ND,t = (1 − δ ) [ND,t−1 + NE,t ]

Parameterization

• Pareto Distribution: G (z ) = 1 −(zminz

)k. Several simplifying

implications• k indexes dispersion of productivities: high values of k imply

productivities are more concentrated towards zmin (drawpicture)

• zD = vzmin , zX,t = vzX,t and NX,t

ND,t= 1 − G (zX,t ) =

(vzminzX,t

)kwhere v =

k

[k−(θ−1)]

1θ−1

• Zero export prot condition for cuto rm: dX,t (zX,t ) = 0implies

• dX,t = (θ − 1)(vθ−1

k

)fX,twt

Zt

Real Exchange Rate

• Welfare-based price indexes don’t correspond exactly to the CPI

• They can be break into Pt = N1

1−θt Pt where Nt = ND,t + N

∗X,t .

• We can re-write a dierent price index as:

Pt =

ND,t

ND,t + N∗X,t

[(pD )]1−θ+

N∗X,t

ND,t + N∗X,t

[p∗x

]1−θ

11−θ

• Using this denition, it is possible to re-dene the RER

Qt =εt P∗tPt

Qt = Qt

(NtN∗t

) 11−θ

• It can be the case that if the product variety in the home marketis large enough, we can have, for instance, that Qt > 1 whileQt < 1

RER and TOT

• The terms of labor measures the relative cost of eective unitsof labor of one country in terms of the other: TOLt =

ε (W ∗t /Z

∗t )

(Wt/Zt )

• A decrease in TOL, implies that labor in the home country hasbecome relatively more expensive.

• The RER is aected by TOL as described by:

Q1−θt =

[N∗D,t

N∗t(TOLt )

1−θ +NX,t

N∗t(τt

zDzX,T

)1−θ]

[ND,t

Nt+

N∗X,t

Nt(TOLtτ ∗t

zDz∗X,T

)1−θ]

Changes in the RER

1 Changes in TOL translates into changes in home and domesticprices (potential source of dierences for prices of nontraded

goods across countries) (P∗D,t

(z )

PD,t (z )=

θθ−1

W ∗tZ∗t z

θθ−1

WtZt z

)

2 Changes in tradable prices ( either by changes in taris orexport productiviy cut-os)

3 Expenditure switching between domestic and import varities

Calibration

Parameter Value Target/Sourceβ 0.99 standard RBC choiceθ 3.8 US plant and Macro Trade Dataγ 2 standard RBC choicek 3.4 Std. Dev. log of US plant salesδ 0.025 US job destruction ratesfE 1 Only fX

fEmatters

fX 21% of US plants export.

HBS Eect

Real Exchange Rate

• Welfare-based price indexes don’t correspond exactly to the CPI

• They can be break into Pt = N1

1−θt Pt where Nt = ND,t + N

∗X,t .

• We can re-write a dierent price index as:

Pt =

ND,t

ND,t + N∗X,t

[pD,t (zD )]1−θ+

N∗X,t

ND,t + N∗X,t

[p∗X,t (z

∗X,t )

]1−θ

11−θ

• Using this denition, it is possible to denitionthe RER, closer

to the CPI-measured one. Qt =εt P∗tPt

Qt = Qt

(NtN∗t

) 11−θ

• It can be the case that if the product variety in the home marketis large enough, we can have, for instance, that Qt > 1 whileQt < 1

ReferencesHopenhayn, H. A. (1992). Entry, exit, and rm dynamics in long run

equilibrium. Econometrica: Journal of the Econometric Society,1127–1150.

Melitz, M. (2003). The impact of trade on intra-industry reallocationsand aggregate industry productivity. Econometrica 71(6),1695–1725.