Heterogeneous Firms - University of Oxfordusers.ox.ac.uk/~econ0211/teaching/trade-mphil/... ·...

Heterogeneous FirmsNotes for Graduate Trade Course

J. Peter Neary

University of Oxford

January 30, 2013

J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 1 / 29

Plan of Lectures

1 Empirical Background

2 Overview of the Melitz Model

3 Equilibrium in Autarky

4 Effects of Trade

5 Extensions


Empirical Background

Plan of Lectures




4 Effects of Trade

5 Extensions




The Data Revolution: Micro-Data on Firms.

Evidence totally orthogonal to traditional theory:

Exporting firms are:

Rare!: Very few firms export;... and those that do sell most of their output domestically;

Larger;More productive;... ex ante (”selection into exporting”), not ex post (”learning by

exporting”) (Clerides et al. QJE 1995; Bernard-Jensen JIE 1999).OlderPay higher wages

Effects of trade liberalisation:

Forces least productive firms to exit (Bernard and Jensen, JIE 1999).Encourages market share reallocation towards more productive firms;... and so raises aggregate productivity (Pavcnik REStud 1999,

Bernard-Jensen-Schott JIE 2006).


Overview of the Melitz Model

Plan of Lectures


2 Overview of the Melitz ModelDynamic Industry EquilibriumContinuum CES Preferences


4 Effects of Trade

5 Extensions


Overview of the Melitz Model Dynamic Industry Equilibrium

Dynamic Industry Equilibrium

Monopolistic competition with CES preferences; so what’s new?

Melitz (2003), Helpman-Melitz-Yeaple (2004) [HMY]

Population of ex ante identical firms.

Firms face two sources of uncertainty:1 Uncertain productivity ϕ / cost c ; quality another interpretation.

Drawn from a known distribution with pdf g(

1c

)with positive support

over (0, ∞) and associated cdf G(

1c

). [g

(1c

)= G ′

(1c

)]

To learn its c, a firm must pay a sunk cost of entry fe .

2 Uncertain lifetime if it chooses to enter.

Exogenous probability δ of ”death” i.e., a bad shock that will cause itto exit.HMY and Chaney (AER 2008) ignore the second source, assuming thata successful entrant produces for one period only.This simplifies the model a lot without affecting its main predictions,and many authors have followed them; but it is insightful to solve thefree entry case in full.


Overview of the Melitz Model Continuum CES Preferences

Continuum CES Preferences

CES preferences with a continuum of goods: little new:

U =[∫

ω∈Ω q (ω)θ dω]1/θ

,

0 < θ = σ−1σ < 1, σ = 1

1−θ > 1

⇒ Optimal consumption: q (ω) =[p(ω)P

]−σQ Q = U

Optimal expenditure: r (ω) ≡ p (ω) q (ω) =[p(ω)P

]1−σR

R = PQ =∫

ω∈Ω r (ω) dω

Price index: P =[∫

ω∈Ω p (ω)1−σ dω] 1

1−σ


Equilibrium in Autarky

Plan of Lectures



3 Equilibrium in AutarkyProductionEntryEquilibrium Selection in Autarky: FigureProductivity of EntrantsAverage Productivity, Prices and ProfitsZCP (Zero Cutoff Profit) ConditionFE (Free Entry) ConditionIndustry Equilibrium: FigureEquilibrium in Autarky: RecapGeneral Equilibrium in Autarky: Details

4 Effects of Trade

5 Extensions


Equilibrium in Autarky Production

Production

Firms have different productivities ϕ (inverse of variable costs c).

All infra-marginal firms make positive profits; otherwise little new:

Labour the only factor, with w = 1: TC (c) = f + cq (c)

Profit maximisation implies: p (c) = σσ−1 c , q (c) =

(σ

σ−1 c)−σ

Pσ−1R

Price-cost margin: p (c)− c = 1σ−1 c = 1

σ p (c)

Revenue: r (c) =(

σσ−1 c

)1−σPσ−1R =

(σ−1

σ1c P)σ−1

R

Variable profit:

r (c)− cq (c) = [p (c)− c ] q (c) = 1σ r (c) =

(σ−1

σ1c P)σ−1

Rσ

Total profit: π (c) = 1σ r (c)− f =

(σ−1

σ1c P)σ−1

Rσ − f

Higher productivity firms have higher output, revenue and profits.They also charge lower prices.Ratios (rankings) of p, q, and r depend only on productivities:

p(c1)p(c2)

= c1c2

q(c1)q(c2)

=(c2c1

)σand

r (c1)r (c2)

=(c2c1

)σ−1


Equilibrium in Autarky Entry

Entry

To be or not to be? Depends on firm’s expected value given ϕ:

V (ϕ) = max

0,

∞∑0(1− δ)t π (ϕ)

= max

0, 1

δ π (ϕ)

.

This is steady-state analysis; no learning-by-doing etc.Probability of death acts just like a discount factor.

So: Entry occurs IFF V (ϕ) ≥ 0⇔ π (ϕ) ≥ 0⇔ ϕ ≥ ϕ∗ where:

ϕ∗ : π (ϕ∗) = 0 (1)

So far, very like homogeneous-firms model; but there:π = 0 holds for all firms (since they are homogeneous);This pins down q for all firms

... and market-clearing pins down mass of firms n.

Here: π (ϕ∗) = 0 is one equation in ϕ∗ and (through P) n.It determines q (ϕ∗) but there are many other q (ϕ) ...So: we need more equations ...


Equilibrium in Autarky Equilibrium Selection in Autarky: Figure

Equilibrium Selection in Autarky

c

11 c1*)(c

f

EnterExit EnterExit


Equilibrium in Autarky Productivity of Entrants

Productivity of Entrants

Equilibrium distribution of firm productivities:

Ex ante, it continues to be g (ϕ).Ex post, distribution of entrants’ productivities is different: µ (ϕ).Probability of a bad draw is G (ϕ∗); so probability of entry is1− G (ϕ∗)Hence distribution of ϕ on [ϕ∗, ∞), conditional on successful entry, is:

µ (ϕ) =

g (ϕ)

1−G (ϕ∗) if ϕ ≥ ϕ∗

0 otherwise(2)

Aggregate price: P =[∫ ∞

0 p (ϕ)1−σ Mµ (ϕ) d ϕ] 1

1−σ

= M1

1−σ p (ϕ) M: mass of firms.

where: ϕ (ϕ∗) ≡[∫ ∞

ϕ∗ ϕσ−1µ (ϕ) d ϕ] 1

σ−1

Average productivity; (strictly, a ”weighted symmetric mean”).


Equilibrium in Autarky Average Productivity, Prices and Profits

Average Productivity, Prices and Profits

Proof that P = M1

1−σ p (ϕ)

P =[∫

ω∈Ω p (ω)1−σ dω] 1

1−σ=[∫ ∞

0 p (ϕ)1−σ Mµ (ϕ) d ϕ] 1

1−σ

Recall:p(ϕ1)p(ϕ2)

= ϕ2ϕ1

so: p (ϕ) = p (ϕ)ϕϕ

⇒ P = M1

1−σ

[∫ ∞0 p (ϕ)1−σ ϕ1−σ

ϕ1−σ µ (ϕ) d ϕ] 1

1−σ

= M1

1−σ p (ϕ) ϕ[∫ ∞

0 ϕσ−1µ (ϕ) d ϕ] 1

1−σ

= M1

1−σ p (ϕ) from defn. of ϕ (Careful with exponents!) QED

ϕ is a ”sufficient statistic” for the industry.In particular: Average profits π = π (ϕ);

Proof: π ≡∫ ∞

ϕ∗ π (ϕ) µ (ϕ) d ϕ; BUT: π (ϕ) = 1σ r (ϕ)− f

= 1σ

∫ ∞ϕ∗ r (ϕ) µ (ϕ) d ϕ− f ; BUT:

r (ϕ)r (ϕ)

=(

ϕϕ

)σ−1

= 1σ r (ϕ)

(1ϕ

)σ−1 ∫ ∞ϕ∗ ϕσ−1µ (ϕ) d ϕ− f ; BUT: Recall defn. of ϕ.

= 1σ r (ϕ)− f QED


Equilibrium in Autarky ZCP (Zero Cutoff Profit) Condition

ZCP (Zero Cutoff Profit) Condition

Given π = 1σ r (ϕ)− f and ϕ (ϕ∗), π is a function of ϕ∗.

It’s even neater than that, recalling thatr (ϕ)r (ϕ∗) =

[ϕ(ϕ∗)

ϕ∗

]σ−1;

⇒ π = 1σ r (ϕ∗)

[ϕ(ϕ∗)

ϕ∗

]σ−1− f ; BUT: π (ϕ∗) = 1

σ r (ϕ∗)− f = 0;

⇒ π =

[ϕ (ϕ∗)

ϕ∗

σ−1

− 1

]f [ZCP] (3)

Slope in π, ϕ∗ space depends on two competing effects:1 Higher ϕ∗ raises average productivity, and so profits, of surviving firms:

ϕ′ (ϕ∗) > 0.A selection effect, not a firm-level productivity effect

2 Higher ϕ∗ means tougher competition: profits are decreasing in rivals’productivity.

(2) dominates for many distributions: ZCP downward-sloping.e.g., fat-enough tails: lognormal, exponential, etc.

(1) and (2) exactly cancel for Pareto: G (ϕ) = 1−(

bϕ

)k: ZCP flat.


Equilibrium in Autarky FE (Free Entry) Condition

FE (Free Entry) Condition

Expected PV of profits must equal sunk cost of entry:∫ ∞0 v (ϕ) g (ϕ) d ϕ = fe ; BUT: v (ϕ) = 1

δ π (ϕ) for ϕ ≥ ϕ∗;otherwise = 0;⇒ 1

δ

∫ ∞ϕ∗ π (ϕ) g (ϕ) d ϕ = fe ; BUT: Recall distribution of entrants’

productivities;⇒ 1

δ [1− G (ϕ∗)]∫ ∞

ϕ∗ π (ϕ) µ (ϕ) d ϕ = fe ; BUT: Recall defn. of π;

⇒ π =δfe

1− G (ϕ∗)[FE] (4)

So: Average industry profits rise with δ (think Grim Reaper), fe (thinklarger firms) and ϕ∗ (higher productivity cutoff).

(3) and (4): Two equations in two unknowns, π and ϕ∗;

So they are determined by fe , f , δ, and g (ϕ) only.

Melitz shows that FE must be cut by ZCP only once from above;

So equilibrium is unique and stable.


Equilibrium in Autarky Industry Equilibrium: Figure

Industry Equilibrium in Autarky

FE

ef

ZCP

*


Equilibrium in Autarky Equilibrium in Autarky: Recap

Equilibrium in Autarky: Recap

The story so far:

p (ϕ) = σσ−1

1ϕ P = M

11−σ p (ϕ) = M

11−σ σ

σ−11ϕ

π (ϕ) = 1σ r (ϕ)− f π = 1

σ r (ϕ)− f

r (ϕ) =(

σ−1σ ϕP

)σ−1R

r (ϕ) =(

σ−1σ ϕP

)σ−1R =

(M

11−σ

)σ−1R = R

M

This implies that: π = 1σRM − f

σ, f given; π and ϕ∗ determined by (3) and (4).

Finally: R determined by aggregate budget constraint: R = wL = L.

N.B. This is the first, and only, place where GE appears: see next page.BUT: it is crucial: without induced rise in real wage in GE, theselection effects of trade would not arise in the model.

Only remaining unknown is M, which is therefore: M = L(π+f )σ .

Compare Krugman: M = Lcy+f from LME; = L

f σ from CES.

Since y = (σ− 1) fc → cy + f = (σ− 1)f − f .


Equilibrium in Autarky General Equilibrium in Autarky: Details

General Equilibrium in Autarky: Details

Stationary equilibrium: Aggregate variables stay constant.

So, mass of new entrants each period Me must be such that mass ofsuccessful entrants equals mass of exiters: [1− G (ϕ∗)]Me = δM.Successful entrants and failing incumbents have the same productivitydistribution;So: equilibrium distribution µ (ϕ) is not affected by simultaneous entryand exit.

Finally, how come R = L? What happened to profits?

The trick is that sunk entry costs also require labour.So: L = Lp + Le ; aggregate labour used for production and investment(in entry).But: wLp = R −Π;

while: wLe = wMe fe = w δM1−G (ϕ∗) fe = Mπ = Π

So, with w = 1, L = R −Π + Π = R.


Effects of Trade

Plan of Lectures




4 Effects of TradeTrade CostsFirms in Home and Export MarketsEquilibrium Selection in Trade: FigureThe ZCP Locus with TradeIndustry Equilibrium in Autarky and Trade: FigureAdjustment to Trade LiberalisationComparing Trade and AutarkyComparing Trade and Autarky (cont.)

5 ExtensionsJ.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 19 / 29

Effects of Trade Trade Costs

Trade Costs

Now: replicate the home economy without trade costs:

n: # foreign countries; all identical, so FPE holds, w = 1 in all.Budget constraint is now: R = (n + 1) L.All firms will export, trade does not affect average productivity.(3) and (4) continue to determine the same equilibrium π and ϕ∗.

[Think Krugman 1979: n + 1 > 0 ⇒ M = n + 1.]

So, we need trade costs; 2 kinds in fact:

1 Iceberg variable costs: τ2 Fixed cost of exporting fx ; incurred after firm learns its ϕ.


Effects of Trade Firms in Home and Export Markets

Firms in Home and Export Markets

Firms charge constant mark-ups in both domestic and exportmarkets.

So: Domestic revenue: rd (ϕ) =(

σ−1σ ϕP

)σ−1R as before

Revenue in export market j : rX (ϕ) = τ1−σ(

σ−1σ ϕPj

)σ−1Rj

Symmetry ⇒ Total exporter revenue:(1 + nτ1−σ

)rd (ϕ)

Profits on domestic sales: πd (ϕ) =rd (ϕ)

σ − f .This = 0 determines threshold ϕ∗ as before.

Profits in an export market: πX (ϕ) =rX (ϕ)

σ − fX = τ1−σrd (ϕ)σ − fX .

This = 0 determines a new threshold ϕ∗X .

Threshold ratio:rd(ϕ∗X )rd (ϕ∗) = τσ−1 fX

f BUT:rd(ϕ∗X )rd (ϕ∗) =

(ϕ∗Xϕ∗

)σ−1

⇒ ϕ∗X = τ

(fXf

) 1σ−1

ϕ∗ (5)

i.e., sorting as in data (ϕ∗X > ϕ∗) IFF τ(fXf

) 1σ−1

> 1⇔ τσ−1fX > f .

We assume this holds from now on. (A little unsatisfactory ... )


Effects of Trade Equilibrium Selection in Trade: Figure

Equilibrium Selection in Trade

c

cx

11 c1*)(c 1* )( xc )( x

f

f ExportHome SalesOnlyxf

EnterExit

pOnly

EnterExit


Effects of Trade The ZCP Locus with Trade

The ZCP Locus with Trade

Probability of entry = 1− G (ϕ∗) as before;

Probability that an entrant exports: pX =1−G(ϕ∗X )1−G (ϕ∗)

Average expected profits of an entrant:π = πd (ϕ) + pX nπX (ϕX )

Here: ϕX is the weighted average productivity of exporters; so:

π(ϕ∗)=

[ϕ(ϕ∗)

ϕ∗

σ−1

−1

]f +pXn

[ϕX (ϕ∗)ϕ∗X (ϕ∗)

σ−1

−1

]fX [ZCPt ] (6)

This is clearly greater than in autarky; i.e., π (ϕ∗) > πa (ϕ∗) for anyarbitrary ϕ∗.i.e., ZCP shifts up relative to autarky.Finally: FE locus is unaffected.So: ϕ∗ > ϕ∗a and π (ϕ∗) > πa (ϕ∗a)


Effects of Trade Industry Equilibrium in Autarky and Trade: Figure

Industry Equilibrium in Autarky and Trade

FE)( *a

*

)( a

)( *t

ZCP - Trade

)( *aaa

efZCP - Autarkyy

**a

*t


Effects of Trade Adjustment to Trade Liberalisation

Adjustment to Trade Liberalisation

Result that ϕ∗ > ϕ∗a matches the data:

Trade causes marginal firms to exit;Selection effect of trade;Why? NOT a competition effect through demand side.

Remember: CES: Firm size and therefore π fixed by costs.Ans.: Labour market adjustment is crucial (though in the background).Increase in profitable opportunities for relatively more productive firms⇒ more entry⇒ Increase in labour demand⇒ Increase in real wage w

P ; i.e. fall in P⇒ Least productive firms in autarky can no longer make profits.


Effects of Trade Comparing Trade and Autarky

Comparing Trade and Autarky

Mass of active home firms falls: M < Ma; proof:

Total revenue of domestic producers: R = LAverage revenue: r =

∫ ∞0 r (ϕ) µ (ϕ) d ϕ

= σ (π + f + pX nfX )

BUT: R = Mr ⇒ M = Lσ(π+f +pXnfX )

< Ma

(Recall that Ma = Lσ(πa+f )

and π > πa.

What about total number of varieties?

# firms selling in any one market: Mt = (1 + npX )M

Likely (except for very high τ) that Mt > Ma

i.e., gains from variety.


Effects of Trade Comparing Trade and Autarky (cont.)

Comparing Trade and Autarky (cont.)

ϕ∗ > ϕ∗a ⇒ rd (ϕ) = σ(

ϕϕ∗

)σ−1f < ra (ϕ) = σ

(ϕϕ∗a

)σ−1f

i.e., all firms earn less on home market.However: ra (ϕ) < rd (ϕ) + nrX (ϕ) for ϕ > ϕ∗X (harder to prove ... )i.e., exporting firms gain.


Extensions

Plan of Lectures




4 Effects of Trade

5 Extensions


Extensions

Extensions

Quadratic preferences ⇒ Variable mark-ups, competition effects:Melitz-Ottaviano (REStud 2007); but, partial equilibrium.

Combine with HO: Bernard-Redding-Schott (REStud 2008).

Quality: Baldwin-Harrigan (AEJ 2011) and many more:Predict that price rises not falls with productivity.

Zeroes in the Trade Matrix: Helpman-Melitz-Rubinstein (QJE 2008):Link with gravity model; avoids prediction that every firm serves everyexport market.

Other types of sorting:FDI: HMY (AER 2004)R&D: Bustos (AER 2011)Wages: Egger-Kreickemeier (IER 2009), Helpman-Itskhoki-Redding(Em 2010).In all cases: Trade-off between fixed and variable costs;

“Only more productive firms select into higher fixed-cost activity.”Or, is that true? See next file and Mrazova-Neary (2012) . . .


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