Stanford Econ 266: PhD International Trade
— Lecture 14: Gravity Models (Theory) —
Stanford Econ 266 (Dave Donaldson)
Winter 2016 (Lecture 14)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
1 / 67
Today’s Plan
1
The Simplest Gravity Model: Armington
2
Gravity Models and the Gains from Trade: ACR (2012)
3
Beyond ACR’s (2012) Equivalence Result: CR (2013)
4
Beyond gravity: Adao, Costinot and Donaldson (2016)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
2 / 67
Today’s Plan
1
The Simplest Gravity Model: Armington
2
Gravity Models and the Gains from Trade: ACR (2012)
3
Beyond ACR’s (2012) Equivalence Result: CR (2013)
4
Beyond gravity: Adao, Costinot and Donaldson (2016)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
3 / 67
The Armington Model
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
4 / 67
The Armington Model: Equilibrium
Labor endowments
Li for i = 1, ...n
CES utility ⇒ CES price index
Pj1−σ =
∑i =1 (wi τij )1−σ
n
Bilateral trade flows follow gravity equation:
Xij =
In what follows ε ≡ −
Trade balance
(wi τij )1−σ
wL
1− σ j j
∑nl=1 (wl τlj )
d ln Xij /Xjj
d ln τij
= σ − 1 denotes the trade elasticity
∑ Xji = wj Lj
i
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
5 / 67
The Armington Model: Welfare Analysis
Question:
Consider a foreign shock: Li → Li0 for i 6= j and τij → τij0 for i 6= j.
How do foreign shocks affect real consumption, Cj ≡ wj /Pj ?
Shephard’s Lemma implies
d ln Cj = d ln wj − d ln Pj = − ∑i =1 λij (d ln cij − d ln cjj )
n
with cij ≡ wi τij and λij ≡ Xij /wj Lj .
Gravity implies
d ln λij − d ln λjj = −ε (d ln cij − d ln cjj ) .
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
6 / 67
The Armington Model: Welfare Analysis
Combining these two equations yields
d ln Cj =
∑ni=1 λij (d ln λij − d ln λjj )
.
ε
Noting that ∑i λij = 1 =⇒ ∑i λij d ln λij = 0 then
d ln Cj = −
d ln λjj
.
ε
Integrating the previous expression yields (with x̂ ≡ x 0 /x )
Ĉj = λ̂jj−1/ε .
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
7 / 67
The Armington Model: Welfare Analysis
In general, predicting λ̂jj requires (computer) work
We can use “exact hat algebra” as in DEK (Lecture #3)
Requires data on initial levels of data {λij , Yj }, and ε
Where to get ε? Gravity equation suggests natural (but not the only)
way would be to estimate (via OLS):
ln Xij = αi + αj − ε ln tij + νij
With αi as fixed-effects and tij some observable and exogenous
component of trade costs (that satisfies τij = tij νij ).
But predicting how bad it would be to shut down all trade is easy...
In autarky, λjj = 1. So
CjA /Cj = λ1jj/ε
Thus gains from trade can be computed as
GTj ≡ 1 − CjA /Cj = 1 − λ1jj/ε
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
8 / 67
The Armington Model: Gains from Trade
Suppose that we have estimated the trade elasticity using the gravity
equation (as described above).
Central estimate in the literature is ε = 5; see Head and Mayer (2013)
Handbook chapter.
Using World Input Output Database (2008) to get λjj , we can then
estimate gains from trade:
Canada
Denmark
France
Portugal
Slovakia
U.S.
Stanford Econ 266 (Dave Donaldson)
λjj
0.82
0.74
0.86
0.80
0.66
0.91
Gravity Models (Theory)
% GT j
3.8
5.8
3.0
4.4
7.6
1.8
Winter 2016 (Lecture 14)
9 / 67
Cheese, really?
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
10 / 67
Today’s Plan
1
The Simplest Gravity Model: Armington
2
Gravity Models and the Gains from Trade: ACR (2012)
3
Beyond ACR’s (2012) Equivalence Result: CR (2013)
4
Beyond gravity: Adao, Costinot and Donaldson (2016)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
11 / 67
Motivation
New Trade Models
Micro-level data have lead to new questions in international trade:
How many firms export?
How large are exporters?
How many products do they export?
New models highlight new margins of adjustment:
From inter-industry to intra-industry to intra-firm reallocations
Old question:
How large are the gains from trade (GT)?
ACR (AER, 2012) question:
How do new trade models affect the magnitude of GT?
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
12 / 67
ACR’s Main Equivalence Result
ACR focus on gravity models
PC: Armington and Eaton & Kortum ’02
MC: Krugman ’80 and many variations of Melitz ’03
Within that class, welfare changes are (x̂ = x 0 /x)
Ĉ = λ̂1/ε
Two sufficient statistics for welfare analysis are:
Share of domestic expenditure, λ;
Trade elasticity, ε
Two views on ACR’s result:
Optimistic: welfare predictions of Armington model are more robust
than you might have thought
Pessimistic: within that class of models, micro-level data do not matter
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
13 / 67
Primitive Assumptions
Preferences and Endowments
CES utility
Consumer price index,
Pi1−σ =
Z
ω ∈Ω
pi (ω )1−σ d ω,
One factor of production: labor
Li ≡ labor endowment in country i
wi ≡ wage in country i
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
14 / 67
Primitive Assumptions
Technology
Linear cost function:
1
1− β
Cij (ω, t, q ) = qwi τij αij (ω ) t 1−σ + wi
|
{z
} |
variable cost
β
wj ξ ij φij (ω ) mij (t ),
{z
}
fixed cost
q : quantity,
τij : iceberg transportation cost,
αij (ω ) : good-specific heterogeneity in variable costs,
ξ ij : fixed cost parameter,
φij (ω ) : good-specific heterogeneity in fixed costs.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
15 / 67
Primitive Assumptions
Technology
Linear cost function:
1
1− β
Cij (ω, t, q ) = qwi τij αij (ω ) t 1−σ + wi
β
wj ξ ij φij (ω ) mij (t )
mij (t ) : cost for endogenous destination specific technology choice, t,
t ∈ [t, t ] , mij0 > 0, mij00 ≥ 0
Heterogeneity across goods
Gj (α1 , ..., αn , φ1 , ..., φn ) ≡ {ω ∈ Ω | αij (ω ) ≤ αi , φij (ω ) ≤ φi , ∀i }
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
16 / 67
Primitive Assumptions
Market Structure
Perfect competition
Firms can produce any good.
No fixed exporting costs.
Monopolistic competition
Either firms in i can pay wi Fi for monopoly power over a random good.
Or exogenous measure of firms, N i < N, receive monopoly power.
Let Ni be the measure of goods that can be produced in i
Perfect competition: Ni = N
Monopolistic competition: Ni < N
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
17 / 67
Macro-Level Restrictions
Trade is Balanced
Bilateral trade flows are
Xij =
Z
ω ∈Ωij ⊂Ω
xij (ω ) d ω
R1:For any country j,
∑i 6=j Xij = ∑i 6=j Xji
Trivial if perfect competition or β = 0.
Non trivial if β > 0.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
18 / 67
Macro-Level Restrictions
Profit Share is Constant
R2: For any country j,
Πj /
∑
n
X
i =1 ji
is constant
where Πj : aggregate profits gross of entry costs, wj Fj , (if any)
Trivial under perfect competition.
Direct from CES preferences in Krugman (1980).
Non-trivial in more general environments.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
19 / 67
Macro-Level Restrictions
CES Import Demand System
Import demand system defined as:
(w, N, τ ) → X
R3:
0
εiij
≡ ∂ ln (Xij /Xjj ) ∂ ln τi 0 j =
ε<0
0
i = i 0 6= j
otherwise
Note: symmetry and separability.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
20 / 67
Macro-Level Restrictions
Comments on CES Import Demand System
The trade elasticity ε is an upper-level elasticity: it combines
xij (ω ) (intensive margin)
Ωij (extensive margin).
R3 =⇒ complete specialization.
R1-R3 are not necessarily independent
E.g., if β = 0 then R3 =⇒ R2.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
21 / 67
Macro-Level Restrictions
Strong CES Import Demand System (AKA Gravity)
R3’: The IDS satisfies
χij · Mi · (wi τij )ε · Yj
Xij = n
ε
∑i 0 =1 χi 0 j · Mi 0 · (wi 0 τi 0 j )
where χij is independent of (w, M, τ ).
0
Same restriction on εiij as R3 but, but additional structural
relationships (R3’ ⇒ R3 but converse not true)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
22 / 67
Welfare results
State of the world economy:
Z ≡ (L, τ, ξ )
Foreign shocks: a change from Z to Z0 with no domestic change.
Effects of a domestic change in Li would be different between PC and
MC models (due to the home-market effect which is at work in the MC
models but not in the PC models).
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
23 / 67
Equivalence (I)
Proposition 1: Suppose that R1-R3 hold. Then for any foreign
shock it must be true that
cj = b
W
λ1/ε
jj .
Implication: b
λjj acts as (along with elasticity ε) sufficient statistics for
fully global, GE welfare analysis
Note that it is still true that for any of these models we could
estimate ε from OLS gravity regression:
ln Xij = αi + αj − ε ln tij + νij
New margins affect structural interpretation of ε
...and composition of gains from trade (GT)...
... but size of GT is the same.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
24 / 67
Gains from Trade Revisited
Proposition 1 is an ex-post result (based on seeing b
λjj ).
A simple ex-ante result (for the case of going to autarky):
Corollary 1: Suppose that R1-R3 hold. Then
cjA = λ−1/ε .
W
jj
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
25 / 67
Equivalence (II)
A stronger ex-ante result for variable trade costs (and things that
act like variable trade costs, like productivity shocks; but not for other
types of foreign shocks) under R1-R3’:
Proposition 2: Suppose that R1-R3’ hold. Then
b 1/ε
cj = λ
W
jj
where
b
λjj =
h
∑i =1 λij (ŵi τ̂ij )ε
and
w
bi =
∑
n
j =1
n
i −1
,
λij ŵj Yj (ŵi τ̂ij )ε
.
Yi ∑ni0 =1 λi 0 j (ŵi 0 τ̂i 0 j )ε
cj (ex-ante) from τ̂ij , i 6= j.
So: ε and {λij } are sufficient to predict W
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
26 / 67
Taking Stock
ACR consider models featuring:
(i ) CES preferences;
(ii ) one factor of production;
(iii ) linear cost functions; and
(iv ) perfect or monopolistic competition;
with three macro-level restrictions:
(i ) trade is balanced;
(ii ) aggregate profits are a constant share of aggregate revenues; and
(iii ) a CES import demand system.
Equivalence for ex-post welfare changes and GT
under R3’ equivalence carries to ex-ante welfare changes
So if R3’ is satisfied, then models in this ACR class agree on all
implications of changes in variable trade costs
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
27 / 67
A note on methodology
ACR set out to answer the question of whether different trade models
predict different GT
In some of these cases, tempting to think that it is easy to rank GT
across these models.
E.g., following Melitz and Redding (AER, 2015), note that since
Krugman is a strictly nested special case of Melitz for which the firm
productivity distribution is degenerate, the Melitz model is Krugman
plus an additional margin of adjustment.
And since both models are efficient (i.e. correspond to planner’s
problem, so equilibrium is maximizing something), an additional margin
of adjustment guarantees that the damage done by a negative shock
(e.g. move to autarky) is less bad in Melitz than in Krugman.
But note that this is not the ACR thought experiment.
ACR’s thought experiment is to ask about GT (or any other
counterfactual), conditional on the data {λij , Yj } and elasticity ε we
have.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
28 / 67
Today’s Plan
1
The Simplest Gravity Model: Armington
2
Gravity Models and the Gains from Trade: ACR (2012)
3
Beyond ACR’s (2012) Equivalence Result: CR (2013)
4
Beyond gravity: Adao, Costinot and Donaldson (2016)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
29 / 67
Departing from ACR’s (2012) Equivalence Result
Costinot and Rodriguez-Clare (Handbook of International Econ,
2013) has nice discussion of general cases.
Other Gravity Models:
Multiple Sectors
Tradable Intermediate Goods
Multiple Factors
Variable Markups (ACDR 2012)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
30 / 67
Multiple sectors, GT
Nested CES: Upper level EoS ρ and lower level EoS ε s
Recall gains for Canada of 3.8%. Now gains can be much higher:
ρ = 1 implies GT = 17.4%
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
31 / 67
Tradable intermediates, GT
Set ρ = 1, add tradable intermediates with Input-Output structure
Labor shares are 1 − αj,s and input shares are αj,ks (∑k αj,ks = αj,s )
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
32 / 67
Tradable intermediates, GT
Canada
Denmark
France
Portugal
U.S.
Stanford Econ 266 (Dave Donaldson)
% GT j
3.8
5.8
3.0
4.4
1.8
% GT MS
j
17.4
30.2
9.4
23.8
4.4
Gravity Models (Theory)
% GT IO
j
30.2
41.4
17.2
35.9
8.3
Winter 2016 (Lecture 14)
33 / 67
Multiple sectors: a combination of micro and macro
features
In Krugman, free entry ⇒ scale effects associated with total
employment
In Melitz, additional scale effects associated with sales in each market
In both models, trade may affect entry and fixed costs
All these effects do not play a role in the one sector model
With multiple sectors and traded intermediates, these effects come
back
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
34 / 67
Gains from Trade
“MS” = multiple sectors; “IO” = tradable intermediates
......................................
Aggregate
Stanford Econ 266 (Dave Donaldson)
Canada
3.8
China
0.8
Gravity Models (Theory)
Germany
4.5
Romania
4.5
Winter 2016 (Lecture 14)
US
1.8
35 / 67
Gains from Trade
“MS” = multiple sectors; “IO” = tradable intermediates
......................................
Aggregate
MS, PC
Stanford Econ 266 (Dave Donaldson)
Canada
3.8
17.4
China
0.8
4.0
Gravity Models (Theory)
Germany
4.5
12.7
Romania
4.5
17.7
Winter 2016 (Lecture 14)
US
1.8
4.4
36 / 67
Gains from Trade
“MS” = multiple sectors; “IO” = tradable intermediates
......................................
Aggregate
MS, PC
MS, MC
Stanford Econ 266 (Dave Donaldson)
Canada
3.8
17.4
15.3
China
0.8
4.0
4.0
Gravity Models (Theory)
Germany
4.5
12.7
17.6
Romania
4.5
17.7
12.7
Winter 2016 (Lecture 14)
US
1.8
4.4
3.8
37 / 67
Gains from Trade
“MS” = multiple sectors; “IO” = tradable intermediates
......................................
Aggregate
MS, PC
MS, MC
MS, IO, PC
Stanford Econ 266 (Dave Donaldson)
Canada
3.8
17.4
15.3
29.5
China
0.8
4.0
4.0
11.2
Gravity Models (Theory)
Germany
4.5
12.7
17.6
22.5
Romania
4.5
17.7
12.7
29.2
Winter 2016 (Lecture 14)
US
1.8
4.4
3.8
8.0
38 / 67
Gains from Trade
“MS” = multiple sectors; “IO” = tradable intermediates
......................................
Aggregate
MS, PC
MS, MC
MS, IO, PC
MS, IO, MC (Krugman)
Stanford Econ 266 (Dave Donaldson)
Canada
3.8
17.4
15.3
29.5
33.0
China
0.8
4.0
4.0
11.2
28.0
Gravity Models (Theory)
Germany
4.5
12.7
17.6
22.5
41.4
Romania
4.5
17.7
12.7
29.2
20.8
Winter 2016 (Lecture 14)
US
1.8
4.4
3.8
8.0
8.6
39 / 67
Gains from Trade
“MS” = multiple sectors; “IO” = tradable intermediates
......................................
Aggregate
MS, PC
MS, MC
MS, IO, PC
MS, IO, MC (Krugman)
MS, IO, MC (Melitz)
Stanford Econ 266 (Dave Donaldson)
Canada
3.8
17.4
15.3
29.5
33.0
39.8
China
0.8
4.0
4.0
11.2
28.0
77.9
Gravity Models (Theory)
Germany
4.5
12.7
17.6
22.5
41.4
52.9
Romania
4.5
17.7
12.7
29.2
20.8
20.7
Winter 2016 (Lecture 14)
US
1.8
4.4
3.8
8.0
8.6
10.3
40 / 67
Today’s Plan
1
The Simplest Gravity Model: Armington
2
Gravity Models and the Gains from Trade: ACR (2012)
3
Beyond ACR’s (2012) Equivalence Result: CR (2013)
4
Beyond gravity: Adao, Costinot and Donaldson (2016)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
41 / 67
Adao, Costinot and Donaldson (AER, 2016)
Without access to (much) quasi-experimental variation, traditional
approach in the field has been to model everything: demand-side,
supply-side, market structure, trade costs
E.g. #1: “Old CGE”: GTAP model [13,000 structural parameters]
E.g. #2: “New CGE”: EK model [1 key parameter]
Question: Can we relax EK’s strong functional form assumptions
without circling back to GTAP’s 13,000 parameters?
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
42 / 67
ACD (2016): 4 Steps
1
For many counterfactual questions, neoclassical models are exactly
equivalent to a reduced factor exchange economy
Reduced factor demand system sufficient for counterfactual analysis
2
Nonparametric generalization of standard gravity tools:
Dekle, Eaton and Kortum (2008): exact hat algebra
Arkolakis, Costinot, and Rodriguez-Clare (2012): welfare gains
Head and Ries (2001): trade costs
3
Reduced factor demand system is nonparametrically identified under
standard assumptions about data/orthogonality
4
(Not today) Empirical application: What was the impact of China’s
integration into the world economy in the past two decades?
Departures from CES modeled in the spirit of BLP (1995)
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
43 / 67
Neoclassical Trade Model
i = 1, ..., I countries
k = 1, ..., K goods
n = 1, ..., N factors
Goods consumed in country i:
qi ≡ {qjik }
Factors used in country i to produce good k for country j:
lijk ≡ {lijnk }
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
44 / 67
Neoclassical Trade Model
Preferences:
ui = ui (qi )
Representative consumer (driven by data from “country” i)
Technology:
qijk = fijk (lijk )
Non-increasing returns to scale. No joint production.
Extensions in paper to include (global/domestic) input-output linkages
and tariffs/taxes/subsidies.
Factor endowments:
νin > 0
Defined as the (set of imperfectly substitutable) inputs to production
that are in fixed supply.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
45 / 67
Competitive Equilibrium
A q ≡ {qi }, l ≡ {li }, p ≡ {pi }, and w ≡ {wi } such that:
1
Consumers maximize their utility:
qi ∈ argmaxq̃i ui (q̃i )
∑ pjik q̃jik ≤ ∑ win νin for all i;
n
j,k
2
Firms maximize their profits:
lijk ∈ argmaxl˜k {pijk fijk (l˜ijk ) − ∑ win l˜ijnk } for all i, j, and k;
ij
3
n
Goods markets clear:
qijk = fijk (lijk ) for all i, j, and k;
4
Factors markets clear:
∑ lijnk = νin for all i
and n.
j,k
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
46 / 67
Reduced Exchange Model
Fictitious endowment economy in which consumers directly exchange
factor services
Taylor (1938), Rader (1972), Wilson (1980), Mas-Colell (1991)
Reduced preferences over primary factors of production:
Ui (Li ) ≡ maxq̃i ,l˜i ui (q̃i )
q̃jik ≤ fjik (l˜jik ) for all j and k,
∑ l˜jink ≤ Lnji for all j and n,
k
Easy to check that Ui (·) is strictly increasing and quasiconcave.
Not necessarily strictly quasiconcave, even if ui (·) is.
Example: H-O model inside FPE set.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
47 / 67
Reduced Equilibrium
Corresponds to L ≡ {Li } and w ≡ {wi } such that:
1
Consumers maximize their reduced utility:
Li ∈ argmaxL̃i Ui (L̃i )
∑ wjn L˜nji ≤ ∑ win νin for all i;
j,n
2
n
Factor markets clear:
∑ Lnij = νin for all i
and n.
j
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
48 / 67
Equivalence
Proposition 1: For any competitive equilibrium, (q, l , p, w ), there
exists a reduced equilibrium, (L, w ), with:
1
2
3
the same factor prices, w ;
the same factor content of trade, Lnji = ∑k ljink for all i, j, and n;
the same welfare levels, Ui (Li ) = ui (qi ) for all i.
Conversely, for any reduced equilibrium, (L, w ), there exists a
competitive equilibrium, (q, l , p, w ), such that 1-3 hold.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
49 / 67
Equivalence
Comments:
Proof is similar to First and Second Welfare Theorems. Key distinction
is that standard Welfare Theorems go from CE to global planner’s
problem, whereas RE remains a decentralized equilibrium (but one in
which countries fictitiously trade factor services and budget is balanced
country by country).
Key implication of Prop. 1: If one is interested in the factor content of
trade, factor prices and/or welfare, then one can always study a RE
instead of a CE. One doesn’t need direct knowledge of primitives u and
f but only of how these indirectly shape U.
Stanford Econ 266 (Dave Donaldson)
Gravity Models (Theory)
Winter 2016 (Lecture 14)
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Reduced Counterfactuals
Suppose that the reduced utility function over primary factors in this
economy can be parametrized as
Ui (Li ) ≡ Ūi ({Lnji /τjin }),
where τjin > 0 are exogenous preference shocks
Counterfactual question: What are the effects of a change from
(τ, ν) to (τ 0 , ν0 ) on trade flows, factor prices, and welfare?
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Reduced Factor Demand System
Start from factor demand = solution of reduced UMP:
Li (w , yi |τi )
Compute associated expenditure shares:
χi (w , yi |τi ) ≡ {{xjin }|xjin = wjn Lnji /yi for some Li ∈ Li (w , yi |τi )}
Rearrange in terms of effective factor prices, ωi ≡ {wjn τjin }:
χi (w , yi |τi ) ≡ χi (ωi , yi )
Stanford Econ 266 (Dave Donaldson)
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Reduced Equilibrium
In this notation, RE is:
∑
xi
n
xij yj
∈ χi (ωi , yi ), for all i,
= yin , for all i and n
j
Gravity model (i.e. ACR): Reduced factor demand system is CES
χji (ωi , yi ) =
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(ωji )e
, for all j and i
∑l (ωli )e
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Exact Hat Algebra
Start from the counterfactual equilibrium:
xi0 ∈ χi (ωi0 , yi0 ) for all i,
∑(xijn )0 yj0 = (yin )0 , for all i
and n.
j
Rearrange in terms of proportional changes:
{x̂jin xjin } ∈ χi ({ŵjn τ̂jin ωjin }, ∑ ŵin ν̂in yin ) for all i,
n
∑ x̂ijn xijn (∑ ŵjn ν̂jn yjn ) = ŵin ν̂in yin , for all i and n.
j
n
Stanford Econ 266 (Dave Donaldson)
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Counterfactual Trade Flows and Factor Prices
A1 [Invertibility]. In any country i, for any x > 0 and y > 0, there
exists a unique vector of factor prices (up to normalization) such that
x ∈ χi (ω, yi ), which we denote χi −1 (x, y ).
Sufficient conditions:
A1 holds if χi satisfies connected substitutes property (Arrow and
Hahn 1971, Howitt 1980, and Berry, Gandhi and Haile 2013)
χi satisfies connected substitutes property in a Ricardian economy if
preferences satisfy connected substitutes property
Stanford Econ 266 (Dave Donaldson)
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Counterfactual Trade Flows and Factor Prices
Proposition 2 Under A1, proportional changes in expenditure shares
and factor prices, x̂ and ŵ , caused by proportional changes in
preferences and endowments, τ̂ and ν̂, solve
{x̂jin xjin } ∈ χi ({ŵjn τ̂jin ωjin }, ∑ ŵin ν̂in yin ) ∀ i,
∑
x̂ijn xijn (
j
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∑
ŵjn ν̂jn yjn )
=
n
n n n
ŵi ν̂i yi
∀ i and n.
n
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Counterfactual Trade Flows and Factor Prices
Proposition 2 Under A1, proportional changes in expenditure shares
and factor prices, x̂ and ŵ , caused by proportional changes in
preferences and endowments, τ̂ and ν̂, solve
{x̂jin xjin } ∈ χi ({ŵjn τ̂jin (χnji )−1 (xi , yi )}, ∑ ŵin ν̂in yin ) ∀ i,
n
∑
x̂ijn xijn (
j
Stanford Econ 266 (Dave Donaldson)
∑
ŵjn ν̂jn yjn )
=
ŵin ν̂in yin
∀ i and n.
n
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Welfare
Equivalent variation for country i associated with change from (τ, ν)
to (τ 0 , ν0 ), expressed as fraction of initial income:
∆Wi = (ei (ωi , Ui0 )) − yi )/yi ,
with Ui0 = counterfactual utility and ei = expenditure function,
ei (ωi , Ui0 ) ≡ minL̃i
∑ ωjin Lnji
Ūi (L̃i ) ≥ Ui0 .
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Integrating Below Factor Demand Curves
To go from χi to ∆Wi , solve system of ODEs
For any selection {xjin (ω, y )} ∈ χi (ω, y ), Envelope Theorem:
d ln ei (ω, Ui0 )
= xjin (ω, ei (ω, Ui0 )) for all j and n.
d ln ωjn
Budget balance in the counterfactual equilibrium
ei (ωi0 , Ui0 ) = yi0 .
Stanford Econ 266 (Dave Donaldson)
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Counterfactual Welfare Changes
Proposition 3 Under A1, equivalent variation associated with change
from (τ, ν) to (τ 0 , ν0 ) is
∆Wi = (e ({(χnji )−1 (xi , yi )}, Ui0 ) − yi )/yi ,
where e (·, Ui0 ) is the unique solution of (1) and (2).
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Application to Neoclassical Trade Models
Suppose that technology in neoclassical trade model satisfies:
fijk (lijk ) ≡ f¯ijk ({lijnk /τijn }), for all i, j, and k,
Reduced utility function over primary factors of production:
Ui (Li ) ≡ maxq̃i ,l˜i ui (q̃i )
q̃jik ≤ f¯jik ({l˜jink /τ nji }) for all j and k,
∑ l˜jink ≤ Lnji for all j and n.
k
Change of variable: Ui (Li ) ≡ Ūi ({Lnji /τjin }) ⇒ factor-augmenting
productivity shocks in CE = preference shocks in RE
NB: τ̂ cannot depend on k.
However, τ can do so freely.
And can always allow for τ̂jink by defining a new factor that is specific
to sector k (with arbitrage condition across sectors).
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Taking Stock
Given data on expenditure shares and factor payments, {xjin , yin }, if one
knows factor demand system, χi , then:
1
Can compute exact, fully GE change due to counterfactual move from
(τ, ν) to (τ 0 , ν0 )) in:
Factor prices
Factor content of trade (and hence total volume of trade)
Welfare
2
Proposition 1 ⇒ will get same answer as any neoclassical economy
with the same χi .
3
Standard gravity “toolkit”—developed for CES factor
demands—extends nonparametrically to any invertible factor demand
system
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Identification Assumptions: Shocks
Data generated by neoclassical trade model at different dates t
At each date, preferences and technology such that:
k
ui (qi ,t ) = ū ({qji,t
/θji }), for all i,
k
k
k
nk
n
fij,t (lij ,t ) = f¯i ({lij,t /τij,t
}), for all i, j, and k.
A2. [Price heterogeneity] In any country i and at any date t, there
n θ τ n }, such that factor demand can be expressed
exists ωi ,t ≡ {wj,t
ji ji,t
as χ̄(ωi ,t , yi,t ).
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Identification Assumptions: Exogeneity
Observables:
1
2
3
xjin,t : factor expenditure shares (normal FCT data in principle; but
non-trivial aggregation bias issues in practice)
yin,t : factor payments
zjin,t : factor price shifters (e.g. observable shifter of trade costs)
n :
Effective factor prices, ωji,t , unobservable, but related to zji,t
n
n
n
n
+ eji,t
, for all i, j, n, and t
lnωji,t
= ln zji,t
+ ϕnji + ξ j,t
A3. [Exogeneity] E [ei ,t | ln zi ,t , di ,t ] = 0, where di ,t is a vector of
importer-exporter-factor dummies and exporter-factor-year dummies
with i = importer and t = year for all dummies.
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Identification Assumptions: Completeness
Following Newey and Powell (2003), we conclude by imposing the
following completeness condition.
A4. [Completeness] For any g (xi ,t , yi,t ),
E [g (xi ,t , yi,t )| ln zi ,t , di ,t ] = 0 implies g (xi ,t , yi,t ) = 0.
A4 = rank condition in estimation of parametric models.
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Identifying Factor Demand
A1 ⇒
n
ωji,t
= (χnji )−1 (xi ,t , yi,t ).
n :
Taking logs and using definition of eji,t
n
n
n
eji,t
= ln(χnji )−1 (xi ,t , yi,t ) − ln zji,t
− ϕnji − ξ j,t
.
A2 ⇒
n
n
n
eji,t
= ln(χ̄nj )−1 (xi ,t , yi,t ) − ln zji,t
− ϕnji − ξ j,t
.
Using A3, we obtain the following moment condition
n
n
E [ln(χ̄nj )−1 (xi ,t , yi,t ) − ln zji,t
− ϕnji − ξ j,t
| ln zi ,t , di ,t ] = 0.
A4 ⇒ unique solution (χ̄nj )−1 to (3) (up to a normalization)
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Identifying Factor Demand
Once the inverse factor demand is known, both factor demand and
effective factor prices are known as well, with prices being uniquely
pinned down (up to a normalization).
Proposition 4 Suppose that A1-A4 hold. Then effective factor prices
{ωi ,t } and the factor demand system χ̄ are identified (up to a
normalization).
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