Short Run Gravity∗
James E. Anderson
Boston College and NBER
March 7, 2016
Abstract
Short run gravity is a geometric weighted average of long run gravity and a bilateral
capacity variable, where the weight on long run gravity is the passthrough elasticity.
Bilateral capacity adjusts over time toward an efficient level. The model features (i)
bilateral trade costs endogenous to volume, (ii) interdependence of bilateral trade costs,
(iii) long run gravity as a limiting case of efficient investment in capacity on bilateral
links, (iv) tractable models of short run and long run action on the extensive margin
of trade, and (v) time invariance of estimated short run distance and border effects.
JEL Classification:
Contact information: James E. Anderson, Department of Economics, Boston College, Chestnut Hill, MA 02467, USA.
∗
This is a very preliminary version of a paper that will have an important empirical application testing
whether the theory fits the data. Comments welcome.
Estimated gravity equations do not vary much over time, once free trade agreements
and other policy changes are appropriately controlled for (e.g., Anderson and Yotov, 2016).
Time invariance poses an empirical and conceptual puzzle. Ocean shipping rates and delivery
lags are well known to move with business cycle and secular movements in demand; why is
there no effect on gravity parameter estimates? Bilateral variation over time seems implied
by imperfectly correlated national expenditure variation over the business cycle along with
big secular shifts in locations of economic activity. Time invariance also appears to conflict
with the richly patterned adjustment over time in bilateral trade links emphasized by Chaney
(2014, looking at French firm export market behavior) and Besedes and Prusa (2006, looking
at the flickering on and off of US 10 digit bilateral trade). This paper offers a theoretical
solution to the puzzle: a tractable structural model of short run gravity that admits the
preceding dynamic phenomena in a general equilibrium of distribution. Short run gravity
is a geometric weighted average of long run gravity and a bilateral capacity variable, where
the weight on long run gravity is the passthrough elasticity. The bilateral capacity variable
adjusts over time toward an efficient level reached in the long run. The steady state of
the dynamic model is the long run model previously used to interpret structural gravity
equations.
Short run gravity as developed here features (i) endogeneity of bilateral trade costs with
respect to volume, (ii) interdependence of bilateral trade costs, (iii) long run gravity nested
as a limiting case of efficient investment in capacity on bilateral links, (iv) tractable models
of short run and long run action on the extensive margin of trade, and (v) time invariance
of estimated distance and border effects. Demand and supply size effects and bilateral
investments in capacity help determine bilateral trade costs and thus both the extensive and
intensive margin volumes of bilateral trade.
The key innovation is combining CES gravity with production cum distribution based on
the specific factors continuum technology of Anderson (2011). Allocation of variable factors
across sectors in Anderson (2011) becomes allocation across destinations from a given origin
in the present paper, extended to allow for fixed costs of serving any destination. The
elegant log-linear simplicity of structural gravity still obtains in the short run reduced form.
Estimated gravity elasticities are interpreted as reduced form short run elasticities, with
their volume invariance and observed constancy over time consistent with any amount of
short run variation of observable trade costs.
In the benchmark case of perfectly efficient origin-destination-specific investment, the
model converges to the standard gravity equation, a manifestation of the envelope theorem.
Long run trade elasticities exceed short run elasticities. A model of adjustment toward efficient allocation is developed. The adjustment model combined with the static gravity model
offers a test of the short run vs. long run interpretation of gravity. When some trade cost
information is directly available (e.g. tariffs bound by WTO obligations), it permits identification of both the elasticity of substitution and the elasticity of transformation associated
with allocation of ex post variable factors between service to different destinations. The
adjustment model can explain a scale effect in cross-border trade documented by Anderson, Yotov and Vesselovsky (2016): bilateral links form more speedily within than between
countries.
The model provides a tractable way to handle two prominent features of bilateral trade
data on the extensive margin: flickering (active entry and exit) one the one hand, and
on the other hand persistence (many zeroes in disaggregated bilateral trade remain zero).
The mechanism avoids issues with the two current models of zeroes. The first model of
zeroes (Helpman, Melitz and Rubinstein, 2008) is based on the assumption that firms draw
productivities from a Pareto distribution that has an upper bound. The upper bound can
result in no firms at a particular origin foreseeing a draw high enough to cover the fixed cost
of entering a particular destination market. While the Pareto productivity distribution itself
has empirical support, the bounded support appears to some critics as a non-economic deus
ex machina to ‘explain’ zeroes (since eliminating the upper bound implies that all markets
are entered). The other model to explain zeroes uses demand systems with choke prices
2
such as the quadratic (Melitz and Ottaviano, xxxx) or translog (Novy, 2013). While choke
prices are realistic, such systems are more awkward to use than the CES and (according to
Novy’s results) they appear to do worse at fitting the data than the standard CES gravity
equation. In contrast, the model here has an endogenously determined extensive margin.
In the long run it depends only on technology and taste parameters in general competitive
equilibrium. Short run entry and exit flickering on the extensive margin are induced by
demand or technology shocks that can have any distribution.
Another useful feature of the model is scalability. The basic trading world setup can be
applied to sectors or firms. Estimated elements of the model can readily be used to construct
consistent aggregates. The elements can be embedded in a wide class of general equilibrium
production models.
The model is applied to 2 digit world production and trade date from UNIDO from xx
to xx.
Key results are xxx
1
A Model of Joint Trade Costs
An origin region produces and ships a product to potentially many destinations. Distribution on each link requires multiple resources in variable proportions, generalizing the usual
notion of iceberg-melting trade costs but preserving the essential feature that distribution
multiplicatively amplifies production costs. The scalar trade cost factors differ across destinations due to endogenous as well as exogenous geographical features of the world economy.
Cost-minimizing resource allocation implies that bilateral trade destinations are imperfect
substitutes.
Shipment of an output from an origin to any destination requires variable labor services and one unit of managerial labor for each destination served. Shipment also requires
destination-specific capital and origin-specific capital that is committed (sunk) before allo-
3
cation of the other inputs, though in principle it is variable ex ante. The capital variable
stands in for embodied investment best thought of as a combination of human capital (exostructure) at origin and destination that specializes bilaterally, combined with and physical
infrastructure at origin and destination. Think for example of a worker or group of workers
that invest in building a network link. The details of origin-destination specialization, while
interesting and important (e.g., Chaney, 2014), are outside the scope of this paper. Labor,
including the managerial labor, is mobile across destinations from any origin.
Bilateral production and trade service of the generic region and sector to destination
z is modeled as a Cobb-Douglas function of labor L(z) − 1 and capital K(z): x(z) =
(1/t(z))K(z)1−α (L(z) − 1)α . One unit of labor is designated to serve as the manager, while
the functional form implies that the manager is required.1 x(z) is delivered product. t(z) > 1
is the technological iceberg-melting parameter from the given origin to destination z, a
penalty imposed by ‘nature’ relative to the frictionless benchmark t = 1. t(z) also reflects
a productivity penalty in the usual sense that would apply to all destinations z uniformly.
With all inputs variable (in the long run), the production function exhibits economies of
scale. In the short run with K(z) fixed, decreasing returns dominate except when L(z) is
small.2
The supply of labor in the generic region to the generic sector is L =
Pn
0
L(z) where
n is the extensive margin sector. Labor is efficiently allocated across destination activities
with value of marginal product equal to the common wage. Managers are drawn from the
common labor pool but command a premium in the form of a share of the residual return
to an active sector. The share can be rationalized with a simple bargaining model of the
interaction of the manager with the owners of specific capital. On the extensive margin
1
How the manager is chosen is outside the model. Normalization of managerial input to 1 unit of labor
is a harmless convention. A useful extension empirically introduces heterogeneity across destinations in the
amount of managerial input required. At this level of generality, the production function has the Stone-Geary
form.
2
To see this, consider a scalar expansion of K(z), L(z). The elasticity of x(z) with respect to the scalar
µ is equal to 1 − α + αL(z)/[µL(z) − 1 in the long run and equal to just the second term in the short run.
At µ = 1 the long run elasticity exceeds 1 while the short run elasticity is less than 1 for α < 1 − 1/L(z).
The equilibrium entry condition below ensures that diminishing returns characterize active destinations.
4
the premium falls to zero and the marginal manager prefers ordinary labor due to a small
disutility of managing.
Destination specific capital K(z) is allocated for service in production and delivery to
P ∗
n∗ ≥ n potentially active destinations, accounting for n0 K(z) units of capital. The allocation shares of capital are given by λ(z) = K(z)/K. (In the full model, K(z) is decomposed
into origin and destination infrastructure components along with the contribution of origindestination specific human capital, a decomposition that is not needed for present purposes.)
The price for delivered output at destination z is p(z). The delivered price covers costs
in competitive equilibrium, w[L(z) − 1] + (1 − π)r(z)K(z) + [w + πr(z)K(z)] = p(z)x(z),
where r(z) is the realized (residual) return on the specific capital for delivery to destination
z and π is the manager’s premium share. The first term on the left hand side of the equation
is the wage bill, the second term the payments to sector specific capital and the third term
the payment to the manager. Payments by end users cover all costs. Destination z is not
served if r(z) = 0 since managers can be paid no premium and we assume at least some small
premium is required to assume managerial responsibilities. (Destination z is also not served
when λ(z) = 0, no destination-specific capital is allocated. This possibility is addressed
below when efficient allocation of capital is analyzed.)
The setup here extends the specific factors continuum production model of Anderson
(2011) by introducing fixed managerial cost.3 The value of production at delivered prices in
the generic region and sector is the sum over destinations z of the payments to labor and to
managerial capital and the payment to non-managerial capital given by
Y = (L − n)α K 1−α C.
(1)
n
X
C≡[
λ(z)(p(z)/t(z))1/(1−α) ]1−α .
(2)
where
0
3
In Anderson (2011) the environment is a GDP function where z denotes a sector in a continuum of fixed
size, t(z) is a productivity penalty and there is no fixed cost.
5
The real activity level
R = (L − n)α K 1−α
is multiplied by a price index C embedding efficient allocation of the joint activity to delivered output at the many destinations. The setup thus yields a Constant Elasticity of
Transformation (CET) joint revenue function for delivered output. Interestingly, this joint
production property arises even though the underlying technology for production and delivery from each origin to each destination is independent. What permits the joint production
property is the identical production and delivery technology (up to the gravity coefficients
t(i, z)). The separability of the price index from the real activity level R(i) in (1) implies
that the model is about distribution technology and efficient allocation of variable inputs to
distribution in the presence of origin-destination specific factors. The level of production in
i can be take as exogenously determined, showing up as a scaling term y(i) (effectively an
endowment) multiplying R(i). Thus it is not necessary to believe that production y(i) is a
Cobb-Douglas function of labor and specific human capital.
The equilibrium share of sales to each destination z that is served, by Hotelling’s Lemma
(applied to (1) using (2)), is
s(z) = λ(z)
p(z)/t(z)
C
1/(1−α)
.
(3)
It is convenient for later purposes to sort destinations by rank order, beginning with the
largest (so in equation (3) z ∈ [0, n]). The local delivery market is s(0) by convention
because empirically it is almost universally the largest.
The extensive margin n is determined by the indifference condition of managers, who
weakly prefer their outside option of the common wage rate over managing when the premium
falls to zero. This implies that the extensive margin is efficient in the sense of maximizing
(1) with respect to n. For simplicity, temporarily think of a continuum of destinations (with
‘shares’ being densities) and differentiate (1) with respect to n. The first order condition
6
yields
w = αY /(L − n) = (1 − α)s(n)Y.
(4)
The second equation in (4) implies that a manager paid w (if he would actually manage at
this price) exhausts the entire residual payment in sector n. Concentrate for simplicity on
this case where the disutility of managing is equal to zero.4 Ordering sectors by decreasing
size, the extensive margin is the smallest market that can be served. The wage = αY /(L−n)
is rising in n, while (1 − α)s(n)Y decreasing in n; hence the equilibrium extensive margin
exists and is unique. Eliminating Y , the implication is
L−n=
α
(1 − α)s(n)
(4) readily rationalizes the flickering on and off of bilateral trade that is observable in highly
disaggregated data: capacity is in place but insufficient revenue to cover the overhead cost
of management indicate temporary shutdown. Section 2.1 considers the longer run extensive
margin of installed capacity.
An important implication of the model of zeroes in bilateral trade based on (4) is that
volume equations for positive trade are not subject to selection bias. That is because firms
are identical, in contrast to heterogeneous firms setup of Helpman, Melitz and Rubinstein
(2008) and others. When the model is extended to the firm level (sketched below in Section
??, the action on the extensive margin implied in (4) allows for a realistic heterogeneity
of behavior: different firms from the same origin will move differently on their extensive
margins, entering or exiting different destinations depending on their idiosyncratic history
of specific factor allocations. The main alternative model allows for idiosyncrasy in firms’
productivity draws from a bounded Pareto distribution, a structure which is both more
restrictive and less economic.
For any market that is served, the equilibrium delivered price p(z) is endogenously deter4
If u is the disutility of managing, the manager’s indifference condition is w + u = w + πr(z)K(z), hence
(4) becomes π + αY /(L − n) = (1 − α)s(n)Y .
7
mined by the supply side forces described in (1)-(3) interacting with demand forces described
by Constant Elasticity of Substitution preferences or technology (in the case of intermediate
goods). The intuitive notion of a bilateral trade cost corresponds to p(z)/p(0), a clear idea
when t(0) = 1, so bilateral trade cost is endogenous. In practice, this is a dangerous simplification because internal delivery costs are not zero, differ across countries and are endogenous
just as the bilateral costs are endogenous.
Description of the demand side of the market requires an expansion of notation to designate the location of the originating sector. The CES expenditure share for goods from origin
i in destination z is given by
X(i, z)
=
E(z)
β(i)p(i, z)
P (z)
1−σ
.
(5)
Here, X(i, z) denotes the bilateral flow at end user prices, E(z) denotes the total expenditure
in destination z on goods from all origins serving it, β(i) is a distribution parameter of the
CES preferences/technology, σ is the elasticity of substitution, and P (z) is the CES price
index for destination z.
The market clearing condition for positive bilateral trade from i to z is
Y (i)s(i, z) = X(i, z).
(6)
Using (3) and (5), the equilibrium price p(i, z) is given by
E(z)P (z)σ−1 β(i)1−σ [t(i, z)C(i)]η
p(i, z) =
Y (i)λ(i, z)
1/(η+σ−1)
,
(7)
where η = 1/(1 − α) > 1. The short run equilibrium price in an active origin-destination
pair in (7) is an intuitive constant elasticity function of demand shifters E(z)P (z)σ−1 , supply
shifters Y (i) and C(i), and the exogenous bilateral friction components in t(i, z) and the
contemporaneously exogenous bilateral capacity λ(i, z).
8
Passthrough of trade costs is incomplete: the passthrough elasticity is d ln p(i, z)/d ln t(i, z) =
η/(η +σ −1) ≡ ρ. The passthrough elasticity plays a key role in the gravity representation of
the model, so it is worthwhile to examine its deep microfoundation. It is intuitively a combination of the demand elasticity parameter σ and the transformation elasticity parameter
η, itself microfounded in the general equilibrium of distribution based on a Cobb-Douglas
specific factors model.
(7) can in principle account for substantial variation in prices across time and space.
Rents to the sector specific factor similarly vary (see Section 2.1). Rents are competitive in
this version of the model, so pricing to market behavior in the usual sense is not implied.
(The model can be straightforwardly extended to monopolistic competition by treating each
origin i as a firm. With CES demand, markups are constant when firms shares are small.)
2
Gravity Representation
The gravity representation reduces the somewhat forbidding complexity of endogenous price
(7) to yield a short run gravity equation that remains a log linear combination of origin
fixed effects, destination fixed effects, and bilateral effects comprising bilateral frictions and
bilateral specific capacity. The origin and destination fixed effects comprise aggregate activity and multilateral resistances as in the standard gravity model, with the multilateral
resistances retaining their interpretation as sellers’ and buyers’ incidence of exogenous trade
costs. Short run gravity trade flows are a geometric weighted average of long run gravity
flows and the bilateral specific capacity investment where the geometric weight on long run
gravity is equal to passthrough elasticity.
The gravity representation is obtained by first solving a global market clearing condition
for shipments Y (i) as if to a single world market with a CES world expenditure share equation
represented by a version of (5) with outward multilateral resistance Π(i). First, substitute
(7) into (5), then multiply by E(z) and sum to obtain the global market clearance condition
9
for Y (i). The result is
Y (i) = [β(i)C(i)]ρ(1−σ) Y (i)ρ(σ−1)/η
X
[E(z)P (z)σ−1 ]ρ t(i, z)ρ(1−σ) λ(i, z)ρ(σ−1)/η .
z
Divide both sides by world sales from all origins Y , use Y = Y (σ−1)/(η+σ−1) Y η/(η+σ−1) on the
left hand side and solve for i’s sales share Y (i)/Y . The result is
Y (i)
= [β(i)C(i)]1−σ Π(i)1−σ
Y
(8)
where
"
1−σ
Π(i)
=
X E(z) ρ t(i, z) (1−σ)ρ
z
Y
P (z)
#1/ρ
1−ρ
λ(i, z)
(9)
and passthrough elasticity ρ is substituted for η/(η + σ − 1). The left hand side of (8) is
recognized as a CES share equation for a hypothetical world buyer on the world market,
with a world market price index for all goods equal to 1. Then for each origin i the outward
multilateral resistance Π(i) is effectively the equilibrium sellers’ incidence of the trade frictions consistently aggregated across all bilateral pairs. Π(i) is homogeneous of degree one in
{t(i, z)} for given {P (z)}.
The next step is to substitute (7) in to the CES price definition P (z)1−σ =
P
1−σ
.
i [β(i)p(i, z)]
Then use (8) to substitute for [β(i)C(i)]1−σ in the resulting equation. After simplification
this gives
P (z)
1−σ
E(z)
=
Y
−(1−ρ) X
i
Y (i)
Y
(t(i, z)
Π(i)
(1−σ)ρ
λ(i, z)1−ρ .
(10)
Thus the buyers’ price index P (z) is the product of the effect of market size of z times
the size invariant buyers’ incidence a uniform weighted CES function of the set of bilateral
buyers’ incidences: {[t(i, z)/Π(z)]η }. Compared to long run gravity, higher demand in the
short run raises the buyers’ price index for any given value of size invariant buyers’ incidence,
as is intuitive. As η → ∞, ρ → 1 and the buyers’ market size effect vanishes.
The final step in deriving short run gravity is to substitute the right hand side of (7)
10
for p(i, z) in (5) and use (8) to substitute for [β(i)C(i)]1−σ in the resulting expression. Also
use (10) and its decomposition in (5). After simplification using passthrough elasticity
ρ = η/(η + σ − 1), this gives
X(i, z) =
1−σ !ρ
t(i, z)
Y (i)E(z)
λ(i, z)1−ρ .
Y
Π(i)P (z)
(11)
The structural gravity model consists of (11) where (9) and (10) define Π(i) and P (z).
It is a reduced form in which exogenous frictions t(i, z) and exogenous (modularly) activity
levels Y (i), E(z) generate endogenous bilateral trade costs and multilateral resistances. Short
run structural gravity in logs in (11) is a geometric weighted average of long run gravity and
short run bilateral capacity λ(i, z) where the weight on long run gravity is the passthrough
elasticity. As η → ∞, ρ → 1 and short run gravity converges to long run gravity.
Intuition about short run gravity system (9)-(11) is aided by considering a uniform change
in all bilateral trade costs t(i, z): t1 (i, z) = µt0 (i, z). Intuitively, bilateral trade flows should
be unchanged because no relative price changes. Checking the system, this follows because
P 1 (z) = µρ P 0 (z), Π1 (i) = µ1−ρ Π0 (i); ∀i, z and thus X 1 (i, z) = X 0 (i, z) ∀i, z. More formally,
{P (z)} is homogeneous of degree ρ in {t(i, z)} and {Π(i)} is homogeneous of degree 1 − ρ
in {t(i, z)}, hence {X(i, z)} is homogeneous of degree zero in {t(i, z)}. These results are
elegantly intuitive: the passthrough elasticity gives the buyers’ incidence of bilateral trade
cost change in partial equilibrium in (7) and also the buyers’ incidence of uniform trade cost
change in general equilibrium. In contrast, the long run general equilibrium incidence of a
uniform change on {P (z)} and {Π(i)} is indeterminate, just as it is for the long run partial
equilibrium incidence.
Short run gravity equation (11) gives insight into why cross-section gravity estimates
of bilateral frictions typically show little variation over time. Setting aside the role of the
evolution of destination-specific investment (treated in Section 3.1), (11) is a reduced form
structural gravity equation of the now standard type. If the exogenous component of bi-
11
lateral trade costs t(i, z) is time invariant, estimation of a time series of cross sectional
gravity equations will produce invariant trade elasticities. Gravity estimation controls for
substantial time variation of the incidence of time-invariant trade costs represented by inward and outward multilateral resistance (Anderson and Yotov, 2010) with origin-time and
destination-time fixed effects.
Bilateral trade costs themselves are nonetheless endogenous. A standard notion of relative
trade cost is the ratio of external to local equilibrium prices determined by (7):
η
1/(η+σ−1)
E(z)P (z)σ−1 t(i, z)
λ(i, i)
p(i, z)
=
.
T (i, z) ≡
p(i, i)
E(i)P (i)σ−1 t(i, i)
λ(i, z)
(12)
The fixed effect controls in estimated gravity equations control for the relevant part of variation of short run bilateral trade cost (12). Expenditure shocks in destination z relative to expenditure in i raise trade costs T (i, z) from i to z in (12) with elasticity ∂ ln T (i, z)/∂ ln[E(z)/E(i)] =
1/(η + σ − 1). Thus (12) accommodates readily observable variation in congestion costs over
time and across origin-destination pairs. Relative expenditure shocks also shift relative inward multilateral resistance P (z)/P (i) with trade cost elasticity (σ −1)/(η +σ −1). Relative
supply shocks Y (i)/Y affect bilateral trade costs from i via their effect on {P (z)/P (i)} in
(12). The differences between (11) and (12) suggest that trade cost inferences from price
comparisons should differ from trade cost inferences from trade flows, especially with regard
to time series behavior.
The volume effect of short run bilateral cost (12) relative to the volume effect of long run
bilateral cost in (11) is
(1−σ)/(η+σ−1)
T (i, z)1−σ
E(z)P (z)σ−1
−η λ(i, i)
=
t((i, i)
.
t(i, z)η(1−σ)/(η+σ−1)
E(i)P (i)σ−1
λ(i, z)
When elements of T (i, z) are observable and its volume effects can be inferred using values of
σ, then its relationship to inferred volume effects of t(i, z) can be used with (12) to quantify
the impact on T (i, z) of expenditure variation and the other time varying components of
12
(12). Observability is presumably improved with highly disaggregated data where price
comparisons are credible.
Equation (12) is a model of average bilateral trade costs that abstracts from many interesting details. In particular the average behavior may not reflect the individual behavior of
cost components such as transportation, trade credit and other intermediary services. The
rents received by the bilaterally specific factors are likely to be unevenly split among the
bilateral components. In potential drilling down to build component models, the model in
(12) presumably acts as an aggregate constraint on the sum of individual cost component
behavior.
2.1
Benchmark Efficient Allocation
The allocation of capital to destinations z is outside the static model developed above under
the realistic assumption that investment is predetermined and generally inefficient relative to
current realizations of random variables. It is nevertheless useful to construct a benchmark
efficient allocation as an aid to intuition and to learning something about the inefficiency of
allocation. A key, though in some sense obvious result is that the general gravity model under
inefficient investment nests the standard iceberg trade cost model as a special case of efficient
allocation. The difference between actual and hypothetical efficient allocation is presumably
due to un-modeled frictions hampering investment in the face of various risks and imperfect
information about realizations of natural bilateral resistance and other components of the
realized equilibrium. The development of the benchmark allocation provides structure to the
econometric application that generates inferred differences between actual and benchmark
allocations as residuals. A second stage regression affords an opportunity to learn something
about the pattern of the residuals.
Economic intuition suggests that standard iceberg trade costs should emerge as a reduced
form of all efficient equilibrium production and distribution models because it is consistent
with the envelope theorem in the allocation of all relevant resources in distribution. Effec13
tively, geography dictates the allocation of capital as well as the distribution of goods given
that efficient allocation of capital. The development of the benchmark special case is useful
in demonstrating just how this works.
Efficient capital allocation achieves equal returns on investment in each destination
served. The average return on investment for the generic sector and economy of Section
1 is given by r̄ = YK = (1 − α)Y /K. The return on capital relative to the average for destination z is given by s(i, z)/λ(i, z) (Anderson, 2011). If investors perfectly foresee bilateral
natural trade costs and the extensive margin, then λ(i, z) = s(i, z) in the capital allocation equilibrium actually realized. Then λ∗ (i, z) = s(i, z) = λ∗ (i, z)[p(i, z)/t(i, z)C(i)]η ⇒
p(i, z) = t(i, z)C(i). Combine this restriction with equation (7) for the market clearing price
to solve for the efficient allocation
E ∗ (z)
λ (i, z) = ∗
Y (i)
∗
β(i)C ∗ (i)t(i, z)
P ∗ (z)
1−σ
E ∗ (z)
= ∗
Y (i)
t(i, z)
∗
Π (i)P ∗ (z)
1−σ
.
(13)
Note that (13) is a general equilibrium concept: the multilateral resistances imply that
all origins solve for efficient allocation simultaneously. Efficient allocation share λ∗ (i, z) is
decreasing in the cross section of trade pairs in natural trade friction t(i, z), increasing in
destination market potential E(z)P (z)σ−1 and origin utility weight β(i)1−σ . Each of these
effects is intuitive. It is also increasing in the ‘average economic distance’ of the origin from its
markets, Π∗ (i), implying that for markets actually served, relationship-specific investments
must be larger to overcome the average resistance.
Notice that η = 1/(1 − α) plays no role in the efficient allocation equilibrium. This
arises because no short run reallocation of labor is needed; the trade flows have converged
to the standard gravity model pattern. The ‘long run’ trade cost elasticity is 1 − σ, which
exceeds in absolute value the ‘short run’ elasticity (1 − σ)η/(η + σ − 1) = (1 − σ)ρ, a familiar
implication of the envelope theorem. A convenient approximation of the convergence to
14
efficient allocation is the parametric result
lim (1 − σ)η/(η + σ − 1) = 1 − σ.
η→∞
Thus the short run gravity model of Section 1 is effectively related to the long run model
as if the elasticity of transformation became infinite through the mechanism of efficient
reallocation of specific investment. (Opening the black box of convergence here requires
developing the dynamics of λ(i, z), discussed in Section 3.1.)
2.1.1
Efficient Extensive Margin
In efficient equilibrium, the extensive margin is determined by equation (4) with s(i, n) =
λ∗ (i, n). Order the destinations with ordering Z(i) such that the efficient destination investment shares defined by (13) are decreasing in z: Z(i) = P({z}) : dλ∗ (i, z)/dz < 0 where
P({z}) denotes a permutation of the ordering of {z}. Then n(i)∗ is defined by
n∗ (i) = z : L(i) − z −
α
= 0; z ∈ Z(i).
(1 − α)λ∗ (i, z)
Existence and uniqueness of the fixed point n∗ (i) is guaranteed because
L(i) − z −
α
(1 − α)λ∗ (i, z)
is decreasing in z. More intuitively,
α
1
n (i) = L(i) −
∗
1 − α E (n∗ )
∗
t(i, n∗ )
Π∗ (i)P ∗ (n∗ )
σ−1
.
(14)
Evidently the efficient equilibrium extensive margin n(i)∗ is increasing in origin size L(i)
and increasing in origin average economic distance Π∗ (i), the latter effect because it reduces
the relative difficulty of entering the marginal market t(i, z)/Π∗ (i). A more steeply rising
profile of bilateral trade costs for an origin country reduces its extensive margin. Destination
15
size distribution (market potential) E(z)P (z)σ−1 affects all exporters equally; as markets are
more equal, more are served by every origin. The intuitive results on origin and destination
size accord with observed characteristics of the extensive margin of trade. The more subtle
implications of (14) remain to be explored in applications. Notice that this theory of the
extensive margin imposes no structure on the distribution of productivity that contributes
implicitly to the variation of t(i, z).
2.2
Infrastructure
Expanding the model to include a role for infrastructure appears to be important, since
its omission from most gravity treatments seems anomalous. But the omission of infrastructure investment is harmless whether origin-destination-specific investment in each origin
and destination country is efficient or not, provided that infrastructure is non-discriminatory.
Demonstrating this claim is the purpose of this section
Assume that domestic infrastructure D(i) is available in each region i, needed for distribution within i. D(i) is used as a public good by the trading activity in getting exports
out of origin i and D(z) is used as a public good in getting the exports to locations within
destination z. H(i, z) is now the skilled labor needed to connect i to z with
K(i, z) = D(i)ψ D(z)φ H(i, z)1−ψ−φ .
This expanded model leaves all the essentials as before in Section 2.1. The necessary
P
P
exceptions are that λ(i, z) ≡ H(i, z)1−ψ−φ / z H(i, z)1−ψ−φ , H(i)1−ψ−φ ≡ z H(i, z)1−ψ−φ
and the real activity is R(i) = L(i)α H(i)(1−α)(1−ψ−φ) . The exogenous bilateral trade cost
is t̃(i, z) ≡ t(i, z)/[D(i)ψ D(z)φ ]1−α . The joint activity R(i) now has diminishing returns to
scale. The implied rent to the activity is (1 − α)(ψ + φ)Y (i). This rent might be paid for
access to the public infrastructure assets if competitively priced, or (more realistically) split
between the owners of the activity and the public authorities that collect fees for access to
16
the public goods D(i) and D(z).
In this setup, infrastructure investment is irrelevant to the gravity model, whether investment is efficient or not. The intuition is that infrastructure as modeled here acts like a
trade productivity shifter in t̃(i, z) everywhere in the extended version of (9)-(11).
3
Econometric Application
Estimation of (11) is conveniently done with origin and destination fixed effects and controls
for bilateral frictions such as bilateral distance. With efficient investment (the standard
case previously) the size effects have unitary elasticity, a structure that can be imposed by
transforming the dependent variable into size-adjusted trade X(i, z)Y /E(z)Y (i). The short
run model (11) has size elasticities not equal to one, presenting an inference problem from the
origin and destination fixed effects that combine size effects with the influence of multilateral
resistance. The role of size effects can be identified in a two stage procedure that utilizes the
full structure (11)-(10). The origin and destination fixed effects absorb the constant term,
so alternatively it is sensible to drop one such fixed effect and have the constant term absorb
all other influences as well as the dropped fixed effect. This normalization of fixed effects is
required in any case by the structure of (9) and (10), as noted by Anderson and van Wincoop
(2003). The new issue posed by estimation of (11) is controls for inefficient investment λ(i, z).
This is discussed in Section 3.1, after which Section 3.2 returns to estimation.
3.1
Gravity with Evolving Investment
An application of the model structure when panel data is available is the investigation of
evolution of the allocations λ(i, z). In the absence of a fully developed theory of the determinants of the investment trajectory,5 our treatment of λs is necessarily limited. A guiding
5
Such a theory seems a chimera. The investments that plausibly form parts of Λ(i, z) are the result
of uncoordinated decisions of many actors. The information network theories of links suggest search and
random matching conditioned on anticipated probabilities of a match.
17
intuition is that some evolutionary process is gradually approaching efficiency because selection operates on each potential i, z link. Uncoordinated buyers’ and sellers’ agents grope
forward experimentally to form links. Over time they reduce or shut down currently more
inefficient allocations and maintain or increase more efficient ones.
Suppose that investment in pair-specific capital moves the current level from its past
level toward the efficient level at some rate of log-linear adjustment that implied by a CobbDouglas function of efficient and past levels. (This adjustment specification follows inter alia
Lucas and Prescott, 1971; Hercowitz and Sampson, 1991; and Anderson, Larch and Yotov,
2015 in the gravity context.) The stock adjustment process is
λ(i, z; τ ) = λ∗ (i, z)1−δ λ(i, z; τ − 1)δ ; δ ∈ (0, 1).
The parameter δ reflects costs of adjustment, the lower is δ the faster the movement to the
efficient level. In the steady state, λ = λ∗ . Agents know past λ(i, z; τ − 1), the gravity
frictions t(i, z) and form expectations of demand E ∗ (z) and multilateral resistances Π∗ (i)
and P ∗ (z), the arguments of λ∗ (i, z) in (13).
Operationalization of the adjustment process requires taking a stand on the agents’
expectations of the steady state. The agents know that efficient allocation implies that
λ∗ (i, z) = s∗ (i, z). Here we choose static expectations for s∗ (i, z): the demand and multilateral resistances are inferred by agents from past values as if by the econometrician using the
structure of the model and its fixed effects estimates. (The alternative of super-humanly rational expectations is implausible considering the extremely high dimensionality where each
bilateral link has potentially many uncoordinated agents and there are very many links with
simultaneous interaction.)
Moving toward operationality, replace the unobservable λ∗ (i, z) = s∗ (i, z) in (15) with
estimated (by the econometrician and the agent) ŝ(i, z, τ ). Similarly replace the unobservable
λ(i, z, τ − 1) with ŝ(i, z, τ − 1). Realized demands differ from expectations, so the short run
18
gravity model applies at each point in time. Then the costly adjustment specification in the
spirit of Lucas and Prescott uses the information on lagged expected shares to infer λs:
λ(i, z, τ ) = ŝ(i, z, τ )1−δ ŝ(i, z, τ − 1)δ .
A technical issue to be resolved is whether the adjustment process converges.
Return now to apply the implications of the adjustment process for an empirical form
of gravity suitable for estimation. Substitute the right hand side of the preceding equation
for the (implicit) contemporaneous value of λ(i, z, τ ) in gravity equation (11) and divide by
Y (i, τ ) to form the sales share of i to destination z. The result is
s(i, z, τ ) = Y (i, τ )
ρ−1
E(z, τ )
Y (τ )
ρ t(i, z)
Π(i, τ )P (z, τ )
(1−σ)ρ
s(i, z, τ )1−δ s(i, z, τ − 1)δ
1−ρ
The presence of the contemporaneous share on the right hand side of (15) implies scale effects
in the reduced form share:
s(i, z, τ ) =
ρ−1
Y (i, τ )
E(z, τ )
Y (τ )
ρ t(i, z)
Π(i, τ )P (z, τ )
(1−σ)ρ
!1/[ρ+δ(1−ρ)]
δ(1−ρ)
s(i, z, τ − 1)
(15)
All else equal (in the cross section), a rise in E(z, τ )/Y (τ ) raises s(i, z, τ ) less than proportionately when δ > 0. A rise in Y (i, τ )/Y (τ ) will raise X(i, z, τ ) less than proportionately
when δ > 0.
A richer lag structure may be an appropriate alternative specification.6
6
Evidence for a relative scale effect [consistent with a contemporaneous share in the expectation of efficient
investment levels leading to (15)] is provided by Anderson, Yotov and Vesselovsky (2016) in a study of the
inter-provincial and cross-border trade of Canadian provinces. In terms of specification (15), their results
may be interpreted as a value of δ that is larger for cross-border trade (region z is in another country) than
for trade within Canada (i and z are both regions within the country).
19
3.2
Estimation
The econometric reduced form of (15) can be estimated with standard gravity methods to
control for multilateral resistance and time-invariant trade cost proxies. Potential issues arise
from control of serial correlation with a lagged dependent variable. First solve (15) for the
reduced form value of s(i, z, τ ) and then add a Poisson error term representing measurement
error and all other sources of deviation from theory. The result is
ω0
s(i, z, τ ) = Y (i, τ ) E(z, τ )
ω1
t(i, z)
Π(i, τ )P (z, τ )
ω2
s(i, z, τ − 1)ω3 c(τ ) + (i, z, τ )
(16)
Here c(τ ) is a time varying constant term controlling for Y (τ ) and any other pure time
effects. Time varying origin and destination fixed effects control for multilateral resistance
and absorb the size effects as well.
Some elements of structure can be identified easily. ω3 > 0 if the short run model applies,
readily tested with a standard t-test. The structural interpretation of ω3 is:
ω3 = δ(1 − ρ)/[ρ + δ(1 − ρ)],
(17)
If a direct measure of trade cost is available (e.g. tariffs), the substitution parameter σ is
identified from σ = (1 − ω2 − ω3 )/(1 − ω3 ). Moreover, the theory implies that ω1 = 1 − ω3 ,
hence P (z, τ ) can be identified from P (z, τ )−ω2 = µ̂(z, τ )/E(z, τ )ω1 where µ̂(z, τ ) is the
estimated fixed effect used in regression (16).
The full set of structural parameters are identifiable using the estimated regression and
the restrictions of theory along with equations (9) and (10). Note ω̂1 = 1 − ω̂3 . Then
µ̂(z, τ ) = E(z, τ )1−ω̂3 P (z, τ )−ω̂2 . Substitute in (9) to form:
Π̂(i, τ )(1−σ)ρ =
X
γ̂(i, z)µ̂(z, τ ).
z
Then identify passthrough elasticity ρ̂ and hence adjustment cost parameter δ̂ = ω̂3 ρ̂/[(1 −
20
ω̂3 )(1 − ρ̂)] from fitting the second stage regression:
X
ln Y (i, τ ) − ln[
γ̂(i, z)µ̂(z, τ )] = (1/ρ) ln χ̂(i, τ ) + ν(i, τ )
(18)
z
Iterate using ρ̂ from (18) in (16) and (17) until convergence. Bootstrap for standard errors.
The approach in (15) is a first cut at a difficult problem. It leaves out plausibly important
effects. Investment in bilateral trade may be systematically affected in a differential fashion
by a number of variables reflecting allocations subject to credit constraints. Exchange rate
risk’s effects on trade flows can be hedged for many sectors with minimal cost, but hedging
over longer intervals appropriate for fixed commitments is expensive. This suggests that
bilateral exchange rate volatility may significantly affect investment in bilateral trade but not
variable trade cost. Similarly, bilateral covariance of business cycles may affect investment
but not variable trade cost. Such refinements are beyond the scope of this project.
3.3
Empirics of the Extensive Margin
In the short run, the extensive margin is endogenous because managers must be paid a premium that the marginal destination is just able to provide. Higher costs or lower demand
than anticipated when the specific factor allocation was made lead to lower than average
premia, hence sufficiently bad realizations will fall below the cutoff for managerial participation.
Let N be the number of destinations and n(i) be the proportion of destinations served
by origin i. For each origin i, order the destinations in decreasing share size, and scale z to
be in the unit interval. Then the marginal sector is s(i, n(i)) such that
n(i)N = L(i) −
1
α
.
1 − α s(i, n(i))
(19)
The time dimension is implicit here for simplicity. In the cross-section, the observables are the
proportion of sectors served n(i), the labor supply L(i) and the share of the marginal market
21
s(i, n(i)). Equation (19) can in principle be used to estimate α. But this procedure relies on
the simplifying assumption that 1 unit of labor is required to manage each destination. More
generally, assume that management in any destination market requires m units of labor. m
is identified from empirical forms based on (19) if α is observed or identified from estimation
of (15) and application of the structural restrictions.
Now reintroduce the time dimension. For a single origin i, equation (19) across time
describes the movement of the extensive margin as driven by size L(i, τ ) and underlying
determinants of equilibrium sales share s(i, n(i, τ ), τ ). The short run extensive margin modeled here is within the long run extensive margin set by positive investment λ in destinations
previously served.
Empirically modeling the long run extensive margin itself requires facing the difficulties
suggested in Section 2.1.1. A suggested approach to this problem is to use (19) in combination
with projected destination shares for destinations never previously served. The first line of
estimated (15) for destinations previously served generates the projections. Effectively, the
projection is taken as the steady state of process (15), as an approximation to the theoretical
(14). Projected shares are then ordered along with the observed lagged shares. The successful
fit of structural gravity suggests that projected shares are likely to be reasonably accurate.
(See Anderson, Borchert, Mattoo and Yotov, 2015, for projection of missing data in services
trade.)
4
Extensions
The model developed here can be applied at the firm level, where indeed its assumption
of a common Cobb-Douglas distribution ‘production’ function is most natural. This extension suggests more carefully modeling the ‘manager’ input, introducing heterogeneity across
markets and also across modes of organization: arms length contracting, joint ventures or
horizontal integration in a multinational structure.
22
Another potential extension is to the explanation of income inequality. The rents to
destination-specific managerial labor may be linked to inefficient investment and costs of
adjustment, inducing income inequality within firms and across firms in a sector.
The model also extends upward to embedding in a multi-sector general equilibrium setting. The stock of labor and human capital is simply the sectoral allocation, possibly with
differentials due to search costs.
23