Variable elasticity of substitution and price of capital in
asymmetric trade model
Sergey Kichko, Sergey Kokovin, Evgeny Zhelobodko
NRU HSE, NSU, Sobolev Institute of mathematics, Russia
Classic questions in the new-trade theory include the firms sizes in different countries, specialization of
countries, trade volume and value, Home Market Effect, and benefits to factor owners. How countries
asymmetry and decreasing trade costs (globalization) affect all these equilibrium outcomes? In addition
to such questions rised in many studies (see books by Helpman and Krugman, 1987; Brakman and
Heijdra, 2004), we are interested in dumping (see Brander and Krugman, 1983; Greenhut et al., 1987):
what leads firms to practice market segmentation?
Positioning our approach among the vast trade literature involves several dimensions. It has some
common questions with the classic (Hecksher-Ohlin) trade theory: comparative advantages of the
factor-endowed countries, countries' specialization (trade flows), factor's prices (non-)equalization. On
the other hand, instead of classic constant returns to scale, we share the common belief with the New
trade tradition: firm's increasing returns to scale and monopolistic competition that explains intraindustry trade, see Krugman (1979, 1980, 1981), reviews in Brakman and Heijdra (2004), and review of
heterogeneous models by Melitz and Redding (2012). However, we differ from this tradition in applying
the non-specified utility instead of CES or quadratic specifications used by most papers (see also CARA
functions suggested by Behrens and Murata (2007) and translog suggested by Feenstra (2008)). We
follow the attempt to restore the initial generality of the monopolistic competition (Dixit and Stiglitz,
1978; Krugman, 1979), undertaken in Zhelobodko et al. (2012) for a closed economy. It means deriving
the equilibria properties without the closed-form demands, as theorems for broad classes of functions.
Now this approach is applied to trade, to target the impact of capital endowments on pricing, firm's size,
trade flows, and Home Market Effect.
Our model combines disparities in the factor' endowment (as in Hecksher-Ohlin) and monopolistic
competition a la Martin and Rogers (1995). However, our novelty is the non-specified variable elasticity
of substitution (VES) utility in studying such questions. To motivate such approach, we would say that,
CES hypothesis is related to constant-markup outcome that is not consistent with the empirical
evidence. First, mark-ups vary with market size (Syverson, 2007). Second, firms do price discriminate
across destinations, using either dumping or reverse dumping (Martin, 2009; Manova and Zhang, 2009;
Bernard et al., 2007). Last, firms located in countries endowed with a large amount of human or physical
capital charge prices higher than firms located in poorly endowed countries (Schott, 2004; Hummels and
Klenow, 2005; Hallak, 2006; Hallak and Schott, 2008). The aim of this paper is to obviate some of these
drawbacks by studying trade flows and firms' pricing in a monopolistic competition setting with nonhomothetic preferences. To be more precise, instead of using the CES or any other specific preferences,
we use general additive preferences.
Specifically, we consider a trade model with two endowments-asymmetric countries, namely, capitalrich Home and poorer Foreign. Unlike the model developed by Krugman and Helpman (1987) who
assume that the relative endowment of labor and capital is the same in both countries and capital is
substitutable with labor, we assume changing share of labor and changing capital share between the
countries and look on the comparative statics. We combine these two main types of asymmetries with
non-homothetic preferences. Namely, the quasi-linear utility includes the numerarie (agriculture) and
one (easily extended to several) differentiated sector called manufacturing, gross utility from
manufacturing being embedded into an upper-tier concave function to model non-trivial substitution
with agriculture. The reason for using the numerarie is two-fold. First, it simplifies the analysis;
otherwise, under our possibly non-homothetic preferences we would have wage non-equalization
problem and income effects interfering into all effects that we want to see. Second, as soon as we want
to focus our attention on the difference between increasingly-elastic and decreasingly-elastic demands,
such per-industry question looks compatible with our partial equilibrium setting.
Our main results rely on the notion of the inverse demand elasticity (i.e., the inverse elasticity of
substitution among the varieties, or relative love for variety, RLV), as in Zhelobodko et al (2012). RLV,
which measures the concavity of elementary utility, can either increase or decrease with individual
consumption, CES being just the degenerate borderline between these two broad classes of
utilities/demands. Increasingly/decreasingly elastic inverse demands (IED/DED case) are called subconvex (super-convex) in Mrazova and Neary (2011), making a distinction for some qualitativelydifferent market outcomes. In particular, IED case is usually considered as most realistic, and guarantees
in many settings that the product price decreases with the number of firms, though the opposite case
cannot be excluded also.
In our setting some effects are independent from this demand characteristic. Speaking of individual
equilibrium consumptions of each variety, under any utilities, the purchase-size of a domestic variety in
Foreign is larger, than in Home, which in turn is larger than the size of imported variety in Home and the
smallest is imported in Foreign. These comparisons, naturally increasing with the trade cost, indirectly
manifestate the character of international competition here. Recall that RLV is just the markup (Lerner
index) in monopolistic competition and monopoly. So, under IED case the price is higher where the
consumption is higher (intensity of competition is lower) and these relations, inverted into the price
dimension, mean that the highest price is for domestic variety in Foreign, somewhat lower is the
domestic price at Home, then goes import in Home and import in Foreign is the cheapest. Quite
opposite is the ordering under DED. Moreover, both orderings keep holding even when we divide the
prices of import by the trade coefficient. This means dumping pricing by firms in both countries under
IED and reverse-dumping under DED, both effects being stronger for Foreign, which has less capital.
Importantly, these market outcomes here have nothing to do with strategic interactions or cooperative
behavior of producers, often used as explanations. Instead, the demand elasticity is the only driving
force of such market segmentation, and CES case is a borderline one, showing zero effects. Additionally,
the difference in the mill prices (cheaper manufacturing in capital-rich country) explain unequal
dumping effects between Home and Foreign, and we show how prices change with the trade cost and
capital share. Here the number of firms remains proportional to the capital endowment (as in traditional
models), so, the individual consumption of each variety decreases with the capital share. However, this
decreasing can be outweighed by increasing number of varieties and falling prices, so, under IED case
the worker's welfare increases with her country richness in capital (which is not easy to prove under
DED).
Some weak analogue of the Home Market Effect in the immobile-capital version of the model is the
value (and unit volume) of Home export exceeding that of the poor Foreign country.
Another topic is the firm sizes (outputs) and capital prices. Our simulations show that in Home country
both magnitudes behave generally (with small exception) monotonically increasing with the trade cost
and capital asymmetry. The reason is that the market-crowding effect from very numerous firms of
Home matters more for (few) Foreign firms than their market access to Home, and globalization is
harmful for them (though not for welfare). By contrast, Home firms generally suffer from big trade cost,
i.e., benefit from globalization, both in output and in capital price, because the access to (insufficiently
served domestically) Foreign market is important for them. Such effects are more subtitle under small
trade costs, which are not so small in our examples: less than 15% of the production cost. Here the
indirect effects of competition may outweigh the direct effects, Home outputs and capital price behave
non-monotonically. In general, we provide criteria for increasing (decreasing) firm's sizes and capital
pries that depend on asymmetry in countries' sizes, capital endowments, and concavity of both utilities-- upper-tier and elementary one (Propositions 5, 6). Moreover, under some particular classes of utilities
IED markets may behave oppositely to DED markets, CES case again being the borderline. Difference in
capital prices always increases with trade cost at least for small trade cost. This is an accelerator for
Home Market effect arising.
Finally, we allow for capital migration, which means the model becoming a Footloose Capital one. The
classic Home Market Effect (proportion of firms/population is disproportionally greater in a more
populated country) always arise. The qualitative nature of HME is the same for IED or DED situations
(sub-convex or super-convex demands), only the degree changes: sub-convex demands show stronger
HME and earlier switch to agglomeration in the globalization evolution. Our simulations illustrate this
result: not less than 70% of firms gather in the country having 60% of the world population even under
very big trade coefficient 3, and the coefficient less than 1.14 is sufficient to drop this analogue of the
Core-Periphery model into complete agglomeration: no firms in the smaller country, which is supplied
by manufacturing from abroad.
Summarizing, the patterns of trade in sectors with two (complimentary) production factors largely
depend on sub- or super-convexity nature of demands, on the relative sizes of the trading partners and
change with globalization in understandable directions even when we do not specify the functional
forms of the demands.