Jean-Luc Rouas

Isabel Trancoso

Céu Viana

Mónica Abreu

PORTUGUESE VARIETY IDENTIFICATION ON BROADCAST NEWS

Keywords: Automatic Language Identification, Portuguese Varieties, Broadcast News

This paper describes an accent identification system for Portuguese, that explores different type of properties: acoustic, phonotactic and prosodic. The system is designed to be used as a pre-processing module for the Portuguese Automatic Speech Recognition system developed at INESC-ID. In terms of variety identification, the overall rate of correct identification is 69.0% if all 7 varieties are considered, and the best results are obtained for Brazilian Portuguese, also the variety that proved easiest to identify in perceptual experiments. When distinguishing between European, Brazilian and African Portuguese, the identification rate goes up to 94.7%. The fact that the prosodic system alone can achieve an identification rate of 77% is also worth investigating.

INTRODUCTION

One of the problems encountered by the Automatic Speech Recognition (ASR) system developed at INESC-ID when applied to automatic captioning of broadcast news (BN) is the presence of different languages and different varieties of Portuguese. The presence of varieties other than European Portuguese (EP) may severely degrade the performance of the recognizer. In fact, whereas the word error rate (WER) of an ASR trained for EP is around 24% for this variety, for African Portuguese (AP) it can go from 30% to 38%, and for Brazilian Portuguese (BP), it may exceed 60%. This motivated the need for a variety identification module.

The orthographic differences are minor, which justifies similar out-of-vocabulary rates for the three varieties (1.4%, 2.0%, and 1.8%, for EP, AP and BP, respectively). Syntactic differences can be found in the use of prepositions, the position of clitics, and the alternative use of infinitive/gerundive verb forms. The lack of number agreement can be also found in BP and specially AP. However, the most striking differences concern pronunciation, namely vowel reduction, which is much more extreme in EP than in BP [START_REF] Mateus | The Phonology of Portuguese[END_REF], [START_REF] Barbosa | Brazilian Portuguese -illustrations of the IPA[END_REF]. Concerning prosody, whereas comparative studies of BP and EP can already be found [START_REF] Fernandes | Subject localization constraints in BP and EP[END_REF], as far we know, such studies are inexistent for African varieties. However, we strongly believe that they will play a crucial role in distinguishing between themselves.

Dialect/accent identification is a somewhat harder topic that language identification and has not yet been as much investigated [START_REF] Wu | Improving the discrimination between native accents when recorded over different channels[END_REF] [5] [6] [START_REF] Rongqing | Gaussian mixture selection and data selection for unsupervised spanish dialect classification[END_REF] although one can find a growing number of references on a related problem -foreign accent identification. Many approaches use language identification (LID) systems applied to native dialect identification.

This was the approach that we also followed and that will be described in section 2. The next section is dedicated to the corpus used in our variety identification experiments. The results are discussed in section 4, and compared with a human benchmark test.

VARIETY IDENTIFICATION SYSTEM

Our system is a fusion of 3 subsystems: Acoustic (section 2.2), Phonotactic or PRLM (section 2.2), and Prosodic (section 2.4). These 3 subsystems share a common audio pre-processing module (APP) as represented in Figure 1. For the time being, the fusion

PRLM System Acoustic System Prosodic System Audio signal F u s i o n

Decision

Preprocessing Fig. 1. Overview of the language identification system. method is only a simple weighted addition of the log-likelihoods generated by each system. The weights have been computed on the train part of the corpus described in the next chapter. The method is clearly non-optimal. Hence, it will not be described in detail and is only mentioned to give an idea of the performances that could be achieved using the three subsystems together.

Audio pre-processing

The APP module is part of our speech recognition system [START_REF] Meinedo | A stream-based audio segmentation, classification and clustering pre-processing system for broadcast news using ANN models[END_REF]). It integrates five components: three for classification (Speech/Non Speech, Gender and Background), one for speaker clustering and one for acoustic change detection. These models are composed of artificial neural networks of the type feed-forward fully connected multi-layer perceptron, and were trained with the back-propagation algorithm on a Portuguese BN corpus of over 60 hours [START_REF] Meinedo | Audio segmentation, classification and clustering in a broadcast news task[END_REF]. Two of the modules of this pre-processing stage are specially interesting for variety identification: the speech/non speech detection, as we do not want to treat non-speech parts, and the speaker clustering, as we assume that each speaker speaks a single variety and make the identification decision on a speaker by speaker basis.

Acoustic system

A generic acoustic language identification system is displayed in Figure 2. The system works in two phases: a learning procedure to create the models, and a testing procedure. The acoustic features Fig. 2. Generic acoustic language verification system. extracted from the audio signal are 12 MFCC plus delta, resulting in a 24-dimensional vector. The models used are Gaussian Mixture Models (as in [START_REF] Zissman | Automatic language identification using gaussian mixture and hidden markov models[END_REF]), learnt with the classic VQ and EM algorithms. As explained above, the PRLM system is based on a single Portuguese phone-recognizer. A synoptic of the system is given in Figure 3.

PRLM system

The phone recognizer is part of the AUDIMUS system [START_REF] Meinedo | Audimus.media: a broadcast news speech recognition system for the european portuguese language[END_REF], a hybrid recognizer that combines the temporal modeling capabilities of hidden Markov models with the pattern discriminative classification abilities of multi-layer Perceptrons. This phonetic decoding is applied to all the languages in the training database, resulting in Portuguese-phones sequences which are then modeled for each language by n-grams, using the SRI language modeling toolkit [START_REF] Stolcke | Srilm -an extensible language modeling toolkit[END_REF].

Prosodic system

The prosodic system is the same as used in [START_REF] Rouas | Automatic prosodic variations modelling for language and dialect discrimination[END_REF]. It is based on two different aspects: the definition of relevant units (pseudo-syllables) and the separate processing of the variations of macro-and microprosodic components (long-and short-term models). A synoptic of the system is displayed in Figure 4.

Segmentation, Vowel detection and Pseudo-syllables

The pseudo-syllable unit is defined as a cluster of consonants ending with a vowel, corresponding to the most frequent syllable structure in the world [START_REF] Dauer | Stress-timing and syllable-timing reanalysed[END_REF].

Three baseline procedures lead to relevant consonant, vocalic and silence segment boundaries: automatic speech segmentation [START_REF] André-Obrecht | A new statistical approach for automatic speech segmentation[END_REF], vocal activity detection [START_REF] Pellegrino | Vocalic system modeling: A VQ approach[END_REF] and vowel localisation (see [START_REF] Pellegrino | Vocalic system modeling: A VQ approach[END_REF] for more details). Labels "V", "C", or "#" are used to qualify each segment. Then, all the consonantal segments are merged until the next vocalic segment, which ends the pseudo-syllable.

Prosodic coding

Two models are used to separate the long-term and short-term components of prosody. The long-term component characterizes prosodic movements over several pseudo-syllables while the shortterm component represents prosodic movements inside a pseudosyllable. The fundamental frequency processing is divided into two phases, representing the phrase accentuation and the local accentuation, as in Fujisaki's work [START_REF] Fujisaki | Prosody, information and modeling -with emphasis on tonal features of speech[END_REF]. The phrase accentuation is used for the long-term model while the local accentuation is used for the short-term model. Fundamental frequency and energy are extracted from the signal using the SNACK Sound toolkit [START_REF] Sjölander | The snack sound toolkit[END_REF].

The long-term coding uses the pseudo-syllable segmentation as a time-base. The coding is described in Figure 5. The "baseline" is a The short-term coding is detailed in Figure 6. The short-term Fig. 6. Short-term coding.

coding use the "C", "V" and "#" segments as a time base. The local accentuation, named here residue, is represented by the difference between the original F0 contour and the baseline. This residue is then approximated on each segment by a linear regression. The F0 variation on voiced parts gives the label (Up or Down). Unvoiced parts are labelled "#". In parallel, the energy curve is computed and also approximated by linear regressions on each segment. The process is the same as the one used for the residue coding. The Up and Down labels are used to describe the variations while very short segments (e.g. <20ms) are labelled "#". Duration labels are also computed on the segment units. The "s" (short) and "l" (long) labels are assigned considering the mean duration of each kind of segment (vocalic, consonantic or silence). These three coding are used conjointly to form the short-term coding. Hence, for each segment, the label is then composed of three symbols.

Prosodic N-gram Modeling

To model the prosodic variations, we use classical n-gram language modelling provided by the SRI-LM toolkit [START_REF] Stolcke | Srilm -an extensible language modeling toolkit[END_REF]. As the best results for other databases were obtained with 3-grams [START_REF] Rouas | Automatic prosodic variations modelling for language and dialect discrimination[END_REF], this setting has been kept for these experiments. For each system -long-and shortterm -each target variety is modeled by an n-gram model during the learning procedure. During the test phase, the most likely variety is picked according to the model which provides the maximum likelihood.

CORPORA

For the variety identification task, we used the EP subset of the COST 278 corpus [START_REF] Vandecatseye | The COST 278 pan-european broadcast news database[END_REF], recorded from a public channel (RTP), complemented with BN shows transmitted from Portugal to Brazilian (TV Record) and African (Reporter Africa) speakers. For EP, the corpus includes 6 shows of different types. For BP, the corpus includes 6 40-minute shows (publicity segments were later excluded).

For AP, we have recorded 16 30-minute shows and manually labeled the varieties spoken by reporters in Angola, Cabo Verde, Guinea-Bissau, Mozambique, São Tomé and Príncipe. The number of automatically detected speakers and the duration (in minutes) for each variety is shown in Table 1.

VARIETY IDENTIFICATION EXPERIMENTS

The first experiment aimed at discriminating between all the Portuguese varieties at the same time. The tests were made according to the cross-validation procedure: First, one speaker was selected for testing. All the remaining data was used for learning the variety models. After the test was completed, a new speaker was used for testing. This procedure was iterated until all the speakers of the corpus had been used for testing. The global variety identification results are shown in Table 2. The average identification rate is 69.0%. The best recognised variety is Brazilian (97.9%), followed by European Portuguese (88.4%). The varieties from Guinea-Bissau and São Tomé and Principe are the worst recognised (0% and 7.8%). The lack of data for the African varieties -as compared with European and Brazilian Portuguese -may explain the poor performance achieved on this data by the system. In order to provide a more balanced experiment, we next addressed the identification of broad varieties by regrouping all the African varieties into one class. Using the grouping, we have almost the same amount of data for each broad variety (194 minutes for AP, 190 minutes for BP and 194 minutes for EP). Thus, the aim of this experiment was to identify if the test speaker speaks African, Brazilian or European Portuguese. The designed system performs quite well on this data, with a global identification rate of 94.7%. Detailed results (Table 3) show that the best identified variety is European Portuguese (97.1%). This result is obtained using the fused system. It is however noticeable that the prosodic system alone achieves an identification rate of more than 77%.

After identifying that a speaker speaks African Portuguese, the third experiment aimed at finding which African Portuguese variety is actually spoken. The global identification rate (see Table 4) is 60.1%. The most clearly identified varieties are Portuguese from Guinea-Bissau (73.7%) and Angola (71.2%). The weakest identification rate is for São Tomé and Príncipe.

Human benchmark experiment

In order to compare the performance of our automatic variety identification system with a manual one, we conducted a human benchmark. For this purpose, we selected 8 stimuli from each of the 7 varieties. In this selection, we avoided sentences that could give an indication either by lexical, syntactical or semantical cues of the origin of the speaker. The sentences (or segments from sentences) ranged in duration between 1.6 and 23.4 seconds. Most of the sentences were extracted from spontaneous speech (64%), in order to avoid easily identifiable journalists or politicians. Participants were asked to classify each stimulus as one of the 7 varieties, but they also had an option to mark it as African Portuguese (AP). In very few cases they forgot to (or could not) mark their preference (no answer -NA).

The test involved 65 participants: 44 participants from Portugal, 7 from Brazil and 14 from Africa (8 from Angola, 4 from Cape Verde and 2 from Mozambique). Table 5 shows the confusion matrix.

The results very clearly show that, as in the automatic test, Brazilian Portuguese is the least confusable variety. They also show It was interesting to notice that practically all Portuguese participants correctly identified BP and (although not so clearly) EP sentences, and most could correctly identify African varieties as such but, even if they have some suspicion about the African country of origin, namely if they have lived there, they were often reluctant to discriminate. Some Brazilian participants had no familiarity at all with African varieties, tending to confuse them with EP. Hence, the bad results for EP identification. Most African participants correctly identified BP and EP varieties, but they also tried to discriminate between African varieties more often. Their general opinion was that identifying African varieties in BN was much more difficult than identifying the varieties of the African people they meet everyday, most probably because in BN, many speakers (reporters, politicians and people involved in cultural events) have a higher level of education and/or familiarity with EP.

If these results are analyzed using only three broad classes (AP, BP and EP), as shown in Table 6, the average ratio of correct identification is 96.2%. Just for comparison purposes, we have also run an experiment aimed at investigating the behavior of the automatic system with these stimuli. The number of files is too small to get any significant results, and some of the files are too short, but still the automatic 3-class system yielded reasonably good results (above 70%).

CONCLUSIONS

Our accent identification system achieved an average correct identification ratio of 69.0%. The least confusable variety was by far BP (97.9% correct identification). EP was next. African varieties were the hardest to discriminate. That led us into trying to build a system with only 3 broad classes: AP, BP or EP. The average ratio achieved by this system was 94.7%.

The results of these experiments were compared with the ones of a human benchmark test, which basically revealed a very good capacity for detecting BP and, although not so easily, EP, and similar difficulties in discriminating African varieties, although they could also be easily identified as such. The average 3-class identification ratio was 96.2%.

Fig. 3 .

 3 Fig. 3. PRLM System overview.

Fig. 4 .

 4 Fig. 4. Prosodic system overview.

Fig. 5 .

 5 Fig. 5. Long-term coding.

Table 1 .

 1 Portuguese varieties.

	Code Country	#spks Dur(min)
	EP	Portugal	296	192
	BP	Brazil	404	190
	AN	Angola	86	41
	CV	Cape Verde	81	37
	GB	Guinea-Bissau	86	43
	MO	Mozambique	69	44
	ST	São Tomé and Principe	70	29

Table 2 .

 2 Identification

			of Portuguese varieties -Confusion matrix
	(%) using 7 varieties
		AN	BP	CV	EP	GB MO ST
	AN 31.2 26.0	1.3	37.7 0.0	2.6	1.3
	BP	0.0	97.9	0.5	1.6	0.0	0.0	0.0
	CV	0.0	5.3	36.0 53.3 0.0	5.3	0.0
	EP	0.7	1.7	0.7	88.4 7.6	1.0	0.0
	GB	6.3	11.4	3.8	74.7 0.0	3.8	0.0
	MO	7.5	16.4	0.0	40.3 0.0 35.8 0.0
	ST	9.4	17.2	4.7	53.1 0.0	7.8	7.8

Table 3 .

 3 Identification of Portuguese varieties -Confusion matrix (%) using 3 broad classes (African, Brazilian and European Por-

	tuguese).
		AP	BP	EP
	AP 93.0	6.3	0.7
	BP	5.5	94.5	0.0
	EP	2.3	0.6	97.1

Table 4 .

 4 Identification of African Portuguese varieties (%).

		AN	CV	GB	MO	ST
	AN 71.2	6.8	10.9	6.8	4.1
	CV	2.8	60.6 22.5	9.8	4.2
	GB	9.2	5.2	73.7	9.2	2.6
	MO 20.3	3.1	14.0 59.4	3.1
	ST	32.3 14.5 11.3 11.3 30.6

Table 5 .

 5 Human benchmark results (% of correct identification).

	Variety	AN	BP	CV	EP	GB	MO	ST	AP	NA
	AN	20.0	0.6	7.5	0.0	7.3	9.2	8.1	47.3	0.0
	BP	0.0	99.2	0.4	0.0	0.0	0.0	0.2	0.2	0.0
	CV	11.0	0.4	16.5	4.8	4.0	10.4	6.7	45.8	0.4
	EP	1.9	0.6	1.3	88.7	0.4	1.0	0.8	5.4	0.0
	GB	17.7	0.2	8.3	2.1	10.0	8.7	7.7	45.2	0.2
	MO	13.7	0.2	5.4	1.5	7.7	14.6	9.4	47.1	0.4
	ST	14.4	1.2	10.4	2.5	8.1	10.2	9.2	43.8	0.2
	that European Portuguese is next and that African varieties are easily
	confused with each other. Among these varieties, ST was the hardest
	to identify.

Table 6 .

 6 Human benchmark results with only 3 broad classes (%).

	Variety	AP	BP	EP	NA
	AP	97.1	0.5	2.2	0.2
	BP	0.8	99.2	0.0	0.0
	EP	10.8	0.6	88.7 0.0


Matthew T Cole
email: colemat@gmail.com

Keywords: Intra-industry trade, Trade policy, Firm heterogeneity, Monopolistic competition JEL classification: F10; F12;

There has been great focus in the recent trade theory literature on the introduction of firm heterogeneity into trade models. This introduction has highlighted the importance of the entry/exit decision of firms in response to changes in trade barriers. However, it is typical in many of these models to use iceberg transport costs as a general form of trade barriers that can be interchangeable with ad valorem tariffs. I show that this is not always an appropriate conclusion. Specifically, I illustrate that profit for an exporter is more elastic in response to tariffs than iceberg transport costs, which affects the entry/exit decision of firms. This has implications for welfare analysis and empirical specifications.

Introduction

There has been great focus in the recent trade theory literature on the introduction of firm heterogeneity into trade models beginning with Jean (2002) and Melitz (2003). These models, among many others, have provided a significant advancement in the literature on intraindustry trade since its conception with Krugman (1979, 1980). This comes, in part, from highlighting the entry and exit mechanism of firms. In fact, Chaney (2008) illustrates how properly accounting for the extensive margin, in a model with firm heterogeneity, actually reverses the Krugman (1980) prediction that the elasticity of substitution magnifies the sensitivity of trade flows to trade barriers. Additionally, the entry and exit of firms has direct ramifications for the number of varieties in equilibrium. Since consumers in these models show a love of variety, this has important welfare implications; if more low productivity domestic firms exit in response to lower trade barriers than foreign exporting firms enter, the domestic country actually loses varieties from freer trade. This is indeed interesting since all the gains from trade in the "New Trade Theory" stemmed purely from gains in variety. [START_REF] Arkolakis | Market Penetration Costs and the New Consumers Margin in International Trade[END_REF] The more recent trade theory models still find gains from trade. However, the effect on product variety has less consensus. In Melitz (2003), the effect on the total mass of varieties in a particular country is left ambiguous. Baldwin and Forslid (2010) address this issue and find that decreased trade restrictions, in fact, have a counterintuitive anti-variety effect for the importing country. However, Melitz and Ottaviano (2008) find that decreased trade restrictions have a pro-variety effect. In all three models (as in most models dealing with such issues), trade restrictions are modeled as the standard iceberg transportation cost. [START_REF] Baldwin | Hysteresis in Import Priced: The Beachhead Effect[END_REF] Although iceberg trade costs are equivalent to ad valorem tariffs in some settings, they are not equiva- [START_REF] Arkolakis | Market Penetration Costs and the New Consumers Margin in International Trade[END_REF] Broda and Weinstein (2006) show that growth in product variety from US imports has been an important source of gains from trade.

2 "Iceberg" transport costs are defined as a firm needing to ship more than one unit of good in order for one unit to arrive; the additional units "melt" away. lent in the case of monopolistic competition. Therefore one cannot take the lessons learned from the existing literature and blindly apply them to changes in tariffs. [START_REF] Baldwin | Trade Liberalization with Heterogeneous Firms[END_REF] A key contribution of this paper is to show iceberg transport costs affect firm profits and consequently the entry/exit decision differently than ad valorem tariffs in a monopolistic competition setting.

At first glance, this may seem to be a minor point. But, in models of monopolistic competition, the entry/exit decision affects total product variety and the aggregate price index; both of which determine welfare. In some sense, this is not unlike the literature contrasting the effects of ad valorem with per-unit transport costs. [START_REF] Becker | Taxation of Foreign Profits with Heterogeneous Multinational Firms[END_REF] However, the difference between iceberg transport costs and tariffs are more subtle because the two trade barriers are both ad valorem and the prices consumers pay are identical under each restriction. It is this subtlety that makes it so important to bring these differences to light.

To accomplish this, I provide a highly tractable model of heterogeneous firms that allows for asymmetric changes in three types of trade barriers; iceberg transport costs, ad valorem tariffs, and the additional fixed cost to become an exporter. Chaney (2008) uses a model with asymmetric iceberg transport costs and country sizes to investigate the effects of the elasticity of substitution on both the intensive and extensive margins of trade. He finds that the elasticity of substitution always dampens the impact of variable trade costs on trade flows. In particular, the decreased sensitivity of the extensive margin outweighs the increased sensitivity of the intensive margin. However, variable trade costs are modeled only as iceberg transport costs and not ad valorem tariffs. Thus, in this regard, my paper complements Chaney (2008) as it illustrates how the entry/exit decision of a firm is different depending on whether one models trade barriers as iceberg transport costs or ad valorem tar-iffs. In a general equilibrium model, this difference will affect the extensive margin directly and the intensive margin indirectly. [START_REF] Broda | Globalization and the Gains from Variety[END_REF] In order to provide the most tractable baseline model, I make various key assumptions that greatly ease the analysis of a situation with heterogeneous firms and endogenous entry. First, I assume firms are heterogeneous across fixed cost. Though the majority of the recent models assume firms are heterogeneous across marginal cost, there is a growing literature that assumes firms differ across fixed cost, e.g. Schmitt and Yu (2001), Jørgensen and Schröder (2006, 2008), and Davies and Eckel (2007). [START_REF] Chaney | Distorted Gravity: The Intensive and Extensive Margins of International Trade[END_REF] Jørgensen and Schröder (2008) provide a very nice motivation for the use of fixed cost heterogeneity. For instance, they point out that fixed cost heterogeneity is more appropriate with so-called "original brand name manufacturers" that differ in the power of their brand name -a result of marketing and other fixed cost activities. Arkolakis (2008) also incorporates marketing into a model with heterogeneous firms. Though firms can differ in expenditures on marketing, the main source of heterogeneity is from marginal cost in this model. This coincides well with recent empirical results that suggest there is heterogeneity in fixed as well as marginal cost. For instance, Cole, Elliott, and Virakul (2009) find that sunk costs (which are identical to fixed costs in my static model) and firm characteristics are important factors in explaining Thai manufacturing firm's decision to export. [START_REF] Chor | Subsidies for FDI: Implications from a Model with Heterogeneous Firms[END_REF] I have chosen to use fixed cost heterogeneity for three reasons: using marginal cost heterogeneity will not change the qualitative results; marginal cost heterogeneity will complicate the comparative statics significantly; and there is evidence that firms do differ across fixed costs. [START_REF] Cole | Firm Heterogeneity, Origin of Ownership and Export Participation[END_REF] Second, I assume the representative consumer has Dixit-Stiglitz preferences embedded in a quasi-linear utility function. The cost of this assumption is that the income elasticity of demand for the heterogeneous good is zero. Despite the cost, this assumption is not entirely uncommon in the literature. Chor (2009) uses a similar technique to investigate the merits of subsidizing foreign direct investment (FDI) in a model with heterogeneous firms. [START_REF] Cole | Distorted Trade Restrictions: A Comment on "Distorted Gravity[END_REF] Moreover, some models use more general utility functions, but then make other simplifying assumptions that mitigate income effects. Demidova and Rodríguez-Clare (2009) utilize a small country assumption to eliminate any income feedback effects. Similarly, Chaney (2008) makes a small/remote country assumption to ensure changes in transport and fixed costs have no significant impact on the general equilibrium. [START_REF] Davies | Tax Competition for Heterogeneous Firms with Endogenous Entry: The Case of Heterogeneous Fixed Costs[END_REF] In this paper, quasi-linear preferences prove useful beyond the simplifications they provide. My goal is to compare the differences between iceberg transport costs and ad valorem tariffs. Tariffs generate income and transport costs are often assumed to be lost. Thus, tariffs would create an income effect whereas iceberg transport costs would not, clouding the difference I focus on. Alternatively, I could model a transport sector that generates income, but I would need to take up a stance on which country the transport sector resides in as the income would affect demand. [START_REF] Demidova | Trade Policy Under Firm-Level Heterogeneity in a Small Economy[END_REF] Finally, in addition to the typical barriers to trade (tariffs and transport costs), I consider the effect of "foreign beachhead costs", that is, those fixed costs necessary to switch to engage in exporting. [START_REF] Eaton | An Anatomy of International Trade: Evidence from French Firms[END_REF] This is often a minor consideration, but with the rapid technological growth and service industries being created to facilitate business operations, these beachhead costs are becoming increasingly important. Friedman (2007:173) explains, "...UPS also has entry and exit decision. However, this does not eliminate the differences between iceberg transport costs and ad valorem tariffs as trade barriers. [START_REF] Cole | Distorted Trade Restrictions: A Comment on "Distorted Gravity[END_REF] See also Becker (2009). 10 Chaney (2008) points out that relaxing this assumption would reinforce his results. [START_REF] Demidova | Trade Policy Under Firm-Level Heterogeneity in a Small Economy[END_REF] It should be noted that income does change in response to changes in trade barriers and this income change affects welfare. However, it will all be through changes in consumption/production of the numeraire and not affect the heterogeneous goods sector. [START_REF] Eaton | An Anatomy of International Trade: Evidence from French Firms[END_REF] The term 'beachhead' costs was coined by Baldwin (1988).

a financing arm -UPS Capital -that will put up the money for the transformation of your supply chain, particularly if you are a small business and don't have the capital...UPS is creating enabling platforms for anyone to take his or her business global or vastly improve the efficiency of his or her global supply chain". This has direct implications for these particular beachhead costs and needs to be considered in conjunction with investigating changes in other trade restrictions, as they may have conflicting results.

The paper proceeds as follows. Section 2 sets up the model and characterizes the equilibrium. Section 3 analyzes the results including a discussion of the results under alternative modeling assumptions including marginal cost heterogeneity. Section 4 concludes.

The Model

There are two countries labeled k and j. Country k ( j) is endowed with Lk ( L j ) units of labor which is the sole factor of production. Without loss of generality, let Lk ≥ L j . There are two sectors. Sector 1 is the numeraire and consists of a homogeneous good (y) that is produced under constant returns to scale, freely traded, and sold in a perfectly competitive market. Sector 2 consists of a continuum of differentiated goods, each variety of which is indexed by i. As is standard in the Melitz model, this is produced under increasing returns to scale in a monopolistically competitive market with free entry. Unlike sector 1, this market may face both transportation costs and tariff barriers. With the exception of different labor endowments and (potentially) tariff rates, countries are identical. Therefore, analyzing the situation for country k informs us of the analogous situation for country j, and I will refer to country k as the domestic country to ease discussion.

Sector 1

The price of good Y is normalized to 1 in each market. Assuming that one unit of labor is needed for production, this will normalize the wage in each country to unity. Finally, I assume that in equilibrium a positive amount of good Y is produced in each country.

Consumers

The representative consumer in country k has quasi-linear preferences with an embedded Dixit-Stiglitz utility function which displays love for variety over the heterogeneous good;

U k = µ ln(X k ) +Y k , X k = N k 0 x k (i) α di 1 α , µ > 0 (1)
where ε = 1/(1 -α) > 1 is the elasticity of substitution, N k is the total mass of varieties in country k, Y k denotes aggregate consumption of the numeraire, and X k can be interpreted as the amount of a composite good comprised of the different varieties of the heterogeneous good x k (i). Quasi-linear utility will isolate the decision whether to become an exporter or not without any income feedback effects; providing a model that allows for asymmetric changes in trade restrictions to be easily analyzed. Moreover, this specification allows me to compare the differences between an ad valorem tariff and iceberg trade costs on productivity and variety without having to account for the income effects of the tariff or the "wasteful" costs of iceberg transport costs. Finally, I assume that income in each country is sufficiently large that both y and x goods are consumed.

Consumers maximize utility subject to their budget constraint:

N k 0 p c k (i)x k (i)di +Y k ≤ I k (2)
where p c k (i) is the price of variety i paid by consumers and I k is aggregate income in country k. [START_REF] Friedman | The World is Flat: A Brief History of the Twenty-first Century[END_REF] The solution to this problem yields a demand function for the heterogeneous good of variety i in country k:

x k (i) = p c k (i) -ε µ N k 0 p c k (i) 1-ε di . (3)
Since preferences are identical across both countries, it follows that the total expenditure on the heterogeneous good is equal to µ in both foreign and domestic markets.

Heterogeneous Firms

There are a continuum of firms, each of which holds a unique position on an index, where each point i represents a unique variety and productivity level. [START_REF] Irarrazabal | The Tip of the Iceberg: Modeling Trade Costs and implications for Intra-Industry Reallocation[END_REF] Armed with this index the firm decides whether to serve the domestic market and/or the overseas market. To serve a given market, the firm must incur a fixed cost. These costs are referred to as 'beachhead' costs and can be interpreted as forming a distribution and servicing network. To serve its domestic market, a firm with index i must hire f (i) units of labor (making the fixed cost of serving this market f (i)). If a firm chooses to serve the foreign market, it can do so through exports and pay an extra γ f (i). I assume that γ > 1; f ′ (i) > 0 and f ′′ (i) ≥ 0, i.e. the mapping from the index to the labor required for beachhead costs is increasing and convex in the index. [START_REF] Jean | International Trade and Firms' Heterogeneity under Monopolistic Competition[END_REF] Thus, firms requiring fewer workers to cover beachhead costs have a lower index i. These fixed cost differences are the source of firm heterogeneity. A firm, therefore, faces the menu of fixed costs (measured in units of labor) illustrated by Table 1:

Table 1 Fixed Cost Menu Firm Type Fixed Cost domestic only f (i) domestic and exporter (1 + γ) f (i)
Goods that are exported from country k to country j are subject to melting-iceberg transport costs, σ = 1 + s ≥ 1, where a firm must ship σ units in order for one unit to arrive at its destination. I assume that transport costs are symmetric and thus omit country subscripts. [START_REF] Jørgensen | Tariffs and Firm-Level Heterogeneous Fixed Export Costs[END_REF] I do not investigate the effect of a per-unit transport cost; since marginal costs are identical for all exporting firms, this would have the same effect as iceberg transport costs. Additionally, an exporting firm from country k is subject to an ad valorem tariff τ j , where I define t j ≡ 1 + τ j . Furthermore, I assume that a government is unable to distinguish a particular firm's type, so any tariff is an across-the-board tariff applied to all exporters. Note that tariffs can differ across countries.

The decision to become a firm and which market(s) to service depends on the associated profit for each type. Recall that the numeraire yields wages equal to one in both countries, thus the operating profits from serving the domestic market are

π k D (i) = p k (i)q k (i) -q k (i) -f (i). (4)
heterogeneous good are not too different, this ensures that will never happen. Moreover, it does so by allowing profits in both countries to be additively separable, which is quite attractive. Relaxing this assumption, but restricting the firm to sell at home before exporting would only complicate the model without changing the qualitative results. [START_REF] Jørgensen | Tariffs and Firm-Level Heterogeneous Fixed Export Costs[END_REF] This assumption is only done for notational ease. In order to investigate asymmetric changes in transport costs, one only needs add a country subscript to σ .

Given the nature of monopolistic competition, the price will be a constant mark-up over marginal cost and be equal to [START_REF] Arkolakis | Market Penetration Costs and the New Consumers Margin in International Trade[END_REF] α . From market clearing, set q k (i) = x k (i), and the firm has the following profit function for supplying to the domestic market only:

π k D (i) = B k -f (i) (5)
where

B k = 1 εα 1-ε µ P 1-ε k and P k = N k 0 p c k (i) 1-ε di 1 1
-ε is the aggregate price index of the heterogeneous good.

The decision to become an exporter stems purely from the additional profits from serving the foreign market. [START_REF] Jørgensen | Fixed Export Cost Heterogeneity, Trade and Welfare[END_REF] The main contribution of this paper is to illustrate that there is an important distinction between modeling trade restrictions as iceberg transport costs or ad valorem tariffs. Consequently, I will explicitly derive the additional profit function from exports for the firm in country k exporting to country j. This function is:

π k X (i) = t j p j (i)x j (i) -τ j p j (i)x j (i) -σ x j (i) -γ f (i). (6)
It can easily be seen by t j p j (i)x j (i) -τ j p j (i)x j (i) = p j (i)x j (i), that imposing a tariff on the firm is analogous to imposing it on the consumer. Recalling that p c j (i) is the price consumers pay, the demand for variety i is

x j (i) = p c j (i) -ε µ N j 0 p c j (i) 1-ε di = [t j p j (i)] -ε µ N j 0 p c j (i) 1-ε di . (7)
Thus ( 6) can be written as

π k X (i) = [p j (i) -σ ] [t j p j (i)] -ε µ N j 0 p c j (i) 1-ε di -γ f (i) (8)
Note that the presence of a tariff is just a monotonic transformation of the profit function, so the firm's optimal price setting rule is unaffected by the tariff (it is still a constant markup over marginal cost). However, the price paid by the consumer, p c , is affected. Therefore, the exporting firm's optimal price is

p j (i) = - εσ 1 -ε = σ α (9)
and the price consumers pay for exported variety (imported from their perspective) is

p c j (i) = t j σ α . (10)
Thus, regardless of whether one chooses to model trade restrictions as iceberg transport costs, ad valorem tariffs, or some more general term encompassing them both, ς = tσ , the effect on the price consumers pay is the same -the restriction is completely passed through onto them. However, the effects on firm profits are, in fact, different and this is important when dealing with a general equilibrium model and firm entry. To see this, insert the price, (9), into the firm profit function (8)

π k X (i) = σ α -σ t j σ α -ε µ P 1-ε j -γ f (i). (11)
The underbraced term is the key here. Due to the monopolistic nature of the model, firms charge a markup over marginal cost and transport costs are included in marginal cost. [START_REF] Krugman | Increasing Returns, Monopolistic Competition, and International Trade[END_REF] Recall σ = 1 + s. If a firm ships one unit, it loses s units in transport, but gains s α from it's ability to charge a price higher than marginal cost, for a net gain (excluding demand effects) of αε > 0. Conversely, a tariff results only in decreased demand, which as seen by ( 10) is identical to that of iceberg transport costs. Thus, not only are profits higher with iceberg transport costs compared to an identical ad valorem tariff, but the sensitivity of profits to changes in these two different forms of trade restrictions differs as well; i.e. the variable profit will be more elastic with respect to tariffs than iceberg transport costs. [START_REF] Krugman | Scale Economies, Product Differentiation, and the Pattern of Trade[END_REF] Essentially, through monopolistic power, the firm is able to recoup a portion of its losses in transport;

whereas tariff revenue is completely captured by the domestic government. The markup over marginal cost drives a wedge between the effect of iceberg transport costs and an ad valorem tariff.

The effect of trade restrictions on product variety is an important welfare consideration and is determined by the extent domestic varieties enter to replace imported foreign varieties.

Since the choice of trade restrictions affects the variable profit elasticity and consequently the foreign firm's decision to enter or exit, this has implications with regard to product variety. For notational ease, I will write profits from exports as:

π k X (i) = t -ε j σ 1-ε B j -γ f (i). (12)
Again, note the different exponents on tariffs (t j ) and transport costs (σ ), a difference at the heart of the differing variety effects.

Relaxing Modeling Assumptions

In this section, I briefly describe the effects of two specific assumptions on the results' generality. To begin, suppose firms were additionally heterogeneous across marginal costs, a(i). This means that firms will charge a different price; specifically

t j σ a(i) α
for an exporter. [START_REF] Lawless | Where Do Firms Export, How Much, and Why[END_REF] Additionally, let M j (p c j , I j (•)) be the expenditure on the heterogeneous good in country j, which is a function of the vector of prices and income. Therefore, an exporting firm is faced with the following general profit function:

π k X (i) = t -ε j σ 1-ε a(i) 1-ε M j ε i jD 0 a(i) 1-ε di + i kX 0 (t j σ a(i)) 1-ε di -γ f (i) (13)
i.e. the B j term becomes more complex.

For a baseline model, I assume that a(i) = 1 for all i, and M j = µ. It can be seen that allowing marginal cost to differ across firms will have an affect on the results. However, since the t j is raised to a different exponent than σ in the underbraced term, differences still arise for different trade barriers. Furthermore, using a different utility function will obviously affect M j (p c j , I j (•)). [START_REF] Melitz | The Impact of Trade on Intraindustry Reallocations and Aggregate Industry productivity[END_REF] In particular, tariffs generate income where iceberg transport costs are generally assumed to be wasted. There are two points to be made with regard to this: One, in order for tariffs to have the same affect on exporting firm profits as iceberg transport costs, the utility function would have to result in M j (p c j , I j (t j , •)) ≡ t j M j (p c j , I j (•)), which is more restrictive than assuming quasi-linear preferences; and two, as mentioned in the introduction, there is a transportation sector that does generate income. To be completely rigorous, I would need to model this sector. However, it would seem to be a very special case, in which tariffs and transport costs affect income in such a way to offset the differences highlighted by the underbraced term in equation (13). Therefore, the result that ad valorem tariffs affect exporting profits differently than iceberg transport costs is not driven by my simple baseline model.

Equilibrium

Firms will enter each market as long as there are positive profits, that is, until equations ( 5) and ( 12) are driven to zero. Thus, define the cut-off firms as the firms that draw the values in the index (i) that solves the following equalities:

B k = f (i kD ) ( 14
)
B j γt ε j σ ε-1 = f (i kX ) (15)
B j = f (i jD ) (16)
B k γt ε k σ ε-1 = f (i jX ). (17)
The indices i kD and i jD represent the firms that are indifferent between producing the heterogeneous good and not producing at all in country k and j respectively. The indices i kX and i jX represent the firms that are indifferent between serving both the domestic and foreign markets and serving only the domestic market. Furthermore, the terms on the left-hand side of the equalities represent the variable profit for a particular firm and are functions of the total mass of firms (domestic and foreign). [START_REF] Melitz | Market Size, Trade, and Productivity[END_REF] Figure 1 illustrates the profits, with zero variable trade costs, of firms in country k including those who export and those who only sell domestically. [START_REF] Schmitt | Economics of Scale and the Volume of Intra-industry Trade[END_REF] It can be seen that the greater the index i, the greater the fixed cost to enter a market, and thus the lower the profits.

The intersection with the horizontal axis represents the index in which profits are zero for operating in that particular market. Note that the line representing export profits defines the profits from exporting in addition to serving the domestic market. In other words, firms with an index i ∈ [0, i kX ] make profits from exporting and serving the domestic market, and firms with an index i ∈ (i kX , i kD ] make profits from only serving the domestic market. Firms with an index i > i kD do not produce. After careful inspection of the equilibrium conditions, it can be seen that there are, in fact, two systems of two equations and two unknowns: equations ( 14) and ( 17) and equations (15) and (16). [START_REF] Yu | Entrepreneurship and Intra-Industry Trade[END_REF] Moreover, due to the symmetry it is sufficient to only focus on one country. I will focus on the output market in country k, and thus equations ( 14) and (17). For future use, it will be helpful to rewrite the equilibrium conditions, ( 14) and (17), in the following manner:

f (i kD ) = µ ε(i kD + (t k σ ) 1-ε i jX ) ( 18
)
f (i kD ) = γt ε k σ ε-1 f (i jX ). ( 19
)

Changes in Equilibrium

Although, I cannot explicitly solve for the cutoff values without assuming a functional form of the fixed cost mapping f (i), I am still able to characterize the comparative statics. Totally differentiating this system of equations ( 18) and ( 19) yields the following comparative statics:

∂ i jX ∂t k = - 1 t k ψ ε f (i kD )[1 + δ (i kD )] + i jX f ′ (i kD ) (t k σ ) ε-1 < 0 (20) ∂ i jX ∂ σ = (1 -ε) f (i kD ) σ ψ [1 + δ (i kD )] < 0 ( 21
) ∂ i jX ∂ γ = - f ′ (i kD ) f (i kD )ψγ µ ε + f (i kD ) 2 f ′ (i kD ) < 0 ( 22
) ∂ i kD ∂t k = εγ f (i jX ) ψ 1 + αδ (i jX ) > 0 ( 23
) ∂ i kD ∂ σ = (ε -1)γt k f (i jX ) σ ψ 1 + δ (i jX ) > 0 ( 24
) ∂ i kD ∂ γ = t k f (i jX ) ψ > 0 ( 25
)
where

ψ ≡ f ′ (i kD ) f (i kD ) f ′ (i jX ) f (i jX ) µ ε + f (i kD ) 2 f ′ (i kD ) + γt k f (i jX ) , and
δ (z) = z f ′ (z) f (z) > 0.
The term δ (z) represents the elasticity of fixed costs with respect to the index i, evaluated at z. Equations ( 20) through (22) represent the effect of changes in trade restrictions (either through a tariff, transport cost, or foreign beachhead cost) on the cutoff firm serving the foreign market. It follows that increases in trade restrictions decrease this cutoff, or in other words the mass of exporting firms has decreased. By decreasing the mass of exporting firms and the foreign firms still producing now charging a higher price relative to domestic producers, there is less competition in the domestic market. This decreased competition makes being a domestic firm more profitable, thereby increasing the mass of domestic firms -illustrated by equations ( 23) through (25). The fact that increased trade restrictions, in general, have these results is not surprising. What is important is that different trade restrictions correspond to different magnitudes in firm cutoff changes.

It is not sufficient to simply show that the magnitudes of the comparative statics are different in order to show that there is a meaningful difference between iceberg transport costs and ad valorem tariffs. If there is a mapping between t k and σ that equates the comparative statics, then we simply have a monotonic transformation. In fact, there does exist a (t k , σ )

pair that equates the comparative statics ( 20) with ( 21) and ( 23) with (24). However, this pair is not identical for both equalities. Note the following:

∂ i jX ∂t k = ∂ i jX ∂ σ ⇔ αt k σ = 1 + (t k σ ) 1-ε i jX f ′ (i kD ) ε f (i kD )[1 + δ (i kD )] > 1 (26) ∂ i kD ∂t k = ∂ i kD ∂ σ ⇔ αt k σ = 1 + αδ (i jX ) 1 + δ (i jX ) < 1 (27)
As can be seen, the (t k , σ ) pair that equates (20) with ( 21) is not the same pair that equates (23) with (24). This reinforces the fact that iceberg transport costs are not isomorphic to ad valorem tariffs. I illustrate this in more detail throughout the following sections.

Variety Effect

As just shown, different trade barriers affect the entry and exit decision by firms in different ways. This is important for two main reasons; the effect on total variety is part of welfare and if two or more barriers are changing at the same time, it is critical to understand these differences to know whether these changes will amplify or negate each other. Therefore, I now investigate how each trade barrier affects the equilibrium mass of varieties. The corresponding effects on the mass of varieties in country k are as follows:25

∂ N k ∂t k = εγ f (i jX ) ψ 1 + αδ (i jX ) -(t k σ ) ε-1 [1 + δ (i kD )] - i jX εi kD δ (i kD ) (28)
∂ N k ∂ σ = (ε -1)γt k f (i jX ) σ ψ 1 + δ (i jX ) -(t k σ ) ε-1 1 + δ (i kD ) (29)
∂ N k ∂ γ = 1 ψγ 1 (t k σ ) ε-1 -1 f (i kD ) - f ′ (i kD ) f (i kD ) µ ε < 0 (30)
It can be seen from equations ( 28) and ( 29) that the effect of tariffs and iceberg transport costs have an ambiguous effect on equilibrium total variety. The following proposition pins down the condition that ensures a pro-variety effect associated with decreases in iceberg transport costs.26

Proposition 1 There is a pro-variety effect associated with decreases in iceberg transport costs if and only if

1 (t k σ ) ε-1 < 1 + δ (i kD ) 1 + δ (i jX )
.

Proof Proof is by direct calculation. This is a sufficient and necessary condition. A more restrictive condition for a pro-variety effect, although one that is perhaps more intuitive, is if the elasticity of f (i) with respect to the index is nondecreasing in i. Examples would include both linear, exponential, and power functions of i.

It is difficult to compare the magnitudes of the variety effects from changes in iceberg transport costs and ad valorem tariffs because these magnitudes depend on the actual values of t k and σ . 27 However, I can comment on the direction of these variety effects.

Corollary 1 If ∂ N k ∂ σ ≤ 0 then ∂ N k ∂t k < 0. Proof Let ∂ N k ∂ σ = 0, then (t k σ ) ε-1 [1 + δ (i kD )] = 1 + δ (i jX ).
Plugging this into (28) yields:

∂ N k ∂t k = εγ f (i jX ) ψ 1 + αδ (i jX ) -1 -δ (i jX ) - i jX εi kD δ (i kD ) = -γ f (i jX ) ψ δ (i jX ) + i jX i kD δ (i kD ) < 0. It is straightforward that ∂ N k ∂ σ < 0 ⇒ ∂ N k ∂t k < 0.
It is obvious that the contrapositive to Corollary 1 is also true, that is

∂ N k ∂t k ≥ 0 ⇒ ∂ N k ∂ σ > 0.
The fact that trade restrictions can have an ambiguous effect on total product variety in a country is not surprising or new. What is surprising and new is that it is possible for there to be an anti-variety effect associated with lower transport costs while there also being a provariety effect associated with lower tariffs. Thus, under certain specifications a reduction in both trade barriers could lead to no change in total product variety. Again, these differences 27 For purposes of comparison, one logical choice would be to evaluate the comparative statics when the trade restrictions are equal. Let σ = t k = ρ ≥ 1, then

∂ N k ∂t k σ =t k =ρ = ∂ N k ∂ σ σ =t k =ρ + γ ρ ∂ N k ∂ γ σ =t k =ρ .
are driven by how iceberg transport costs affect profit differently than an ad valorem tariff.

Changes in a firm's variable profit (vπ) are the reason for entry and exit and the variable profit is more elastic with respect to tariffs than iceberg transport costs:

t k vπ ∂ vπ ∂t k - σ vπ ∂ vπ ∂ σ = -f ′ (i jX ) f ′ (i kD )µ + ε f (i kD ) 2 f ′ (i jX ) f ′ (i kD )µ + ε f (i kD ) 2 + εγt k f (i jX ) 2 f ′ (i kD ) < 0.
This, in turn, affects the elasticity of the firm cutoffs -the cutoffs for an exporting and a purely domestic firm are more elastic in response to a change in an ad valorem tariff than iceberg transport costs. Turning now to the elasticity of total variety, N k , which I define as ρ t k and ρ σ for tariffs and transport cost respectively:

ρ t k = εγt k f (i jX ) N k ψ 1 + αδ (i jX ) -(t k σ ) ε-1 [1 + δ (i kD )] - i jX εi kD δ (i kD ) ρ σ = (ε -1)γt k f (i jX ) N k ψ 1 + δ (i jX ) -(t k σ ) ε-1 [1 + δ (i kD )] .
Comparing the two elasticities yields:

ρ t k -ρ σ = t k γ f (i jX ) N k ψ 1 -(t k σ ) ε-1 [1 + δ (i kD )] - i jX i kD δ (i kD ) < 0.
One has to be careful in interpreting this result given the results of Corollary 1. If there is a pro-variety effect associated with a decrease in iceberg transport costs, then the total product variety is more elastic (but is negative) with respect to tariffs. 28 If there is an antivariety effect associated with a decrease in ad valorem tariffs, then the total product variety is less elastic (but is positive) with respect to tariffs. 29 Finally, there are scenarios in which the two trade barriers have opposite effects on variety resulting in ambiguity as to the relative elasticities.

Welfare Effect

Though the number (mass) of varieties is a significant part of welfare, it is not the only part;

price changes also affect welfare. In this section, I analyze how modeling trade barriers as iceberg transport costs affects welfare (through both variety and price changes) differently then by using an ad valorem tariff. The indirect utility of the representative consumer, and my measure of national welfare, is

V k = µ ln (X k ) + I k -µ. (31)
Income is equal to labor income plus profits from domestically owned firms (for purposes of comparison, I will omit tariff revenue and a transport sector as sources of income):

I k = L k + i kX 0 π k X (i)di + i kD 0 π k D (i)di.
There are many potential ways to compare the different effects on welfare from changes in the different trade barriers. In order to be as succinct and clear as possible, I make a few further assumptions. As mentioned, I omit the different avenues of income from trade barriers; i.e. through tariffs or a transport sector. Furthermore, although this model can easily analyze asymmetric changes in trade barriers, I will focus on symmetric changes in identical tariff levels (t k = t j = t). Note that these assumptions, along with quasi-linear utility, result in a completely symmetric equilibrium. Thus, for brevity, I will drop country subscripts.

Differentiating national welfare (31) of country k with respect to the tariff and iceberg transport costs yields: 30

∂V ∂t = (C D + σC X ) -µ P ∂ P ∂t - σC X tα < 0, ( 32
) ∂V ∂ σ = (C D + σC X ) -µ P ∂ P ∂ σ -C X < 0, (33)
where

∂ P ∂t = P ε α 1-ε (1 -ε) ∂ i D ∂t + (1 -ε) (σt) 1-ε t i X + (σt) 1-ε ∂ i X ∂t > 0, and
∂ P ∂ σ = P ε α 1-ε (1 -ε) ∂ i D ∂ σ + (1 -ε) (σt) 1-ε σ i X + (σt) 1-ε ∂ i X ∂ σ > 0.
The term C X = i X 0 x(i)di represents a country's aggregate consumption of imported varieties of the differentiated good and I define C D = i D 0 x(i)di as a country's aggregate consumption of the domestically produced varieties of the differentiated good.

For purposes of comparison, let the tariff level be identical to iceberg transport costs;

i.e. t = σ = ρ. The differences in the welfare effect is:

∂V ∂t - ∂V ∂ σ = (C D + σC X ) -µ (ε -1) (σt) 1-ε f ′ (i D ) ρψ - C X (ε -1) < 0. (34)
Therefore, excluding income effects from tariff or transport revenue, using the lessons learned from lowering iceberg transport costs underestimates the welfare benefits of trade liberalization from a reduction in tariffs. This suggests that, given the choice, it may be more welfare improving to focus efforts on tariff liberalization than lowering transport costs. Since like firms (domestic or exporting) have the same marginal cost and consequently the same price, the extensive margin is the driving force behind the change in the price index. Further, in 30 Detailed derivations are available upon request.

this particular situation, iceberg transport costs are identical to per-unit costs since marginal costs are homogeneous and normalized to one. Therefore, in this setting, there are greater welfare gains from a reduction in an ad valorem tariff than a reduction in per-unit transport costs. Irarrazabal et al. (2010) find that the welfare gains from reducing per-unit frictions are higher than that of reducing iceberg frictions. Their model has heterogeneity across marginal costs which emphasizes within-market reallocation as an additional channel of gains from trade; this is obviously a different model. However, my model suggests that if ad valorem tariffs were used instead of iceberg transport costs, this gap in welfare gains would at the very least be dampened.

Conclusion

It is common in the recent trade literature to simply assume iceberg transport costs as a general proxy for many types of trade restrictions (in particular ad valorem tariffs). When perfect competition is assumed the two trade barriers are analogous. However, in the often used model of monopolistic competition, this is no longer the case. I have provided a simple model of trade with monopolistic competition and heterogeneous firms that illustrates how changes in iceberg transport costs affect profit differently than ad valorem tariffs. This, in turn, affects the elasticity of entry and exit differently. This elasticity of entry and exit is important as it affects the equilibrium number of varieties and the aggregate price index; both of which affect welfare. Furthermore, I have shown that although iceberg transport costs and ad valorem tariffs have an ambiguous effect on total variety, they do not necessarily have the same effect; i.e. it is possible for there to be an anti-variety effect associated with lower transport costs while there also being a pro-variety effect associated with lower tariffs.

Finally, I have shown that using the lessons learned from lowering iceberg transport costs underestimates the welfare benefits of trade liberalization from a reduction in tariffs. Thus, there are consequences in how one chooses to model trade restrictions. These consequences can be over(under)-emphasizing the effects of trade liberalization or a misspecified empirical equation. For instance, Cole (2010) illustrates using the Chaney (2008) framework that the elasticity of trade flows with respect to ad valorem tariffs is not a constant, like that of iceberg transport costs, but is a function of the elasticity of substitution. This has implications for how one would specify their estimating equation when using a gravity model.

Fig. 1

 1 Fig. 1 Profits from production in country k with free trade

Similarly, it may not be appropriate to simply "waste" tariff revenue in order to model iceberg transport costs as[START_REF] Jørgensen | Fixed Export Cost Heterogeneity, Trade and Welfare[END_REF] does.

See, for example,[START_REF] Irarrazabal | The Tip of the Iceberg: Modeling Trade Costs and implications for Intra-Industry Reallocation[END_REF] who show that welfare gains from reducing per-unit frictions are higher than that of reducing iceberg frictions.

See[START_REF] Cole | Distorted Trade Restrictions: A Comment on "Distorted Gravity[END_REF] for an explicit description of how my paper complements[START_REF] Chaney | Distorted Gravity: The Intensive and Extensive Margins of International Trade[END_REF].

A key difference betweenJøgensen and Schröder (2006, 2008) and my model is that all firms, purely domestic and those who export, are heterogeneous across fixed cost, whereJørgensen and Schröder (2006, 2008) only allow the fixed cost to export to differ.

See also Eaton et al. (2008) who use a Melitz-type model calibrated to a French data set, and Lawless and Whelan (2008) who explain trade flows for Irish owned firms.

The use of fixed cost heterogeneity results in all firms of the same "type" (either pure domestic or exporting) to charge the same price. This obviously affects firm demand and profits which in turn affects the

Note that if tariffs are set to zero or the firm is domestic the prices, p k (i) = p c k (i), are equivalent. Furthermore, recall that under perfect competition, the price of y is equal to one.

[START_REF] Irarrazabal | The Tip of the Iceberg: Modeling Trade Costs and implications for Intra-Industry Reallocation[END_REF] One interpretation of the model is that firms are owned by entrepreneurs and that firm profits accrue to these entrepreneurs. In my representative agent setting, these profits would simply enter national income in the same way that wages do, therefore I discuss the model in terms of firms to avoid needless jargon. This interpretation is similar to that of[START_REF] Yu | Entrepreneurship and Intra-Industry Trade[END_REF]. Additionally, it is common in heterogeneous firm models to have entrepreneurs draw from a distribution of productivities (often at a cost). The advantage to that approach is that it permits multiple varieties to have the same productivity. The cost, however, is one of added complexity and additional assumptions since modelers are often forced to parameterize this distribution (the Pareto distribution is a common choice). Here, my assumption of unique variety/productivity combinations aids greatly in the presentation of my results in the simplest, most tractable

fashion.[START_REF] Jean | International Trade and Firms' Heterogeneity under Monopolistic Competition[END_REF] The assumption that γ > 1 is fairly standard (e.g.[START_REF] Melitz | The Impact of Trade on Intraindustry Reallocations and Aggregate Industry productivity[END_REF]) and important. It is rarely seen that a firm (particularly not a multinational) that sells abroad but not at home and as long as expenditure on the

Since preferences are identical across both countries, it follows that the total expenditure on the heterogeneous good is equal to µ in both markets. Furthermore, recall that technologies and the mass of entrepreneurs are also identical across countries. This, along with γ > 1, is sufficient to ensure that a firm which exports will always serve the domestic market.

Note that, in perfect competition, price equals marginal cost and the standard result of iceberg costs having the same effect as an ad valorem tariff still holds.

This will be shown later.

Recall equation[START_REF] Davies | Tax Competition for Heterogeneous Firms with Endogenous Entry: The Case of Heterogeneous Fixed Costs[END_REF].

Note though, that t j and σ affect p c j in the exact same way, as shown by equation[START_REF] Davies | Tax Competition for Heterogeneous Firms with Endogenous Entry: The Case of Heterogeneous Fixed Costs[END_REF].

Note that since t and σ enter B identically, it can be seen by equations (15) and (17) that, ceteris paribus, iceberg transport costs and ad valorem tariffs will lead to different firm cutoff values.

For numerical simulations, I assume that the function f (i) is linear.

This nice simplification stems from the utility specification used.

Note that N k = i kD + i jX .

Note that this is the case when ∂ N k /∂ σ < 0.

Recall, from Corollary 1, that if ρ σ ≤ 0 ⇒ ρ t k < 0. Thus, in this case, if (ρ t k -ρ σ ) < 0 then |ρ t k | > |ρ σ |.

Again, from Corollary 1, it follows that if ρ t k ≥ 0 ⇒ ρ σ > 0. Thus, in this case, if (ρ t k -ρ σ ) < 0 then |ρ σ | > |ρ t k |.


H F Zhang

Y Gao

N Wang

J Q Li

E G Zhao

G Royer

Double magic nuclei with Z > 82 and N > 126

The 'island of stability' of superheavy nuclei due to shell effects is explored and the α decay half-lives of these nuclei are predicted. The calculations of the binding energies within a new macroscopic-microscopic model (MMM) are performed and compared with the experimental data for heavy nuclei from Md to the Z=118 element. The agreement is excellent. The data show that the 270 Hs is a deformed double sub-magic nucleus beyond 208 Pb. The features of α decay energies and one proton separation energies from the MMM reveal that the next double magic nucleus after 270 Hs should be 298 114. The potential energy surfaces calculated within the constrained relativistic mean field theory confirm that the 270 Hs is a deformed double magic nucleus and 298 114 is a spherical double magic nucleus. The α decay half-lives are determined using a generalized liquid drop model and the Qα of the MMM for Hs and Z=114 isotopes respectively.

The existence of an 'island of stability' of superheavy nuclei (SHN) is predicted in the remote corner of the nuclear chart around the superheavy elements 114 to 126 due to shell effects. The recent discovery of new elements with atomic numbers Z ≥ 110 has brought much excitement to the atomic and nuclear physics communities. The experimental efforts have been focused on the direct creation of superheavy elements in heavy ion fusion reactions, leading to the production of elements up to proton number Z=118 up to now [1][2][3][4][5][6][7]. The half-life of the new synthesized isotope 287 114 ( several seconds) is several times shorter than that of the previously observed heavier isotope 289 114 ( T α ≈ 20 s), formed in the reaction 48 Ca + 244 Pu [6,7]. Such a trend is expected to be associated with a decrease of the neutron number. The observed radioactive properties of the new nucleus 287 114, together with the data obtained earlier for the isotope 289 114 and the products of its α-decay (namely, the isotopes 283 Cn and 285 Cn) can be considered as experimental proof of the approach of the 'island of stability' of superheavy elements around Z=114.

Theoretically it had been concluded that the existence of the heaviest nuclei with Z > 104 was primarily determined by the shell effects in 1960s [8][9][10]. These early calculations predicted that the nucleus with Z=114 and N=184 is the center of an island of long-lived SHN. Recently, the detailed spectroscopic studies were performed [11,12] for nuclei beyond fermium (Z=100), with the aim of understanding the underlying single-particle structure of superheavy elements. A study of the Nobelium isotope 254 No was accomplished [13], finding three excited structures, two of which are isomeric and one of these structures is firmly assigned to a two-proton excitation. These states are highly significant as their location is sensitive to single-particle levels above the gap in shell energies predicted at Z=114, and thus provide a micro-scopic benchmark for nuclear models of the superheavy elements. The microscopic models are, however, still uncertain when extrapolating in Z and the mass number A. In particular, there is no consensus among theorists with regard to what should be the next doubly magic nucleus beyond 208 Pb (Z=82, N=126). In the SHN the density of single-particle energy levels is fairly large, so small energy shifts, such as those, for instance, due to poorly known parts of nuclear interaction, can be crucial for determining the shell stability. So an alternative choice is to develop the theoretical calculations in taking into account all the recent experimental data to give reliable predictions for the properties of the SHN.

Very recently, the macroscopic-microscopic method (MMM) was developed, the isospin and mass dependence of the model parameters being investigated with the Skyrme energy density function [14]. A very good improvement is that the macroscopic and microscopic parts in the proposed mass formula are closely connected to each other through the coefficient a sym of the symmetry energy. It is a main advantage to provide reasonable mass extrapolations for exotic and heavy nuclei. The number of model parameters (13 independent parameters) is considerably reduced as to be compared with the finite-range droplet model (FRDM) in which the number of parameters is about 40 [15]. The root-mean-square (rms) deviation with respect to 2149 measured nuclear masses is reduced to 0.441 MeV (the corresponding result with FRDM is 0.656 MeV), which should be one of the best results actually. Another most impressed improvement is that the rms deviation of α-decay energies of 46 SHN is reduced to 0.263 MeV (the corresponding result with FRDM is 0.566 MeV), which allows us to give reliable predictions of α-decay half-lives for SHN. It is meaningful to use the present data from the MMM to explore the features of SHN around the proposed 'island First, we compared the binding energy of the MMM with the up-to-date nuclear data [16,17]. As shown in the upper panel of Fig. 1, the agreement between the MMM [14] calculations and the experimental results [16,17] is excellent for all the known nuclei from Md to Z=118 isotopes. This gives us full confidence to explore the α-decay energies coming from the binding energy :

Q = E D b + E α b -E P b
, where E D b , E α b and E P b are the binding energies of daughter nucleus, α particle and parent nucleus, respectively. The MMM α-decay energies and the experimental values are shown in the lower panel of Fig. 1. The agreement between the two data is good. The lowest α decay energies are located at N = 162 and 184. If we check the results more carefully, one observes that from Md to Hs isotopes the shell effect at neutron number N=162 is increasing, then decreasing, and nearly disappearing after the Z=115 isotopes. For N=184, the shell effects increase from the Ds to Z=114 isotopes, then decrease till the isotope Z=118. From Md to Z=114 isotopes, N=162 is the magic neutron number and from Ds to Z=118 isotopes N=184 is the magic neutron number. It is interesting to explore the proton magic number from the systematic properties of the SHN.

The one proton separation energy and α-decay energy of the MMM [14] and experimental data [16,17] shown in Fig. 2 for N=162 and N=184 isotones to find the proton magic number. S 1p generally decreases with increasing Z with obvious even-odd effect from the upper panel. With careful observation it can be found that at Z = 108, and 114 the values of S 1p are above the general trend, indicating that these nuclei are more stable. The results obtained by the MMM and the experimental data show clearly that the proton number Z=108 is a magic proton number for N=162 isotones and the calculated one proton separation energy of the MMM confirmed that Z=114 is a proton magic number for N=184 isotones. The α-decay energies for N=162 and 184 isotones are shown in the lower panel of Fig. 2. Again we find the kinks of α-decay energy curves at Z=108 and 114. The conclusions that both 270 Hs and 298 114 are double magic nuclei after 208 Pb are verified again and it is very interesting to study the ground state deformations of the two nuclei.

In fact, most of superheavy nuclei found experimentally are known to be deformed. It is worthy to investigate the potential energy surfaces in order to see the validity of the lowest equilibrium deformation. It is well known that the relativistic mean field calculation gives a good description of the structure of nuclei throughout the periodic table [18][19][20]. In this paper, the potential energy surfaces of possible double magic nuclei are obtained by using the deformation-constrained relativistic mean field theory [21] and the pairing correlations are coped with the Barden-Cooper-Schrieffer (BCS) approximation [22]. The deformation parameter β 2 is set to the expected deformation to obtain high accuracy and reduce the computing time. The potential energy surfaces have been calculated for 270 Hs and 298 114 with the successful parameter set NL3 [23].

In Fig. 3, the potential energies of the nuclei 270 Hs and 298 114 are presented versus the deformation. For nucleus 298 114, there is a local spherical minimum (β 2 ∼0). For nucleus 270 Hs, the spherical minimum has completely disappeared while a well-deformed local minimum appears at β 2 ∼ 0.26. So we can draw the conclusion that the nucleus 270 Hs is a deformed double sub-magic nucleus and 298 114 is a spherical double magic nucleus.

The main decay mode of SHN is the α emission. Recently the α-decay half-lives have been calculated within a tunneling effect through a potential barrier determined by a generalized liquid drop model ( GLDM ) [24,25] and the Wenzel-Kramers-Brillouin (WKB) approxima- tion. The penetration probability is estimated by

P = exp[- 2 h Rout Rin 2B(r)(E(r) -E(sphere)) dr] ,
(1) Where two approximations ae used : R in = R d + R α and B(r) = μ where μ is the reduced mass, and R out is simply e

2 Z d Z α /Q α .
The decay constant is :

λ = P α ν 0 P, (2)
where P α is the α particle preformation factor and ν 0 the assault frequency [25]. Then the half-life can be calculated by T α = ln2 λ . The α-decay half-lives calculated by taking the experimental α-decay energies and theoretical MMM ones are shown by small triangles and circles in Fig. 4 respectively. The experimental α-decay half-lives are also presented by black dots for comparison. The calculated α-decay half-lives from experimental Q α coincide with the experimental ones almost perfectly, implying that as long as we have the right Q α , the presently used method can give precise results for α-decay halflives. The calculated α-decay half-lives with Q α from MMM are reasonably consistent with the experimental data which tells us that the present method can be used to predict the α-decay half-lives. The calculations of αdecay half-lives for Hs and Z=114 isotopes are performed. The α-decay half-life of the deformed double magic nucleus 270 Hs calculated by a phenomenological formula is 22 s [3], 23.33 s by our calculations using the MMM Q α , and 15.14 s by using the experimental Q α ( 9.02 MeV [3] ). For the spherical double magic nucleus 298 114, the α-decay half-life is 1537588 s ( about 18 days) with Q α of MMM. It would not exist on earth at all if it was not constantly being produced.

The investigation of the properties of these nuclei is extremely intriguing for exploring the position of the predicted island of stability of the super heavy nuclei, and understanding some new, unexpected features of nuclear structure. There is unlikely super heavy nucleus existing in nature, and the extreme difficulties to synthesize the super heavy nuclei greatly restrict the experimental studies on it, so the theoretical studies are very important.

As a conclusion, a fundamental prediction of modern nuclear theory is the existence of an 'island of stability' among the largely unstable superheavy elements. Different models have predicted different magic numbers, and up to now, this island of stability has not yet been localized experimentally. The central goal of the present work is to find some decisive evidences for localizing this island. With this in mind, we investigate the position of the 'island of stability' in a way which is closely connected with the experimental data. The latest experimental average binding energies are compared with the recent calculations by the MMM for the heavy nuclei from Md to Z=118 elements, and the agreement with the available data is excellent. Both the two data show that the 270 Hs is a double sub-magic nucleus after 208 Pb. The features of α decay energies and one proton separation energies of the MMM reveal that the next double magic nucleus after 270 Hs should be the 298 114 nucleus. The potential energy surfaces are calculated within the CRMF theory and the results confirm that the 270 Hs is a deformed double magic nucleus and the 298 114 is a spherical double magic nucleus. The α decay half-lives are predicted within a generalized liquid drop model and the WKB method and the Q α of the MMM for Hs and Z=114 isotopes respectively.

FIG. 1 :

 1 FIG. 1: Comparison of the experimental binding energy E b /A (upper panel) and α-decay energies Qα (lower panel) with the theoretical results. The vertical dotted lines indicate the magic neutron numbers in the lower panel.

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FIG. 2 :

 2 FIG. 2: (Color online).Comparison between the experimental one proton separation energies (upper panel) and α-decay energies (lower panel) with the theoretical results for N=162 and N=184 isotones.

FIG. 3 :

 3 FIG.3: Potential energy calculated in the constrained relativistic mean field (CRMF) theory with effective interaction NL3 for 270 Hs (upper panel) and 298 114 (lower panel) respectively.

FIG. 4 :

 4 FIG. 4: (Color online).Comparison between the experimental α-decay half-lives and the theoretical results.

Acknowledgements.

The work was supported by the National Natural Science Foundation of China (Grant Nos.10875152, 10775061, 10805061,10825522 and 10975064, 11105035 and 11175074), the Knowledge Innovation Project of the Chinese Academy of Sciences (Grant No. KJCX2-EW-N01).