Fabien Giovannelli
email: fabien.giovannelli@univ-tours.fr.

Alphonsine Ngo Ndimba

Pablo Diaz-Chao

Mikael Motelica-Heino

P I Raynal

Cecile Autret

Fabian Delorme

Synthesis of Al doped ZnO nanoparticles by aqueous coprecipitation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

ZnO has focused much attention for electronic and optoelectronic applications as a large band gap material [START_REF] Özgür | A comprehensive review of ZnO materials and devices[END_REF][START_REF] Pearton | Recent progress in processing and properties of ZnO[END_REF][START_REF] Klingshirn | ZnO: Material, physics and applications[END_REF]. In order to manufacture ZnO-based optoelectronic devices,both n-and ptype ZnO are needed. It is easy to obtain good conductive n-type ZnO by mono-doping III group elements as donors. Al doped n-type ZnO, in particular, exibits high electrical conductivity and optical transparency in the visible region. It is a transparent conductive oxide which could replace the more expensive indium tin oxide [START_REF] Minami | Highly transparent and conductive rare earth doped ZnO thin films prepared by magnetron sputtering[END_REF][START_REF] Lee | Low temperature Al doped ZnO films on a flexible substrate by DC sputtering[END_REF][START_REF] Cohen | Transparent conducting Zn 1-x Mg x O:(Al,In) thin films[END_REF][START_REF] Beyer | Transparent conducting oxide films for thin film silicon photovoltaics[END_REF]. Morever the Al doped ZnO is also highly promising as a potential high-temperature thermoelectric material [START_REF] Tsubota | Thermoelectric properties of Al-doped ZnO as a promising oxide material for high temperature thermoelectric conversion[END_REF][START_REF] Jood | Al-doped zinc oxide nanocomposites with enhanced thermoelectric properties[END_REF][START_REF] Guilmeau | Thermoelectric Oxides: Effect of Doping in Delafossites and Zinc Oxide[END_REF].

Moreover, among inorganic semiconductors, reported morphologies of ZnO are the richest. Indeed, a wide variety of ZnO micro/nano-structures including wires, rods, belts, tubes, cages, rings or flowers have been synthesized by many methods, such as sol-gel processing, vapor deposition, hydrothermal syntesis, thermal decomposition or microwave synthesis [START_REF] Wang | ZnO nanowire and nanobelt platform for nanotechnology[END_REF][START_REF] Djurisic | ZnO nanostructures for optoelectronics: Material properties and device applications[END_REF][START_REF] Ye | Zinc Oxide Nanostructures: Morphology Derivation and Evolution[END_REF][START_REF] Wang | Semiconducting and piezoelectric oxide nanostructure induced by polar surfaces[END_REF][START_REF] Wang | Zinc oxide nanostructures: growth, properties and applications[END_REF][START_REF] Giovannelli | Synthesis of ZnO microwires and tetrapods by optical furnace[END_REF].

Precipitation from colloidal solutions is an usual method for producing transition metal oxide nanoparticles [START_REF] Jolivet | Design of metal oxide nanoparticles : control of size, shape, crystalline structure and functionalization by aqueous chemistry[END_REF][START_REF] Baes | The hydrolysis of cations[END_REF][START_REF] Giovannelli | Synthesis of manganese spinel nanoparticles at room temperature by coprecipitation[END_REF]. Two reactions are generally used to describe the precipitation process:

Hydrolysis and water condensation. Hydrolysis consists in the formation of an insoluble hydroxide which can be converted into the oxide compound by heat assisted dehydratation. Indeed, in order to obtain ZnO, many studies used a thermal treatment after the precipitation step [START_REF] Raoufi | Synthesis and microstructural properties of ZnO nanoparticles prepared by precipitation method[END_REF][START_REF] Moezzi | Zinc hydroxyacetate and its transformation to nanocrystalline zinc oxide[END_REF][START_REF] Chand | Structural and optical properties of ZnO nanoparticles synthesized at different pH values[END_REF][START_REF] Sambath | Morphology controlled synthesis of ZnO nanostructures by varying pH[END_REF]. However, the water condensation phenomenom could also appear directly in the solution. ZnO precipitation in a solution heated to 80°C has been already reported [START_REF] Wahab | Low temperature synthesis and characterization of rosette-like nanostructures of ZnO using solution process[END_REF][START_REF] Rodrıguez-Paez | Controlled precipitation methods: formation mechanism of ZnO nanoparticles[END_REF][START_REF] Jia | Evidence of the formation mechanism of ZnO in aqueous solution[END_REF][START_REF] Wahab | Synthesis, characterization and effect of pH variation on zinc oxide nanostructures[END_REF].

Oliveira et al. [START_REF] Almeida De Oliveira | Controlled precipitation of zinc oxide particles at room temperature[END_REF] have proposed a precipitation under double-jet conditions to produce micrometric ZnO particles at room temperature. Moreover, recently, Yang et al. [START_REF] Yang | Influence of mixing ways of reactants on ZnO morphology[END_REF] have reported the obtention of ZnO at 25°C by adding soda to a Zn sulfate solution. However, in many applications ZnO doping is necessary in order to reach the best properties, especially Al doping.

In this study, the influence of different parameters (temperature, hydrolysis duration, reagent concentration and time) on the precipitation of Al-doped ZnO was studied as well as the reaction mechanisms.

Materials and methods

100 ml of cationic solution was obtained by dissolving Zn(NO 3 ) 2 ,6H 2 O (Aldrich 99.99%) and AlCl 3 (Acros 99%) in demineralized water. The concentration of Zn is 0.1M and Al concentration corresponds to 2 at% of Zn one. Precipitation was made by the dropwise addition of 125ml of NaOH solution at 1 mol/l (NaOH solid from Scharlau (99%)) during 25 minutes. The precipitate obtained was centrifuged at 4000 rpm during 5 min and washed with demineralized water. This step was repeated five times. The solid was finally dried at 80°C.

The following synthesis parameters were modified: influence of reactant concentration (0.5M and 1M), Zn/OH molar ratio (0.1, 0.2 and 0.6), soda addition time (10 s) and temperature (25 °C and 60°C).

The morphologies of ZnO powders were analyzed on a ZEISS Ultra Plus field-emission scanning electron microscope (FE-SEM) operated at an acceleration voltage of 2 kV. The samples were platinum-coated prior to observation using a GATAN Precision Etching and Coating System (PECS™). Transmission electron microscopy samples were prepared by adding the powder to n-butanol and the small crystallites in suspension were deposited on a holey carbon film, supported by a copper grid. The electron diffraction and the high resolution electron microscopy were carried out with a JEOL 2100F electron microscope equipped with a tilting and rotating goniometer (200 kV and point resolution of 1.8 Å) and coupled with EDX analysis (Oxford).

Powder X-ray diffraction (XRD) patterns were acquired on a Bruker D8-Advance diffractometer in Bragg/Brentano geometry using Cu-K  radiation and operating at 40kV and 40 mA at room temperature. Scans were recorded from 5° to 85° (2) with a step of 0.02 and a counting time of 1s per step.

Laser ablation-inductively coupled plasma-mass spectrometer (LA-ICP-MS) analysis was made on the samples. An element XR Thermo (Thermo Finnigan) ICP-MS, following the procedure described in previous studies [START_REF] Gratuze | Mass spectrometry with laser sampling: A new tool to characterize archaeological materials[END_REF][START_REF] Aries | Direct determination of lead isotope ratios by laser ablation-inductively coupled plasma-quadrupole mass spectrometry in lake sediment sample[END_REF][START_REF] Motelica-Heino | Comparison of UV and IR Laser Ablation ICP/MS with Silicate Reference Materials and Implementation of Normalization Factors for Quantitative measurements[END_REF] was used in combination with a VG UV laser probe, laser ablation sampling device. Solid samples were prepared as pressed pellets. Powder samples of 350 mg were put under a pressure of 8 metric tons, without added binder, to form 13 mm (diameter) x 1 mm (height) pellets. The repetition rate of the 266 nm wavelength laser was fixed to 10 Hz. Calibration took place with a certified artificial glass, NIST-610.

Measurements were replicated four times to validate the analytical precision of the technique.

Results and discussion

When NaOH is added to the zinc solution, as described in the materials and methods section, the evolution of pH exhibits both important variations from pH 3 to 6 and then from pH 6.5 to 10.5. The solid precipitation appears with the first soda addition. The colour of the final solid precipitate is white. The final pH is higher than 12. The XRD pattern of the resulting powder exhibits all the peaks of ZnO (PDF card 01-070-8070) (Figure 1). However, the peak intensity of (002) planes is enhanced compared to the PDF card (01-070-8070). The SEM observations show the presence of two kinds of particles (Figure 2A): on the one hand, nanoparticles exhibiting an isotropic shape of several tens of nanometers and on the other hand, platelets like particles with a size of several hundreds of nanometers and a thickness of several tens of nanometers. The (002) peak intensity enhancement could be linked to the presence of these platelets in the powder. The analysis of Al concentration leads to a content of 0.1 at% which is far less than the initial content of 2 at. %.

Hydrolysis and water condensation depend on the reactant's concentrations and on the ratio Zn/OH. The influence of Zn concentration was evaluated by increasing the 0.1M initial concentration to 0.5 M and 1 M without changing the ratio Zn/OH=0.2 and the initial aluminum atomic percent. The XRD pattern of the 1 M sample only exhibits peaks related to ZnO (Figure 1). SEM observations still show the presence of the two kinds of particles (Figure 2B). However, the number of platelets increases. The analysis of Al concentration leads to a content of 0.28 at%. So the increase of the initial Zn concentration leads to an increase of the Al content in ZnO powder. This value is close to 0.3 at% which is the limit of solubility of Al in ZnO [START_REF] Shirouzu | distribution ans solubility limit of Al in Al2O3 doped ZnO sintered body[END_REF][START_REF] Serier | Al-doped ZnO powdered materials: Al solubility limit and IR absorption properties[END_REF]. Moreover, TEM analyzes analysis confirms these results, showing homogeneous compositions of the particles (Al concentrations close to 0.3 at %) regardless of their morphology. The 0.5M sample presents a similar behavior: XRD pattern only exhibits peaks related to ZnO and SEM observations show the presence of the two particle morphologies. The comparison of XRD data show that the full width at half maximum (FWHM) of all the peaks of the 0.1 M sample is larger than for the two other concentrations (Figure 1). This is consistent with SEM images, which show that in this case, the proportion of small particles is greater (Figure 2A). Therefore, the increase of the initial Zn concentration leads to a coarsening of particles and an increase in the aluminum content within the particles.

In order to observe its influence, the initial Zn/OH ratio (Zn/OH = 0.2) was initially decreased to 0.1 with an initial Zn concentration of 1 M. No significant difference was observed (Figure 3). Afterwards the ratio Zn/OH was increased to 0.6. The final pH was 7.9. In this final case, new peaks appeared on the XRD pattern which could be attributed to different phases (Figure 3) : zinc hydroxide salts structures (PDF-00-24-1460 and PDF 00-052-0627), Al(OH) 3 gibbsite (PDF-00-054-0036) and a Zn-Al hydrotalcite-like structure (PDF 00-057-0528) (Figure 3). These phases are layered hydroxides which are closely related to the brucite structure [START_REF] Nowacki | Die kristallstruktur von zinkhydroxychlorid II, Zn 5 (OH) 8 Cl 2 • 1H 2 O[END_REF][START_REF] Stählin | The crystal structure of zinc hydroxide nitrate, Zn 5 (OH) 8 (NO 3 ) 2 .2H 2 O[END_REF][START_REF] Delorme | Hydroxides : Synthesis, Types and Applications[END_REF]. The brucite structure is based on CdI 2 layers: divalent metal cations occupy the centers of OH -octahedra joined by their edges to form infinite layers. The layers are held each other by weak intermolecular forces or hydrogen bondings. In this case of a hydrotalcite like structure, the substitution of a divalent cation by a trivalent cation in brucite structure induces the presence of an interlayer anion in order to conserve the electroneutrality of the structure. Gibbsite presents a brucite structure with one third empty octahedral sites, as aluminum has an oxidation state of three. The zinc hydroxide nitrate structure is formed by brucite layers presenting one quarter empty octahedra. Tetrahedrally coordinated zinc atoms are located above and below the empty octahedra. These layered hydroxides can present several kinds of crystallographic structures. The first key parameter is the nature of the layers, whereas the second one is their stacking, which can generate different polytypes. Moreover, XRD patterns exhibit two more small peaks that are not identified at 13.3 and 16.3 °2Figure 3) These small peaks could be related to another layered hydroxide due to the large variety of structures in the family [START_REF] Delorme | Hydroxides : Synthesis, Types and Applications[END_REF]. The high Zn/OH ratio leads to the presence of hydroxide in the resulting powder. However, the maximum intensity peaks in XRD is due to ZnO. The final Al concentration in the powder is 6 at% which is three times more than the initial solution content. It means that all of the aluminum is in different solid phases and that Zn has not yet totally precipitated at this pH value. The amphoteric character of Al explains the fact that when the pH increases to 12 the Al bearing phases are dissolved and the final content of Al is close to the solubility limit of Al in ZnO. This is consistent with dissolutioncrystallization of preexisting hydroxides in ZnO when the pH increases. This is consistent with previous reports as hydroxide compounds have already been proposed by several authors as intermediary phases [START_REF] Hosono | Non-basic solution routes to prepare ZnO nanoparticles[END_REF][START_REF] Caillaud | Effect of the solution on the deposition of zinc oxide films by spray pyrolysis[END_REF][START_REF] Li | Control preparation of zinc hydroxide nitrate nanocrystals and examination of the chemical and structural stability[END_REF][START_REF] Newman | Comparative study of some layered hydroxide salts containing exchangeable interlayer anions[END_REF]. However, the type of anions present in the solution as well as the experimental procedure seem to strongly influence the intermediary phases. Indeed, Hosono et al. [START_REF] Hosono | Non-basic solution routes to prepare ZnO nanoparticles[END_REF], in a non-basic solution route to prepare ZnO, described the polymerization of zinc acetate in layered Zn hydroxide acetate which is then hydrolyzed in order to obtain ZnO, whereas by adding soda in aqueous Zn acetate solution, Caillaud et al. [START_REF] Caillaud | Effect of the solution on the deposition of zinc oxide films by spray pyrolysis[END_REF] observe the appearance of -Zn(OH) 2 . Nevertheless, the transformation of a hydroxide phase in another hydroxide compound is possible: indeed, Zn hydroxy-nitrate in the presence of acetate or in basic medium leads to the formation of -Zn(OH) 2 [START_REF] Li | Control preparation of zinc hydroxide nitrate nanocrystals and examination of the chemical and structural stability[END_REF][START_REF] Newman | Comparative study of some layered hydroxide salts containing exchangeable interlayer anions[END_REF].

When the soda addition time in our experimental procedure was reduced to 10 seconds instead of 25 minutes, the XRD exhibited the presence of ZnO and -Zn(OH) 2 phases (Figure 4). The most intense peaks corresponded to -Zn(OH) 2 . Moreover, the FWHM of -Zn(OH) 2 was smaller than ZnO one. This could be linked to different particles sizes. This interpretation is consistent with the SEM images which show the presence of two kinds of particles (Figure 5A): well crystallized micrometric particles of -Zn(OH) 2 and submicrometric agglomerate of needle like particles of ZnO. These experimental conditions lead to a formation of particles in a basic solution (final pH>12) whereas in the conventional method crystallization occurs from pH = 3 to 12. Yang et al. [START_REF] Yang | Influence of mixing ways of reactants on ZnO morphology[END_REF] have also obtained -Zn(OH) 2 when they added the cationic solution to the soda solution. In order to investigate the relationships between ZnO and -Zn(OH) 2 , a part of the slurry was aged 24 hours under vigorous stirring. The resulting XRD pattern also exhibited -Zn(OH) 2 and ZnO peaks (Figure 4 B). However the relative intensities indicate that the presence of ZnO is greater. SEM images still show the welldefined micrometric particles and submicrometric agglomerates of needle like particles (Figure 5B). However, the surfaces of the well-defined micrometric particles appear degraded by small holes and the presence of needle like particles is more important which is consistent with XRD patterns. In these experimental conditions, the Zn(OH) 2 particles are transformed slowly into ZnO nanoparticles by a non topotactic reaction, i.e. dissolution-recrystallization.

The way of mixing the cationic and soda solutions leads to different intermediary phases and therefore ZnO nanoparticles with different morphologies and sizes. Even if the final product is pure ZnO, the careful selection of the synthesis process is a route to tune the particles size and morphology.

The effect of the Al content was also studied from pure ZnO samples up to 5 at % Al. Pure ZnO was obtained at room temperature. In the case of pure ZnO, the powder exhibits a sand rose morphology with micrometric size platelets (Figure 6A). With 5 at % of Al, the powder morphology was essentially formed by small isotropic particles close to 30 nanometers (Figure 6B). The measured aluminum content in the powder only reaches 0.3 atomic percent which is the limit of solubility of Al in ZnO [START_REF] Shirouzu | distribution ans solubility limit of Al in Al2O3 doped ZnO sintered body[END_REF][START_REF] Serier | Al-doped ZnO powdered materials: Al solubility limit and IR absorption properties[END_REF]. The increase of initial aluminum content allows increasing the final aluminum content up to the solubility limit and also enables the nucleation centers of ZnO to increase leading to a drop in the number of large platelet like particles. Indeed, the trivalent aluminum cation precipitates at lower pH than divalent cation leading to amorphous aluminum hydroxide which acts as nucleation centers for ZnO when the pH reaches higher values [START_REF] Cavani | Hydrotalcite-type anionic clays: preparation, properties and applications[END_REF][START_REF] Seron | Synthesis of layered double hydroxides (LDHs) with varying pH: A valuable contribution to the study of Mg/Al LDH formation mechanism[END_REF][START_REF] Boclair | Layered double hydroxide stability. 1. Relative stabilities of layered double hydroxides and their simple counterparts[END_REF].

Finally, the effect of temperature was also studied. The experiment was carried out at 60°C. The XRD pattern exhibited only the peaks of ZnO (Figure 7). It is worth noting that relative intensities of XRD pattern peaks are similar to ZnO PDF card whereas in all the other experiments peak intensity of (002) plane are enhanced compared to PDF card. SEM images show that this sample is only composed of isotropic nanoparticles of several tens of nanometers in size (Figure 8). Therefore, the enhanced peak intensity of (002) plane can be correlated to the presence of the platelets in the samples.

Conclusions

Al-doped ZnO nanoparticles were synthesized with a simple precipitation process. A nontopotactic reaction mechanism, involving firstly the precipitation of various hydroxide compounds depending on the route (low or high pH), followed by the dissolutionrecrystallization of the hydroxide species into ZnO, was demonstrated. When the initial pH is low, different Zn hydroxide salts precipitate whereas when the initial pH is high, the intermediary hydroxide phase is Zn(OH) 2 . The Al concentration in the final ZnO nanopowders does not exceed 0.3 at% which corresponds to the reported solubility limit of Al in ZnO. The different experimental conditions allow the morphology of ZnO particles to be controlled from isotropic nanoparticles of several tens of nanometers, platelets of several hundreds of nanometers or agglomerates of needle like particles.

Figure 1 :

 1 Figure 1: X-ray diffraction patterns of samples with different initial Zn concentrations.

Figure 2 :

 2 Figure 2: SEM images of samples with initial concentration of (A) 0.1M and (B) 1M.

Figure 3 :

 3 Figure 3: X-ray diffraction patterns of samples with different Zn/OH ratios

Figure 5 :

 5 Figure 5: SEM images of obtained powders with (A) fast soda addition and (B) fast soda addition followed by 24h ageing.

Figure 6 :

 6 Figure 6: SEM images of (A) pure ZnO and (B) 5 at % Al doped ZnO.

Figure 7 :

 7 Figure 7: X-ray diffraction pattern of Al doped ZnO synthesized at 60°C.

Figure 8 :

 8 Figure 8: SEM image of Al doped ZnO synthesized at 60°C.


Guy Meunier
email: guy.meunier@polytechnique.edu

Jean-Pierre Ponssard
email: jean-pierre.ponssard@polytechnique.edu

Catherine Thomas
email: c.m.thomas@lse.ac.uk.

Christian Belzil

Wenjie Chen

Amit Khandelwal

Bruce Kogut

Benjamin Lock- Wood

Dmitri Nizovtsev

Hendrik G Van Oss

Peter Schott

Demand uncertainty is thought to inuence irreversible capacity decisions. Suppose local demand can be sourced from domestic (rigid) production or from (exible) imports. This paper shows that the optimal domestic capacity is either increasing or decreasing with demand uncertainty depending on the relative level of the costs of domestic production and imports. This relationship is tested with data on the U.S. cement industry, where, because cement is costly to transport over land, the dierence in marginal cost between domestic production and imports varies across local U.S. markets. Industry data for 1999 to 2010 are consistent with the predictions of the model. The introduction of two technologies to the production setone rigid and one exible is crucial in understanding the relationship between capacity choice and uncertainty in this industry because there is no relationship at the aggregated U.S. data. The analysis presented in the paper reveals that the relationship is negative for coastal districts, and signicantly more positive in landlocked districts.

The relationship between uncertainty and investment decisions has been the subject of academic debate since the early work of [START_REF] Jorgenson | Econometric studies of investment behavior: a survey[END_REF]. As summarized in [START_REF] Abel | Options, the Value of Capital and Investment[END_REF], theoretical arguments can be made to ensure either a positive or a negative relationship between demand uncertainty and investment. A mean-preserving increase in the variance of demand may induce a positive eect on the value of a marginal unit of capital, and, hence, on investment, due to the increased probability of high demand states. There may also be a counteracting negative eect when there is an option to delay investment until uncertainty is partly resolved [START_REF] Dixit | Investment Under Uncertainty[END_REF]. The ndings in the empirical literature reect the ambiguity of these theoretical results [START_REF] Carruth | What do we know about Investment under Uncertainty[END_REF].

We build on the framework developed by [START_REF] Rothschild | Increasing risk ii: Its economic consequences[END_REF] in a model adapted to characteristics of the cement industry, and then explore the theoretical predictions in data from the U.S. in the early 2000s. In this industry, local demand for cement can be met by the output from two technologies: capital-intensive local production or imports from abroad. Imports are a more exible and less capital-intensive alternative source of production to the output from local capacity, and the ability to import to a market aects rms' local investment decisions. The main contribution of the paper is to explicate the role of the production set in the relationship between uncertainty and investment.

There are three main reasons why the U.S. cement industry is an attractive industry in which to study this relationship: First, capacity decisions are major rm-level decisions in this industry because cement production is very capital-intensive. Second, the industry is regionally segmented in terms of supply and demand, and the market structure is quite concentrated within each region. At the start of the 2000s there were 114 active cement plants operating across the U.S. Regions vary in the extent of local demand uncertainty because it is aected by both the general business cycle and the local cycles typical of the construction industry. Third, long-haul maritime imports are responsive to uctuations in U.S. domestic demand, and regional demand is often met by a mix of local production capacity and imports from overseas controlled by domestic cement producers.

1

We develop a theoretical model which captures these three characteristics. Each rm in a local market has to make two decisions in sequence under imperfect Cournot competition.

First, it decides its local capacity. Second, after the level of demand in the following period is revealed, the rm decides its production mix from its domestic capacity and imports. In the context of the model, domestic capacity and imports can be considered substitutable inputs, and they play similar roles to capital and labor in Rothschild and Stiglitz's model [START_REF] Rothschild | Increasing risk ii: Its economic consequences[END_REF]. We extend their results and show that the domestic capacity choice is either increasing or decreasing in the level of uncertainty, depending on the relative marginal cost of the domestic versus the import technology. Specically, capacity is increasing with uncertainty if the cost of imports is relatively large, and decreasing if the cost of imports is relatively small. Our empirical analysis of the US cement industry between 1999 and 2010 conrms this contingent property: The nature of the relationship between demand uncertainty and investment is related to local access to the exible production technologyimports from abroad.

An increase in local demand uncertainty is associated with a signicant decrease in production capacity and average plant size only in coastal districts, and signicantly more positive in landlocked districts. We also show that, at the country-level, the data reveal no clear aggregate relationship between uncertainty and investment. These results suggest that rms respond to an increase in uncertainty about future returns from an investment by choosing to make smaller irreversible investments only when imports are relatively cheap.

The signicance of our empirical contribution comes from the fact that there is a monotonic relationship between uncertainty and investment only when accounting for variation in production-set exibility across geographic markets. The model provides a theoretical rationale for this fact, and the empirical evidence reveals that, without controlling for production set exibility, the role of demand uncertainty would be obscured.

The rest of the paper is organized as follows: Section 2 develops the analytical model.

Section 3 reviews the literature related to both the model and empirical work in this industrial setting. It also includes a calibration of the model to some key industry facts. Section 4 describes the data used in the paper. Section 5 develops the methodology employed and gives the empirical results. Section 6 discusses some of the implications of these results and concludes.

2 An Analytical Model

Set Up

The inverse demand function for a given market is p(q, θ), in which p is the price and q the quantity sold in the market. Uncertainty is introduced through the random variable θ, which is assumed to be distributed on the interval [θ; θ], where the cumulative distribution of θ is given by F , assumed to be dierentiable. The inverse demand function p(q, θ) is assumed to be twice dierentiable and strictly decreasing w.r.t. the quantity q when q is positive. We also make the standard assumption that

∂p ∂q + q ∂ 2 p ∂q 2 < 0 (A1)
which, when the market is served by a monopoly producer, ensures that the revenue of the rm is concave w.r.t. to its production. When the market is served by an oligopoly, this assumption implies that the revenue of a rm is concave whatever the production of its competitor, and that this marginal revenue is decreasing w.r.t. to the production of its competitor.

2

To ensure that both the revenue and the marginal revenue are increasing with respect to the draw θ from the distribution F , where θ can be interpreted as a demand shock, it is assumed that:

∂p ∂θ (q, θ) > 0 and ∂p ∂θ + ∂ 2 p ∂θ∂q q > 0 (A2)
This assumption will hold whenever uncertainty is additive or if uncertainty pertains to, for example, incomplete information about market size.

3

Turning to the supply side, a rm's cost function for the home technology consists of two terms: a linear per unit investment cost c k for a capacity choice denoted k, and a linear per unit production cost c h . 4 The rm is unable to produce more than its capacity with the home technology. For the foreign technology, there is no unit investment cost. This technology is assumed to have a linear per unit production cost c f that varies across local markets. In the case of no uncertainty, the home technology is preferred to the foreign, that 2 This ensures the uniqueness of the Cournot equilibrium in the oligopoly case.

is, it is assumed that: c h + c k < c f . Finally, it is assumed that local demand is high enough to make some domestic investment worthwhile, so θ θ p(0, θ)dF > c h + c k , and that, in all realized demand states, it is worth producing with the home technology: p(0, θ) > c h , ∀θ.

The game has three stages, with rm decisions being made at the rst and third stage.

First, the rm decides its local capacity k relative to the home technology. Second, uncertainty is resolved, and the realized value of θ is revealed to the rm. Third, the rm makes production decisions (q h , q f ) using the home and foreign technologies respectively, subject to the constraint q h ≤ k. The optimal capacity is denoted k * and is a function of both the distribution of demand states and the local relevant import cost c f .

Demand Uncertainty and Optimal Local Capacity for a Monopoly

Firm

We rst look at the capacity choice of a rm that has a production monopoly in the local market and then generalize to the oligopoly setting. The short-term prot of a monopoly rm is:

π(k, θ) = max q h ≤k,q f [p(q, θ)q -c h q h -c f q f ]
and the expected long-term monopoly prot is the expected short-term prot minus the investment cost:

Π(k) = θ θ π(k, θ)dF (θ) -c k k Since c h + c k < c f ,
the optimal capacity in the case of no uncertainty is simply the monopoly capacity with marginal cost c h + c k . Production is equal to capacity, both are independent of c f , and there are no imports.

With suciently large uncertainty, however, production and capacity become uncoupled.

In low realized demand states, the rm has excess capacity, and production is determined by the home plant's variable cost. For high realized demand states, the rm imports to satisfy demand in excess of capacity, and the quantity sold is determined by the import cost. Capacity is equal to production only over a range of intermediate levels of the realized demand shock.

The bounds of this range are denoted θ -and θ + respectively. At θ -and θ + , the marginal revenue of local production at capacity is equal to the home variable production cost and to the import cost, respectively:

p(k, θ -) + ∂p ∂q (k, θ -)k = c h and p(k, θ + ) + ∂p ∂q (k, θ + )k = c f . (1)
The relationship between k, home production and imports in the short term for dierent ranges of the realized demand shock θ, are given by Table (1), which also shows, in the second row, the corresponding marginal ex post prot associated with additional capacity for each range:

θ † θ ≤ θ ≤ θ - θ -≤ θ ≤ θ + θ + ≤ θ ≤ θ q h , q f q h < k, q f = 0 q h = k, q f = 0 q h = k, q f > 0 ∂π/∂k 0 p(k, θ) + ∂p ∂q (k, θ)k -c h c f -c h †
The two thresholds θ -, θ + are dened by equation ( 1).

Table 1: The level of demand, θ, determines whether the rm is at full capacity and whether it imports; and also the short-term marginal prot of an additional capacity.

Capacity choice is derived from maximizing the expected long-term prot given the dis-tribution of possible demand states, F . The derivative of expected long-term prot with

respect to k is ∂Π ∂k = θ θ ∂π ∂k -c k dF (θ) (2)
The eect of an increase in risk à la Rothschild and Stiglitz (1970) on the long-term marginal prot depends on the shape of ∂π/∂k with respect to θ. If the short-term marginal prot is concave (resp. convex) with respect to θ then an increase in risk reduces (resp. increases) the optimal investment. 5

In the present framework, and as summarized in Table (1), the short-term marginal prot is neither concave nor convex and depends on the realized value of θ. For (θ < θ -), shortterm marginal prot is constant (equal to zero); for (θ

-< θ < θ + ), short term marginal
prot is increasing in capacity; for (θ

+ < θ), it is once again constant.
Without further assumptions on either the revenue function or the distribution of demand states, an increase in risk has an ambiguous eect on the equilibrium capacity. However, with a uniform distribution of demand states, the eect of an increase of uncertainty on the optimal capacity is clear, and depends on the magnitude of relative costs.

Proposition 1 If θ is uniformly distributed over [-λ, λ], then the equilibrium capacity is increasing (resp. decreasing) with respect to

λ if c f > c h + 2c k (resp. c f < c h + 2c k )
Proof. Using the expressions in the second row of Table 1, and including the capital costs which are relevant in the long term, we can write the long-term marginal prot as:

∂Π ∂k = θ - θ -c k dF (θ) + θ + θ - ∂r ∂q (k, θ) -c h -c k dF (θ) + θ θ + (c f -c h -c k )dF (θ) (3) 0 = ∂Π ∂k = 1 2λ θ - -λ -c k dθ + θ + θ - p(k, θ) + ∂p ∂q (k, θ)k -c h -c k dθ + λ θ + (c f -c h -c k )dθ
Then, the derivative of ∂Π ∂k with respect to λ at k * (λ) is:

∂Π 2 ∂λ∂k = - 1 2λ ∂Π ∂k + 1 2λ [(c f -c h -c k ) -c k ] = 1 2λ (c f -c h -2c k ).
Therefore, the sign of ∂k * /∂λ is the sign of c f -c h -2c k .

The simplicity of the condition on the cost components is appealing; the precise nature of the demand function does not aect the result. With uniformity, the weight of interior demand states in which the rm produces at capacity is constant, and the relative weights of each range of demand states do not change when uncertainty increases.

The nature of the impact of the import cost on the direction of the relationship between capacity and demand uncertainty can be made more general in the following way: On the one hand, when the import cost is so high that the rm never imports, an increase in uncertainty increases the incentive to invest to satisfy high demand realizations. On the other hand, the lower the import cost, the more exible is the rm in its ability to face uncertainty. This exibility intuitively leads to a negative relationship between uncertainty and investment.

This intuition, while developed above using the uniform class of distribution functions, is likely to hold for a broader set of possible demand distribution functions.

The Oligopoly Case

We now extend the analytical results to dierent market structures. Specically, we move to an oligopolistic setting, and then include a competitive fringe of importers. Together with the monopoly case, these market structures describe the set of local markets considered in our empirical application.

In a standard static Cournot model, each rm's best response can be viewed as the monopoly response to a residual demand curve. This interpretation remains true in our dynamic model as long as we make the assumption that the production decisions of any given rm do not depend on the capacity of its competitors. This rules out preemptive motives for capacity investment.

6

Making the further assumption that rms have similar cost structures we can show that while the number of rms in the market does inuence the magnitude of the relationship between uncertainty and capacity, it does not aect its sign. The proof of this Lemma is in Appendix 1.

The oligopoly result is, however, obtained assuming an exogenous market structure, and may not generalize to settings where the number of rms is endogenous to the level of demand uncertainty. However, the market size relative to economies of scale tends to be the primary determinant of market structure in a capital intensive industry, and demand uncertainty as such certainly plays a limited role. As a consequence, the contingent property of the model may be tested as long as the observed market structure is relatively stable. In our review of literature we provide indirect conrmation that market structure is relatively static during the time period under study.

8

Altogether, we think that Proposition 1 can be seen as a relatively robust property that can be evaluated empirically at the market level, rather than the rm level. Figure 2.3 illustrates how the optimal capacity k * is predicted to depend on both demand uncertainty and the relative cost of imports. Suppose that for a landlocked district, we have c f ≥ 2c k + c h , while the reverse is true for a coastal district. The line AB traces out the predicted relationship between capacity and demand uncertainty in landlocked districts, and the line CD does the same for coastal districts. Proposition 1 states that the slope of AB is positive and the slope of CD is negative. 3 Relevant literature and Simple Model Calibration

Review of the literature

This study relates to several distinct literatures. First, we review the relevant literatures on option value and on capacity and uncertainty in industrial organization, pointing out the contribution made by this paper. Then, we review the papers that are specically about the cement industry, and use some empirical ndings from these papers to discuss the relevance of our model.

The result of Proposition 1 can be directly interpreted in terms of options [START_REF] Dixit | Investment Under Uncertainty[END_REF][START_REF] Abel | Options, the Value of Capital and Investment[END_REF]. There are two valuable options in this model when demand is uncertain. Investing in an additional unit of capacity creates the put option to produce less than installed capacity for a benet of c h but eliminates the call option to import at a cost of c f . An increase in uncertainty increases the value of each of these two options, and the overall eect on investment depends on the comparison of these changes. 9

9 Our framework could be compared with the one presented in [START_REF] Abel | Options, the Value of Capital and Investment[END_REF]. The possibility of producing less than capacity in our model could be compared to the possibility of divesting capacity in their model, and the possibility of importing is comparable to the possibility of (immediately) expanding capacity in their model. [START_REF] Abel | Options, the Value of Capital and Investment[END_REF] points out the ambiguity of the eect of uncertainty on irreversible

Our model can also be related to two theoretical industrial organization papers. [START_REF] Demers | Investment under Uncertainty, Irreversibility and the Arrival of Information Over Time[END_REF] analyzes capacity choice in a dynamic oligopolistic Markov model and shows that the equilibrium capacity is decreasing with uncertainty. In his model, the rm is constrained to always produce as much as its earlier capacity commitmentpossibly more with a penalty costbut never less. [START_REF] Gabszewicz | Demand Fluctuations and Capacity Utilization under Duopoly[END_REF] consider a two-stage game and show that rms invest more with uncertainty. In their framework, rms can produce less and not more than their capacity. Capacity is increasing with uncertainty. Our model combines these two models into one unied framework rms may produce more than capacity (through imports) or produce less than capacity and extends their results, so that the relative level of the domestic production and import costs explains when capacity is increasing or decreasing with uncertainty.

Previous empirical studies have reached mixed conclusions about the inuence of dierent types of uncertainty on investment. [START_REF] Goldberg | Exchange rates and investment in United States industry[END_REF] shows that there is a negative relationship between investment and exchange rate variability in some sectors in the United States, but then [START_REF] Campa | Investment in manufacturing, exchange rates and external exposure[END_REF] nd that exchange rate variability has no signicant eect on investment levels in manufacturing. [START_REF] Bell | Irreversible Investments and Volatile Markets[END_REF] nd no relationship between product demand uncertainty at the country level and capacity investment in the chemical processing industry. Ghosal andLoungani (1996, 2000) nd a negative relationship between investment and uncertainty, focusing on the role of concentration ratios and whether industries are dominated by small rms. Our ndings from the U.S. cement industry support the suggestion made by [START_REF] Carruth | What do we know about Investment under Uncertainty[END_REF] that production set exibility is a possible explanation for the ambiguous results obtained in previous studies.

We now come to the papers that directly model the cement industry, focusing on the U.S.

investment in the general case, while we provide a clear-cut comparison with a specic distribution.

game in the tradition of [START_REF] Ericson | Markov-perfect industry dynamics: A framework for empirical work[END_REF], where the stage game involves an investment phase (entry or exit, and choice of capacity for the next stage), followed by a Cournot competition production phase under capacity constraints for incumbents. This model is estimated using data from geographic areas which are similar to ours (U.S. districts). [START_REF] Fowlie | Market-Based Emissions Regulation and Industry Dynamics[END_REF] extends the Markov framework developed in [START_REF] Ryan | The costs of environmental regulation in a concentrated industry[END_REF] to include a competitive fringe of importers. Perez-Saiz ( 2011) also draws on [START_REF] Ryan | The costs of environmental regulation in a concentrated industry[END_REF] but this time allows for mergers and acquisitions. These papers note that the U.S. market structure at the district level has a lot of inertia, and [START_REF] Ryan | The costs of environmental regulation in a concentrated industry[END_REF] introduces the possibility of heterogenous rms, but the estimation shows that there are no signicant dierences in cost functions across rms. Our model shares the same underlying features (investment and production with capacity constraint, symmetric rms). It directly assumes an exogenous market structure with identical cost functions and is robust to the introduction of a competitive fringe. These three papers neither allow for demand uncertainty nor for the capability of domestic rms to import, which are the two characteristics we study here.

It is worth noting that, according to our model, these two characteristics are the primary factors to explain the disproportionate response of cement imports to market demand uctuations. We shall come back to this point in Section 4.

Since the rms in U.S. cement markets are often large multinationals that produce cement in many markets and also often own the major import terminals in the U.S., our paper also relates to the theory model set out in [START_REF] Rob | Foreign Direct Investment and Exports with Growing Demand[END_REF]. In their model, as in ours, rms have the choice between home and foreign production to satisfy an uncertain demand.

10 Salvo (2010) studies the cement industry in Brazil, where, unlike in the U.S., there are limited imports.

This paper also models importers as a competitive fringe and tests the potential threat of fringe on the domestic market even in the absence of imports.

Their paper focuses on the optimal strategy mix between local production and imports as local demand grows over time. In contrast, we focus on the optimal strategy mix under demand uncertainty and varying costs of imports. In both models, domestic production and imports may co-exist.

11 Our model provides another explanation to the observation that under some circumstances rms serve a given market through both imports and local production [START_REF] Blonigen | In Search of Substitution between Foreign Production and Exports[END_REF]).

Analytical Model Calibration

Before evaluating the empirical predictions generated in the analytical model, we make a rough calibration to test its empirical relevance for the U.S. cement industry. Our main source of information is provided by the industry analysts Jeeries, in their Industrials Building Materials Report (August, 2012). The model's prediction is based on two inequalities. The rst one states that building home production is cheaper than import, c k + c h < c f . The second one says that the sign of the relationship between demand uncertainty and local capacity hinges on the sign of c f -c h -2c k . In coastal markets the sign is expected to be negative while it is expected to be positive in landlocked markets.

Taking each variable in turn: For import cost, insurance, and freight value (CIF value) for cement arriving at U.S. ports Jeeries provides an estimate of $70 to $80 per ton (page 158).

The additional costs incurred for transporting the cement to nal markets are very dierent by region. We rely on Lafarge, the largest global cement producer, to have estimates of these costs to various U.S. districts. On average, we arrive at $10 per ton to coastal markets and at $40 per ton for inland markets. This generates estimates of c f in the range [80,90] for coastal regions and c f in the range [110,120] for landlocked regions.

Second, to estimate the costs of local production, c h , Jeeries provide EBITDA margins of 20 to 30 percent of price per ton (p 186). They estimate price per ton at $70 to $80, which gives us a local production cost estimate of c h of between $49 and $64 per ton.

Third, for capital costs, the Jeeries Report estimates an investment cost per ton of $250 (page 153). At an annual cost of capital of 8 percent, this gives us an estimate of c k of $20.

Turning now to the two inequalities underlying the prediction. We rst note that c h +c k < c f for landlocked districts is well satised, while in some easily accessible coastal locations it may be not be protable to build new capacity (c h + c k = 84 > c f = 80). These extreme cases should not endanger our analysis. Secondly, we note that for coastal regions, this calibration gives an estimate of c f -c h -2c k of between -24 and 1. For landlocked regions, the estimate of c f -c h -2c k is between 16 and 41. Altogether we think that this calibration gives some reassurance about the relevance of our model to move to the empirical analysis.

Data

We now evaluate whether the empirical predictions of the model developed in Section 2 are consistent with investment in capacity in the U.S. cement industry over the 2000s, when construction activity cycles, as mentioned in the introduction, led to substantial localized demand volatility. Our model contributes to the explanation of the disproportionately large fall in imports in the cement industry, where imports and local production are perfect substitutes: When demand falls in coastal regions, a larger share of that decreased demand can be satised with local production rather than imports.

12

The empirical analysis presented in the paper examines localized variation in these aggregate patterns. The three main variables of interest are: local capacity, the relative marginal cost of imports, and demand uncertainty. We discuss each of these in turn:

12 To the best of our knowledge, this mechanism has not been previously discussed as a contributory factor to The Great Trade Collapse. Other papers, as summarized in [START_REF] Bems | The Great Trade Collapse[END_REF], have studied the roles of product durability, inventory adjustments, and trade nancing constraints.

Capacity and Capacity Investment

The USGS publishes an annual Minerals Yearbook containing detailed data about the cement industry at the region, or district, level. District-level capacity is measured in the data as the nish grinding capacity in thousands of metric tons, and is based on the grinding capacity required to produce the district's plants' normal output mix, including both portland and masonry cement, allowing for downtime for routine maintenance. Production, in thousands of metric tons, includes cement produced using imported clinker. The USGS Minerals Yearbook also reports data on the number of active plants by district, which permits a measure of the average plant size for each district in each year. 16

In each year, there are also imports to Detroit, MI and other northern Midwestern districts from Canada.

The nal column of Table 2 indicates our classication of districts into landlocked or coastal regions. This classication is based on overland distance from the coast and, specifically, distance from a port where cement is imported.

17 Figure 3 represents this classica- tion graphically. The empirical analysis investigates whether or not the relationship between investment and demand uncertainty diers signicantly between the two district groups.

Import costs are not included as explanatory variables for two main reasons: Firstly, import terminals were built prior to the time period of analysis, so the landlocked-coastal classication is quite robust over the data period. Secondly, as described in Section 3.2, interviews with managers at Lafarge suggested that local investment decisions across the U.S. depended critically on whether there was nearby access to a deep water harbor. This suggests that it is not the precise value of the import cost that matters for the investment decision but whether the district is coastal or landlocked.

The relative cost of local production is also, of course, aected by any district-level 16 This list is not comprehensive. Annual statistics can be found in

Demand Uncertainty

One of the key variables in the theoretical model set out in Section 2 is market-level demand uncertainty at the time when capacity investment decisions are made. In the model, this variable is the parameter that measures the variance in demand over the productive lifetime of the investment. In taking this to the data, we seek to measure managers' expectations about future demand variability in a district.

Cement is an intermediate input into dierent types of construction. The Portland Cement Association constructs a demand index for cement by construction activity, taking into account both the amount of activity in that sector and the intensity with which the sector 32 percent of the demand for cement; non-residential construction made up 31 percent; and the remaining 37 percent was in public construction. These gures varied considerably in the years preceeding 2010 as residential construction is strongly pro-cyclical, and public expenditure counter-cyclical. The total quantity of cement used in a district is an endogenous outcome reecting local demand conditions, and local supply conditions, including capacity, which is treated in our model as another endogenous variable.

To focus in on the expected variance in demand, we turn to measures of local construction activity, over the recent past, current and future years. Using data from the Occupational Employment Statistics, published by the Bureau of Labor Statistics, we collect the state-level number of construction laborers from 1999 to 2010.

19 Of the dierent types of employment in this sector, we anticipate that laborers are the most exible part of the construction workforce, often employed on short-term contracts on an as needed basis. As a proxy for district-level cement demand, we aggregate the state-level data on the number of construction laborers employed across states to the district-level, where the districts are as listed in Table 2. As a proxy for district-level demand uncertainty, we calculate the variance in the demand measure in each district over the current and past four years.

We make two further adjustments to this uncertainty measure: We use de-trended data to account for changes in employment levels that are consistent with patterns that are arguably predictable and would, otherwise, lead us to overstate uncertainty in fast-growing, or fast-shrinking, districts. Specically, we regress employment by district over the past ve years on a constant term. The larger the residuals from this regression on a trend, the 18 See http://www.cement.org/econ/pdf/CementInvue102010.pdf 19 See http://www.bls.gov/oes/tables.htm less informative are recent employment levels in predicting current demand. To measure the average diculty of predicting current employment demand using the data from recent years, we take the standard deviation of the residual values over the current year and the prior four years.

Second, since this measure of demand variation is increasing in the level of employment in a district, we also normalize the standard deviation by the mean employment level over the ve years in question. This normalized standard deviation measure summarizes the extent of recent employment volatility across districts, adjusting for dierences in district size. Our intent is to capture the plant manager's view about the diculty in predicting the local demand level in any one upcoming year using information about past construction activity.

20

We note that this measure of demand uncertainty is backwards-looking since it is constructed using district-level data from the current and last four years.

21 Any increase in volatility in a given year within a district is due to the level of demand in the current year being less similar to the level of demand for the past three years than is the level of demand four years ago to the prior three years.

While this measure of recent demand uncertainty has been adjusted to account for dierences in the average growth rates by district by de-trending, predictable demand growth, or decline, is also likely to have an independent eect on investment decisions and, hence, on capacity levels within a district. We construct a measure of recent employment growth within a district at any point in time as the average percentage change in the level of construction 20 Section 6 presents a series of robustness tests measuring cement demand volatility in alternative ways:

using dierent data on construction activity, and using dierent models of expectation formation.

21 [START_REF] Carruth | What do we know about Investment under Uncertainty[END_REF] contain a discussion about the relative merits of dierent measures of uncertainty. [START_REF] Guiso | Investment and Demand Uncertainty[END_REF] is one of very few studies that uses survey data on manager's certainty about future demand as a measure of rm-level uncertainty.

employment over the prior four years. This measure is included as a control variable in our main regressions.

Empirical Analysis

In this empirical setting, as described in Section 3, the investment in local cement production capacity incurs high xed costs and relatively low variable production costs. The alternative production technology of importing cement has low xed costs, and a variable production cost that varies by geography. We now investigate whether observed investment decisions in local cement production capacity in the U.S. over the 2000s are consistent with the model's prediction that the relationship between local investment decisions and demand uncertainty depends on the relative marginal costs of the capital intensive and alternative production technologies.

Estimation Strategy

We exploit the panel nature of the data to investigate the relationship between capacity and uncertainty. The estimated equation is:

y i,t = α + (β + γL i ) (V i,t + X i,t ) + µD i + i.Y ear t + ε i,t (4)
where i denotes one of the 21 districts, and t denotes the year. The key variable of interest on the right hand side of equation ( 4) is V i,t , the measure of demand uncertainty, measured as the level of recent demand volatility in the number of construction laborers in district i in year t, as described in Section 4.3. L i indicates whether district i is landlocked, so that the association between demand uncertainty and the dependent variable y i,t is allowed to vary with whether or not the district is landlocked.

X i,t is a vector of time-varying district-level control variables, namely demand growth and the number of plants. The association of these variables with the dependent variable is also allowed to depend on whether the district is landlocked. α is a constant term, D i are district xed eects, and Y ear t are year xed eects.

The dependent variable in equation ( 4) is one of three measures of capacity: The rst dependent variable is the installed capacity in the district. The second is a weighted measure of installed capacity that is intended to take account of dierences in size across districts. It is calculated as the installed capacity in each year minus the mean district-level production over the entire time period and then divided by installed annual capacity. It is a measure of capacity in excess of the quantity locally produced in a typical year. We note that as a consequence of using mean production in the denominator, variation over time in this excess capacity measure within a district is due entirely to changes in capacity, that is, due to the investment decisions that we want to study. The third dependent variable used is the average plant size in a district.

As is common in the analysis of panel data, the observations are likely to be correlated within groups, in our case, within districts. In addition to the clustering problem arising from the fact that observations within a district are likely to share some unobserved variable, our measure of demand uncertainty in any year is based on the variance in local demand levels over that year and the past four years. This introduces serial correlation in the observations from a given district. We estimate the xed eects and rst dierences specications using OLS regressions and report Newey-West standard errors with a maximum lag order of correlation within a district of four years [START_REF] Newey | A Simple, Positive Semi-denite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix[END_REF]. This correction addresses the serial correlation in the errors resulting from how we measure local demand uncertainty.

Having estimated the coecients in each specication, we then test whether linear combinations of the estimated coecients are signicantly dierent from zero, and signicantly dierent from each other, in ways that are consistent with the predictions of the model.

Specically, to test whether a change in demand uncertainty is associated with a change in capacity in coastal districts, we examine the signicance of the coecient estimate for demand uncertainty. To test whether a change in demand uncertainty is associated with a change in capacity in landlocked districts, we test whether the linear combination of the coecients on demand uncertainty and the interaction of demand uncertainty and the landlocked indicator is signicantly dierent from zero. We then test whether changes in demand uncertainty have signicantly dierent eects on capacity in coastal versus landlocked districts.

Results

The results of estimating equation ( 4) are given in Tables 4a, 4b and 4c. In Table 4a, the dependent variable is the level of capacity. Column 1 reveals that, when we do not allow for dierences between landlocked and coastal districts, there is no signicant relationship between changes in demand uncertainty and investment in capacity across all U.S. districts over this time period.

Columns 2 and 3 presents the rst test of the predictions of the model by allowing this relationship to depend on whether the district is landlocked or coastal. There is a negative association between demand uncertainty and investment in coastal districts in Column 2, although the estimated coecients is insignicantly dierent from zero. Column 3 includes the control variables of demand growth interacted with the landlocked indicator variable.

After including these controls, the negative coecient on demand uncertainty becomes signicantly dierent from zero. The coecient on the interaction of landlocked and demand uncertainty is now positive and signicant. This suggests that the relationship between capacity and uncertainty does vary with proximity to the coast.

Columns 4 to 6 of Table 4a add further control variables. First, we include the districtlevel coal price. We allow the relationship between local coal price and capacity to vary with whether or not a district is landlocked. Including these controls does not aect the nature of the relationship between uncertainty and capacity investment, which is negative in coastal districts and positive in landlocked districts.

We next include the number of plants as an additional control. This variable is intended to control for any changes in market structure within a district over time. As mentioned in Section 2, we would like to control for changes in the number of rms, but in the absence of this data, the number of plants in a district in each year takes account of any market-level consolidation or expansion. Column 5 includes this control, and allows its role to vary with whether the district is landlocked. Column 6 includes both the number of plant controls and the coal price controls from Column 4. In each case, the main result remainsthe coecient on demand volatility in coastal districts is negative and signicant and the coecient on the interaction of the landlocked indicator variable and demand volatility is positive and signicant, both at the one-percent level.

The rst two columns of Table 4a, Panel B report the estimated change in local capacity associated with a one standard deviation increase in local demand uncertainty, using the coecient estimates from Column 3 of Panel A. In coastal districts, this increase in demand volatility is associated with a decrease in capacity of 260 thousand metric tons. The average district capacity (Table 2) is 5.2 million metric tons. Hence, this corresponds to a decrease of around ve percent. In landlocked districts, the same increase in demand uncertainty is associated with an increase in capacity of 113 thousand metric tons, although this number is not signicantly dierent from zero at conventional signicance levels. Column 3 of the panel shows that the dierence in the response to changes in uncertainty between landlocked and coastal districts is signicant at the one-percent levelthe relationship is signicantly more positive in landlocked districts than in coastal districts.

Table 4b shows the results for the same analysis with the measure of excess capacity as a dependent variable.

22 As in Table 4a, Column 1 of Table 4b shows that there is no signicant association between local demand uncertainty and excess capacity without accounting for geographic dierences. Allowing the relationship to vary between landlocked and coastal districts, in Columns 2 to 6, reveals that there is a negative and signicant association in coastal districts and a positive and signicant association in landlocked districts.

Panel B of Table 4b demonstrates that the relationship is signicantly dierent from zero in each case, and signicantly dierent between coastal and landlocked districts. The mean level of the excess capacity variable over all districts between 2002 and 2010 is 1.41. Hence, the coecient of -0.07 suggests a one standard deviation increase in uncertainty is associated with a ve-percent reduction in excess capacity in coastal regions. The coecient of 0.04 in the next column suggests that the same increase in uncertainty is associated with around a three-percent increase in excess capacity in landlocked districts.

Table 4c repeats the key analyses with average plant size as the dependent variable.

Across all districts, there is no discernable relationship between plant size and uncertainty.

In coastal districts, however, the relationship is negative and signicant and, in landlocked relationship diers signicantly between the coastal and landlocked districts. In this specication too, a one standard deviation increase in uncertainty is associated with a ve-percent reduction in the capacity measure, in this case, average plant size.

Together, these results show that investment is negatively associated with an increase in demand uncertainty, but only in coastal districts. With reference to Figure 2.3, the ndings establish that the line AB has a slope that is not signicantly dierent from zero, although positive in sign. This line is the elasticity of capacity choice with respect to demand uncertainty. In contrast, the elasticity of capacity choice with respect to demand uncertainty in coastal districts, given by line CD, has a negative slope that is signicantly dierent from zero. Moreover, the slope of the line AB is signicantly more positive than the slope of the line CD. This suggests that the exibility oered by a choice between two dierent production technologies (in this case, local production and imports), where the technologies dier in the amount of investment required, has a signicant role in determining the overall relationship between demand volatility and investment.

6 Robustness Tests

Demand Uncertainty

We rst examine whether the results in Section 4 are robust to dierent measures of districtlevel uncertainty. As a reminder, uncertainty, in the model in Section 2, is the variance of the possible demand realizations, θ. In Section 4, we make the assumption that the managers of the cement rms making investment decisions form a belief about demand volatility in future demand using the variance in construction activity in recent history: the last four years and current year.

Table 5 presents an investigation of robustness about this assumption of how manager form beliefs about upcoming demand uncertainty. We rst give them a shorter memory: We measure local demand uncertainty using the variance in the number of local construction laborers over the last two years and current year, that is, over a three-year period.

24 The rst three columns of Table 5 show the results of the specication given in Column 3 of Table 4a, 4b, and 4c, but with this shorter time period taken into consideration. In each case, the coecient on demand uncertainty for coastal districts is negative and signicant and positive and signicant in the case of demand uncertainty in landlocked districts. Moreover, these coecients are signicantly dierent from each other, all at the one-percent level. That is, assuming managers are have shorter memories when considering the nature of future demand leads to results that are also consistent with the empirical predictions in the model.

Columns 4 to 6 of Table 5 allow managers some foresight. Demand uncertainty in these columns is measured using ve years of employment data, centered on the current year in question. That is, it assumes managers anticipate ongoing demand uncertainty that reects the volatility seen around the current year. In these specications, the coecients related to uncertainty and investment have the same signs as in the earlier tests; negative for coastal regions and positive for landlocked regions. However, these coecients are not signicantly dierent from zero. Columns 7 to 9 assume managers are able to anticipate construction activity over the current year and the following four years and use this to approximate market uncertainty when making investment decisions. In this case, the data do not provide any support for the empirical predictions of the model.

In sum, Columns 4 to 9 suggests that managers' investment decisions are either inconsistent with the predictions of the model or that managers decisions did not correctly anticipate future market-level demand uncertainty. We feel it is reasonable to think that managers did not fully anticipate the ex post observed level of volatility in construction activity during the 2000s. Hence, the fact that the ndings based on forward-looking expectations of uncertainty do not support the model does not, therefore, undermine the validity of the model, which is supported with backwards-looking expectation formation.

Landlocked classication

Our second set of robustness tests redenes how districts are classied into landlocked and coastal districts. We do three sets of tests, each one relaxing a dierent assumption made in Sections 3 and 4.

First, we reclassify the Great Lakes regions as coastal. Michigan and Wisonsin, and Ohio districts, are able to import cement from Canada via the Great Lakes. While Canadian cement is likely to have a local production cost that is similar to U.S. Cement, the relative import cost may vary over time to make imports viable. The results for this reclassication are given in Table 6, Columns 1 to 3. Each column replicates the specication Column 3 from Table 4, with the three dierent dependent variables (capacity, excess capacity, and average plant size). The ndings are strengthened, relative to the base case.

Next, we relax the assumption that each of the 21 districts is a well-dened segmented market. In particular, in the USGS data, Michigan and Wisconsin are classied as a single market in the base case, despite the fact that they are separated by Illinois and and Indiana, which are recorded as two separate districts. In addition, Missouri and Illinois are treated as separate districts, when much economic and, hence, construction activity, takes place along their shared border. We redene the district classication in our data, to group Michigan and Wisconsin with Illinois and Indiana, and also with Missouri, as one large landlocked district. This leaves us with 18 districts in total. Columns 4 to 6 of Table 6 replicate again the specication from Column 3 of Table 4. There remains a signicant dierence between the landlocked and coastal districts, in all columns.

Finally, we explore variation among the set of coastal markets in the relative price of imports. Of the 8 districts classied as coastal in Table 1, we divide them into those with large import terminals, where the local cost of imports, c f is likely to be particularly low, and those without large terminals. The import terminal districts are California, Florida and Texas. In this set of robustness tests, we dene only these three districts as coastal, and group the remaining ve districts with the landlocked districts. Repeating the estimation given in Column 3 of Table 4 yields results that are very similar to the base case. The estimated coecients take on the same signs, and while the dierence in the relationship is not signicant at conventional levels for capacity levels, there remains a signicant dierence in the relationship between excess capacity and average plant size and local demand uncertainty between landlocked and coastal regions. The results for this specication are presented in Table 7. In each case, the data continue to provide empirical support for the model's main prediction: There is a signicant dierence between the relationship between uncertainty and investment in coastal and landlocked districts, with the relationship being signicantly more positive in landlocked districts. In contrast to the xed eects specications, the slope for landlocked districts is positive and signicantly dierent from zero. The slope for coastal districts is negative, but not signicantly dierent from zero.

With reference to Figure 2.3, this specication suggests that line AB is positive and line CD is negative. In the main xed eects specications, AB is not signicantly dierent from zero and CD is negative. Nonetheless, the specication oers further support for the prediction that the slope of AB is signicantly more positive than the slope of CD. That is, the relationship between capacity and uncertainty is signicantly more positive relationship in landlocked districts.

Concluding Comments

This paper contributes to the literature on the theory of irreversible decisions under uncertainty. While the theory has elaborated general conditions under which the relationship between investment and uncertainty can be expected to be increasing or decreasing, these conditions remained to be empirically tested at the industry level. The US cement industry provides a unique opportunity to carry on the test. We have developed a model in line with the theory and show that the empirical ndings are consistent with our model. The key factor in our analysis is the existence of a production set involving two technologies: a rigid one and a exible one. The rigid technology is domestic production and the exible technology is imports. The fact that cement transportation is high over land routes while it is low over maritime routes generates signicant changes in the opportunity cost of the two technologies as one consider landlocked versus coastal markets. The relationship is shown to be decreasing in coastal markets and signicantly more increasing in landlocked markets.

The paper provides a number of directions for future research. The model remains simple. The two stage setting could be extended to a Markov framework such as the one introduced in Ryan ( 2012). This would allow for endogenous market structures. Another direction would be to allow for some asymmetry among rms, in particular relative to their importing capabilities. Data sets for the US cement industry are pretty exhaustive so that the empirical analysis could encompass some of these added characteristics. It would be interesting to explore our nding in this extended framework.

On the application side, the ideas developed in this paper have direct implications to understand the leakage risk associated to a unilateral climate regulation in a sector such as cement. Indeed cement is a highly carbon intensive industry, a unilateral climate policy can be expected to aect trade patterns. This sector is a recurrent topic in the leakage literature [START_REF] Droege | Tackling leakage in a world of unequal carbon prices, a study of the Greens/EFA Group[END_REF]. Building on the model presented here, Meunier and

Ponssard [START_REF] Meunier | Capacity decisions with demand uctuations and carbon leakage[END_REF]) dierentiate between short term and long term leakage, depending on whether or not the investment decisions have taken into account the change in the climate policy. Without capacity adaptation there is no leakage, but there is with capacity adaptation since the increased in domestic cost will reduce the future capacity.

The larger the demand uncertainty the larger this decrease. These considerations remain to be integrated in empirical analysis. For instance, using the Markov framework, Fowlie et al. [START_REF] Fowlie | Market-Based Emissions Regulation and Industry Dynamics[END_REF]) empirically explore the leakage risk for the US cement industry.

Introducing demand uncertainty in this analysis, an uncertainty which is large in the case of the US market, would certainly magnify the risk leakage they identify.

FigureFigure 1 :

 1 Figure'1:'Empirical'Predictions

  Figure 2(a) shows the U.S.-wide levels of cement consumption, capacity, production and imports for each year between 1998 and 2010. Consumption increased in each year up to 2007, at which point it fell o dramatically.The gure reveals that imports also saw a big reduction from 2007 onwards. We note that exports of cement production were very low throughout the period, less than one twentieth of import levels when imports were at their lowest. For this reason, we abstract from any incentive to invest in local production to serve export markets, in the model and in the empirical work.

Figure 2

 2 Figure 2(b) plots each of the same series as in Figure 2(a) relative to each of their

FigureFigure 2 :

 2 Figure)2:)U.S.)Cement)Industry,)1998?2010)

6. 3

 3 Estimation approach: First dierences Our last set of robustness tests relates to the econometric specication of equation (4). The estimated equation in the benchmark specication, as presented in Section 4, controls for any non-time varying observable and unobservable characteristics correlated with capacity in a district and demand uncertainty by including district xed eects. It also controls for time-varying factors, such as changes in average import costs over time, with year xed eects. As an alternative specication, we also estimate equation (4) in rst dierences, where the estimated coecients measure the association between yearly changes in capacity and yearly changes in demand uncertainty.

  The district-level capacity is given for dierent local regions, dened mostly by state-lines and consisting of groups of states. To account for changes in the market-level boundaries over time, we group the data into 23 dierent districts, where the outer boundaries of the 23 district groupings are constant over time. In some cases, these districts are divided dierently into two or more separate regions at some point during the data, and the set of 23 allows for consistency over time in the states that are included in each. In addition, since the USGS divides California and Texas into two districts in each case, and we want to match the capacity data with state-level data about demand and other control variables, we group up the capacity data to the state level. This leaves us with 21 districts altogether, as listed in the rst column of Table2.

13

13 We note that the USGS also divides Pennsylvania into two districts, Eastern and Western. We have matched the state-level variables for New Jersey to the USGS capacity-related data for Eastern Pennsylvania and the state-level variables for Pennsylvania to the USGS capacity-related data for Western Pennsylvania. This is a more accurate matching than grouping Eastern and Western Pennsylvania together and matching it with demand and other variables in Pennsylvania as a whole.

  Table 2 summarizes the levels of capacity in each district in 2002, which is the rst year of capacity data that we use as the dependent variable in the analysis. Californiahad the largest installed base of cement capacity, with over 13 million tons, and the district containing Alaska, Hawaii, Oregon and Washington had the lowest level of capacity, at 2.5 million tons. Michigan and Wisconsin had the largest plants, at an average of 1.3 million tons each, and the plants in the district containing Idaho, Montana, Nevada and Utah averaged just over half a million tons each. Over the eight years from 2002 to 2010, 19 of these 21 districts experienced both annual increases and annual decreases in total installed capacity.

	In the early 2000s, according to industry sources, global cement rms such as Cemex,
	Holcim, Lafarge, and Lehigh (Heidelberger) operated import terminals located on the East
	Coast, and Lafarge, Lehigh (Heidelberger) and Taiheiyo on the West Coast. The USGS
	breaks down total imports of cement and clinker into the U.S. by customs district. Major
	import terminals include Tampa, FL, New Orleans, LA, Los Angeles, CA, Miami, FL, and
	Houston-Galveston, TX. Smaller import terminals are spread out over the East Coast of the
	US and include Baltimore, MD, New York City, NY, Norfolk, VA, and Philadelphia, PA.
	Each of the 21 districts experienced both increases and decreases in average size from
	year to year.	14
	4.2 Relative Marginal Costs of Flexible Production
	Imports of cement to areas such as Florida, California, New York, and Texas had increased
	steadily since improvements in shipping technology in the 1970s, with imports coming from
	South America, Europe, and Asia. The industry association Cembureau now estimates that
	it is now less costly for cement to cross the Atlantic Ocean than to truck it 300km overland. 15

Base'Case'Classification'of'Districts'into'Coastal'and'Landlocked

  Table 18 of the Cement Yearbook.

				Introduction
				Theoretical Model		Estimation Strategy
			The U.S. Cement Industry		Results
				Empirical Analysis		Robustness Tests
			Discussion and Conclusions
	Variable DeÖnition II
			DISTRICT CLASSIFICATION INTO LANDLOCKED AND COASTAL
	Washington
	Oregon	Idaho	Montana	North Dakota South Dakota	Minnesota	Wisconsin		Vermont New York	Maine New Hampshire Massachusetts
				Wyoming						Michigan	Connecticut Rhode Island
	California	Nevada	Utah	Colorado	Kansas Nebraska		Iowa Missouri	Illinois	Kentucky Ohio Indiana	Virginia Pennsylvania DC West Virginia	New Jersey Maryland Delaware
										Tennessee	North Carolina
			Arizona	New Mexico	Oklahoma	Arkansas			South Carolina
										Alabama	Georgia
								Mississippi
					Texas		Louisiana		Coastal
											Districts
	Alaska									Florida	Landlocked Districts
				Hawaii			0			500 Miles	No Plants
							0			500 KM
			AK & HI not to scale
			Meunier, Ponssard, Thomas		The Role of Imports in the US Cement Industry
	Figure 3: Base Case Classication of Districts into Coastal and Landlocked

17 Section 6 includes a set of robustness tests to dierent denitions of district groupings and classications. Figure'4:'production cost shocks. The largest variable cost in cement production is energy costs, and the majority of U.S. kilns are coal-powered (USGS, 2011). The U.S. Energy Information Administration publishes data on coal prices delivered to Other Industrial Plants (other than Electric Utilities) in each state in each year. We include nominal coal price data, aggregated up across states to the geographic district boundaries, as a control variable in the anaysis.

The USGS notes that, in the U.S. ...since the early 1990s, the majority of cement imports have been con- trolled by domestic cement producers, and they import only as needed to make up for production shortfalls.(

USGS, 2006, p.166).

This assumption rules out the possibility that a monopolist would reduce its output under a higher draw of θ. Note that if p(q, θ) = p(q/θ), the second part of A2 is equivalent to A1.

For simplicity, we assume there is no xed component to investment or production costs. Introducing xed costs does not aect the predictions of the model as long as investing remains protable, which we assume all through the model.

This is a possible denition of an increase in risk: any risk-averse decision maker, with a concave utility function, prefers a less risky distribution. Intuitively, an increase in risk increases the probability of extreme demand states, with a concave function the weight of lower states are larger than the weight of higher states, so that an increase in risk reduces the expectation.

Consider now the introduction of a competitive fringe of price-taking importers: The oligopolists continue to face a residual demand function and the analytical results developed so far continue to hold. Thus, the contingent nature of the relationship between uncertainty and capacity, depending on the relative cost of imports, is robust to the local market structure.6 In other words, this does not allow the possibility that rms operate with excess capacity to deter entry. There have been several theoretical contributions that could motivate such strategic behavior, such as[START_REF] Spence | Entry, capacity, investment and oligopolistic pricing[END_REF] and[START_REF] Dixit | The role of investment in entry-deterrence[END_REF]. A number of empirical studies have tested this hypothesis in specic industry studies; see, for example,[START_REF] Ghemawat | Capacity expansion in the titanium dioxide industry[END_REF] and[START_REF] Mathis | Excess capacity as a barrier to entry in the US titanium industry[END_REF]. As noted by[START_REF] Lieberman | Excess capacity as a barrier to entry: An empirical appraisal[END_REF], the empirical results provide limited supporting evidence of this type of behavior.7 Note that this symmetry assumption extends to the rms's technology mix. We conjecture that the relationship stated in Proposition 1 should be restated as a progressive shift from being negative to positive when the fraction of rms with importing capability in the market increases. Since the investment cost associated to an import cement terminal is low as compared to the investment cost in a cement plant we may assume that there are no entry barrier for terminals so that this fraction is large in our empirical analysis.

We do not have data on the number of rms, but our data do contain the number of plants in each district in each year. We include the number of plants in each market as a control variable in our main analysis, controlling for changes in market structure resulting from the entry of new plants and changes in market structure due to ownership consolidation. We also see that the number of plants within a district is uncorrelated with demand variability in the data overall.

[START_REF] Kogut | Operating Flexibility, Global Manufacturing, and the Option Value of a Multinational Network Management Science[END_REF] show how the value of joint ownership of production facilities in more than one country can be related to the operating exibility this oers under exchange rate uncertainty. When rms have local capacity constraints, their model can also generate the predictions that rms simultaneously export to and produce in a given market.

While regulation may play a role in capacity decisions, it is more likely to act as a very local constraint (at the city or town level) and is unlikely to matter dierently in landlocked and coastal districts.

See: http://www.cembureau.be/about-cement/cement-industry-main-characteristics

This specication is intended to take account of the large dierences in size across districts since this variable can be interpreted as the average percentage of installed capacity that is unused in a year.

coecients reect plant-level, rather than district-level, decisions.

This robustness test also allows us to include one more year of data in the analysis, since it allows us to construct a measure of recent demand volatility for 2001. This adds 21 observations to the analysis.

Appendix 1: Proofs

The Oligopoly Case

We consider that there is a set I of n rms. The decision process takes place in two steps.

First, each rm i ∈ I chooses both k i and a production plan q i h (θ), q i f (θ) with q i h (θ) ≤ k i , then θ i becomes known and each rm produces.

The short-term prot of a rm i depends on its production and the sum of its rivals' production, denoted q -i :

For any vector of capacity (k i ) i∈I , in any state θ rms play a Cournot game with capacity constraints. Thanks to our assumption A1, there is a unique equilibrium in all demand states.

The production of each rm is increasing with respect to θ (this is due to assumption A2 and cost symmetry). In low demand states, the equilibrium is symmetric, each rm has excess capacity; as θ increases each rm's capacity starts to bind, in increasing order (the smallest rm rst). For each rm we can dene two thresholds θ i-and θ i+ as function of the vector of capacities, such that q i h < k i i θ < θ i-and q f > 0 i θ > θ i+ .

Proof. We rst show that any equilibrium of the game is symmetric; then, we prove uniqueness of the equilibrium.

• To show symmetry we proceed by contradiction. Let us assume that there is an equilibrium in which two rms A and B have dierent capacity k A < k B . In that case, θ A-< θ B- and θ A+ < θ B+ , the comparison between θ A+ and θ B-is ambiguous.

Let us show that, in each demand state, the marginal short-term prot of rm A is larger than that of rm B because:

-in low demand states in which both rms have excess capacity and in high demand states in which both import, their marginal revenues are equal;

-in intermediary states in which both produce at full capacity p + ∂p ∂q k A > p + ∂p ∂q k B because p is decreasing with respect to q.

-if rm A has excess capacity but rm B does not, or rm A imports but rm B does not, the marginal revenue of rm A is strictly higher than that of rm B.

Therefore, the expected marginal short-term prot of rm A is strictly larger than that of rm B and since, at equilibrium, both should be equal, this is a contradiction.

• Then we show uniqueness by showing that at a symmetric equilibrium the total capacity is a solution of an equation that admit a unique solution.

At a symmetric equilibrium, with total capacity k * , in low demand states, rms play a Cournot game with identical marginal cost c h . All rms capacity constraint start being binding simultaneously in a state θ -such that the individual marginal revenue is equal to the marginal cost: p(k * , θ -) + ∂p ∂q (k * , θ -) k * n = c h ; and they start importing simultaneously in the same state θ + in which the individual marginal revenue is equal to the import price; in higher demand states each rm total production is equal to the Cournot production with marginal cost c f . Consequently, the equilibrium total capacity k * is the solution of the rst order condition:

By assumption (A1), there is a unique solution to this equation.

• Finally, existence is proved by construction. The solution of the above equation exists and this capacity together with Cournot quantities are an equilibrium.

Corollary 1 With n rms, if θ is homogeneously distributed over [-λ, λ], the equilibrium capacity is increasing (resp. decreasing) with respect to

Proof.

The proof is similar to the monopoly case, rewriting the rst order condition for an homogeneous distribution: