Corentin Ribeyre

Marie Candito

Djamé Seddah

Semi-Automatic Deep Syntactic Annotations of the French Treebank

We describe and evaluate the semi-automatic addition of a deep syntactic layer to the French Treebank (Abeillé and Barrier [1]), using an existing scheme (Candito et al. [6]). While some rare or highly ambiguous deep phenomena are handled manually, the remainings are derived using a graphrewriting system (Ribeyre et al. [22]). Although not manually corrected, we think the resulting Deep Representations can pave the way for the emergence of deep syntactic parsers for French.

Introduction

Syntactic parsing has been the focus of intensive international research over the last decades, leading to current state-of-the-art parsers to provide quite high performance (on well-formed English text at least). However, extracting more semanticallyoriented information from syntactic parses, be them of high quality, is as not straightforward, given the abundant syntax-semantic divergences and the idiosyncratic nature of syntax itself. "Deep syntax" is generally intended as an intermediate representation between what is actually observable (surface syntax) and a semantic representation, which abstracts away from syntactic variation, such as diathesis alternations or non-canonical word order, and which can thus serve as an easier basis for semantic analysis. Such view forms, for example, the basis of the Meaning-Text Theory, MTT, (Melčuk [16]).

Several initiatives have been proposed to obtain "deep" syntactic treebanks, with various meanings attached to the term "deep". For instance for Spanish, the AnCORA-UPF multi-layer corpus (Mille et al. [START_REF] Mille | AnCora-UPF: A Multi-Level Annotation of Spanish[END_REF]) includes a deep syntactic layer, inspired by the MTT. For English, the Penn Treebank (PTB, (Marcus et al. [START_REF] Marcus | The penn treebank: annotating predicate argument structure[END_REF])) contains a certain amount of "deep" annotations (such as traces for subjects of infinitives, long-distance dependencies and so on). Initially encoded with traces and co-indexes through constituent structures, the processing and recovery of these phenomena entailed complicated algorithms and methods. Nevertheless, the emergence of various conversion algorithms and enrichment processes from the PTB phrase structures to deep syntax representation (e.g LFG F-Structures as in (Cahill et al. [4]), HPSG feature structures (Miyao et al. [START_REF] Miyao | Corpus-oriented grammar development for acquiring a head-driven phrase structure grammar from the penn treebank[END_REF]), or CCG complex lexical types and derivations (Hockenmaier and Steedman [START_REF] Hockenmaier | Ccgbank: a corpus of ccg derivations and dependency structures extracted from the penn treebank[END_REF]) ) have made these complex syntactic phenomenon more straightforwardly available.

Recently, more semantically oriented "Deep" treebanks have been made available ( Čmejrek et al. [START_REF] Čmejrek | Prague czech-english dependency treebank: Syntactically annotated resources for machine translation[END_REF], Flickinger et al. [START_REF] Flickinger | DeepBank: a dynamically annotated treebank of the wall street journal[END_REF]) and their use was popularized through the Semeval 2014 broad semantic shared task (Oepen et al. [START_REF] Oepen | Semeval 2014 task 8: Broad-coverage semantic dependency parsing[END_REF]) which simplified these data set by providing mostly graph-based predicate-argument structure instances of these treebanks (Miyao et al. [START_REF] Miyao | In-house: An ensemble of pre-existing off-the-shelf parsers[END_REF]). It worth noting that providing access to different representation layers of the same source, each having a different degree of granularity in term of syntax-to-semantic interface was, among others such as the MTT, formalized through the Prague Dependency Bank line of work (Böhmová et al. [3], Hajic et al. [START_REF] Hajic | Prague dependency treebank 2.0. CD-ROM, Linguistic Data Consortium[END_REF]). Inspired by the LFG theory, the various Stanford dependency schemes (De Marneffe and Manning [START_REF] De | Stanford typed dependencies manual[END_REF], de Marneffe et al. [START_REF] De Marneffe | More constructions, more genres: Extending stanford dependencies[END_REF]) are also a milestone in making deep syntax structures easily processable for further downstream semantic processing.

For French, which is the language we focus on, the annotations of the largest treebank available for French (the French Treebank (Abeillé and Barrier [START_REF] Abeillé | Enriching a French treebank[END_REF]), hereafter FTB) are surface-only. However, earlier attempts at deriving deeper representations were carried out by Schluter and Van Genabith [START_REF] Schluter | Dependency parsing resources for french: Converting acquired lexical functional grammar f-structure annotations and parsing f-structures directly[END_REF] within a treebank-based LFG framework, using an heavily modified subset of the initial FTB release. Focusing on delivering a free data set based on structures as close as possible from the current FTB, Candito et al. [START_REF] Candito | Deep Syntax Annotation of the Sequoia French Treebank[END_REF] have defined a deep dependency syntactic annotation scheme for French, and added a deep syntactic layer to the Sequoia Treebank (Candito and Seddah [START_REF] Candito | Le corpus Sequoia : annotation syntaxique et exploitation pour l'adaptation d'analyseur par pont lexical[END_REF]), a freely available corpus, made of 3,099 sentences . Although this resource can be used to train statistical deep parsers for French, we anticipate that its size will be insufficient to train accurate models given the additional complexity of deep syntax with respect to surface syntax. 1 We have thus undertaken to semi-automatically annotate the FTB with deep syntactic annotations, leading to a "silver" deep treebank of 18,500 sentences.

In the following, we start by describing the Deep Syntactic Representations (hereafter DSRs) of (Candito et al. [START_REF] Candito | Deep Syntax Annotation of the Sequoia French Treebank[END_REF]) in section 1, and the methodology used to obtain such representations for the sentences of the FTB. Section 3 is devoted to the tool we designed to convert surface dependency trees into such deep syntactic representations: we describe both the graph-rewriting system (section 3.1) and the hand-craft rules (section 3.2). We provide an evaluation of the DSRs obtained using this tool in section 4, and conclude.

Target Deep Representations

In order to describe the Deep Syntactic Representations (DSRs) that we target, we sum up their description by Candito et al. [START_REF] Candito | Deep Syntax Annotation of the Sequoia French Treebank[END_REF]. As mentioned in the introduction, deep representations are intended to be an intermediary step between surface syntax and semantic representations. The DSRs make explicit three major types of information with respect to the surface representations:

• First, DSRs make explicit the deep syntactic arguments of verbs and adjectives and "subjects" of adjectives (predicates with other part-of-speech are left for future work). The deep syntactic arguments include those arguments that are syntactically dependent of another head (e.g. the subject of infinitival verbs) or that appear as the surface governor of the predicate (e.g. in the case of an attributive participle: des personnes parlant italien ((some) people speaking italian)).

• Second, following Relational Grammar (Perlmutter [START_REF] Perlmutter | Studies in Relational Grammar 1. Studies in Relational Grammar[END_REF]), predicates are taken to subcategorize for dependents with certain canonical grammatical functions, potentially different from their effective final functions. The deep arcs are thus labeled with both canonical and final functions (at least for the grammatical functions that can be involved in syntactic alternations). For instance in Figure 1, while Jean is both the final and canonical subject of semble, it is the final subject and canonical object of the passive form respecté (written with a suj:obj label).

• Third, the semantically-void tokens are discarded, and dependencies coming in or out from these tokens are shifted to semantically full tokens (e.g. semantically void prepositions or complementizers are bypassed, auxiliaries are discarded and replaced by features on full verbs).

In order to capture syntactic alternations, DSRs make use of the distinction between canonical grammatical function (canonical GF) and final grammatical function (final GF) 2 , and between canonical subcategorization frames (canonical SF) and final subcategorization frames (final SF). The final SF of an occurrence of a verb is defined as the list of GFs associated to its expressed arguments, plus the GFs that would be associated with the linguistic expressions that would appear as argument, if the verb were used in finite mode and in a non elliptical construction. This formulation accounts for the subject of infinitives, the subject of coordinated verbs or more generally any argument shared by several predicates. For instance, in Figure 1, the final SF both for compter (to matter) and for respecté (respected) is [subject=Jean].

The deep syntactic arguments of a verb are defined as the set of linguistic expressions that bear a final GF with respect to that verb, and that are not semantically empty. Syntactic alternations are viewed as redistributions of the grammatical functions associated to the syntactic arguments. Following Relational Gram-mar (Perlmutter [START_REF] Perlmutter | Studies in Relational Grammar 1. Studies in Relational Grammar[END_REF]), the final SF is considered as resulting from the application of 0 to n redistributions to a canonical SF. A simple case is for instance a passive occurrence of a transitive verb: the final SF is [subject, (by-object)] while the corresponding canonical SF is [objet, (subject)]. So for instance in Figure 1, the canonical SF of respecté is [object=Jean]. This is shown in the figure with double labels on the arcs of the form final_function:canonical_function (hence the label suj:obj between Jean and respecté).

Candito et al. [START_REF] Candito | Deep Syntax Annotation of the Sequoia French Treebank[END_REF] only considered redistributions that are morpho-syntactically marked (for instance with an auxiliary for passives, or a void reflexive clitic se for middle or neuter alternations). Unmarked redistributions are not accounted for, because disambiguating them resorts to semantic analysis. For instance, for the verb couler ('to sink'), the non-marked causative/inchoative alternation gives rise to two canonical SFs: the two constructions X coule Y (X sinks Y) and Y coule (Y sinks) are not related in the deep syntactic representation. They get the two distinct canonical SF [subject, object] and [subject] respectively, and for both occurrences, the canonical SF is identical to the final SF. On the contrary, the neuter and middle alternations, which are marked by a void reflexive clitic se, are represented using redistributions. For instance, for both (Paul cassa la vase) Paul broke the vase and le vase se brisa (litt. the vase SE broke for the vase broke), vase is canonical object.

From the formal point of view, the DSRs are graphs, whose nodes are the nonvoid tokens of the sentence. The arcs are labeled using the canonical functions (hence for instance, the suj:obj arc between Jean and respecté is labeled with obj only in the DSR). The DSRs may contain cycles, for instance in the case of an adjective or participle modifying a noun : the modifier dependency is retained in the deep representation, and an inverse arc is added (the noun is a deep syntactic argument of the modifying adjective or participle). In order to obtain pseudo-gold DSRs for the FTB, we used as starting point the surface dependency version of the FTB, as released for the SPMRL Shared Task (Seddah et al. [START_REF] Seddah | Overview of the spmrl 2013 shared task: A crossframework evaluation of parsing morphologically rich languages[END_REF]), which contains 18, 535 newspaper sentences. To obtain DSRs for these sentences, we partially re-used the annotation methodology of the Deep Sequoia Treebank, which consisted of three steps (Candito et al. [START_REF] Candito | Deep Syntax Annotation of the Sequoia French Treebank[END_REF]):

Jean

(i) Manual annotation of certain phenomena, (ii) Automatic pre-annotation using two independently designed graph-rewriting systems (Grew (Guillaume et al. [START_REF] Guillaume | Grew : un outil de réécriture de graphes pour le TAL[END_REF]) and OGRE (Ribeyre et al. [START_REF] Ribeyre | A Linguistically-motivated 2-stage Tree to Graph Transformation[END_REF]))

(iii) Double manual correction plus adjudication of the divergences.

We applied this methodology on the FTB, but we skipped the last step, which currently seems out of reach given the corpus' size. We retained the dichotomy between manual annotation of certain phenomena (step (i)) and automatic annotation (step (ii)), this time using OGRE only.

The focus of this paper is on the evaluation of step (ii) mainly, so we only briefly list the phenomena that were manually annotated during step (i) 3 : longdistance dependencies, impersonal subjects, causative alternations, cleft sentences, and finally the status of the se clitic, which can either be part of a lexicalized pronominal verb (like s'apercevoir (to realize)), or mark a reflexive construction (as in Anna se soigne tout seule (Anna cures herself on her own)), or mark a middle diathesis alternation (Ces livres se vendent bien (These books sell well)) or a neuter diathesis alternation (Le vase s'est rompu (The vase broke)). All these phenomena are either highly ambiguous (clitic se) and/or rare (long-distance dependencies, causatives, cleft sentences), and their disambiguation resorts to semantic properties that are notoriously difficult to capture in a rule-based approach. By contrast, phenomena which exhibit more systematic syntactic properties, such as raising and control or the passive alternation, are handled automatically at step (ii).

We can now turn to the description of the graph rewriting system and the handcraft rules used at step (ii).

3 Surface to deep tool 3.1 OGRE OGRE (for Optimized Graph Rewriting System) is a two-stage graph rewriting system (Ribeyre et al. [START_REF] Ribeyre | A Linguistically-motivated 2-stage Tree to Graph Transformation[END_REF]). The first stage is based on the Single Pushout approach described at length in (Rozenberg [START_REF] Rozenberg | Handbook of Graph Grammars and Computing by Graph Transformation: Volume I. Foundations[END_REF]) while the second has his roots in the constraint programming paradigm (Rossi et al. [START_REF] Rossi | Handbook of Constraint Programming (Foundations of Artificial Intelligence)[END_REF]).

OGRE uses a set of rules, applied in two stages. A rule is defined by a subgraph pattern called a match, a set of rewriting commands (performed at first stage) The rule contains a Negative Application Condition (NAC), which blocks the application if the x node depends on the y node.

and/or a set of "triggers" 4 (instantiated at first stage, but activated in the second stage). In addition, a rule may contain negative application conditions (NAC defined in Lambers et al. [START_REF] Lambers | Conflict detection for graph transformation with negative application conditions[END_REF]) which block matches based on certain conditions. An example of rule is given in Figure 2.

During the first stage, rules are applied sequentially. The rewriting commands can add, modify and remove properties on nodes (token, features, POS, ...) and edges (labels, surfacic or deep status), and remove or add edges. In the surfaceto-deep rules, removal of edges or features is not used though. Importantly, the match is always performed on the input graph only, independently of the added arcs/features, so the order of the rules does not matter. The set of rewriting commands is applied to the input graph and triggers are instanciated on pairs of nodes, to be used during the second stage.

In the second stage, triggers instantiated at first stage apply until no more edges are added. In the surface-to-deep rules, we only use one of the several trigger types available in OGRE, namely the share triggers 5 , which add edges. A share trigger share(l) is defined for a pair of nodes (y, z) and a label l. It states that if a y l -→ x arc belongs to the current graph (i.e. if it either belongs to the modified graph from the first stage, or was added by a share trigger), then the arc z l -→ x should be added to it. For example in Figure 3, three share triggers share(suj) sequentially add the orange, purple and green edges, in that order, each new edge triggering the applicability of the subsequent trigger. As will be seen in the next section, this system allows to express in a compact way general linguistic constraints such as cascaded control or raising verbs.

Termination of the second stage is guaranteed by the absence of multi-edges with the same label. Moreover, the iterative process combined with the fact that triggers can only add edges ensures the confluence of the system.

Jean

Rules

The rules for surface to deep syntax conversion are organized into five modules, namely sets of rules, designed to be applied sequentially. The rules are partitioned into modules so that arcs or features added within one module can serve as matches for the rules of a subsequent modules : while the rules within a module need not be ordered, the modules themselves are sequentially ordered. The first module makes verbal tense and mood explicit, converting tense auxiliaries into appropriate features on the lexical verb. For instance, in example 1, the verb respecté is a past participle at the surface level, but it bears infinitival mood and past tense at the deep level. This normalization facilitates the writing of rules in subsequent modules.

The second module marks the final subjects of non finite verbs (and by extension, of adjectives also, whether used as predicative complements or noun modifiers). It uses the constraint propagation system of OGRE to handle embeddings involving cascades of predicates and/or coordination. For instance the rule for raising or subject control verbs introducing infinitives contains a share-constraint stating that their subject should also be the final subject of the infinitive. This constraint instantiates for two pairs of nodes in Figure 3 (the orange and purple constraint instantiations), which add Jean as final subject of vouloir and in turn as final subject of compter. We extracted control and raising verbs from the Dicova-lence lexicon (van den Eynde and Mertens [START_REF] Van Den Eynde | Le dictionnaire de valence DICOVALENCE : manuel d'utilisation[END_REF]), and subsequently extended the list during rule tuning on the DeepSequoia dev corpus.

VP coordination is handled through another constraint, stating that for two coordinated verbs, if a final subject exists for the first conjunct, then it must also be added as the final subject of the second conjunct (provided the latter does not initially have a final subject). This is displayed as the green constraint in Figure 3, which adds the final subject of respecté as soon as compter gets a final subject.

Syntactic alternations are mainly handled in the third module, which identifies canonical functions for arguments of verbs (whether these arguments were already in the surface tree, or added by the second module). For instance, a rule states that if a passive verb has a final subject, then that is its canonical object. Such a rule applies in sentence 1 to identify Jean as the canonical object of the passive verb respecté. This module interacts with some manual annotations performed at step (i) : while passive verbs are automatically identified, other highly ambiguous alternations are first manually identified at step (i) (causatives, impersonal, middle and neuter alternations), then the rules interpret the manual annotations to correctly derive the canonical functions of the arguments (including the cases of alternation interaction such as impersonal passives). A clear-cut separation between the module for final subjects and the module for syntactic alternations is not possible in particular because of control verbs specificities. The syntactic generalization applicable to control verbs mixes canonical and final functions. Indeed, a given control verb imposes which of its canonical argument is the final subject of the infinitive. For instance, the verb condamner (to condemn) is an object-control verb, meaning that its canonical object is the final subject of the infinitival clause it introduces. So, in La cour a condamné Jean à être incarcéré (The court condemned Jean to be incarcerated), the object Jean is the final subject of the passive verb incarcéré. When condamner is passive, then the controler of the infinitive, it still holds that its canonical object (but final subject) is the final subject of the infinitive, as in Jean a été condamné à être incarcéré (Jean was condemned to be incarcerated). We resolved this interaction by explicitely distinguishing rules for active and passive control verbs in the final-subject module.

A fourth module handles comparative and superlative constructions, mostly. It also adds morphological features such as definiteness in case of determiners, and identifies the clause types (interrogative, imperative...). Finally the last module exclusively deals with the removal and bypassing of semantically empty words. Incoming and outgoing edges of these words are attached to semantically full words.

To give an idea of the degree of complexity of the system, the five modules contain 19, 40, 21, 39 and 36 rules respectively, for a total of 155 rules. While being a reasonable figure, we must admit that the understanding and maintenance of this rule set requires training.

Evaluation

Quantative analysis

We now turn to the evaluation of the Surface-to-deep conversion module and of the quality of the DSRs we obtain for the FTB. The current version of the surface-todeep conversion module was designed in two stages. As mentioned in section 2, the conversion module was first designed to pre-annotate the DeepSequoia, which has been subsequently manually corrected. More precisely, in order to build the DeepSequoia, the two research teams who produced the DeepSequoia (namely ours and the Sémagramme team) first manually annotated a subset of 247 sentences (called the MINIREF), and then both tuned a conversion tool on this subset. Then two pre-annotated versions of the treebank could be produced, manually corrected by each team, and finally conflicts were manually adjudicated. For the current work, we subsequently improved our conversion rules using another subset of the DeepSequoia. More precisely, we split the DeepSequoia into four parts, as shown in Table 1. We used the DEV2 set to improve the rules' coverage, while setting aside a training set for future experiments, and a test set for final evaluation. Further, in order to evaluate the quality of the DSRs obtained for the FTB, we manually annotated the first 200 sentences from the FTB development set. The top part of Table 2 concerns the evaluation of the conversion rules on the DeepSequoia test set. We report labeled and unlabeled precision, recall and F-measure when considering either the set of deep edges (first row of Table 2) , or the set of deep-only edges (second row). Performance on this test set is rather high, although a little lower on the deep-only edges.

The bottom part of the table concerns the evaluation on the 200 sentences from the FTB. We proceeded as follows: we applied the surface-to-deep rules on the reference surface trees, augmented with the deep manual annotations (cf. step (i) mentioned in section 2), and obtained predicted DSRs (hereafter Predicted Deep 1). We manually corrected these predicted DSRs, and also manually corrected some errors in the reference surface trees. We thus obtained corrected deep representations and corrected surface trees. The line "REFERENCE vs CORRECTED SURFACE" in Table 2 shows the evaluation of the reference surface trees against the corrected surface trees. The next line provides an evaluation of the Predicted Deep 1 representations against the corrected deep ones. It shows that the overall quality of the resulting deep syntactic corpus is rather good. It can be anticipated that the DSRs obtained for the FTB will have sufficient quality to serve as training data.

Yet, while the evaluation of Predicted Deep 1 (penultimate row) provides an es-timation of the quality of the full set of predicted DSRs for the whole FTB, it mixes errors due to the rules, and errors in the reference surface trees. In order to evaluate the former more precisely, we applied the conversion rules on the corrected surface trees, and obtained a second version of predicted DSRs (hereafter Predicted Deep 2). The results are shown in the last row of Table 2. We obtain no drop in performance with respect to the evaluation of the DeepSequoia, which indicates that our rule set has a good coverage, and generalizes well to other corpora.

Qualitative analysis

We checked the errors on the 200 sentences from the FTB. A qualitative evaluation reveals that some phenomena are not (properly) handled by the rules, because of their complexity and ambiguity. For example, nominal predicative complements in sentences such as C'est une femme Capitaine. (It's a female captain.), where an arg edge should be added between femme and Capitaine, are not automatically annotated. Elliptic coordination is another unhandled phenomena, in particular head gapping and argument clusters. Finally, automatic annotation of infinitive subjects leads to the highest rate of errors. We can distinguish two types: (i) Control or raising verbs not present in our lexical resources: annoncer (to announce) or continuer de (to continue to) are two examples (continuer was present, but with preposition à). The same goes, for "control nouns". For instance, the noun idée (idea) was missing in the rules, which thus fail to assign the possessive as subject of the infinitive verb in D'où son idée de calmer le jeu. (Hence his idea to calm things down). (ii) Arbitrary control, for certain modifying prepositions introducing infinitive clauses, the rules arbitrarily choose the subject of the main verb as subject of the infinitive, though it is clear that such a simple and systematic rule will fail in some cases. For instance, in Ils ont re cu les élèves pour visiter le fournil (they received the pupils to visit the bakery), the subject of visiter (to visit) is not properly found.

Conclusion

In this paper, we described the methodology we used to add a deep syntax annotation layer to the French Treebank. Based on the work carried out by Candito et al. [START_REF] Candito | Deep Syntax Annotation of the Sequoia French Treebank[END_REF] to develop and the DeepSequoia treebank, we enhanced the conversion process from surface trees to obtain state-of-the-art results in term of expected quality as shown by our evaluation on a small gold standard we built from the FTB. Furthermore, we manually corrected a reduced set of difficult constructions. This evaluation suggests that the resulting new data set, a deep syntax version of the FTB, can be used as pseudo-gold data to train deep syntactic parsers, or to extract syntactic lexicons augmented with quantitative information. The Deep French Treebank will be released with the paper (following the original license).

Figure 1 :

 1 Figure 1: A dependency graph containing both the surface syntactic tree and the deep syntactic representation. The black arcs belong to both representations. The red arc belongs to the surface representation only. The blue arcs below the sentence are "deep-only": they belong to the DSR only.

  rule add_suj_edge{ match{ x <-[label:"suj"]-[] -[label:"obj"]-> y[cat:"VINF"] } negative{ y -[]-> x } commands{ add_edge(y, x, [label:"suj"] } } (a) Textual form of the rule. Subgraph pattern (match).

Figure 2 :

 2 Figure 2: Example of rule which adds a suj edge, in text format (a), with graphical format for the match pattern (b) and for the resulting graph after application (c).

Figure 3 :

 3 Figure 3: Share-triggers (dotted arcs) for a cascade of raising/control verbs and coordinated verbs. Each share-trigger instance adds the deep-only arc of the same color.

Table 1 :

 1 Experimental Split

	Sets	#Sent. #Tokens #Deep
				Tokens
	TRAIN	2,202	47,415	40,792
	DEV-1 (Miniref)	247	5,852	5,038
	DEV-2	250	5,360	4,606
	TEST	400	8,411	7,264

Table 2 :

 2 Rules' evaluation (Labelled/Unlabelled recall, precision, F-measure).

	DeepSequoia (test set)	# gold edges	LP	LR	LF	UP	UR	UF
	DEEP EDGES	8259	99.5 99.2 99.4	99.5	99.3 99.4
	DEEP ONLY EDGES	1806	98.1 97.3 97.7	98.3	97.5 97.9
	FTB (200 sent. dev.)	# gold edges	LP	LR	LF	UP	UR	UF
	REFERENCE vs CORRECTED SURFACE	6170	98.7 98.0 98.4	100.0 99.4 99.7
	PREDICTED DEEP 1	6012	97.5 97.1 97.3	98.9	98.4 98.7
	PREDICTED DEEP 2	6012	99.5 99.3 99.4	99.6	99.4 99.5

As shown by the mixed level of performance obtained by Ballesteros et al.[START_REF] Ballesteros | Deepsyntactic parsing[END_REF] on a comparable parsing task for Spanish.

We use the term canonical instead of the Relational Grammar term initial.

The manual annotations were mainly performed by the second author of this paper, and other colleagues. We hope to be able to describe the manual annotations in another publication.

In (Ribeyre et al.[START_REF] Ribeyre | A Linguistically-motivated 2-stage Tree to Graph Transformation[END_REF]), the term "constraints" was used instead of "triggers".

Formerly defined as share constraints.


Leo Kaas
email: leo.kaas@uni-konstanz.de

Patrick A Pintus
email: patrick.pintus@univ-amu.fr

Simon Ray
email: simon.ray@banque-france.fr

Pierre Cahuc

Edouard Challe

Etienne Lehmann

Matthias Hertweck

Olivier Loisel

Land Collateral and Labor Market Dynamics in France *

Keywords: JEL classification: E24, E32, E44 Financial shocks; Labor market frictions

The value of land in the balance sheet of French firms correlates positively with their hiring and investment flows. To explore the relationship between these variables, we develop a macroeconomic model with firms that are subject to both credit and labor market frictions. The value of collateral is driven by the forward-looking dynamics of the land price, which reacts endogenously to fundamental and non-fundamental (sunspot) shocks. We calibrate the model to French data and find that land price shocks give rise to significant amplification and hump-shaped responses of investment, vacancies and unemployment that are in line with the data. We show that both the endogenous movements in the firms' discount factor and the sluggish response of the land price are key elements that drive the results.

Introduction

Recent evidence suggests that real estate collateral is an important determinant of firms' employment decisions. Using administrative data for French firms, [START_REF] Chaney | Real estate collateral and labor demand[END_REF] show that shocks to the value of real estate have sizeable effects on employment and investment. The impact of collateral on investment appears to be larger for French firms than for the US firms covered by the Compustat database [START_REF] Chaney | The collateral channel: How real estate shocks affect corporate investment[END_REF]). 1 Moreover, investment appears to be considerably more sensitive to cash-flow variations in France, which suggests that French firms may be more financially constrained. To better understand how the labor market responds to shocks to the firms' collateral value, this paper builds a canonical macroeconomic model with financially constrained firms and search frictions in the labor market. We calibrate the model to France and relate its business-cycle features to those in the data. In particular, we find that the model response to a shock to collateral value features amplification and propagation that are in line with those obtained from a VAR analysis. Our business-cycle model features collateral constraints and forward-looking land prices as in [START_REF] Kiyotaki | Credit cycles[END_REF] and [START_REF] Kocherlakota | Creating business cycles through credit constraints[END_REF], on the one hand, and labor market frictions similar to [START_REF] Mortensen | Job reallocation, employment fluctuations and unemployment[END_REF], on the other hand. Firms hold productive land and capital, and they borrow in an international credit market, using land and (a fraction of) capital as collateral. 2 In the labor market, firms post vacancies and they are matched with unemployed workers who are hired for production in the subsequent period. To uncover the business-cycle relationships between collateral prices, investment and labor market dynamics, we introduce three types of shocks into our model: (i) TFP shocks; (ii) shocks to the collateral constraint ("financial shocks"); and (iii) shocks to the transaction price of land ("land price shocks"). We think of (ii) as any disruptions in financial markets that change the amount of funds channeled from lenders to borrowers. Although shocks (i) (as in, e.g., [START_REF] Kocherlakota | Creating business cycles through credit constraints[END_REF]) and (ii) alone induce endogenous movements in land prices, this amplification is quantitatively too small to account for the observed volatility in the data. We think of shock (iii) as a short-cut to describe movements in the demand for land, without making the underlying mechanism explicit. This allows us to distinguish explicitly between those events that change the value of land collateral and other types of financial shocks that 1 Specifically, a 1$ increase in collateral value raises investment for US firms by 0.06$, whereas the effect for French firms is about three times as large. Of course, the two firm samples are not directly comparable, since Compustat does not cover private (and smaller) firms which are more bank-dependent. Further studies on the role of financial constraints on employment based on US firm-level data are [START_REF] Benmelech | Financing labor[END_REF] and [START_REF] Chodorow-Reich | The employment effects of credit market disruptions: Firm-level evidence from the 2008-9 financial crisis[END_REF].

2 Our focus on land (as opposed to real estate) builds on [START_REF] Davis | The price and quantity of residential land in the United States[END_REF], who show that fluctuations in US firms' real estate are largely driven by those in the land value.

directly affect the intermediation between borrowers and lenders, as in the recent literature (e.g., [START_REF] Kiyotaki | Winners and losers in housing markets[END_REF], Liu et al. (2013b), [START_REF] Justiniano | Household leveraging and deleveraging[END_REF]).

When we calibrate our model to the French economy, we find that the dynamic response to shocks of type (iii) is especially relevant and exhibits large amplification and hump-shaped responses of vacancies, unemployment and investment. The responses differ notably because the land price shock significantly relaxes the credit constraint and thus takes an impact on the firms' discount factor. We show that the dynamics of the firms' discount factor plays a key role in the response of hiring which is a new amplification channel relative to the existing macro-labor literature. The explanation is that constrained firms react to a relaxation of credit constraints by augmenting current profit income at the expense of future income, which raises the discount factor and induces firms to expand investment and hiring faster. Besides this channel, the key factor behind the propagation result is that the unique steadystate equilibrium is locally indeterminate, which means that the land price not only depends on fundamentals but also reacts to self-fulfilling beliefs (sunspots). The intuition for this result is a "pecuniary externality" in the collateral constraint: if firms expect the land price to go up in the future, they expect relaxed credit constraints which induces them to borrow and invest more, and hence to demand more inputs which in turn bids up the land price. As a consequence, the initial expectation is fulfilled. In the absence of sunspot shocks, on the other hand, the land price reacts to fundamental shocks in a sluggish fashion. This sluggish adjustment accounts for a slow build up of collateral capacity and hence to a hump-shaped response of investment and vacancies to a (fundamental) land price shock. In contrast, pure sunspot shocks or shocks to the collateral constraint generate dynamics that are at odds with the data. Our paper departs from the literature reviewed below by showing that the dynamics of investment and labor market variables under fundamental shocks to the land price are empirically relevant when the steady state is locally indeterminate.

Our work relates to a recent literature that incorporates financial frictions into macroeconomic models with search frictions in the labor market. In an early contribution, [START_REF] Wasmer | The macroeconomics of labor and credit market imperfections[END_REF] introduce frictional search in the credit market and obtain a financial accelerator effect, while [START_REF] Dromel | Credit constraints and the persistence of unemployment[END_REF] show how unemployment becomes persistent under credit market frictions in a related model. Petrosky-Nadeau and Wasmer (2013) consider a stochastic version of a similar model, generating a dynamic financial multiplier which amplifies the impact of productivity shocks on labor market tightness significantly. [START_REF] Monacelli | Financial markets and unemployment[END_REF] introduce borrowing constraints into a macroeconomic model with labor search frictions; while they argue that these constraints do not directly prevent firms from hiring, they show that higher debt improves firms' bargaining position, which takes a negative (positive) effect on wages (job creation). [START_REF] Boeri | Labor and finance: Mortensen and Pissarides meet Holmstrom and Tirole[END_REF] let unconstrained firms choose an optimal level of leverage, together with job creation and job destruction; they show that credit market conditions have an effect on job creation and on job destruction. As in our paper, financial conditions take a direct impact on job creation of financially constrained firms in Petrosky-Nadeau (2014) who focuses on the amplification and propagation of productivity shocks but does not consider the dynamics of collateral value. Closely related to our paper is Liu et al. (2013a) who consider a quantitative macroeconomic model with labor and credit market frictions in which shocks to land prices affect the firms' collateral capacity. Estimating their model to US data, they also obtain labor market amplification of land price shocks, although their model does not generate hump-shaped impulse responses for vacancies. Also, different from our model, they include consumption habits as well as shocks to wage setting and to the matching function.3 On the other hand, multiplicity and self-fulfilling beliefs play a key role in the paper of [START_REF] Miao | Stock market bubbles and unemployment[END_REF] who show how a collapsing stock market bubble tightens credit constraints which induces firms to cut hiring. The rest of the paper is organized as follows. The next section reviews the role of firms' land collateral for the business cycle in France. Section 3 introduces the model and Section 4 characterizes the dynamic equilibrium. In Section 5, we calibrate the model to France to analyze its business cycle features.

Land Value and Labor Market Dynamics in France

In this section we discuss the role of firms' land collateral for business cycle dynamics in the French economy. We begin with a descriptive analysis by showing some key correlation and volatility patterns. We then conduct a VAR analysis to understand how shocks to the market value of land held by the firms affect the labor market.

Descriptive Statistics

We gather data from national accounts (output, investment, credit market liabilities, and land at market value for the non-financial business sector) as well as from labor market statistics (employment, unemployment and vacancies). All time series are available at quarterly frequencies, except data on firm's balance sheet which are available at annual frequency. Due to data availability, we restrict the analysis to the period 1978q1-2011q4. We are mainly interested in the role of the market value of land held by non-financial firms; the market value matters in our model as it reflects the liquidation value taken into account by creditors. The French national statistics institution (INSEE) provides annual data since 1978 on the market value of non-financial firms' assets and liabilities and notably land. We derive land at market value at the end of each quarter for French firms by linearly interpolating end of the year values. This is arguably a good approximation as we find, using a quarterly available residential real estate price index on a sub-period (1996q1-2011q4), that the evolution of French real estate is close to linear within a year and shows only few intra-year trend reversals. Furthermore, the volume of land owned by firms at the aggregate level does not change significantly during a year, so that such volume effects are unlikely to drive the series of land at market value. Our key labor market indicator is labor market tightness, defined as the ratio between vacancies and unemployment, which is available for France from 1989 onwards. The aggregate real wage is calculated as the product between labor productivity and the labor income share. All time series are seasonally adjusted and expressed in logs, and all nominal variables are deflated with the GDP deflator. As we cannot reject the hypothesis that our main series have unit roots, we detrend all series using the band pass filter of [START_REF] Fitzgerald | The band pass filter[END_REF] implemented in Stata. 4 More details on the data sources and further descriptive statistics are reported in Appendix A.

Figure 1 shows time series for the firms' land value, investment and labor market tightness. Notably all three series are positively correlated. While investment and land value are about equally volatile, labor market tightness shows considerably higher volatility. These findings are also confirmed in Table 1 which reports correlation coefficients and standard deviations relative to output for several variables. It is notable that the land value as well as unemployment and vacancies are much more volatile than output. Particularly, labor market tightness is almost 9 times as volatile as output, which is comparable to the figure for the US economy. Unsurprisingly, the land value, investment and labor market tightness are highly procyclical, and the land value is strongly positively correlated with vacancies and tightness. The real wage is much more stable than output and acyclical.

It is also interesting to point out some differences with the US economy for which we report comparable statistics in Appendix A. Over the same time period, the business cycle is considerably more stable in France. Relative to output, however, the French economy has generated almost the same labor market volatility as the US economy. While the land value is more volatile in the US, it correlates less strongly with labor market tightness than it does in France. Another noteworthy difference is that the number of vacancies is highly negatively correlated to unemployment in the US, whereas the negative relation is weaker in France (see also [START_REF] Justiniano | The cyclical behavior of equilibrium unemployment and vacancies in the US and Europe[END_REF]). We further observe that credit market liabilities are more correlated to labor market variables in France than in the US. One possible explanation could be the well documented fact that firms in France (as in other European countries) rely more on credit financing than in the US.

VAR Analysis

In order to study the dynamic relationships between firms' land value, vacancies, unemployment and investment, we estimate a vector autoregression (VAR) model with these four variables. Our estimations are performed with the above-described quarterly French data for the period 1989q1-2011q4. This study aims at assessing the impact of a structural shock to the land value on labor market variables and investment. We estimate the reduced-form VAR A(L)x t = ε t , where the 4 × 1 vector x t includes the detrended log values of land, vacancies, unemployment and investment, ε t is a vector of shocks and A(L) is a lag polynomial matrix with A(0) being the identity matrix. We are interested in identifying the structural shock on firms' land value, that is to say isolating the land shock component uncorrelated with the contemporaneous shocks affecting the labor market or investment, and tracing its effect on the three other variables. We argue that this structural shock on firms' land holdings value appropriately reflects the effect of the land price shock introduced in our theoretical framework. We recover the structural shocks through the Cholesky decomposition of the estimates of the variance-covariance matrix. We order the shock to firms' land value first in this decomposition, thereby allowing labor market variables and firms' investment to have feedback effects on the land value through their lagged values but not through their contemporaneous values.5 Impulse response functions to a one-standard-deviation land value shock are presented in Figure 2. Vacancies and investment react positively to a positive land value shock, whereas unemployment reacts negatively. The land value adjustments following the initial shock (leading to a 1% increase in land value) exhibit a hump-shaped pattern, peaking at roughly 9% above its steady-state value four quarters after the shock. Vacancies almost do not react on impact, but in subsequent quarters the variable follows a similar pattern as the land value, peaking at roughly 11% above its steady-state value five quarters after the shock. Unemployment almost does not react on impact either, and the adjustment is also hump-shaped and a bit more sluggish with a trough at roughly -10% about seven quarters after the shock. We observe a slight positive response of investment in the period of the shock and also a more moderate hump-shaped adjustment than for the three other variables. 6 Overall, the results show that structural shocks to firms' land value drive labor market variables and investment and that the shock has significant long-lasting and hump-shaped effects on land value, labor market variables and investment.

The Model

We consider a small open economy comprising a unit mass of firms and a unit mass of workers. Firm owners hold and accumulate capital, buy land, hire workers and borrow in an international credit market at fixed interest rate R, where borrowing is subject to collateral constraints. 7

6 For comparison, we also consider a similar VAR model where labor market tightness replaces vacancies.

The results turn out to be very similar: labor market tightness peaks about 6 quarters after the shock with somewhat higher amplification than vacancies, whereas the responses of the other variables remain unchanged. 7 The assumption of a small open economy is a reasonable abstraction for a country like France. An alternative model framework would be a closed economy in which worker households are more patient than

Firms produce a common consumption/investment goods from the inputs capital, labor and land. While capital can be accumulated, the aggregate supply of land is fixed and normalized to one. Land is traded at price Q t in period t, and the price of capital and consumption goods is normalized to unity. Workers consume their labor income; neither do they hold land, nor are they active in the credit market. While the markets for capital and land are frictionless, the market for labor is subject to search frictions. Finally, there is a government redistributing income from employed to unemployed workers.

Firms

Firms use capital K t , labor N t and land L t to produce final output with concave and constantreturns technology Y t = A t F (L t , K t , N t ) where A t is aggregate TFP. All three inputs are installed in the period prior to production. Firms also enter the period with credit market liabilities B t . Since there are only aggregate shocks, all firms are identical, so we treat the firm as a representative firm and we do not distinguish in our notation between aggregate and firm-level variables.

The owner of the representative firm consumes C t in period t and has preferences

∞ t=0 β t u(C t ) , (1)
where u is a strictly increasing and concave function. In period t the firm faces the budget constraint:

B t+1 + Y t = C t + T t Q t (L t+1 -L t ) + (1 + R)B t + W t N t + I t + φ K I 1+η K t I -η K t-1 + φ N H η N t N 1-η N t , (2)
where I t is the investment flow, H t is the hiring flow, φ K ≥ 0 and φ N ≥ 0 control for investment and employment adjustment costs, and B t is credit redeemed in period t. Adjustment costs are convex in investment growth I t /I t-1 and in the hiring rate H t /N t ; that is, we require η K ≥ 1 and η N ≥ 1. W t is the real wage which is bargained bilaterally between the firm and each individual worker. As we see below, the outcome of this bargain depends only on aggregate quantities but is independent of the firm's individual factor choices (L t , K t , N t ). Hence the firm takes W t as given when it decides about factor inputs. T t is an exogenous "land wedge" variable which affects the transaction costs of land. It plays the same role as a tax on land sales (which is not paid in equilibrium). We later introduce shocks to this variable to capture any events that take an impact on land demand and hence on the land price. In particular, firm owners and provide credit to borrowing-constrained firms. We conjecture that the results would be rather similar.

we investigate the effects of a decline in the land wedge T t , which triggers an increase in land price Q t , on investment and labor market variables.

Borrowing is subject to a collateral constraint of the form

Ψ t Q t+1 L t+1 + ω(1 -δ)K t+1 ≥ (1 + R)B t+1 , (3)
where Ψ t is a collateral constraint parameter which we interpret as financial market conditions and which may be subject to financial shocks. The term in squared brackets represents collateral of the firm which comprises all land holdings and also a fraction ω ∈ [0, 1] of the capital stock that can be pledged to the firm's creditors. Capital and employment respectively adjust according to:

K t+1 = (1 -δ)K t + I t , (4)
N t+1 = (1 -s)N t + H t , (5)
where s is an exogenous separation rate.8 The firm's problem is to maximize the owner's utility (1) subject to (2), ( 3), ( 4) and ( 5). Write λ t , ι t , µ t and ν t for the Lagrange multipliers on these four constraints. The first-order conditions with respect to C t , I t , K t+1 , L t+1 , H t , B t+1 and N t+1 are:

λ t = u ′ (C t ) , (6)
µ t = λ t 1 + φ K (1 + η K ) I t I t-1 η K -βλ t+1 φ K η K I t+1 I t 1+η K , ( 7
)
µ t = β(1 -δ)µ t+1 + βλ t+1 ∂Y t+1 ∂K t+1 + ω(1 -δ)ι t Ψ t , (8)
λ t Q t T t = βλ t+1 ∂Y t+1 ∂L t+1 + Q t+1 T t+1 + ι t Ψ t Q t+1 , (9)
ν t = λ t φ N η N h η N -1 t , (10)
λ t = β(1 + R)λ t+1 + ι t (1 + R) , (11)
ν t = β(1 -s)ν t+1 + βλ t+1 ∂Y t+1 ∂N t+1 -W t+1 + φ N (η N -1)h η N t+1 . ( 12
)
Define M f t+1 ≡ βλ t+1 /λ t as the firm's discount factor and write J t ≡ ν t-1 /(M f t λ t-1 ) for the firm's value of an additional worker in period t in units of the consumption good. With h t = H t /N t denoting the hiring rate, (10) and ( 12) can be rewritten as:

φ N η N h η N -1 t = M f t+1 J t+1 , (13)
J t = ∂Y t ∂N t -W t + φ N (η N -1)h η N t + (1 -s)M f t+1 J t+1 . (14)
Equation ( 13) is a standard job-creation condition which links firms' hiring behavior to the discounted job value. Equation ( 14) shows that the job value is a forward-looking variable which depends on expectations about productivity and wages. The third term on the righthand side reflects the reduction in future hiring costs from an additional employee.

Workers

Workers are able to pool idiosyncratic unemployment risk within a large family. The representative worker family consumes income

C w t = W t N t + τ W t (1 -N t ) -T w t ,
where τ is the wage replacement ratio and T w t are lump-sum taxes that are set by the government to finance these transfers. The worker household enjoys utility

∞ t=0 β t u w (C w t ) , (15)
with strictly increasing and concave u w , possibly different from the utility function of firm owners. Employment adjusts according to

N t+1 = (1 -s)N t + f t (1 -N t (1 -s))
where f t is the job-finding rate for unemployed workers.9 If χ t is the marginal utility value of an additional worker in period t, we have

χ t = λ w t (1 -τ )W t + β(1 -s)(1 -f t )χ t+1 ,
where λ w t = u ′ (C w t ). Defining Ω t ≡ χ t /λ w t for the value of a worker in consumption units and M w t+1 ≡ βλ w t+1 /λ w t for the worker's discount factor, we obtain

Ω t = (1 -τ )W t + (1 -s)(1 -f t )M w t+1 Ω t+1 . ( 16
)

The Labor Market

The labor market is subject to search frictions. All separations (hires) go to (come from) the unemployment pool, so there are neither job-to-job transitions nor transitions into and out of the labor force. Given that the flow of newly employed workers equals the flow of hires in period t, the rate at which any of the 1 -N t (1 -s) unemployed workers finds a job is linked to aggregate hires according to

f t = H t /[1 -N t (1 -s)].
While firms in this model decide the number of hires, we can also think of them as posting vacancies and anticipating how many of their vacancies are filled according to a standard matching process. 10 That is, the aggregate vacancies V t posted by firms are matched with 1 -N t (1 -s) job seekers through a matching technology, such that the job-finding rate is an increasing and concave function of the vacancyunemployment ratio (market tightness)

θ t = V t /[1 -N t (1 -s)], i.e. f t = f (θ t ) with f ′ > 0 and
f ′′ < 0. This implies that market tightness (and therefore vacancies) are determined from

f (θ t )[1 -N t (1 -s)] = H t . (17)
Any worker matched to a firm bargains with the firm about the real wage W t in any period t. To render the bargaining outcome independent of the individual firm's factor choices (K t , N t , L t ), we assume that there is a secondary market for capital and land at which firms can buy and sell these factors of production after wage bargaining but before the production stage. 11 This assumption implies that the marginal product of labor is only determined by aggregate market conditions so that the firm essentially operates under constant returns in the given period. Hence, although our model features multi-worker firms whose production technology has decreasing returns to labor, the usual difficulties with intrafirm bargaining in those situations (e.g. [START_REF] Stole | Intrafirm bargaining under non-binding contracts[END_REF] and [START_REF] Smith | Search, concave production and optimal firm size[END_REF]) disappear. 12 The Nash bargaining problem between the firm and each of its workers is then to maximize Ω t γ J t 1-γ , where γ is the worker's bargaining power. Equations ( 14) and ( 16) imply that

γJ t = (1 -γ)Ω t . ( 18
)
10 A difference between our model (with hiring costs) and standard search and matching models (with vacancy costs) is that only in the latter case employment adjustment costs are linked to aggregate labor market conditions via the aggregate matching function. Such a link is absent in our model.

11 Specifically, the timing within each period t is as follows.

(1) Firms and workers bargain over the wage; (2)

Firms trade capital and land in the secondary market;

(3) firms produce, hire, invest, buy/sell land at price Q t , then repay the outstanding debt and borrow.

12 Formally, because of constant returns,

Y t = N t G( Lt Nt , Kt Nt )
where G is an increasing and concave production function in intensive form. Prices for land and for capital in the secondary market are

P L t = G ′ 1 (l * t , k * t ) + Q t and P K t = G ′ 2 (l * t , k * t ) + 1 -δ with aggregate l * t = Lt Nt , k * t = Kt
Nt taken as given by the firm. It is then optimal for the firm to hold ℓ * t units of land and k * t units of capital per employed worker. Therefore, the marginal profit per worker is

G(l * t , k * t ) + l * t (Q t -P L t ) + k * t (1 -δ -P K t )
which is identical to the marginal product for labor and independent of individual factor choices. In equilibrium, no trade in the secondary market is required since no wage negotiations fail. See Chapter 1.6 in [START_REF] Pissarides | Equilibrium Unemployment Theory[END_REF] and [START_REF] Cahuc | Does intrafirm bargaining matter in the large firm's matching model?[END_REF] for similar arguments.

Equilibrium

Equilibrium in the land market is simply L t = 1. Goods market equilibrium follows from workers' and firms' budget constraints as

Y t = C t + C w t + (1 + R)B t -B t+1 + I t + φ K I 1+η K t I -η K t-1 + φ N H η N t N 1-η N t . (19)
The right-hand side includes consumption expenditures, net exports (that is, net capital exports) and investment on capital goods and for recruitment expenditures. The left-hand side is final output from market production. Finally, the government's budget is balanced, T w t = τ W t (1 -N t ) so that consumption of the worker household equals aggregate gross labor income C w t = W t N t . Assume that β ≡ 1/(1 + R) > β which implies that the borrowing constraint binds in and around the steady state:

B t+1 = βΨ t Q t+1 + ω(1 -δ)K t+1 . ( 20
)
Definition: Given a shock process for (A t , Ψ t , T t ), a competitive equilibrium with binding borrowing constraints are lists of firm plans, (C t , K t , N t , L t , H t , I t ), wages and land prices (W t , Q t ), vacancy-unemployment ratios (θ t ), and job values for firms and workers (J t , Ω t ), such that for all t ≥ 0:

(i) Firm owners maximize their utility (1) subject to budget constraints (2), borrowing constraints (3), and factor adjustment (4) and ( 5).

(ii) The land and the goods market are in equilibrium: L t = 1 and equation ( 19).

(iii) Labor market flows are consistent: equation ( 17).

(iv) Wages satisfy the Nash bargaining solution (18) where J t and Ω t satisfy ( 14) and ( 16).

In Appendix B we characterize equilibrium by a set of 16 equations in the 16 variables

C t , C w t , λ t , λ w t , µ t , Y t , K t , N t , θ t , J t , Ω t , W t , B t , Q t , h t = H t /N t , I t ,
given the exogenous processes for A t , Ψ t and T t . The predetermined state variables are (I t-1 , K t , B t , N t ). In Appendix B we prove that the model has a unique steady state equilibrium and we show the following:

Proposition 1: A steady state equilibrium is unique. If ω > 0, an increase of the collateral constraint parameter Ψ (or a decrease of the interest rate R = 1/ β -1) has a positive effect on the steady-state values of output Y , employment N, capital K, market tightness θ, the real wage W and the land price Q. Changes of the land wedge T affect the land price, but have no effect on output and on the labor market.

An implication of this proposition is that an improvement of financial condition, expressed by a relaxation of borrowing constraints, leads to a boom in the land price as well as to a tighter labor market and lower unemployment. Put differently, a permanent financial shock triggers a co-movement of the firms' land value and of the labor market. Land price shocks, on the other hand, cannot have permanent real effects. In the next section we explore the business-cycle implications of these different shocks to the firms' collateral constraint. Before we do that, it is instructive to take a closer look at the first-order condition for hiring, (13), rewritten as

φ N η N (h t ) η N -1 = βu ′ (C t+1 ) u ′ (C t ) =M f t+1 J t+1 . (21)
As is well known from the literature on labor market volatility originating from [START_REF] Shimer | The cyclical behavior of equilibrium unemployment and vacancies[END_REF], volatility and propagation in hiring must largely be explained by the dynamics of the expected job value J t+1 , when the discount factor M t+1 is either constant due to a linear utility framework or moves little due to a representative household assumption (as in [START_REF] Andolfatto | Business cycles and labor-market search[END_REF]). Amplification of the job value is difficult to generate, unless wages are rigid and the flow job surplus is small relative to labor productivity. Other authors argue that variations in job creation costs on the left-hand side also account for labor market volatility and that financial frictions contribute to this mechanism (e.g. Petrosky-Nadeau ( 2014)). In our model, financial frictions in combination with sluggish adjustment of land collateral give rise to another channel contributing to labor market volatility and propagation, with much of the action going through movements in the discount factor. When firms have an interest to smoothen profit income (dividend payments) over time, constrained firms react to a relaxation of collateral constraints in period t by an increase in current profit income C t relative to future income C t+1 which raises the discount factor on the righthand side of ( 21) and thus triggers a rise in hiring. This mechanism depends on our modeling assumption that firms are operated by independent owners who desire a smooth dividend stream, which is in line with the finance literature (cf. [START_REF] Lintner | Distribution of incomes of corporations among dividends, retained earnings, and taxes[END_REF], [START_REF] Guttman | Dividend stickiness and strategic pooling[END_REF], [START_REF] Leary | Determinants of dividend smoothing: Empirical evidence[END_REF]) but absent in standard macroeconomic models with a representative household. A similar mechanism would be at work if there is a representative capitalist family, earning aggregate profit income which is more volatile than aggregate consumption.13

To see the discount factor channel in our model more clearly, we can use the Euler equation ( 11) to write the discount factor as follows:

M f t+1 = βλ t+1 λ t = β -ι t λ t , (22)
where λ t = u ′ (C t ) is marginal utility, ι t is the multiplier on the collateral constraint (3) and β = (1 + R) -1 . Equations ( 21) and ( 22) show that, other things equal, when the credit constraint relaxes, that is, when ι t goes down, hirings h t go up. So essentially, one expects that shocks relaxing the credit constraint by a lot should make hirings very responsive. Consistent with the literature (e.g. [START_REF] Kocherlakota | Creating business cycles through credit constraints[END_REF]), the response of the land price to TFP shocks is not quantitatively significant in our model. As a consequence, a shock to TFP has little direct impact on the credit constraint and hence cannot be amplified much through this channel. In contrast, a land wedge shock that has a bigger impact on the land price, has much stronger labor market responses. To the extent that such shocks propagate through several periods, they also raise the job value (which depends positively on future discount factors via ( 14)) which additionally stimulates hiring incentives. This is exactly what happens in our model, as further illustrated with the simulations of the calibrated model that we present in the next section.

Calibration and Quantitative Results

In this section we calibrate the model to the French economy. We explore the reaction of the model economy to different shocks and show that especially shocks to the land price contribute to amplification and propagation of investment and labor market variables that are in line with the data.

Calibration

We choose Cobb-Douglas functional forms

Y t = A t F (L t , K t , N t ) = A t L ξ t K ζ t N 1-ξ-ζ t
for the production function and f (θ t ) = mθ 1-α t for the matching function. Utility functions for workers and firm owners are identical, u(C) = u w (C) = C 1-σ /(1 -σ) with coefficient of relative risk aversion equal to σ = 2. We calibrate the model at quarterly frequency. All real and financial variables are matched to targets obtained from the INSEE data for the non-financial business sector. 14 For more details about the calibration, see Appendix C. All parameter choices are summarized in Table 2.

macro-finance literature (cf. [START_REF] Danthine | Labour relations and asset returns[END_REF], [START_REF] Guvenen | A parsimonious macroeconomic model for asset pricing[END_REF]).

14 In particular, output Y is measured as gross value added. Land QL = Q is land at market value, and

Technology and financial parameters. The share of capital that serves as collateral is set at ω = 0.3 (see further discussion below). The depreciation rate is δ = 2.7% which gives rise to 10.5% annual depreciation. Given the steady-state ratios that are fixed at their actual average values K/Y = 7.31, B/Y = 5.56, Q/Y = 2.35 (see Table 4), we calibrate β = 0.99 (which corresponds to a 4% annual interest rate) and β = 0.98 to match the capital and land shares in output ζ = 0.37 and ξ = 0.02. It follows that Ψ = 1.25. Regarding the investment installation costs, we set the level parameter φ K = 0.1 to target an investment share in output of 22% and we set the convexity parameter η K = 15 to replicate investment volatility relative to that of the land value under land price shocks. It follows that Tobin's q equals 1.13.

Labor market parameters. The matching function elasticity is set to α = 0.5 which is in the range of plausible values for this parameter [START_REF] Petrongolo | Looking into the black box: A survey of the matching function[END_REF]). As [START_REF] Shimer | The cyclical behavior of equilibrium unemployment and vacancies[END_REF], we arbitrarily normalize tightness 15 in steady state to θ = 1 and set the matching efficiency parameter m = 0.214 to target a mean quarterly job-finding rate of f (θ) = 0.21. The quarterly separation rate is set to s = 0.021. Both quarterly worker flow rates correspond to the monthly estimates in Table 2 of [START_REF] Elsby | Unemployment dynamics in the OECD[END_REF], giving rise to a steady-state unemployment rate of 1-N = s(1-f ) s+f (1-s) = 7.2%. 16 We set unemployment income τ at 58.2% of wage income (see Table 1 in [START_REF] Justiniano | The cyclical behavior of equilibrium unemployment and vacancies in the US and Europe[END_REF]), and the wage-to-income ratio at W N/Y = 0.7, which is the labor income share in France (cf. Table 4). Regarding hiring costs, we deliberately choose η N = 1.1 and we calibrate φ N = 0.54 so that recruitment costs per hire in steady state are 14% of quarterly wages (cf. [START_REF] Hall | The limited influence of unemployment on the wage bargain[END_REF]). It follows from the other calibration targets that the worker bargaining power parameter is γ = 0.91 (see Appendix C). One interesting aspect of our calibrated model is that, all other parameter values fixed as in Table 2, the unique steady state is determinate when ω > 0.4 but becomes indeterminate when ω ≤ 0.4. The intuition for why indeterminacy arises is tightly linked to the fact that the collateral constraint depends on the market price of land which is a forward-looking variable that can possibly react to self-fulfilling beliefs. Intuitively, if all firms expect the land price to be larger in future periods, they expect abundant credit which enables them to invest more and hence to demand more inputs which in turn will bid up input prices. As a consequence, the land price goes up and the initial expectation is fulfilled. Of course, the force of this "pecuniary externality" depends on the level of the collateral constraint parameter Ψ. With our calibration credit B are all credit market liabilities. Capital K is structures plus equipment and software at current cost basis. The labor income share is calculated as compensation of employees divided by gross value added net of indirect taxes.

15 The steady-state value of θ plays no role for the log-linearized model dynamics. 16 Since separated workers can search for re-employment within the same quarter, the unemployment rate is slightly lower than its data average. strategy, and given the calibration targets for B/Y , K/Y and Q/Y , a lower share of capital used as collateral, ω, translates into a higher value for Ψ. Essentially, indeterminacy arises in our calibration when firms can borrow more than the market value of the land they possess, which is in line with evidence for the French (as well as for the US) non-financial business sector. There exists, to our knowledge, no source that documents the relative share of land and capital in collateralized lending to French firms, so it is a difficult task to calibrate parameter ω. Given that much of the firms' equipment capital is firm-or industry-specific with low resale value, however, we suspect that the capital share in collateral value is in reality rather low.

In that sense our calibrated value of capital's contribution to collateral of about 30% appears reasonable and conservative.

Although the steady-state is indeterminate when parameters are calibrated according to Table 2, it turns out that pure sunspot shocks to the land price generate responses of investment and labor market variables that are at odds with the data, as we illustrate in the next section. In addition, leverage shocks do not help account for the data either, which is in line with recent literature (for example, see [START_REF] Kiyotaki | Winners and losers in housing markets[END_REF]), Liu et al. (2013b), [START_REF] Justiniano | Household leveraging and deleveraging[END_REF]). This is why our main results, that are presented in the following section, rely on fundamental shocks to the land price, that is, shocks to the land wedge parameter T .

Main Results

We now show that the responses of the linearized model to land price shocks are empirically relevant when the steady state is indeterminate. We report the impulse response functions of the model economy under a negative shock to the land wedge T which follows an AR(1) process with autocorrelation coefficient equal to 0.98.

Figure 3: Impulse responses to a positive land price shock in the indeterminate model.

In Figure 3, investment, vacancies and tightness are procyclical while unemployment is countercyclical, which is in line with the data. In addition, investment and labor market variables respond in an hump-shaped manner, which is broadly consistent with the VAR evidence we present in Section 2.2. Investment and labor market variables reach their peaks around the 5th quarter, except for unemployment which peaks later, as does its estimated VAR response.

The reason why indeterminacy is key to obtain hump-shaped responses is as follows. When the steady state is indeterminate, the land price reacts to fundamental shocks in a sluggish way because those shocks do not affect the price immediately -only current sunspot shocks can affect the land price contemporaneously. If sunspot shocks are shut down, then the land price actually does not move at t = 1, that is, when the land wedge shock hits the economy, as shown in the left panel of Figure 3. At t = 1, however, investment, vacancies and tightness shoot up because next period's land price (that is, at t = 2) goes up and the credit constraint hence relaxes at impact. In subsequent periods, the land price gradually increases further in response to the persistent land price shock, which further relaxes the credit constraint and enhances the incentives both to invest in capital and to create more vacancies. In that sense, the gradual response of the land price to a land wedge shock under indeterminacy is key for our amplification and propagation results.

Complementing the impulse response analysis, we also find that the model with only land price shocks does a reasonably good job at replicating standard deviations and co-movements with output, see Table 3. Unsurprisingly, the pure land price shock generates too much volatility of the land value relative to output, and also investment turns out to be too volatile. We expect that TFP shocks on top of land price shocks can help to improve on these volatilities. As an additional exercise, we feed in a TFP shock simultaneously with a land price shock and find that the standard deviations of the model's moments listed in Table 3 become much closer to their data counterparts. It is also worth noting that our model can account for the wage rigidity that is observed in the data, although wages are strongly procyclical in the model while they are acyclical in French data. Essentially, wage rigidity arises in this model because the marginal productivity of labor turns out to be rather unresponsive to land wedge shocks. For comparison purposes, we report in Figure 4 the economy's responses to a TFP shock in the indeterminate model. 17 Similar to the effect of a land wedge shock, the land price also responds in a sluggish way to an increase in TFP when sunspot shocks are shut down. However, there are key differences between Figures 3 and4. The first difference is that under TFP shocks investment in capital and in hirings crowd out land investment, essentially because investing in capital is subject to highly convex adjustment costs while land investment is not. As a consequence, the land price falls below the steady state in the first periods after the TFP shock, before moving up so that the credit constraint slightly relaxes later on. In addition, under TFP shocks the land price is much less volatile than output. For example, the land price's standard deviation is about half of output's for our calibrated parameters. Hence TFP shocks alone cannot reproduce the observed volatility and co-movement of land prices. Therefore, although the mechanism that makes the land price response sluggish in the indeterminate model is also present under alternative sources of disturbances, we argue that movements in the land investment wedge are empirically more relevant to understand how land price movements affect investment and the labor market in France than TFP shocks.

Figure 4: Impulse responses to a positive TFP shock in the indeterminate model.

To illustrate the role of indeterminacy for our results, Figure 5 shows the responses of the economy when it is subjected to land price or TFP shocks and when calibrated as in Table 2 except that ω = 0.44, which in turn implies that the steady state is now determinate. It can be seen that the determinate model generates monotonic responses of labor market variables to both land price and TFP shocks, in contrast with the indeterminate model. This is because the response of the land price is monotonic from t = 2 onwards in both panels of Figure 5, which essentially implies that the credit constraint relaxes more at impact than in subsequent periods. In particular, the determinate model generates virtually no propagation for vacancies while the sunspot innovations have zero autocorrelation.

and unemployment. While vacancies and market tightness peak on impact, their response flattens out in the quarter after the shock, in spite of more persistent responses of output, debt, land price and investment. As a consequence, the responses of investment and the labor market are not hump-shaped, in sharp contrast with the responses in the indeterminate model shown in Figure 3 that are more in line with the VAR evidence. To further highlight the importance of land price shocks for plausible model dynamics, we now report the economy's responses to pure sunspot and leverage shocks in the indeterminate model. Absent land wedge shocks, the model lacks the amplification mechanism and generates essentially monotonic responses of the labor market variables. This can be seen in Figure 6 which graphs the dynamics triggered by a fundamental shock to the leverage parameter Ψ t (top) and by a sunspot shock (bottom) both hitting the economy at t = 1, under the calibration presented in Table 2. In addition, the corresponding (unreported) moments under leverage and sunspot shocks are at odds with French data. More precisely, the labor market variables are much too volatile compared to output and lack any propagation, while the land price is not volatile enough. In addition, investment and vacancies are negatively correlated with output while tightness is acyclical. In Figure 6, the undershooting of vacancies and tightness in response to leverage or sunspot shocks in the first period can be explained by the fact that firm owners desire faster consumption growth in that period which lowers the discount factor. To make sense of both panels in Figure 6, it is important to keep in mind that investment adjustment costs are more convex than hiring adjustment costs in our calibration, which partially explains why investment crowds out hiring in Figure 6. While none of these shocks alone can account for plausible impulse responses, their joint interaction with land price shocks might still be important to account for overall business cycle dynamics and this should be the object of further analysis that goes beyond examining impulse responses to a single shock.

Finally, it is also worth mentioning that the role that the endogenous land price plays in the credit constraint (3) is important. This means that periods of land price booms are periods in which credit is more abundant because there is more collateral. If we would assume, rather counter-factually, that lenders do not take into account possible changes in the market value of land when deciding how much to lend to firms, so that a constant land price would enter (3), then no relaxation of the credit constraint would happen during land price booms and our quantitative results would disappear.

The main lesson from these results is that the indeterminate model provides a propagation mechanism by which financial shocks to the collateral value induce firms to borrow more, to invest more and to create more vacancies over several quarters after the shock hits the economy. This mechanism, for which the sluggish endogenous land price dynamics is key, gives rise to much more volatile and persistent responses of the labor market than those that can be generated by productivity shocks or by a determinate model in which such endogenous land price propagation is absent. Our main finding is that the indeterminate model generates amplification and hump-shaped propagation of fundamental land price shocks that are empirically relevant. In particular, it replicates the data patterns that investment, vacancies and unemployment follow after a land price shock hits the economy. In contrast, TFP shocks, leverage shocks or pure sunspot shocks generate impulse responses that are at odds with the data.

In unreported analysis, we show that similar results also hold for the same model calibrated on US data. Although the amplification result is broadly in line with Liu et al. (2013a) who estimate a larger model with Bayesian methods, there are key differences that should be stressed. First, there is a new channel that amplifies financial shocks to the labor market via the discount factor of credit-constrained firms. Second, our model generates local indeterminacy, which may not be easily be picked up by standard Bayesian estimation techniques. Third, the indeterminacy is precisely the reason why the labor market responses to land price shocks are hump-shaped and exhibit considerable persistence which is in line with the data. In view of these differences, further research that allows determinate and indeterminate models to compete in the parameter estimations seems promising to uncover how the various shocks affect investment and labor market dynamics. In addition, it remains an open question to what extent these results can be extended to European economies that have been subject to larger real-estate swings, like Spain and the UK, and whether countries with milder land price movements, like Germany, look any different. is highly negatively correlated to unemployment in the US, whereas the negative relation is much weaker in France.

Figure 1 :

 1 Figure 1: Land at market value, investment, and labor market tightness (cyclical components) in France.

Figure 2 :

 2 Figure 2: Impulse response functions to a one-standard-deviation land value shock. Notes: Dotted lines are 95% confidence bands.

Figure 5 :

 5 Figure 5: Impulse responses to a land price shock (top) and to a TFP shock (bottom) in the determinate model.

Figure 6 :

 6 Figure 6: Impulse responses to leverage shock (top) and sunspot shock (bottom) in the indeterminate model.

Table 1 :

 1 Comovement and relative volatilities. All statistics are based on quarterly variables over the period 1978-2011, except credit market liabilities (since 1996), vacancies and market tightness (since 1989). See the text and Appendix A for details.

		Output	Land	Investment	Vacancies Unemployment	Tightness	Credit liabilities	Real wage
	Output	1.000	0.460	0.793	0.692	-0.645	0.840	0.349	-0.011
	Land		1.000	0.570	0.760	-0.402	0.647	0.096	0.082
	Investment			1.000	0.710	-0.689	0.879	0.379	0.124
	Vacancies				1.000	-0.468	0.839	0.057	0.044
	Unemployment					1.000	-0.867	-0.771	0.401
	Tightness						1.000	0.523	-0.164
	Credit liabilities							1.000	-0.075
	Real wage								1.000
	Std. dev. relative	1.000	3.874	3.304	5.228	4.683	8.933	2.940	0.439
	to output
	Notes:

Table 2 :

 2 Parameter values.

	Parameter Value Description
	δ	0.027 Capital depreciation rate
	A	1	Total factor productivity
	σ	2	Firms' and workers' relative risk aversion
	β	0.99 Lenders' discount factor
	β	0.98 Firms' and workers' discount factor
	ξ	0.02 Output elasticity to land
	ζ	0.37 Output elasticity to capital
	s	0.02 Separation rate
	α	0.50 Matching function elasticity
	m	0.214 Matching function scale
	τ	0.58 Wage replacement ratio
	γ	0.91 Workers' bargaining power
	η K	15	Convexity capital installation costs
	φ K	0.10 Scale capital installation costs
	η N	1.13 Convexity hiring costs
	φ N	0.54 Scale hiring costs
	Ψ	1.25 Collateral constraint parameter
	ω	0.30 Share of capital collateralized

Table 3 :

 3 Standard deviations relative to output and correlations with output under land price shocks in the indeterminate model.

		Stand. Deviation	Correlation
	Variable	Model	Data	Model Data
	Land value	6.06	3.87	0.29	0.46
	Investment	5.89	3.30	0.38	0.79
	Debt	3.53	2.94	0.55	0.35
	Vacancies	5.76	5.23	0.57	0.69
	Unemployment	4.65	4.68	-0.90 -0.65
	Tightness	8.66	8.93	0.75	0.84
	Wage	0.72	0.44	0.92 -0.01

Table 4 :

 4 Sample averages. All statistics are based on the period 1978-2011, except credit market liabilities in France (since 1996).

	France	US

Table 5 :

 5 Comovement and volatility in the United States.

		Output	Land	Investment	Vacancies Unemployment	Tightness	Credit liabilities	Real wage
	Output	1.000	0.540	0.897	0.915	-0.945	0.956	0.255	0.597
	Land		1.000	0.577	0.394	-0.612	0.551	0.234	0.311
	Investment			1.000	0.807	-0.916	0.899	0.480	0.698
	Vacancies				1.000	-0.907	0.958	-0.019	0.525
	Unemployment					1.000	-0.987	-0.316	-0.563
	Tightness						1.000	0.230	0.575
	Credit liabilities							1.000	0.646
	Real wage								1.000

Std. dev. relative 1.000 5.901 3.003 5.791 5.000 11.001 1.184 0.417 to output Std. dev. relative 1.869 2.848 1.699 2.071 1.996 2.302 0.753 1.774 to France

Also, unlike our model, households hold land, which contributes to amplification through a "labor channel", induced by wage rigidities due to non-separable utility between consumption goods and housing services. Such a channel is absent in our model.

Specifically, the aim is to remove all fluctuations shorter than 1.5 and longer than 8 years, which is standard practice in the business-cycle literature. A periodogram analysis shows that the Christiano-Fitzgerald filter does a better job than the Hodrick-Prescott filter in that respect. We also note that drifts have been removed from all variables.

The VAR is estimated with lag lengths of four quarters for each equation. Changing the lag lengths has little effect on the results. Note also that the results are not sensitive to the ordering of the three remaining variables.

By focusing on hiring flows, we keep the separation rate fixed for convenience. Given that shocks to the separation rate play a larger role in Europe than in the US (cf.[START_REF] Justiniano | The cyclical behavior of equilibrium unemployment and vacancies in the US and Europe[END_REF]), it should be interesting to investigate how shocks to s t affect the dynamic responses in this model.

It is assumed that a separated worker can search for reemployment within the same period. This is a reasonable modeling assumption for a quarterly calibration.

Relatedly, separate groups of workers and capital owners can help to address asset-pricing puzzles in the

Like the land price shock in Figure3, the shock to TFP is assumed to follow an AR(1) process with autocorrelation coefficient 0.98. We make the same assumption regarding leverage shocks in Figure6below,

ConclusionsIn this paper, we provide a macroeconomic model in which firms are subject to labor and credit market frictions and use land and capital as collateralizable assets in borrowing. When calibrated on French data, a key property of the model is that it generates an indeterminate steady state provided that the share of capital that is accepted as collateral is sufficiently low.

Appendix A: Data Sources and Further Descriptive Statistics This appendix reports the data sources and some further statistics. For comparison with the business cycle in France, we also report some statistics for similarly constructed variables for the United States. For France, data on output (gross value added) and investment for the non-financial business sector are obtained from the INSEE and are available at quarterly frequency. Credit market liabilities are also available at the INSEE but only at annual frequency and since 1996. Quarterly unemployment series are from the INSEE, and quarterly data on vacancies are derived from the INSEE series on the number of job opportunities registered at the French national agency for employment which are only available since 1989q1. The labor income series are quarterly available in the national accounts produced by the INSEE. We derive the income share dividing the labor income by the gross added value minus indirect taxes (the quarterly series are also from the national accounts). The labor productivity corresponds to the gross added value net of indirect taxes divided by the number of employees in the non-financial firms; we can thus obtain quarterly real wages multiplying the labor share by the labor productivity. For the US, quarterly series of output, investment and credit market liabilities series for nonfinancial business are taken from the Flow of Funds Accounts of the Federal Reserve Board. We calculate land at market value series by subtracting structures at current cost to real estate at market value. Quarterly series on the number of unemployed come from the Bureau of Labor Statistics (BLS). The number of vacancies is taken from the Job Openings and Labor Turnover Survey (JOLTS) series built by the BLS combined with the composite Help-Wanted Index of [START_REF] Barnichon | Building a composite help-wanted index[END_REF]. Based on the Flow of Funds Accounts of the Federal Reserve Board and the BLS data, we compute labor share, labor productivity and real wages series in the same way as for France. Table 4 reports some sample averages over the full time period. With respect to these statistics, the two countries are rather similar. One striking difference is the higher credit-to-output ratio in France which reflects that non-financial firms seem to depend more heavily on external finance in this country. Table 5 shows cross-correlations and relative standard deviations for the US economy of our key aggregate variables which are detrended in the same way as the time series for France. The last row in the table shows that the business cycle is considerably more volatile in the US than in France. The correlation coefficients point at some common patterns as in the corresponding Table 1 for France, but also to noteworthy differences. For instance, the number of vacancies Appendix B: Equilibrium and Proof of Proposition 1 Using (11) to solve for ι t = βλ t -βλ t+1 and substitute in the other first-order conditions, we obtain the following set of equilibrium conditions:

Steady-state values satisfy:

The following proposition, stated in section 4 of the main text, is established from the above steady-state relationship.

Proposition 1: A steady state equilibrium is unique. If ω > 0, an increase of the collateral constraint parameter Ψ (or a decrease of the interest rate R = 1/ β -1) has a positive effect on the steady-state values of output Y , employment N, capital K, market tightness θ, the real wage W and the land price Q. Changes of the land wedge T affect the land price, but have no effect on output and on the labor market.

Proof: Uniqueness of the steady state follows by inspection of equations ( 39)-( 54), which can be solved recursively. The hiring rate is given by (50) while the job values J and Ω follow from ( 43) and ( 46). The marginal product of capital F K and the marginal product of labor minus the wage F N -W are determined uniquely from ( 41) and ( 44), where µ/λ (that is, Tobin's q) is determined from (40). From ( 52), the job-finding rate f is a decreasing function of 1/N so that, substituting for f in (45), it follows that the wage W is also a decreasing function of 1/N. Because of constant returns, F K and F N (for given land supply L = 1) are functions of K/N and L/N = 1/N. Both functions are increasing in 1/N while F K (F N ) is decreasing (increasing) in K/N. Hence, F N -W and F K uniquely determine the steady-state levels of N and K. Then market tightness and the real wage follow from ( 52) and ( 45), so that workers' consumption C w and marginal utility λ w follow from ( 54) and ( 53). The land price Q and borrowing B are determined from ( 42) and ( 48). Finally, with output Y determined from ( 51) and investment I given in (49), firms' C and λ follow from ( 47) and (39).

To prove the second and third assertions uses a similar argument. From the previous paragraph, F N is independent of Ψ and R = 1/ β -1, while (41) implies that F K decreases (increases) in Ψ (R) if ω > 0, and F K is independent of these financial variables if ω = 0. Hence, if and only if ω > 0, an increase of Ψ (or a decrease of R) increases both N and K (this follows because F K and F N -W are both increasing in 1/N, F K decreases in K/N and F N -W increases in K/N.). Hence ( 45), ( 51) and ( 52) imply that W , θ and Y are increasing if ω > 0 and independent of Ψ and R otherwise. The increase of N and K (for ω > 0) implies that F L increases. Hence, (42) implies that the land price Q increases for any ω ≥ 0. Finally, marginal products F L , F K and F N are independent of the land wedge T , so that positive changes in the land wedge affect negatively Q through (42) but have no real effects. 2

Appendix C: Calibration Details

In view of equation ( 48), the borrowing constraint parameter follows from

and the calibration targets in Table 4. From the first-order condition for investment (40) we obtain Tobin's q:

We use φ K to calibrate the share of investment cost in output, while η K is used to target investment volatility relative to output volatility.

From the first-order conditions for capital and land, equations ( 41)-( 42) the capital and land shares in output follow from the other targets: where c(H, N) = φ N h η N N is total hiring cost, and we can use that h = s in steady state. Given that

this yields φ N .

In steady state, the first-order condition for vacancies, equation ( 43), implies

Therefore parameter γ follows from (45)-( 46) which can be rewritten in terms of the calibration targets and the other parameters: