An important problem for proving statements in multimodal categorial grammar is that even when using proof nets -which improve upon proof search in natural deduction and sequent calculus by identifying all equivalent proofsthe number of possible axiom links to consider is still enormous. We will propose several efficient strategies to reduce the number of axiom links and will evaluate the resulting combined strategy against a large number of statements in both multimodal and Lambek categorial grammars and find that we eliminate a very large number of the axiom links which do not correspond to any proof net.

Chapter 1 Introduction

The multimodal Lambek calculus [START_REF] Moortgat | Categorial type logics[END_REF]) is a powerful and flexible grammar framework. Unfortunately, it has some sublogics -like the associative Lambek calculus L or the associative, commutative Lambek-Van Benthem calculus LP -which are known to be NP complete [START_REF] Pentus | Lambek calculus is NP-complete[END_REF][START_REF] Kanovich | The multiplicative fragment of linear logic is NP-complete[END_REF].

In this article we look at proof nets for the multimodal Lambek calculus and investigate ways to reduce the number of axiom links we need to perform in order to decide whether a statement Γ ⊢ C is derivable or not.

We then evaluate the effect of these reductions on a large number of randomly generated Lambek calculus sequents as well as on a large number of sequents which have been extracted from a corpus and find that the combined filtering strategies filter our the large majority of the incorrect axiom links, that is, those axiom links which are not part of any proof net, showing that -in spite of the theoretical complexity -in practice we can parse Lambek grammar while making just a small percentage of incorrect axiom links.

The rest of this report is structured as follows.

Chapter 2 gives a short introduction to parsing statements of the multimodal Lambek calculus using proof nets. It discusses proof nets for NL3 R , which is the non-associative Lambek calculus NL extended with unary operators and a set of structural rules R. Several instances of these structural rules as well as their applications are shown. The chapter concludes with an overview of some of the combinatorics of proof nets.

Chapter 3 discusses four strategies for reducing the total number of axioms links. The first two: acyclicity and connectedness (Section 3.1) and first-order approximation (Section 3.2) are fairly well-known and I touch upon them only briefly. The final two strategies are new. We show how to compute the relations between pairs of unary connectives as well as verify the possibility of binary and unary contractions in the presence of some frequently used sets of structural rules using a context-free grammar in Section 3.3 and how to eliminate axiom links which cannot be part of a total linking using methods from constraint logic programming in Section 3.4.

Chapter 4 evaluates the combined algorithm against two sets of derivable sequents. The first set is a set of randomly generated derivable sequents in the Lambek calculus L. The second set is a set of multimodal sequents which has been automatically extracted from a corpus. In both cases, the algorithms proposed will filter out a very important number of underivable sequents.

The final chapter concludes and gives some possiblities for future research.

Chapter 2

Proof nets

We follow [START_REF] Moot | Proof nets for the multimodal Lambek calculus[END_REF] for our presentation of proof nets in this section.

Constructing a proof structure for a statement in the multimodal Lambek calculus is done in three phases:

1. decompose the lexical formulas as well as the goal formula, 2. connect positive and negative atomic formulas, 3. decide whether the resulting proof structure is a proof net, using a correctness criterion.

We look at each of the three phases in turn.

Decomposition

Decomposing the lexical formulas is deterministic and linear in the number of connectives in the statement. Depending on whether a complex formula is a hypothesis (antecedent formula) or a conclusion (succedent formula), only one link can apply: a left link for a hypothesis and a right link for a conclusion. We can simply descend the formula tree until we reach the leaves, which are the atomic formulas.

The full set of links is shown in Table 2.1. There are two links for every connective: a left link for when it occurs as a hypothesis (portrayed at the bottom; antecedents formulas start their unfolding here) and a right link for when it occurs as a conclusion (portrayed at the top; the goal formula starts its unfolding here).

The left link for B\ i A is simply the modus ponens rule, saying it combines with a B on its left to form an A. The right link -for

B\ i A as a conclusion [R/ i ] [R• i ] [R\ i ] [R3 j ] [R2 ↓ j ] I i A A/iB B A•iB A B i i A B B\iA 3j A A j 6 j A 2 ↓ j A A A/iB B i ? i A•iB A B A B B\iA i ? j 3j A A A 2 ↓ j A j [L/ i ] [L• i ] [L\ i ] [L3 j ] [L2 ↓ j ]
Table 2.1: Logical links for the multimodal Lambek calculus -is its exact opposite: it allows us to use a B hypothesis to prove an A. Of course, we'll need to check that the B occurs as the leftmost sister of the A node and this is where the contractions will come in later. which would be a lexical lookup for a sentence like 'Robin wet himself'. The numbers on the atomic formulas are not a part of the proof structure, they serve only to help us refer to the different formulas later. This is a slightly simplified version of the abstract proof structures of [START_REF] Moot | Proof nets for the multimodal Lambek calculus[END_REF]. We have suppressed as much information as possible: the information on the internal nodes, the lexical formula (both of them can be uniquely reconstructed from the unfolding) and the premiss/conclusion distinction of the individual nodes in Figure 2.1 as well (since we need it only when there is just a single antecedent formula).

Axioms

The second stage of constructing a proof net consists of identifying the axiomatic formulas: we select a positive and a negative atomic formula of the same type and identify their vertices, thereby connecting different parts of the graph. For this second step, there are potentially many solutions and the main part of this paper focusses on strategies for reducing these possibilities as much as possible.

Figure 2.1 shows the possibilities for the axioms links by portraying them in a grid: every row represents the axiom link possibilities for a negative formula,

I 0 np 3 0 s 3 np 2 himself 0 s 2 np 6 0 np 1 Robin wet np 5 0 s 1 np 4 0 Goal s 4 np 1 np 2 np 3 np 4 np 5 np 6 s 1 s 2 s 3 s 4 Figure 2.1:
Formula unfolding and axiom link possibilities for np, (np\s)/np, ((np\s)/np)\(np\s) ⊢ s whereas every column represents the axiom link possibilities for a positive formula. A full axiom linking corresponds to putting exactly one mark in every row and column. In graph theory, this is usually called a perfect matching.

Since there are n! different perfect matchings possible, the goal of this paper is to exclude the maximum number of these connections when they cannot be part of a proof net.

Contractions and Structural Rules

The final step is checking if the resulting graph is a proof net, by contracting all par links, drawn with the black center, as shown in Table 2.2. When all par links have been contracted, the result will be a tree -a tensor tree -with the lexical entries used as its leaves.

In case it is impossible to contract a par link, the proof structure we are dealing with is not a proof net and therefore, we know there is no proof of the corresponding sequent.

A typical multimodal categorial grammar has a set of structural rules R in addition to these contractions. Structural rules allow us the replace a substructure of an abstract proof structure which is a tensor tree t 1 by another tensor tree t 2 . Since we operate in a fragment of multiplicative linear logic, we require that t 1 and t 2 share the same leaves. In other words, deletion and copying of subtrees isn't allowed, though we can change the order of the different subtrees. It is this reordering which permits us to push the languages generated by multimodal categorial grammars beyond the context-free languages.

Table 2.3 and 2.4 show the frequently used structural rules of mixed asso-

i ? i τ 2 τ 1 → τ 2 τ 1 i i τ 2 τ 1 → τ 2 τ 1 Table 2.2: Graph contractions z 0 x y 1 → MA l ← MA -1 l y z 0 x 1 ← MC l → MC -1 l y 0 x z 1 Table 2
.3: Mixed associativity and commutativity -left branch ciativity and mixed commutativity of modes 0 and 1. Observe that in Table 2.3 the leaf x always stays on the left branch of a mode 1 structure, whereas in Table 2.4 the leaf z always stays on the right branch of a mode 1 structure. [START_REF] Moortgat | Adjacency, dependency and order[END_REF] use the [MA l ] and [MC l ] rules to give an account of word order in Dutch embedded clauses. A variation of these rules is shown in Table 2.5 and 2.6. In these cases, there is only a single unary mode 0 but a unary branch 0 indicates that an x leaf can move from one left branch to another (as in Table 2.5) or that a z leaf can move from one right branch to another (as in Table 2.6). Moortgat (1999) shows how the right branch extraction rules [MA3 r ] and [MC3 r ] of Table 2.6 give an account of English relativization, whereas the left branch extraction A final set of structural rules involves the communication between binary and unary modes.

x 0 y z 1 → MA r ← MA -1 r x y 0 z 1 ← MC r → MC -1 r y 0 x z 1 Table 2.
x 0 y 0 → K1 ← -1 K1 x y 0 0 ← K2 → -1 K2 y 0 x 0 0 x 0 y 0 ↑ -1 K ↓ K Table 2
.7: Communication between unary and binary modes -K, K1 and K2

[R/ i ] [R• i ] [R\ i ] [R3 j ] [R2 ↓ j ] A/iB B A ] A B A • i B ] B B\ i A A A 3 j A 6 A 2 ↓ j A 6 A/ i B B A ] A B A• i B ] B B\ i A A ] 3 j A A 6 2 ↓ j A A 6 [L/ i ] [L• i ] [L\ i ] [L3 j ] [L2 ↓ j ]
Table 2.8: Logical links for essential nets

Essential Nets

For the acyclicity and connectedness criterion discussed in Section 3.1 as well as for the context-free grammar criterion of Section 3.3, we are interested in an alternative correctness criterion proposed by [START_REF] Lamarche | Proof nets for intuitionistic linear logic I: Essential nets[END_REF]. This criterion is based on a different way of decomposing a sequent, this time into a directed graph, with conditions on the paths performing the role of a correctness criterion. A net like this is called an essential net. The links for essential nets are shown in Table 2.8, though we follow de Groote (1999) in reversing the arrows of [START_REF] Lamarche | Proof nets for intuitionistic linear logic I: Essential nets[END_REF].

In addition -to keep the essential nets as close to the abstract proof structures of before -there are no axiom links and cut links and the main formula of a link can be its hypothesis as well as its conclusion.

As a consequence of this change of perspective, all edges point upwards.

Definition 2.1 Given an essential net E its conclusion is called the output of the essential net and the hypotheses, as well as the B premisses of any [R/ i ] or [R\ i ] link, are called its inputs. When we need to distinguish between these two types of inputs, we will call the hypotheses the lexical inputs and the B premisses of the [R/ i ] and [R\ i ] the auxiliary inputs.

Definition 2.2 An essential net is correct iff the following properties hold.

1. it is acyclic, 2. every path from the B premiss of a [R/ i ] or [R\ i ] link passes through the complex premiss of this link, 3. every path from the inputs of the graph ends at the output of the graph.

np 3 ] np 2 s 3 himself ] np 6 s 2 ] np 1 Robin wet np 5 ] np 4 s 1 ] Goal s 4 np 1 np 2 np 3 np 4 np 5 np 6 s 1 s 2 s 3 s 4
Condition (1) reflects the acyclicity condition on proof nets, whereas conditions ( 2) and (3) reflect the connectedness condition. The formulation of 'every path' exists only to ensure correctness of the negative • link; in all other cases there is at most one path between two points in a correct essential net.

Theorem 2.3 (Lamarche (1994)) A sequent Γ ⊢ C is provable in MILL iff its essential net is correct.
To give an idea of why these three properties need to hold, given that we need to contract to a tensor tree with the hypotheses as its leaves and the conclusion as its root, we remark the following:

Basic Properties

In order to better analyze the properties of the algorithms we propose, we will first take a look at some basic properties of proof nets.

Axiom Links

Since we will be concerned with finding an axiom linking for an abstract proof structure A which will allow a contraction of A into a tensor tree, we first given some bounds on the number of proof structures we will have to consider. Given that the problem we are trying to solve is NP complete, even in the non-commutative case, it is not surprising that these bounds are quite high.

Proposition 2.4 Let P be a proof net and f an atomic formula, then the number of positive occurrences of f is equal to the number of negative occurrences of f . This proposition follows immediately from the fact that every atomic formula has one occurrence as a negative hypothesis (either of the proof net or of a link) and one occurrence as a positive conclusion (again, either of the proof net or of a link).

Proposition 2.5 Every formula unfolding F has O(a!) possible identifications of atomic formulas which produce a proof structure, where a is the maximum number of positive and negative occurrences of an atomic formula in F .

If we have a positive atomic formulas, we have a possibilities for the first one, since all negative formulas may be selected, followed by a -1 for the second etc. giving us a! possibilities.

Proposition 2.6 Every formula unfolding F has O(4 a ) planar axiom linkings which produce a proof structure, where a is the maximum number of positive and negative occurrences of an atomic formula in F .

This follows from the fact that a planar axiom linking is simply a binary bracketing of the atomic formulas and the fact that there are C a-1 such bracketings, where C k , the kth Catalan number, approaches 4 k / √ πk 3/2 .

Proposition 2.7 For every partial proof structure with a atomic formulas which

are not yet identified with an atomic formula of opposite polarity there are O(a 2 ) possible axiom links.

Given that every positive atomic formula can be linked to every negative atomic formula of the same atomic type this gives us a 2 pairs.

Graph Size

Proposition 2.8 For every proof structure S with h hypotheses, 1 conclusion, p binary par links and t binary tensor links, the following equation holds.

p + h = t + 1 = a
Given Proposition 2.4, the number of positive and negative atomic formulas is both a. Suppose we want to construct a proof structure S with h negative conclusions and 1 positive conclusion from these atomic formulas. When we look at the links in Table 2.1 we see that all par links reduce the number of hypotheses by 1 and all tensor links reduce the number of conclusions by 1.

Proposition 2.9 Every essential net E has v = h + 1 + 2(p + t) = O(a) vertices and 2t + p ≤ e ≤ 2(t + p) + a = O(a) edges.
This follows immediately from inspection of the links: all h hypotheses and the single conclusion of the essential net start out as a single vertex and every link adds two new vertices. For the edges: the minimum number is obtained when we have no axiom links and all par links are positive links for \ or / which introduce one edge, the maximum number includes a axiom links and par links which are all negative links for •.

Chapter 3 Filtering Axiom Links

Simple combinatorics shows that there are n! possible axiom linkings for 2n atomic formulas (Proposition 2.5). In Figure 2.1, there are therefore 2 possibilities for s and 6 possibilities for np. This makes exhaustive search prohibitive for all but the most trivial statements.

However, there are several possibilities to rule out axiom links which can never contribute to a contractible proof structure. We discuss some known and some new strategies in the following sections. [START_REF] Danos | The structure of multiplicatives[END_REF] introduce the acyclicity and connectedness criterion for proof nets of multiplicative linear logic. Give that the categorial logics we work with are all sublogics of multiplicative linear logic (in the sense that any derivable sequent in multimodal categorial grammar has a derivable image in multiplicative linear logic) we can use the fact that anything underivable in MLL is underivable in categorial grammar as well. [START_REF] Moot | Graph algorithms for improving type-logical proof search[END_REF] shows how an adaptation of the Floyd-Warshall algorithm can be used to select from the total set of possible axiom links those that produce acyclic and connected proof structures.

Acyclicity and Connectedness

The Floyd-Warshall algorithm computes the transitive closure of a graph by successively eliminating the intermediate vertices c from every path from a to b. Given a vertex c and the paths a → c → b for all a and b we create a direct path a → b if it didn't exist before. That is to say, there is a path from a to b if either there is a path from a to c and from c to b or if there is a path from a to b which we already knew about (Figure 3.1 on the following page). After eliminating c, for every path in the original graph which passed through c there is now a shortcut which bypasses c. After we have created such shortcuts for all vertices in the graph it is clear that the resulting graph has an edge a → b iff there is a path from a to b in the original graph.

The acyclicity condition, condition (1) from Definition 2.2, is easily verified. Verifying conditions ( 2) and (3) from Definition 2.2 is a bit harder. The question we want to ask about each link is: does this link contribute to a connected proof structure? Or, inversely, does excluding the other possibilities for the two atomic formulas we connect mean a connected proof structure is still possible.

To check the conditions we need to verify the following:

1. for every negative input of the net we verify there exists a path to the positive conclusion, 2. for every negative • link we verify that both paths leaving from it reach their destination, 3. for every positive / or \ link we check the existence of a path from its negative premiss to its positive conclusion continuing to the positive conclusion of the essential net.

Given that we are already computing the transitive closure of the graph for verifying acyclicity, we can exploit this by adding additional information to the matrix we use for the transitive closure. There are many ways of storing this extra information, the simplest being in the form of an ordered list of pairs. Given that, for a atomic formulas, each possible connection allows (a-1) 2 other connections (ie. it is agnostic about all possibilities not contradicting this one) but excludes 2(a -1) possibilities. It is therefore more economic to store the connections which are excluded. For example, the ordered set associated to the edge from 1 to 4 will be {1 -5, 1 -6, 2 -4, 3 -4}, meaning "there is an edge from 1 to 4 but not to anywhere else and the only edge arriving at 4 comes from 1".

Note that in the description of the Floyd-Warshall algorithm, we made use only of the logical 'and' and 'or' operators. For ordered sets, the corresponding Note that Equation 3.2 is simply Equation 3.1 with both sides negated, the negations moved inward and set union and intersection in the place of the logical 'or' and 'and' operators.

I 0 np 3 0 s 3 np 2 himself 0 s 2 np 6 0 np 1 Robin wet np 5 0 s 1 np 4 0 Goal s 4 np 1 np 2 np 3 np 4 np 5 np 6 0,1 c,W 2,d X,1 2,Y V,W X,Y V,3 c,d 0
Given that we can implement the union and intersection operations in linear time with respect to the size of the input sets, the total complexity of our algorithm becomes O(v 3 2(a -1)) = O(a 4 ).

Figure 3.2 shows, in dark gray and black, the links which are excluded when we use this condition. We remark that this leaves just one possibility for linking the s formulas.

First-order Approximation and Word Order

Even though the acyclicity and connectedness check is an effective test, it is based on the 'worst case' scenario of a fully associative and commutative logic. A second strategy for removing axiom links which cannot contribute to constructing a proof net is to use first-order approximation to take constraints on word order into account. This has been used at least since LLoré and [START_REF] Lloré | Difference lists and difference bags for logic programming of categorial deduction[END_REF]

(though in a slightly different context). [R/ a ] [R• a ] [R\ a ] [R3 0 ] [R2 ↓ 0 ] I a A A/aB B X -c X -Y Y -c A • a B A B a X -Z Z -Y X -Y a A B B\ i A c -Y c -X X -Y A 3 0 A 0 X -Y X -Y 6 0 A 2 ↓ 0 A X -Y X -Y A A/ i B B a X -Y Y -Z X -Z ? a A• i B A B X -Y X -c c -Y A B B\ i A a Z -X X -Y Z -Y ? 0 3 0 A A X -Y X -Y 2 ↓ 0 A A 0 X -Y X -Y [L/ a ] [L• a ] [L\ a ] [L3 0 ] [L2 ↓ 0 ]
Table 3.1: First-order labelling for Lambek calculus formulas [START_REF] Moot | Linguistic applications of first order multiplicative linear logic[END_REF] propose an embedding of the Lambek calculus using first-order quantifiers. Each formula is assigned a pair of string positions which correspond to the left and right frontier of the string to which the formula corresponds. Lexical formulas are assigned successive string positions: 0 -1 for the first word, 1 -2 for the second up until (n -1) -n for the last word of a sequence of n. The goal formula is assigned 0 -n. Table 3.1 shows the propagation of first-order variables to the other formulas. In every case a c corresponds to a fresh constant and a Z to a fresh variable.

Figure 3.2 shows how we can use these propagation rules to compute the first-order labels assigned to the atomic formulas of our example sequent and -in light grey and black -the axiom links which are excluded using the first order constraints for L, which is justified given that we have a sequent which is derivable even in the non-associative Lambek calculus. In the figure, the numbers 0, 1, . . . correspond to constants referring to string positions, lowercase letters c, d, . . . to constants introduced by the par links and upper-case letters X, Y, . . . correspond to variables.

As long as we make sure that the structural rules permit a subset of the word order possibilities allowed by the first order variables, this strategy is correct. Given that the sequent is derivable in NL, we are justified in using the L firstorder labelling.

In addition to embedding the Lambek calculus, [START_REF] Moot | Linguistic applications of first order multiplicative linear logic[END_REF] show how several linguistic phenomena like quantifier scope ambiguities, wh extraction and island constraints -for which there is no satisfactory treatment in the Lambek calculus -can be analysed using first-order quantifiers as well.

By assigning slightly more subtle first-order variables and constants, we can incorporate these analyses into our labels. An example for the treatment of extraction is shown in Figure 3.3. Here, we simply use the solution of [START_REF] Moot | Linguistic applications of first order multiplicative linear logic[END_REF] for wh words. The first-order variables indicate that the word 'which', when it occurs between string positions 1 and 2, is looking for an n starting at some position X directly to its left and an s ending at position Y directly to its right. Inside this s we can use an np which can take up any position. The final result will then be an n between X (the leftmost position of the n argument) and Y (the rightmost position of the s argument). Note that we still need to add the appropriate structural rules to our grammar in order to derive medial extraction cases (see Moortgat, 1997, for a solution).

Given that we can check the first order constraints simply by unification of variables and constants, meaning O(1) per cell in the axiom matrix, the total complexity of the first-order constraints is O(n 2 ) for 2n atomic formulas.

Context-Free Grammars

The Unary Connectives

The unary connectives are a powerful addition to the multimodal Lambek calculus. They can be used to license structural rules but the relations between the logical formulas they induce can be used to encode linguistic features, as done, for example by [START_REF] Heylen | Types and Sorts: Resource Logic for Feature Checking[END_REF], or to restrict scope possibilities, as done, for example by [START_REF] Bernardi | Generalized quantifiers in declarative and interrogative sentences[END_REF] for the interaction between generalised quantifiers and negation in English, by [START_REF] Moot | Les indices pronominaux du français dans les grammaires catégorielles[END_REF] for generalised object clitics in French and by [START_REF] Bernardi | Partially ordered categories: Optionality, scope and licensing[END_REF] for operators in Hungarian.

In certain cases, we use the unary modalities just for the derivability relations between the different types. Figure 3.4 summarizes the different relations between unary prefixes of up to four. Note that 3232A and 2323A are not

⊢ ⊢ ⊢ ⊢ ⊢ ⊢ ⊢ ⊢ 3322A 32A 3223A A 2332A 23A 2233A
Figure 3.4: Derivability relations using the unary logical rules displayed, given that they are equivalent to 32A and 23A respectively 1 . By extending the unary prefixes further we can generate an intricate hierarchy of formulas, but for many applications the seven formula recipes in Figure 3.4 suffice. However, even for the formulas we've shown here it's not directly clear which pairs of them are derivable.

Fortunately, there is a simply way to check the contractibility of a sequence of unary formulas. First, we convert this sequence to a string as follows (with ǫ being the empty string and . the concatenation operation) Definition 3.1 Let A and B be two formulas which are the identical up to their unary prefixes. σ(A ⊢ B), the string corresponding to A ⊢ B, is defined as A -. B + , where the positive and negative formula are translated as follows.

3B + = B + .m 3A -= l. A - 2B + = B + .r 2A -= m. A - B + = ǫ otherwise A -= ǫ otherwise
The easiest way to see the correspondence between a sequence of unary modes and a string is by turning a page with a formula unfolding 90 degrees to the right and realizing that every arrow pointing left will produce an l, while every arrow pointing right will produce an r. Now, given that we have produced a string corresponding to the two sequences of unary connectives, we can check contractibility of these unary modes using the following context free grammar.

S → ǫ (1) | l S m S (2) | m S r S (3)
Rule (1) corresponds to the fact that it is possible not to have any unary connectives in front of a formula at all. Rule (2) corresponds to the 3 contraction and rule (3) to the 2 contraction.

1 Technically the graph of Figure 3.4 is the transitive reduction of the derivability relation where all equivalent formulas (which would correspond to cycles in the unreduced graph) have been replaced by their smallest element.

Proposition 3.2 A ⊢ B contracts to a single vertex using the unary contractions iff S → σ(A ⊢ B).

Proof (sketch) ⇒ Induction on the number of contractions c. If c = 0 we use rule (1). If c > 0 there we look at the point where the first link is contracted. In order for this contraction to be valid the links between the two contracted links need to contract to a single node and in order for the entire sequence to contract everything after the second link needs to contract to a single node as well. Induction hypothesis allows us to combine these smaller proof nets using either rule (2) or (3).

⇐ Induction on the length of the CFG derivation. 2 To give an illustration of how to use the context free grammar, we show how the derivable sequent 3223A ⊢ 23A translates to lmmlmr, which we can derive as shown below.

S → l S m S → l m S → l m m S r S → l m m S r → l m m l S m S r → l m m l m S r → l m m l m r We can show the inverse statement 23A ⊢ 3223A which would correspond to mlmrrm is underivable simply because we cannot match the final m: there is no r to its right and if we would match it to the single l we would need to derive mrr, the symbols in between, but this is impossible given that it has an odd number of symbols.

The context free grammar is easily extended to the multimodal case, simply by adding different symbols l i , m i and r i to the grammar and adding two new grammar rules for each of the new symbols.

There are limitations to using this system, however. First of all, it requires us to remove all unary branches from the final tree, though we could overcome this limitation in the same way we do for the binary connectives in the next section, though this means adding extra rules and non-terminal symbols to our grammar.

Secondly, we cannot use this strategy if some of the structural rules for the unary modes we're interested in are incompatible with the formula to string translation. Examples would be any inclusion rules between unary modes (though adding grammar rules would again be an option here) or structural rules which move unary modes up or down the tree, like the K, K1 and K2 structural rules of [START_REF] Moortgat | Categorial type logics[END_REF], showing in Figure2.7, which would require changing the translation function. However, as we will explore in Secion 3.3.3, there are cases in which it is possible to incorporate the K1 and K2 postulates into the context-free rules for a combined unary-binary CFG.

Given that parsing a context free grammar is O(n 3 ) and we would have to perform this calculation for all n 2 possible axiom links, the total

O(n 5 ) com- [R/ i ] [R• i ] [R\ i ] [R3 j ] [R2 ↓ j ] A/iB B A ] r 1 l ′ 2 A B A • i B ] m 1 m 2 B B\ i A A l ′ 1 r 2 A 3 j A 6 m A 2 ↓ j A 6 r A/ i B B A ] m 1 m 2 A B A• i B ] l 1 l 2 B B\ i A A ] m 1 m 2 3 j A A 6 l 2 ↓ j A A 6 m [L/ i ] [L• i ] [L\ i ] [L3 j ] [L2 ↓ j ]
Table 3.2: Calculating the strings corresponding to paths in the essential nets plexity is somewhat high. So for grammars which use the unary connectives extensively, it can be beneficial to pre-compute the relations between all sequences of unary connectives occurring in the grammar, after which we can do a simple table lookup to see if contraction is possible. This would reduce the total complexity to O(n 2 ).

Adding the binary connectives

The strategy for the unary connectives extends easily to the binary case. This requires us to look at all paths in the graph. It is most easily seen using the essential nets of Section 2.4. Table 3.2 shows how to calculate the string corresponding to a path in an essential net. The treatment of the unary connectives is unchanged, the only additions are the symbols for the binary connectives. Remark that the paths starting at the B premisses of the [R/ i ] and [R\ i ] links start with a symbol which does not correspond to an edge in the directed graph, but that otherwise every symbol corresponds to an edge.

Definition 3.3 Let E be an essential net which is correct in the sense of Definition 2.2. The set of paths P of E is the smallest set satisfying.

• if i is a lexical input of E and p is a directed path from i to the output o then p ∈ P .

• if i is an auxiliary input of E and p is a directed path from i to the conclusion of the corresponding link then p ∈ P .

Adding structural rules

In some cases, we can go even further and include the structural rules in the context-free grammar.

If we calculate the path starting at x in the K1 rule on both the left hand side and the right hand side of the rule, we see it corresponds to replacing mm 1 by m 1 m. This is a context-sensitive rule, however, and we would like to keep our filtering strategies polynomial. A typical use of the unary modes in combination with the K1 and K2 postulates is in the context of feature checking. In this case we know that the feature needs to be checked by a [R2 ↓ ] rule later. If this is the case, we can -depending on which of the structural rules are available, K1, K2 or both -add one of the following rules to our context-free grammar.

S → m V (m 1 ) * V r S (6a) [R2 ↓ + K1] S → m V (m 2 ) * V r S (6b) [R2 ↓ + K2] S → m V (m 1 |m 2 ) * V r S (6c) [R2 ↓ + K1 + K2]
Technically speaking, it is an abuse of notation to use the regular expression operators * and | in the context-free grammar. It would be more proper to add the regular language rules to the context-free grammar. I have chosen to use regular expression to give the most compact account possible of the generalised contractions as possible. For example, the rule [R2 ↓ + K1] says that in order to perform an [R2 ↓ ] contraction, an arbitrary number of left braches can intervene between the unary tensor and the corresponding par.

A similar strategy is possible for the left and right extraction structural rules ([MA3 l ] and MC3 l ] of Table 2.5, and [MA3 r ] and MC3 r ] of Table 2.6 respectively) provided they only occur in the context of positive A/32 ↓ B or 32 ↓ B\A occurrences. Rules (5a) and (5b) below show the two cases.

V → l ′ 1 l m m 1 V (m 1 |m 2 ) * V (5a) [R\ + L3 + MA3 l + MC3 l ] V → l ′ 2 l m m 2 V (m 1 |m 2 ) * V (5b) [R/ + L3 + MA3 r + MC3 r ]
A limitation of the use of context-free grammars is that it can only be used with structural rules where a single path in the abstract proof structure can be used to decide if the structural rules have been applied correctly. For example, it can't be used to check the correct application of the K postulate given that we'd have to verify both branches to see if there is a unary branch below it.

Régin's Algorithm

Even with all the previous constraints on axiom links in place, we sometimes fail to exclude some axiom links which cannot belong to a total matching. This is in part because an axiom link is regarded more or less in isolation, meaning that we don't exploit the fact the we need to find a total matching. [START_REF] Régin | A filtering algorithm for constraints of difference in CSPs[END_REF] proposes an algorithm for the slightly more general problem of finding solutions for 'all different' constraints in constraint logic programming. His algorithm separates the possible axiom links (in our case) into three cate-2 4 7 8

x 2 x 3 x 4 x 1 2 4 7 8 x 2 x 3 x 4 x 1 M in red
I in blue strong components eliminated edges

x 4 x 3 x 2 x 1 7 4 2 8 x 1 , 8, x 4 , 2
x 2 , 7, x 3 , 4

Figure 3.5: An illustration of Régin's algorithm gories: those which must be a part of any linking, those who are only part of some linkings and those which do not belong to any linking. Régin's algorithm works as follows:

1. find a total matching M of the graph, fail if none exists.

2. using the total matching M , compute the inverse I of the links outside of the matching E -M .

3. using edges which are alternatively in M and in I, compute the strong components.

4. when an edge E connectes two different strong components, do the following.

• if E ∈ M , then E is part of every total linking and all alternatives for both vertices are deleted.

• if E / ∈ M , then remove E from the graph, it cannot be part of any linking.

Régin's formulation includes an 'augmenting path' case to extend the matching, after which we repeat from step 2 with the new matching. However, we don't need this since we are guaranteed that if a matching exists it is necessarily total.

Figure 3.5 shows how Régin's algorithm operates. In the constraint problem, we have four variables x 1 to x 4 which can take four values 2, 4, 7 and 8. The squares marked in black on the left side of the picture are excluded because of the other constraints, whereas the red square are the total matching M computed in step 1 of the algorithm. In the middle, the inverse I of the edges outside of Mcomputed for step 2 of the algorithm -are shown in blue. Remark that these are just the white squares of the figure on the left mirrored at the diagonal. For step 3, we compute the strong components of the graph by alternating between red and blue edges. We have two strong components: x 2 , 7, x 3 , 4 and x 1 , 8, x 4 , 2. This means that the only values possible for x 2 and x 3 are 4 and 7, so we delete the edges between x 2 and 8, between x 3 and 2 and between x 3 and 8. The newly exlcuded edges are indicated in gray on the right of Figure 3.5.

Even though a smart axiom linking strategy (like always linking the atomic formula with the least possibilities first) would not benefit a lot from the reduction in this example, there are cases where it will exploit information like the absence of a total matching to fail directly.

The total complexity for Régin's algorithm is dominated by the cost of the total matching, which we can do at least in O(n 2.5 ) [START_REF] Hopcroft | A n 5/2 algorithm for maximum matchings in bipartite graphs[END_REF]. This makes the total cost of all filtering strategies O(n 4 ) if the context-free grammars are precompiled and O(n 5 ) if they are not.

Chapter 4 Evaluation

Random Sequents

In order to evaluate the combined filtering strategies, we have tested the axiom constraints on randomly generated derivable statements of the Lambek calculus. These statements have been generated using the inductive definition of Lambek calculus proof nets where all duplicate statements have been removed. To make the task as difficult as possible, only a single atomic formula a without unary prefixes has been used. Unfortunately, underivable statements don't have such an easy inductive characterization.

Figure 4.1 shows the amount of statements by the number of atomic formulas per sequent in the sample set, as well as the number of sequents by the number of connectives.

Out of 15.946 possible planar axiom links and 61.524 total possible axiom links, 2.546 correspond to different proofs. Therefore, there are 13.400 planar axioms links and 58.978 total axiom links which do not belong to any proof. Of these, the combined filtering algorithm excludes all but 279, for a total of 2.825 axiom links performed.

This means we eliminated 97.92% of the incorrect planar axiom links and 99.53% of the total number of incorrect axiom links. From the perspective of the correct axiom links, we perform only 10.96% more links than a 'perfect' linking strategy -which through some unknown method would only find links corresponding to proofs in an NP complete logic.

Extracted sequents

For the second evaluation we look at a subset of 5.454 sentences which have been extracted from the Spoken Dutch Corpus (see [START_REF] Moot | Automated extraction of type-logical supertags from the Spoken Dutch Corpus[END_REF] In addition, the sequents tend to have significantly more atomic formulas than the previous experiment: we have a total of 69.810 atomic formulas and 33.383 connectives (divided into 5.355 discontinuous connectives and 28.028 continuous connectives). Figure 4.2 shows the distribution. The median number of atoms is 12 whereas the mean is 12,80. The maximum number of atoms for a sequent in our set is 44. For the connectives, the median is 5, the mean 6,12 and the maximum number of connectives is 23.

For this experiment Grail performs a total of 136.070 axioms out of a total

Plot of performed axioms against optimal axioms

Performed Axioms Optimal Axioms 1 Even though this is still more than double the optimal 61.487 axioms, it is a reduction from a truly astronomic number of potential axioms to something computationally feasible. The median number of axioms performed is 7, which is the median of the optimal number of axioms as well.

The mean number of axioms performed is 24,95 (as opposed to 11,27 for the optimal number of axioms) and the maximum number of axioms performed is 388 (as opposed to 159 optimal). For a total of 3.388 sentences (62,12% of the total) the exact optimal number of axiom links is performed. Figure 4.3 plots the total number of axiom links performed against the minimum number of axiom links required in order to obtain all semantic readings. The figure on the left uses colors to show degradation of performance as the number of atomic formulas increases, whereas the figure on the right shows the degradation of performance as the number of discontinuous connectives increases -deep purple indicating a low number of total axioms or discontinuous connectives and cyan indicating a very high number of them.

Chapter 5

Conclusions and Future Work

We have seen that in spite of the computational complexity of computing all axiom links for a given statement, a combination of constraints on the possible axiom links can reduce the total number of axiom links to just a bit over the optimal number of axiom links to be performed. Some important questions remain unresolved. For example, are there conditions where the filtered axiom links correspond exactly to the axiom links which belong to a proof net? This would mean that after the filtering algorithm has done its job, the resulting axiom matrix would contain all and only those links which would be used for a proof net. In that case, the axiom possibilities would form a sort of shared representation of all proofs for a statement and moreover, it would be computed in O(n 4 ) time. However, analyzing the incorrect axiom links of our experiment doesn't seem to give an easily identifiable handle on the subclasses of multimodal categorial grammar which would have this property.

The results on context-free grammars in Section 3.3 seem to provide an interesting possibility of obtaining polynomial parsing results. It has been observed by [START_REF] Schabes | Deterministic left to right parsing of tree adjoining languages[END_REF] that every path in a derived tree for a tree adjoining grammar is described by a context-free grammar. This would open up the possibility to give an implementation of (fragments of) NL3 R using Embedded Pushdown Automata.

Figure 2 .

 2 1 shows an example unfolding of the sequent np, (np\s)/np, ((np\s)/np)\(np\s) ⊢ s

Figure 2 . 2 :

 22 Figure 2.2: The essential net corresponding to Figure 2.1

  Figure 3.1: Eliminating node c from the path from a to b

Figure 3

 3 Figure 3.2: Links excluded by the acyclicity and connectedness condition

Figure 3

 3 Figure 3.3: First order variables for 'which'

  Figure 4.1: The distribution of the total number of atomic formulas and the total number of connectives in the randomly generated derivable statements

Figure 4

 4 Figure 4.3: Evaluation of the performed versus the minimum number of axiom links

Table 2

 2

	.6: Mixed associativity and commutativity -right branch with unary
	control
	rules [MA3 l ] and [MC3 l ] of Table 2.5 can be used to give an account of Dutch
	relativization.

To give an impression of how enormous this number is, this is a lot more than the number of atoms on Earth, which is 8, 87 10 48 and a bit less than the volume of a small dwarf galaxy like NGC 1705, which has a volume of around 10 55 m 3

The only subtlety in the definition of paths in essential nets is the case for the auxiliary inputs: we want this path to start at the B formula of the [R/ i ] and [R\ i ] and end at the A formulas. Case 2 of Definition 2.2 guarantees there is a path from B to the complex formula. Given the orientation of the arrows, this path must pass through the A formula as well.

The number of different paths of an essential net E is h + p, where h is the number of hypotheses (lexical inputs) in the essential net and p the number of binary par links: the right rules for the implication each add a path because of Definition 3.3 and the left rule for product adds a path by splitting a single path into two.

Definition 3.4 Let p be a path in a correct essential net E the string σ(p) corresponding to p is obtained by traversing the path p from start to finish writing down each symbol corresponding the auxiliary input and each edge we pass.

The grammar rules extend straightforwardly. Because for the binary modes we don't generally require the antecedent to be a single formula and new category V (for 'vertex' since strings of category V contract to a single fortex) has been introduced which corresponds to the category S in the previous section. The category S in the grammar below permits us to pass the m 1 and m 2 letters when necessary.

Lemma 3.5 A ⊢ B contracts to a single vertex using the contractions iff for every path p in the corresponding essential net V → σ(p). Lemma 3.6 Γ ⊢ C contracts to a tensor tree using the contractions iff

• for every path p starting at a lexical hypothesis in the corresponding essential net S → σ(p).

• for every path p starting at an auxiliary hypothesis in the corresponding essential net V → σ(p).


Chouaib Jouf

Asymmetric price transmission along the food marketing chain: A focus on the recent price war

Keywords: Food sector, Price transmission, Asymmetry, Nonlinear ARDL model Classification JEL: C13, C22, Q11, Q13

This paper investigates the price transmission along the food marketing chain in France.

We focus on this transmission at the upstream and downstream levels during the so-called "price war" waged by the retailers in the French market. To this aim, we rely on an asymmetric cointegration approach: the Nonlinear Autoregressive Distributed Lag (NARDL) model. We find that although the asymmetric price transmission is effective along the French food marketing chain, it is more pronounced at the downstream level, illustrating that agri-food companies are the main losers of the recent war price in the food sector.

Introduction

The food sector in France consists of a wide range of companies, from upstream agriculture up to processing industry and retail. The latter (food supply) chain, including around 83,500 companies and generating more than 3 million jobs throughout the country, is strategic for the French economy, as it represents France's second largest employer and its third largest trade surplus. Nevertheless, the food sector is subject to serious competitiveness challenges (fragmented economic structure, adaptation to international demand, cost-competitiveness, investments decrease...), which have become structural. Moreover, since the mid-2000s, regulatory changes (end of quotas under the Common Agricultural Policy), upheavals on world markets, which have become much more volatile, as well as several reforms of the French legislation ("Dutreil law" in 2005, "Chatel law" and "LME law" in 2008, and "EGAlim law" in 2018) increasingly destabilise the sector.

Concerning the market of agricultural goods, gradual abandon by Brussels authorities of any form of market regulation remains its most notable upheavals. Indeed, until 2006, prices of cereals, sugar, dairy products and, to a lesser extent, beef and veal benefited from administered systems and therefore showed a certain stability. Except for a few emergency mechanisms (intervention and storage), quotas abandon in the sugar sector in 2017 can be considered as the end of stabilising policies on the agricultural markets.This implies some new business opportunities, especially for exports, but also challenges such as competition between operators and increasing volatility in food commodity prices, forcing actors of the food chain to adapt to changes. For instance, food commodity prices have risen by almost 50% since 2009, while producer prices increased at a much slower rate (9%), and retail price rises have been even more limited (about 4%). The latter developments reflect an incomplete transmission of changes in prices along the value chain [START_REF] Ferrucci | Food price pass-through in the euro area: Non-linearities and the role of the common agricultural policy[END_REF].

In particular, margins have shrunk at all levels, putting pressure on the most fragmented levels (agricultural and industrial producers).

In recent years, the instabilities of the price of agricultural goods have crystallized.

This, coupled with the recent climatic hazards, has profoundly weakened business opportunities for the sector's stakeholders while highlighting the difficulties associated with managing market volatility. This agricultural crises have significantly increased the cost of production, in a context where retail prices remain sluggish and even falling when considering mass-retail consumer products. For several years now, the market for mass consumption food products is characterized by a "price war", fuelling a deflation of product prices on the shelves, in a context where some raw materials prices are reaching historically high levels (butter, eggs, oils, cocoa, cereals). This situation reinforces the disconnection between input prices and consumer prices, causing a value destructive spiral for the entire food chain.

The link between agricultural prices and consumer prices is not immediate. Indeed, commodities represent only a share of producing costs of food products. However, the relative disconnection between commodity prices rise and retail prices fall may reveal a market failure. This latter is, particularly, linked to the competitive structure at certain stages of the chain as well as inequalities in the balance of power between shareholders at different stages of the chain. In France, the most fragmented sectors concern production (450,000 farms) and processing (more than 17,000 agri-food industries). They find themselves confronted with four purchasing groups that represent more than 80% of food purchases on the French market. These structural differences deepen the divergences in terms of market power, in a sector where each of the major retail players strives to be the most competitive on prices to attract consumers, whose purchasing power has been eroded since the 2008 financial crisis.

The "price war" consequences on the various food chain actors is a point shared by stakeholders. In such a context of high market volatility and erosion of the competitiveness of economic actors, the "Food General Statements (FGS)" was launched on 20 July 2017 by the French Minister for Agriculture and Food. In return, the FGS have motivated the "EGAlim" law of October 2018, aiming at rebalancing the initial relations between the various sector actors and ensuring a better value distribution.

This paper aims at investigating price transmission in the French food sector, by focusing more sophistically on the asymmetric aspect of price transmission at the downstream and upstream levels of the sector during the price war (over the period 01:2011-05:2019). Existing studies on the topic (very few) discuss the vertical price transmission, focusing on sub-sector activities rather than the whole French food sector. We intend to fill this gap considering the whole aggregated sector in one hand, and choosing the "war price" period as the one covered by our data sample. From a methodological viewpoint, in contrast to existing studies relying on usual linear approaches, we rely on a non-linear co-integration specification to investigate the price transmission according to increases or decreases in input prices. Therefore, our approach is based on a non-linear autoregressive distributed lag (NARDL) model, which allows us to test for asymmetry at both long-and short-run and to capture long-run asymmetry effects. Indeed, this specification, proposed by [START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF], consists of a dynamic error correction representation integrating an asymmetric long-run co-integrating regression. More important, contrary to previous literature that estimates the price transmission in first differences and thus ignores its long-run behavior, our specification allows us to capture long-run asymmetry effects.

The remaining of this paper is organized as follows. Section 2 proposes a literature review focused on studies that justify the asymmetric nature of price transmission. Section 3 describes comprehensively the data we use in our empirical analyses. Sections 4 and 5 are respectively devoted to empirical methodology description and our main empirical results. in Section 6, we highlight some potential economic consequences of our results and conclude in Section 7.

Asymmetry in the price transmission

The vertical price transmission has motivated several studies in the agri-food literature, the existence of an asymmetric farm-retail price transmission being the focus of these studies. Thereby, [START_REF] Peltzman | Prices rise faster than they fall[END_REF] argues that the asymmetry in price transmission is the rule, not the exception. The term 'asymmetric', in what follows, refers to the dependence of a price reaction at one level of the chain on the sign of price change at another level.

To explain asymmetric price transmissions, several reasons, mostly related to the downward adjustment (retail level), are pointed out. One of these is the so-called Menu Costs. 1 Blinder (1982) argues that menu cost is the main reason firms do not react to temporary price changes, therefore inducing an asymmetric price transmission. For [START_REF] Ball | Asymmetric price adjustment and economic fluctuations[END_REF], firms facing menu costs and inflation may adjust their costs depending on whether prices are rising or falling. Indeed, a positive shock tends to enlarge the gap between the actual price and the desired price leading to a fast price adjustment, while regarding a negative shock, firms don't need to adjust the prices because the inflation offsets this price cut. In the same perspective, Kovenock and Widdows (1998) explain that menu cost drives the firm's choice of adjusting prices if this latter considers the change as transitory. For instance, to avoid re-engaging customers in search behaviour, firms choose not to inform their clients about changes in the market conditions. A further key element in the firm's adjustment process is inventory management. Indeed, the accounting methods and stock management can lead to an asymmetric price transmission [START_REF] Balke | Crude oil and gasoline prices: an asymmetric relationship?[END_REF]Blinder, 1982). [START_REF] Reagan | Asymmetries in price and quantity adjustments by the competitive firm[END_REF] explain that, when facing a low demand, firms tend to adjust the quantity and increase the inventory instead of decreasing prices.

While in periods of high demand, firms will increase the price. This kind of behaviour creates an asymmetric price transmission.

Public policies and political intervention can also lead to asymmetric price transmission [START_REF] Gardner | The farm-retail price spread in a competitive food industry[END_REF]. In this vein, several studies have discussed the food chain price transmission in the Eurozone and among others the works by Ferrucci et al. (2007), [START_REF] Serra | Price transmission and asymmetric adjustment in the spanish dairy sector[END_REF] and [START_REF] Kinnucan | Asymmetry in farm-retail price transmission for major dairy products[END_REF]. Ferrucci et al. (2007) argue that the Common Agricultural Policy (CAP) plays a significant role in the price transmission, as the CAP subsidy system may lead firms to consider some shock transitory and then react differently to the price changes (increases/decreases). For instance, downstream level firms consider that a negative shock at the farm level will be offset by the CAP system, making this shock transitory. In this context, firms will respond faster to price increases than decreases leading to an asymmetric transmission. Also, governments implement some threshold prices, such as the resale threshold at a loss, inducing asymmetry in the firms' behaviour. 2 For their part, [START_REF] Kinnucan | Asymmetry in farm-retail price transmission for major dairy products[END_REF] show that if the wholesalers and the retailers believe decreases are transitory (due to government intervention), but the increases appear to be more persistent, they will react differently to increases and decreases leading to a price asymmetry. Considering the Spanish Dairy sector, [START_REF] Serra | Price transmission and asymmetric adjustment in the spanish dairy sector[END_REF] argue that the quota system generates some scarcity of milk leading to an asymmetric price transmission. Thus, downstream firms compete to increase their market share, which causes incomplete transmission of increasing the farm prices. A further explanation of the asymmetric price response particularly in the Agri-Food sector is market power. Along the food marketing chain, the stages are more and more concentrated, and firms enjoying market power influence the price adjustments to their advantage to capture profits and welfare. This market power effect is manifested through an oligopoly or oligopsony situation inducing distortions in the process of price adjustment [START_REF] Meyer | Asymmetric price transmission: a survey[END_REF][START_REF] Scherer | International competition policy and economic development[END_REF]Beily and Brorsen, 1989). Studying retail gasoline prices response to crude oil price changes, [START_REF] Borenstein | The competitive effects of transmission capacity in a deregulated electricity industry[END_REF] find results suggesting that oligopolistic co-ordination can cause a downward stickiness of retail prices, leading to an asymmetric price transmission. Another element associated with the market power effect is the collusive behaviour, discussed among others by [START_REF] Balke | Crude oil and gasoline prices: an asymmetric relationship?[END_REF] and [START_REF] Brown | Gasoline and crude oil prices: why the asymmetry?[END_REF]. These authors explain how, in a non-competitive environment, firms aiming to maintain a high profits dynamic engage in collusive behaviour, leading to an asymmetric price transmission. In order to protect their agreement, firms tend to respond to price increases faster than price decreases.

For [START_REF] Blinder | Asking about prices: a new approach to understanding price stickiness[END_REF], the search costs in locally imperfect markets is also a driver of asymmetric adjustments. In the absence of any competition, firms can react more rapidly to increases than decreases in input prices because customers do not have complete information about price changes in other firms. In the same perspective, [START_REF] Benson | An alternative view of pricing in retail food markets[END_REF] argue that the customers take into account the search costs, and their store choice is mainly based on location convenience leading to a locally non-competitive market and, therefore, to an asymmetric price transmission. In the dairy and meat markets, [START_REF] Gohin | Measuring market power for food retail activities: French evidence[END_REF] found that about one-fifth of the wholesale-retail price transmission is attributed to the oligopoly structure in the French food retail market.

Their results reject the competitive behaviour of the French retail firms. Several studies suggest that the presence of market power leads to an asymmetric price transmission, but it is not clear how this price transmission will be skewed.

All these factors may cause the transmission of prices in the Agri-Food sector to be asymmetric in the short-run (asymmetry in speed of price transmission) as well in the long-run (asymmetry in magnitude of price transmission). In particular, the joint dynamics of prices along the marketing food chain can deviate from their historical relationships in the short-run due to menu costs or inventory management (among others...) and in the long-run owing to the structure of retail markets (power market).

Data and Stylized facts

As this paper aims to investigate the price transmission at different levels of the French food marketing chain, we focus on price indices at the agricultural, industrial, and consumer levels. Thereby, it is important to offer a clear insight into the data. This section is devoted to data description.

Data

In this analysis, we intend to use mainly three variables extracted from the Institut National de la Statistique et des Etudes Economiques (INSEE) and Eurostat databases, observed between January 2011 and April 2019 (year basis, 2015). These series are monthly farm producer price index, industrial product price index, and food consumption price index. Hereafter, we provide more details on these three series.

The Farm Producer Price Index (FPPI) relies on a monthly representative sample of transaction prices; the sample covers 158 agricultural products and 46 product groups.

This index is meant to measure the change in prices received by farmers for their products.

The Food Industrial Producer Price Index (FIPP) is calculated from monthly price surveys collected from a representative sample of companies. FIPP measures the change in prices of food goods resulting from industrial food activities. Finally, the Food Consumer Price Index (FCPI) of food and beverage products is used to measure the inflation of those products. Considering two given periods, it measures the average variation in the prices of food products consumed by households. It is based on the observation of a fixed basket of food goods, annually updated. Each product is weighted in proportion to its relative weight in household expenditure dedicated to food consumption.

Glance at the data

This subsection offers a first look at the evolution of price indices at different levels of the marketing food chain in France 3 . Observing Figure 1, we note that before the 2008 financial crisis, the price series showed an upward trend. During the crisis, the farm producer price index settled at a lower level, while the food consumption price index maintained a weak upward trend. For its part, processing prices stabilized over the same period. After the 2008 financial crisis, the different price series returned to their pre-crisis dynamics but with various evolutions. In 2013, the farm producer price index continued to move towards increasingly higher price levels, while the processing and food consumption price indices are part of a stable dynamic despite the steady evolution of farm producer prices. This episode of stability coincides with the beginning in France of the "price war" that retailers are delivering on the French market. In 2015, farm producer prices considerably fell before resuming an upward trend in 2016. Over the same period, processing and 3 Table A.1 in the Appendix presents the descriptive statistics for price series as well as the number of negative and positive price changes. We can clearly see that the evolution of prices, along the agri-food marketing chain, shows as many increases as decreases over the price war period.

food consumption prices continue to move in a context of stability, slightly less pronounced for food consumption prices. The structure of the French food chain is characterized by a funnel shape, thus giving a specific market power growing along the chain. Downstream actors are taking advantage of this market power and waging a race for market shares by exerting pressure on the prices of their suppliers (farmers and processors) to keep their low pricing strategies. 4One of the most striking observations is the disconnection between the evolution of upstream prices and the rest of the actors' prices along the food marketing chain. The period 2013-2016 shows substantial variations in farm producer prices, while processing and food consumption prices remain stable or slightly upwards. This disconnection can have serious consequences on the economic health of the sector. Indeed, the absence of transmission or the low transmission of upstream prices leads to a deterioration in the margins of processors and farmers, since suppliers are charging prices disconnected from the reality of their production costs.

After a long period of stability, food consumption prices recovered slightly in 2017, marking the end of a long period of stability. This recovery is in line with the upward trend in farm producer prices, which also returned to an upward trend over the same period. Processing prices are slowing down and struggling to keep pace with the dynamics of the other players in the sector as they barely recover their upward trend. Moreover, the beginning of 2018 confirms this slowing recovery relative to that of the other players in the French agri-food chain.

Empirical methodology

Empirical literature

A relatively large literature, exploiting various model specification and methods, has been devoted to price transmission along the food marketing chain in the agricultural markets.

Meyer and von Cramon-Taubadel (2004) noticed the first attempt to assess asymmetric price transmission in [START_REF] Tweeten | Positivistic measures of aggregate supply elasticities: some new approaches[END_REF]. The latter used a dummy variable technique to distinguish the increasing and decreasing inputs prices, and in the case where the dummies' effects are statistically different, the symmetry hypothesis is to be rejected.

From this work, a technique including first differences of the prices in the specification was developed (Wolfram; 1971, Houck; 1979) and extended by [START_REF] Ward | Asymmetry in retail, wholesale, and shipping point pricing for fresh vegetables[END_REF] to include lags of the exogenous variables. Thus, the delay effects can be different depending on whether the prices are decreasing or increasing. Thereby, a differentiation can be made between two types of asymmetry, the magnitude and the speed of transmission (Boyd and Borsen, Researches have started paying more attention to the time-series properties of the prices, as many of them are non-stationary. According to Granger and Newbold (1974), with non-stationary variables, a significant relationship can be misleading insofar as this relationship is, in fact, without any economic meaning. Therefore, testing for non-stationarity and eventual cointegration characterize the new generation of models used in this literature. Among others, studies by Von Cramon-Taubadel and Fahlbush (1994); Von Cramon-Taubadel and Loy (1996); Borenstein et al. (1998) and Frost and Bowden (1999) have relied on the cointegration approach in assessing price transmission.

The standard cointegration approach, however, assumes the series to have a symmetric long-run equilibrium relationship, and the impact of positive and negative components is to be the same [START_REF] Schorderet | Revisiting okun's law: an hysteretic perspective[END_REF][START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF], making the transmission asym-metry analysis possible only in the short run. Further, it is not possible to evaluate the price transmission asymmetry with respect to the magnitude. Indeed, asymmetry in the magnitude means that the positive and the negative price changes are permanently driven apart, meaning that the series cannot be co-integrated. An alternative approach is testing for asymmetric long-run adjustment instead of the usual cointegration tests. In this vein, [START_REF] Granger | Hidden cointegration. U of California, Economics Working Paper[END_REF] introduced the "hidden cointegration" concept. Instead of considering the cointegration relationship between the main variables, one considers that the positive and negative components of the variables are co-integrated with the dependant variable, leading to an asymmetric cointegration. This approach makes it possible to test a long-run relationship between positive and negative input price changes and output price [START_REF] Granger | Hidden cointegration. U of California, Economics Working Paper[END_REF][START_REF] Schorderet | Asymmetric cointegration[END_REF].

The model

As previously mentioned, Granger and Young (2002) have shown that the symmetric cointegration is a particular case of the "hidden cointegration". In a first attempt, Schorderet

γ(L)y t = α 0 + β (L)x t + u t (1)
where γ(L) = 1 -p i=1 γ i L i and β(L) = 1 -q j=1 β j L j , with L being the lag operator and (p,q) respectively the lag order of the dependent variable (y t ) and the exogenous vector of variables (x t ) and u t is the error term.

One of the main advantage of this approach is that it performs in detecting cointegration in small samples (Romilly et al., 2001), which is the case in the present paper.

Another advantage of the ARDL approach is that it can be applied without considering the order of integration (I(0) or I(1)), which is not the case of the other cointegration approaches (Pesaran and Shin;1999). In our analysis, we will still control for the order of integration, since this technique is not valid in the case where the variables are I(2).

The approach proposed by Shin et al.( 2011) is an extension of the linear ARDL allowing for the detection of non-linearities in the short and long-run asymmetries. The first step of [START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF] method is to decompose the vector of exogenous variables into positive and negative components:

x t = x 0 + x + t + x - t (2)
where x + t and x - t represent partial sum of positive and negative changes of the exogenous variables, and are defined as follows:

x + t = t j=1 ∆x + j = t j=1 max(∆x j , 0) (3)
x

- t = t j=1 ∆x - j = t j=1 min(∆x j , 0) (4)
For the state of presentation, we consider the following nonlinear long-term regression:

y t = α + x + t + α -x - t + u t (5)
where α + and α -are the long-run parameters, associated with the positive and negative changes in the independent variable. By combining the Eq. ( 5) and the ARDL specification ( Eq. ( 1)), we get the asymmetric error correction model defined as follows:

∆y t = ρy t-1 + η + x + t-1 + η -x - t-1 + p-1 j=1 θ j ∆y t-j + q j=0 (ω + j ∆x + t-j + ω - j ∆x - t-j ) + e t , (6)
where η + = -ρα + and η -= -ρα -. The empirical procedure consists of four steps:

-The first step is the estimation of Eq. ( 6), using Ordinary Least Squares (OLS).

-The second step consists of controlling for the long-run relationship, by testing the joint null hypothesis of ρ = η + = η -using the Bounds-testing procedure proposed by [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF].

-Step three aims to test the long and short-run symmetry using a Wald test. The long-run symmetry test refers to η + = η -= η and the short-run symmetry to

q k=0 ω + k = q k=0 ω - k .
-In the final step, the asymmetric cumulative dynamic multiplier effects are derived of a unit change in the two components of the exogenous variable on the endogenous one. These effects are defined as follows:

m + h = h j=0 δy t+j δx + t and m - h = h j=0 δy t+j δx - t , h = 0, 1, 2, ... (7)

Empirical results

In this section, we estimate a non-linear autoregressive distributed lag (NARDL) model to assess price transmission along the food chain, considering the 2011-2019 period as it coincides with the "price war" in France. Indeed, this period is of a central interest insofar as the retailers have led a war of market shares, applying pressure on prices. Taking advantage of market power and in a race for low prices, retailers have pressured their suppliers, causing some non-linearity in the transmission of prices. Our aim thus is to evaluate this transmission upstream and downstream, particularly the asymmetric nature of the latter.

Although an ARDL specification can be used regardless of the integration order of the series (both I(0) and I(1)), we will perform unit root tests. It allows us to check that the series are not I(2), because in this case, the bounds testing test statistic is invalid.

We implement the Augmented Dickey-Fuller (ADF) and Phillip-Perron (PP) tests, to test for the null hypothesis of unit root. Table 1 summarises the results of unit root tests for the three series. They show that the farm producer price index and the food consumer price index are non-stationary in level but become stationary in the first difference. They are, thus, integrated of order 1. For the processing products price index, the ADF test indicates that it is stationary in level. The PP test confirms our results. 5 Since the unitroot tests suggest that the series are not I(2), we can now proceed to the bounds testing procedure. Next, we accordingly estimate an unrestricted non-linear ARDL specification, for the upstream and downstream levels, as described by the following equations:

∆lf cp t = µ + ρlf cp t-1 + η + lipp + t-1 + η -lipp - t-1 + p-1 j=1 θ j ∆lf cp t-j + q j=0 (ω + j ∆lipp + t-j +ω - j ∆lipp - t-j ) + e t , (8)
∆lipp t = µ + ρlipp t-1 + η + lf pp + t-1 + η -lf pp - t-1 + p-1 j=1 θ j ∆lippi t-j + q j=0 (ω + j ∆lf pp + t-j +ω - j ∆lf pp - t-j ) + e t , (9)
The equation ( 8) describes the downstream specification, with f cp t and ipp t standing for Food Consumption Price and Industrial Product Price, respectively. As for the equation ( 9), it describes the upstream level dynamics, with f pp t standing for Farm Producer Price. All the variables are in logarithms.

The final specification is chosen, based on Akaike Information Criteria, by starting with max p=max and q =12 and dropping all non-significant regressors. 6 The cointegration test applied to the equations ( 8) et ( 9) consists of testing the joint null hypothesis that the coefficients of the lagged variables are equal to zero: ρ = η + = η -= 0.

Table 2 reports the Bounds F-statistics for both models (upstream and downstream models). The results reveal that the two variables, in each specification, co-move in the long-run. Indeed, the F-statistics are respectively equal to 10.57 and 8.76 for the downstream and upstream equations, which are above the critical upper bound. 7 These results allow us to assess the relationship between the food consumption price and the positive and negative changes in industrial product price, on the one hand, and the industrial product price and the positive and negative changes in farm price, on the other hand. Note : The exact specification of the two models are reported in Tables 3 and4.

Following these tests, we estimate the two specifications described by equations ( 8) and ( 9), representing the upstream and downstream dynamics of the agri-food sector.

Downstream specification

To test the non-linear specification adequacy of the relationship between the food consumer price index and the processing price index, we use the Wald test to assess the symmetry of the long and short-run coefficients. The lower part of Table 3 reports the results. In the long-run, The Wald test rejects the null hypothesis of symmetry between the coefficients of the positive and negative components of the processing price index. Indeed, the Wald test shows a p-value of 0.000 below the 5% threshold, suggesting rejection of the null hypothesis, and thus confirming the asymmetric nature of the relationship between the food consumer price index and the positive and negative components of the processing price index. This result supports the hypothesis that a linear specification is inadequate in modeling this relationship in the long-run.

For the short-run dynamic analysis, the Wald test suggests the rejection of the null hypothesis of short-run symmetry: 3). In other words, it rejects the null hypothesis of a weak form additive symmetric adjustment of the processing price index. 8Once asymmetry has been established, we focus on the long-run dynamic of downstream price transmission. As indicated by the results reported in Table 3, the long-run coefficients of the changes in the processing price index are significant. The estimated longrun coefficients of the positive and negative changes in the processing price are -1.62 and -2.027, respectively. The latter are both negative, indicating that, regardless of changes in processing prices, food consumer prices experience a decreasing trend. Moreover, our estimations also show that the greater effect is sourcing from negative changes since the coefficient of negative changes is higher (-2.027) in magnitude than that of positive changes (-1.62). We can also note the presence of asymmetry in the speed transmission of food consumer prices to changes in supplier prices. Indeed, consumer prices react more quickly to the rise than to the fall in processing prices, as shown by the results in Table 3.

q i=0 π + i = q i=0 π - i (Table
We can extend the analysis of the relationship between food consumption prices and processing prices with an analysis of dynamic multipliers. Figure 2 shows the dynamic response of food consumer prices to changes in the positive and negative components of processing prices. On the one hand, we observe that consumer prices react faster to the rise than to the fall in processing prices. On the other hand, we note that asymmetry is relatively unstable in the short-run. Indeed, during the first year, we observe that the asymmetry curve oscillates between positive and negative values.

Our results confirm the asymmetric dynamics of the transmission in the long-run. Indeed, we find that, over the 2011-2019 period, consumer prices always react downwardly to changes in processing prices, whether downward or upward. Our results also provide evidence of a dominant effect of downward trends, meaning that the different upstream price changes are not reflected symmetrically downstream. A possible explanation is the "price war" that the French agri-food environment has experienced. At the beginning of more appropriate insofar as excluding the non-significant stationnary regressors introduces heterogeneity in the lags of the positive and negative components and the dependent variable. Note: This table reports the estimation of Eq.( 9). The estimated long-run coefficients associated to the positive and negative components are defined by η ρ . Long-run effect refers to a negative change in exogenous variable. The retained specification has been determined following the general-to-specific approach (Hendry, 1995).

2013, the retailers waged a market share war. In their races for low prices, they exerted downward pressure on consumer prices, explaining the disconnection between processing prices and food consumption prices. Benefiting from very significant market power, retailers reinforced their negotiation strategies with arguments based on their economic performance on the market, constrained by the aggressiveness of competition in the retail sector. 9 Indeed, under pressure from increased competition, retailers tend to lower their costs, i.e. the purchase prices of industrial products. This strategy puts considerable pressure on their suppliers and tariffs in an economic context already stressed by the high volatility of agricultural and energy input prices. This "price war" has several consequences for the various actors in the agri-food sector. On the one hand, consumers benefit from a relatively low and stable general level of food prices. On the other hand, it puts economic pressure on producers and processors. Indeed, agri-food companies have a declining margin rate that has reached its lowest level in 40 years. Note: Figure 2 shows the predicted dynamic multipliers for the adjustment of food consumption prices to positive and negative changes in food processing prices. The asymmetry curve depicts the linear combination of the dynamic multipliers associated with positive and negative changes. Lower band and upper band for asymmetry indicate the 95% confidence interval. "CI" stands for Confidence Interval. 9 Some recent studies have shown that retailers are enjoying increasing market power over food processors/farmers through the food marketing chain (European Commission 2009; Saitone and Sexton 2012).

Upstream specification

In the following, we investigate the upstream prices' transmission in the marketing chain.

We assess the response of processing prices to positive and negative changes in farm prices. First, we assess the adequacy of the non-linear specification adopted. The Wald test reported in Table 4 indicates that in the long-run, the null hypothesis of symmetry is rejected at the 5% level suggesting asymmetry in the magnitude of price transmission. For the short-run behaviour, the Wald test shows a p-value of 0.063, above the 5% threshold. This last result suggests that at the 5% threshold, the null hypothesis of weak additive symmetric adjustment is not rejected. In other words, processing prices react symmetrically to changes in the positive and negative components of farm producer prices in the short-run, indicating a symmetric speed adjustment of prices. In the end, the dynamics upstream of the supply chain are symmetrical in the short-run, while in the long-run the transmission is asymmetric.

Discussing the long-run dynamics, the results in Table 4 show that the long-run coefficients are significant and also exhibit signs in line with the literature. Indeed, the estimated effects of the negative and positive components are respectively 0.33 and -0.376.

As mentioned above, the Wald test indicates an asymmetry in the transmission of prices upstream of the sector, thus implying that decreases in farm producer prices are relatively more transmitted than increases to processing prices. Figure 3 displays the dynamic effects of farm producer prices on the processing prices, and confirms the absence of asymmetry in the short-run. In the first 12 months, the absolute effect of a decrease in farm prices is greater than the effect of an increase. From the second year onwards, we notice that the effect of asymmetry tends to approach a situation of neutrality. Nevertheless, the persistence of asymmetry remains significant, and we observe that the absolute effect of a decrease dominates the effect of an increase in processing prices. Overall, we note that in the long-run, despite the persistence of asym- 9). The estimated long-run coefficients associated to the positive and negative components are defined by η ρ . Long-run effect refers to a negative change in exogenous variable.The retained specification has been determined following the general-to-specific approach (Hendry, 1995) metry in upstream price transmission, it remains relatively low. In other words, producers can track price changes for their products with slightly more difficulties when it comes to price increases.

During the "price war" period, despite the asymmetric nature of upstream price transmission, this upward rigidity remains relatively low. Thus, food processors, to a certain extent, take into account the economic conditions of their suppliers. At the same time, they are under enormous downstream pressure, which explains the low downward rigidity of the downstream price transmission. In a vice between a fragile upstream and a downstream price war, agri-food companies are, in a way, the adjustment variable in the price transmission process along the agri-food marketing chain. To check that the strong asymmetry in the downstream prices' transmission is specific to the price war period, we have estimated the same specifications over a broader period, namely 2005-2019. 10 First, we note that the upstream price transmission dynamic over the 2005-2019 period is similar to that observed during the price war, and in line with the literature (Meyer and von Cramon-Taubadel, (2004); Vavra and Goodwin, (2005)).

Second, the results of the downstream level, on the broad sample, are utterly opposed to 10 The estimation results are reported in Appendix B (tables B.3 and B.4).

our results for the price war period. These results confirm our central intuition. They confirm that the previous results are, indeed, specific to this period and that these effects are mainly due to the price war that characterized the French food sector over the 2011-2019 period.

Economic consequences

Asymmetry analysis in price transmission along the French food marketing chain, highlighted in this paper, reveals different economic consequences.

Our results show a disconnection between processing prices and food consumption prices downstream of the chain. Indeed, consumer prices are mostly adjusting downwards to the various variations in the prices of industrial suppliers. The price war that retailers have been waging since 2013 mainly explains this situation. Given the market power enjoyed by retailers, 11 on the French market, this asymmetry in price transmission may suggest an economic pressure on the industrial actors of the sector, characterised by a more fragmented structure.

With a buyer power benefiting to downstream actors, the conditions for commercial negotiations are unbalanced. Thus, in the scheme of an annual commercial negotiation, suppliers are unable to transmit their tariffs variations, leading to a disconnection between their negotiated tariffs and their production costs. In a context of high input price volatility, this disconnection has significant consequences on agri-food companies' margins. Indeed, in recent years, the margin rate of agri-food companies has shown a sharp downward trend, reaching historically low levels. Figure 4 compares the changes in margin rates in the manufacturing industry and the agri-food industry. Both series show a downward Note: The margin rate represents the ratio of gross operating margin to added value. The margin rate reflects what remains available to companies -the gross operating surplus -in particular as a return on capital, after deducting remuneration of employees. Source: INSEE trend during the crisis and post-crisis period, and they then recover an upward trend, slightly less pronounced for the agri-food industry. Towards the end of 2018, and after a long "price war" period, the margin rate of agri-food companies reached an unusually low level and for the first time, fell below the margin rate level of the manufacturing industry.

On the one hand, the contraction of agri-food companies' margins leads to a significant reduction in investments. In addition to this quantitative investment deficit, the investment orientation of agri-food companies may often appear defensive. In such conditions, companies would invest less in long-run projects such as innovation or modernization (robotics, digital...), investments that can ensure sustainable value creation. On the other hand, companies are more likely to make investments to comply with standards or to invest in the maintenance and upkeep of the production equipment, especially since the aging of the latter in France is a real brake for industrial growth. From a macroeconomic perspective, the renewal of production capacities is now the main reason for investment, in a context where the aging of the production tool increasingly limits the ability of the agri-food sector to rebound and, more generally, the competitiveness of companies.

The loss of export market share can measure this setback of competitiveness. While it is still largely positive (around 7 billion euros in 2019), the trade surplus of the agri-food industry has been declining steadily in recent years. Also, this surplus is sustained by a few sectors, in particular beverages or milk. Beverages excluded, the agri-food industry has a historically high trade deficit in exports (Fig. 5), particularly against their European competitors.

Conclusion

The present paper investigates the price transmission throughout the agri-food sector marketing chain in France, specifically the downstream transmission during the so-called "war price". This investigation contributes twofold to the literature, first by focusing on the recent price war in the French food sector and second by using a non-linear approach, more precisely the asymmetry, to evaluate the price transmission in the aggregate food sector in France. To this end, we exploit data on prices at the three levels of marketing chain: the food consumption price, the agricultural product price, and the industrial product price indices over the period from January 2011 to April 2019. Firstly, our results suggest the presence of a non-linear downstream price transmission in both the short and long-run. The long-run reaction of food consumption prices to the industrial product price variations is asymmetric. Indeed, regardless of the sign of the industrial price variation, the consumption price reaction is downward, which is a consequence of the price war. Regarding the short-run dynamic, we also evidence an asymmetric behaviour of the price transmission, indicating an asymmetry in the speed of price transmission.

Secondly, we evaluate the upstream price transmission and find that the long-run effects are asymmetric and skewed to the left, indicating that there are increasing rigidities. In the short-run, there is no evidence of asymmetry, which means that the prices' pass-through is relatively the same during increases and decreases episodes. The difference in price transmission elasticities for positive and negative shocks is more significant for the downstream level than the upstream one, indicating that the long-run asymmetry is higher for shocks spread out from the processing level than the ones from the farm level.

These results suggest that the main factor of this asymmetric dynamic is the price war along the food chain, induced by the abuse of market power of the downstream actors. It is also important to mention that other factors might have contributed to the asymmetric behavior of price transmission. Among these factors, we can name the increase of import prices of agricultural and food products, the cost of various structural changes in consumer demand (origin of products, additives, pesticides, etc.). Finally, our results indicate that the linear hypothesis to describe the price transmission in the French food sector could be misleading. We shed light on the relevance of taking into account non-linearity in price behaviour for the aggregated sector. We are aware that a more disaggregated analysis could be interesting, as providing insights into specific sectors. Indeed, the heterogeneity of different food sectors could be of a central interest, and an investigation with sub-sectors-level data will probably be more informative.

Figure 1 :

 1 Figure 1: Food prices at the three marketing chain levels (100 = 2015)

  1988; Punyawadee et al., 1991; Mohanty et al., 1995; Aguiar and Santana, 2002).

(

  2003) proposes with a bivariate regression allowing for asymmetric cointegration. In 2011, Shin et al. (2011) proposed an asymmetric cointegration approach mainly based on the linear Autoregressive Distributed Lag (ARDL) model developed by Pesaran and Shin (1999) and Pesaran et al. (2001). The general form of linear ARDL is defined as follows:

Figure 2 :

 2 Figure 2: Food processing prices cumulative effects on Food consumption prices.

Figure 3 :

 3 Figure 3: Farm producer prices cumulative effects on processing prices

Figure 4 :

 4 Figure 4: Manuafturing and Food industries margin rate.

Figure 5 :

 5 Figure 5: Trade deficit in the French food sector (Beverages excluded).

Table 1 :

 1 Augmented Dickey-Fuller unit root test

		Test statistic		Test statistic 5% critical value
				Trend
	lfpp	-1,43	∆lfpp	-7,09
	lipp	-4,135	∆lipp	-5,917	-3,452
	lfcp	-2,733	∆lfcp	-8,929
			No trend
	lfpp	-1,319	∆lfpp	-7,109
	lipp	-4,864	∆lipp	-5,852	-2,892
	lfcp	-1,794	∆lfcp	-8,945

Note: fpp, ipp and fcp stand respectively for Farm Producer Price, Industrial Product Price and Food Consumption Price. All the variables are in log. Lags orders are chosen using the Akaike Information Criterion

Table 2 :

 2 Bounds testing cointegration test

	95% lower bound 95% upper bound	Result

Table 3 :

 3 Downstream transmission specification

		Coefficients t-value	p-value
	Constant	0.564	4.24	0.000
	Trend	0.0005	5.41	0.000
	lf cp t-1	-0.283	-4.24	0.000
	lipp + t-1	-0.459	-5.24	0.000
	lipp -t-1	0.573	4.60	0.000
	∆lf cp t-10	-0.235	-2.23	0.029
	∆lf cp t-11	-0.208	-2.10	0.040
	∆lipp + t-1	0.787	3.01	0.004
	∆lipp + t-2	0.706	2.60	0.011
	∆lipp + t-7	1.013	3.93	0.000
	∆lipp + t-8	0.564	2.29	0.025
	∆lipp + t-9	0.529	2.12	0.038
	∆lipp + t-10	0.409	2.01	0.048
	∆lipp -t-2	-0.706	-2.88	0.005
	∆lipp -t-4	-0.686	-2.93	0.005
	∆lipp -t-5	-0.623	-2.66	0.010
	∆lipp -t-7	-0.523	-2.03	0.047
	∆lipp -t-12	-0.68	-3.14	0.002
	Long-run coefficients
		Coefficient F-Stat	p-value
	lipp +	-1.620	20.18	0.000
	lipp -	-2.027	36.24	0.000
	Asymmetry test	Diagnostics test
	Long-run	30.91 0.000	χ 2 sc	33.1 0.771
	Short-run	43.26 0.000	JB 1.45 0.484

Table 4 :

 4 Upstream transmission specification

		Coefficients t-value	p-value
	Constant	0.236	3.83	0.000
	lipp t-1	-0.117	-3.81	0.000
	lf pp + t-1	0.038	3.82	0.000
	lf pp -t-1	0.044	4.37	0.000
	∆lipp t-2	0.161	1.74	0.087
	∆lf pp + t	0.081	2.73	0.008
	∆lf pp + t-2	-0.110	-3.48	0.001
	∆lf pp -t	0.073	2.02	0.047
	∆lf pp -t-2	0.135	3.31	0.001
	∆lf pp -t-3	0.073	2.09	0.040
	∆lf pp -t-4	0.084	2.12	0.037
	∆lf pp -t-6	-0.118	-3.2	0.002
	∆lf pp -t-11	-0.060	-1.98	0.051
	Long-run coefficients
		Coefficients F-Stat	p-value
	lf pp +	0.330	31.11	0.000
	lf pp -	-0.376	41.95	0.000
	Asymmetry test	Diagnostics test
	Long-run	6.533 0.013	χ 2 sc	35.22 0.685
	Short-run	3.573 0.063	JB	1.66 0.435

Note: This table repoorts the estimation of Eq.(

"Menu cost" is the cost to a firm resulting from changing its nominal prices, such as reprinting catalogs, labeling, etc.(Barro; 1972)

For instance, the French government increased, in 2018, the resale threshold at loss, to assure a better price transmission and rebalance the trade negociation at the downstream level of the food marketing chain.

The concentration of downstream purchasing groups in France is very high, the four leading purchasing groups control more than 90% (compared to five purchasing groups for about 65% of the market in Germany).

See Appendix A, Table A.2

In practice, the inclusion of insignificant lags induce some inaccuracies and may introduce noise into the dynamic multipliers

See Pesaran, M. H., Shin, Y., and Smith, R. J. (2001): Bounds testing approaches to the analysis of level relationships. Journal of applied econometrics, 16(3), 289-326.

We retain the weak form additive symmetric adjustment in the short run asymmetry test, because it is

Competition authority, document No. 15-A-06 of 31 March 2015 on the purchasing groups and referencing in the mass retail sector.

Appendix A 9). The estimated long-run coefficients associated to the positive and negative components are defined by η ρ . Long-run effect refers to a negative change in exogenous variable.The retained specification has been determined following the general-to-specific approach (Hendry, 1995) 9). The estimated long-run coefficients associated to the positive and negative components are defined by η ρ . Long-run effect refers to a negative change in exogenous variable.The retained specification has been determined following the general-to-specific approach (Hendry, 1995)

Whole sample estimation results

Dynamic cumulative effects