Julien Chevallier

Johanna Etner
email: johanna.etner@univ-paris5.fr

Pierre-André Jouvet
email: pjouvet@u-paris10.fr

Bankable Pollution Permits under Uncertainty and Optimal Risk Management Rules: Theory and Empirical Evidence *

Keywords: Firm behavior, Tradable permits, Policy risk, EU ETS JEL Codes : D21, D80, Q58

The well known economic advantage of tradable permits over command and control obviously vanishes if firms do not trade because of regulatory uncertainty. In fact, uncertainty about political decision changes in the permits program could make firms reluctant to participate in tradable permits markets. Based on a two-period partial equilibrium framework, our results suggest that the banking provisions may be used as a tool of policy risk control and that it is possible to define optimal risk sharing rules in order to respond to political decision changes. Finally, our empirical discussion attempts to put the theoretical results concerning firms' banking and pooling behaviors in the context of the recent development of the European Union Emisions Trading scheme (EU ETS).

Introduction

Pollution permits are now widely considered as efficient instruments for regulating the emissions of pollutants by firms. Their numerous advantages have been extensively discussed in the literature [START_REF] Bohm | Comparative Analysis of Alternative Policy Instruments[END_REF], [START_REF] Pearce | Economic of Natural Resources and the Environment[END_REF], [START_REF] Cropper | Environmental Economics : a Survey[END_REF], [START_REF] Koutstaal | Economic Policy and Climate Change : Tradable Permits for Reducing Carbon Emissions[END_REF], Baumol and Oates (1998)). However, pollution permits may also convey a high level of uncertainty with respect to political decisions enforced by the regulator. While in the case of a tax the political uncertainty concerns the level of the tax, in the case of a pollution permit uncertainty depends not only on its price but also on allocation rules enforced by the regulator.

Hence, the informational efficiency argument1 in favor of pollution permits compared to other classic instruments2 vanishes given this potential higher level of uncertainty at the firm-level linked to the risk of political decision changes 3 . To cope with these political uncertainties, a number of firms may not participate in the permits market and express their fear of an environmental regulation system dependent on such shifts in the regulatory environment [START_REF] Wossink | Environmental Policy Uncertainty and Marketable Permit Systems: the Dutch Phosphate Quota Program[END_REF]). [START_REF] Hahn | Economics prescription for economic problems: How the patiant followed the Doctor's orders[END_REF] first stressed the potential negative effects of political uncertainties of pollution permits systems. He emphasizes that the advantages of permits schemes in terms of pollution control may be undermined by political uncertainties regarding banking and trading provisions. [START_REF] Leston | Investment Decisions and Transferable Discharge Permits: an Empirical Study of Water Quality Management under Policy Uncertainty[END_REF], [START_REF] Stavins | Transaction Costs and Tradable Permits[END_REF] and Ben-David et al. (1999) have also underlined that the performance of pollution permits is critically linked to the clarity of political decisions.

In this article, we examine firms' production decisions subject to the introduction of a pollution permits market as an environmental regulation tool and to the possibility to bank permits forward in a partial equilibrium framework. At the beginning of each period, firms receive an initial permits allocation. Without uncertainty on the next period allocation, firms smooth their emissions between trading periods as already documented in the literature [START_REF] Rubin | A model of Intertemporal Emission Trading, Banking, and Borrowing[END_REF], [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF], [START_REF] Leiby | Intertemporal Permit Trading for the Control of Greenhouse Gas Emissions[END_REF]).

Departing from this benchmark case, the introduction of uncertainty on future allocation provides incentives for firms to bank permits and thus to reduce their emissions of pollutants. This banking behavior also changes the temporal pattern of emissions by decreasing the concentration of emissions on early periods 4 . Since it overcomes potential negative effects, the authorization of banking therefore appears as a decisive feature for the successful implementation of permits systems 5 .

The central question we address is the following: will an increase in the level of uncertainty concerning future allocation impact positively or negatively the amount of banking by firms? We aim at detailing firms' behavior following a variation in the level of uncertainty and whether it is possible to identify an optimal risk sharing rule between firms. That is why we choose to focus our analysis on the banking provisions and consider that permits trading between firms has already occurred.

Compared to previous literature, the main theoretical results of this article are threefold. First, we show that when firms face an increase in the level of risk, the variation of the amount of banked permits is linked to the third derivative of the production function with respect to emissions. This result is close to the notion of prudence derived by [START_REF] Kimball | Precautionary Savings in the Small and in the Large[END_REF]. Second, if the goods used in the production function are complements (substitutes), firms increase (decrease) their inputs consumption as a response to this increase in the risk level. Third, without uncertainty on the total number of permits allocated during the second period, an agency may introduce a Pareto-optimal permits re-allocation between firms. When the regulatory uncertainty concerns the number of permits available during the second period, an optimal risk-sharing rule needs to take into account the sensitivity of firms' marginal productivity to the number of permits, as well as the elasticity of the marginal productivity with respect to the stock of pollution.

Finally, we provide empirical insights into these findings by investigating the banking behavior at the installation level and the pooling behavior at the group level in the light of the emerging European carbon market. We confirm the impact of different allocation rules and overall regulatory uncertainty on the variation of firms' banked permits, and the existence of risks-pooling by parent companies to save penalty and permits purchases costs.

The remainder of the article is organized as follows. Section 2 details the behavior of firms. Section 3 examines risk management strategies between firms and proposes an optimal risk sharing rule. Section 4 provides an empirical discussion. Section 5 concludes.

Behavior of Firms

We analyze in this section the behavior of firms with a two-period timehorizon for production decisions. Since we focus on the effects of uncertainty on firms' banking behavior, we thus assume that permit trading between firms has already occurred. During the first period, t, firms receive a permits allocation noted Pt . This initial allocation may be used for production, but also banked for the next period. During the second period, firms receive a permits allocation noted Pt+1 .

During each period, each firm produces a good with a given production technology by using X t input 6 and P t pollution permits. Thus, following the introduction of pollution permits at time t, each firm uses a quantity of environment P t additional to its input quantity X t in order to produce a good Y t :

Y t = F (X t , P t ) (1)
The quantity of environment P t simply states the number of pollution permits needed by the firm to produce and cover its pollutant emissions level. The production function is strictly concave for each of its arguments and the second non-crossed derivatives are negative (F ii < 0).

Behavior of Firms without Uncertainty

The firm maximizes its intertemporal profit as a function of its inputs, X t and X t+1 , and the choice of using pollution permits P t and P t+1 . Let Pt and Pt+1 be the permits allocated to firms and S t the permits bank computed as the difference between the initial permits endowment and the number of permits used by the firm 7 , S t = Pt -P t . During the second period, the firm may use 6 X t can be a vector of inputs, i.e. X t = {X 1 t , ..., X k t with k ≥ 1. To ease the presentation, we consider the case where k = 1. 7 In this article, we only consider the possibility to bank permits, i.e. at non-negative levels of S t . If S t < 0, then the firm would be allowed to borrow permits. Note that relaxing the non-negativity constraint on borrowing does not change qualitatively the results obtained.

its permits endowment plus its permits bank accumulated during the first period. Noting β the discounting factor used by the firm, the intertemporal profit may be written as:

Π t = π t + βπ t+1 with π t = F (X t , P t ) -R t X t and π t+1 = F (X t+1 , P t+1 ) -R t+1 X t+1 + q t+1 ( Pt+1 + S t -P t+1
) with R t and R t+1 the inputs prices, q t and q t+1 the pollution permits prices.

The optimization program of the firm is :

max Xt,X t+1 ,St≥0,Pt,P t+1    F (X t , P t ) -R t X t +β F (X t+1 , P t+1 ) -R t+1 X t+1 + q t+1 ( Pt+1 + S t -P t+1 )  



With P t = Pt -S t , the first order conditions are:

F Xt (X t , P t ) = R t (2) F X t+1 (X t+1 , P t+1 ) = R t+1 (3)
F Pt (X t , P t ) -βq t+1 ≤ 0 ; = 0 if S t > 0 (4)
F P t+1 (X t+1 , P t+1 ) -q t+1 = 0 (5)
From eq.( 4), we know that the firm will bank permits if the marginal cost of banking is inferior to the anticipated permits price. Combining eq.( 4) and

(5), we have:

F Pt (X t , P t ) = βF P t+1 (X t+1 , P t+1 ) (6)
From eq.( 2), ( 3) and ( 6), we obtain the effects of the variation of the number of permits allocated on the firm's banking behavior during either of the two trading periods. Banking is an increasing function of the first period permits allocation, dS t /d Pt > 0, and a decreasing function of the second period permits allocation, dS t /d Pt+1 < 0. When the number of permits allocated during the first period increases, the firm may increase both the number of permits banked and used, thereby also increasing its present and future production levels. When the number of permits allocated during the second period increases, the firm may increase its production level during both trading periods by using more permits during the second period and banking less during the first period.

In this section, we have developed the basic model underlying our analysis.

In the next section, we study the effects of introducing uncertainty on the number of permits allocated during the second period.

Behavior of Firms under Uncertainty

In this section, we assume a random second period permits allocation, noted Pt+1 with a probability distribution G(.). The randomness reflects changing permits allocation rules by the regulator impacting the second period permits allocation. Only at the beginning of the second period does the firm know its permits allocation. Thus, at time t + 1, the firm knows its amount of permits Pt+1 endowed and may decide on its inputs uses, production level and associated emissions of pollutants. However, at time t, this amount is not known with certainty. We assume here that the firm anticipates an average amount of permits distributed during the second period equal to Pt+1 .

Thus, when there is uncertainty on the second period allocation, the expected intertemporal profit, E Π t = π t + βE π t+1 is:

Π t =    F (X t , P t ) -R t X t +βE F (X t+1 , P t+1 ) -R t+1 X t+1 + q t+1 ( Pt+1 + S t -P t+1 )   
The choice of the firm indeed follows two steps. In a first step, the firm chooses S t and X t by taking into account the uncertainty over the total number of permits distributed in the future. In a second step, the firm chooses X t+1 and P t+1 with P t+1 ≤ Pt+1 + S t given its choices during the first period. Let us solve this program by backward induction.

Choice of X t+1 and P t+1 with S t and Pt+1 given max

X t+1 ,P t+1 π t+1 = β F (X t+1 , P t+1 ) -R t+1 X t+1 + q t+1 ( Pt+1 + S t -P t+1 )
The first order conditions are:

F X t+1 -R t+1 = 0 (7)
and

F P t+1 -q t+1 = 0 (8)
We then obtain the level of profit during the second period, π t+1 as a function of the permits allocation Pt+1 and the bank S t :

π t+1 ( Pt+1 ) = F (X t+1 ( Pt+1 , S t ), P t+1 ( Pt+1 , S t )) -R t+1 X t+1 ( Pt+1 , S t ) +q t+1 ( Pt+1 + S t -P t+1 ( Pt+1 , S t ))
Choice of X t and S t with the introduction of a random permits allocation,

Pt+1 max Xt,St F (X t , Pt -S t ) -R t X t + βE π t+1 ( Pt+1 )
The optimality conditions are:

F Xt = R t (9)
F Pt -βE[q t+1 ] = 0 (10)
Combining eq.( 8) and [START_REF] Ellerman | Over-Allocation or Abatement? A Preliminary Analysis of the EU ETS Based on the 2005-06 Emissions Data[END_REF], we obtain an expected condition similar to the case without uncertainty, i.e.:

-F Pt (X t , Pt -S t ) + βEF P t+1 (X t+1 , Pt+1 + S t )) = 0 (11)
Without a variation of the level of uncertainty, the firm's behavior is simply based on the expected profit and we derive similar results to Section 2.1. Thus, it appears important to investigate the consequences of a change in the risk level associated with the second period permits allocation. To this purpose, we consider an increase in risk in the sense of [START_REF] Rothschild | Increasing Risk: I. A Definition[END_REF] and study the effects of this change in the probability distribution on the firm's banking and inputs consumption choices. The effects of the variation of the risk associated to the banking variable S lead to the following result:

Proposition 1 For a given level of inputs, in response to an increase in risk in the sense of the mean preserving spread, the banking of pollution permits by the firm increases (decreases) if and only if the third derivative of the production function with respect to the emissions, F P P P , is positive (negative).

Proof. Considering a given level of inputs, X t , X t+1 , from eq.( 11) with the distribution of probability, G(.), we obtain:

-F Pt (X t , Pt -S G t ) + βE G F P t+1 (X t+1 , Pt+1 + S G t )) = 0 ≡ H G (S G t )
Considering a distribution of probability K(.), where K(.) is a mean preserving spread of G(.), eq.( 11) gives:

-F Pt (X t , Pt -S K t ) + βE K F P t+1 (X t+1 , Pt+1 + S K t )) = 0 ≡ H K (S K t )
Using the second order condition of the optimization program, we have

S K t > S G t if and only if H K (S G t ) > H G (S G t ). Then, S K t > S G
t if and only if:

E K h(S G t ) > E G h(S G t )
where h(S) = F P t+1 (X t+1 , Pt+1 +S). This inequality is verified if and only if h(S) is convex:

h (S) = F P t+1 P t+1
and h (S) = F P t+1 P t+1 P t+1

The conditions on the third derivative of the production function with respect to emissions indeed relate to the study of the concavity of this function. It is worth noting the similarity between this condition and the conditions associated with the third derivative of utility obtained by introduced by Kimball (1990) 8 when studying individuals' risk behavior while, in our framework, firms are risk-neutral.

Concerning the effects of the variation of the risk level associated with inputs choices, for a given S we have:

dX t = - F XP F XX dS t . (12)
Thus, the variation sense of the input quantity X t depends on the sign of the crossed derivative of the production function (assuming that F XX < 0).

We obtain the following result:

Proposition 2 When banking permits is an increasing function of risk, if the goods used in the production function are complements (substitutes), F XP > 0 (F XP < 0), then the firm increases (decreases) its inputs consumption, X, as a consequence of an increase in risk.

In this section, we have demonstrated that the variations of firm's inputs choices and banked permits depend on its characteristic production function with respect to concavity. Firms with heterogeneous characteristics on their third derivative may adopt dramatically different behaviors in terms of banked permits. We explore in the next section whether heterogeneous firms may pool risks through the intermediation of an agency managing the permits. If such an option exists, then we investigate what may be the optimal risk sharing rule between firms.

8 See the notion of prudence.

Risk Management Strategies

Let us detail first the risks pooling behavior and second the optimal risksharing rule between firms.

The Pooling Behavior

In order to study a risk sharing rule between firms, we assume that in partial equilibrium there exists N firms and Θ states of nature 9 . Note P iθ t+1 the permits allocation that firm i receives in the state of nature θ, θ ≤ θ ≤ θ, with a realization probability µ θ . The optimization program of firm i may be written as:

max Xt,X t+1 ,St,P t+1    F i (X i t , P i t -S i t ) -R t X i t +β Θ θ=0 µ θ F i (X i t+1 , P t+1 ) -R t+1 X i t+1 + q t+1 ( P iθ t+1 + S i t -P i t+1 )   
The pooling behavior implies the introduction of a cooperation agency 10 between firms which is responsible to maximize the sum of firms' profits whatever their states of nature. This agency will thus take into account the sum of firms' permits allocations over the two periods:

i N P i t = i P i t + S t (13)
9 By considering a partial equilibrium framework, we assume that the N firms constitute a sub-sample of the firms subject to the pollution permits market. 10 This agency may either correspond to a "parent agency" with N subsidiaries or to a centralization of production decisions. This latter type of pooling corresponds to common practices for consumers' mutual insurance companies (see [START_REF] Gollier | The economics of risk and time[END_REF]).

and i P iθ t+1 + S t = i P iθ t+1 , ∀θ ∈ Θ (14)
Substituting S t in eq.( 13) and ( 14), we obtain the following constraint for the agency:

i [ P i t + P iθ t+1 ] = i P i t + i P iθ t+1 ≡ P θ , ∀θ ∈ Θ (15)
The agency's program consists in maximizing the sum of profits by choosing firms' inputs levels (X i t and X iθ t+1 ) as well as the use of pollution permits (P i t and P iθ t+1 ) for all states of nature . As the agency takes into account the sum of firms' profits, and given that the sum of permits sales must be equal to the sum of permits purchases, the agency's program may be written as:

max {X i t ,X iθ t+1 ,P i t ,P iθ t+1 } i,θ i    F i (X i t , P i t ) -R t X i t +β Θ θ=0 µ θ F i (X iθ t+1 , P iθ t+1 ) -R t+1 X iθ t+1   
subject to the constraint in eq. [START_REF] Kling | Bankable Permits for the Control of Environmental Pollution[END_REF]. Noting λ θ the Lagrange multiplier of the constraint in the state θ, we obtain the following first order conditions for all i and for all θ ∈ [0, Θ]:

F i X i t (X i t , P i t ) = R t ( 16
)
F i X iθ t+1 (X iθ t+1 , P iθ t+1 ) = R t+1 (17)
F i P i t (X i t , P i t ) = θ λ θ ( 18
)
βµ θ F i P iθ t+1 (X iθ t+1 , P iθ t+1 ) = λ θ (19)
and

λ θ i [ P i t + P iθ t+1 ] - i P i t - i P iθ t+1 = 0 (20)
We may identify Borch's condition applied to firms and the reciprocity principle. At the optimum, the marginal rates of technical substitution of firms i and j between two states of nature, θ 1 and θ 2 , are equal:

F i P iθ 1 t+1 (X iθ 1 t+1 , P iθ 1 t+1
)

F i P iθ 2 t+1 (X iθ 2 t+1 , P iθ 2 t+1 ) = F j P jθ 1 t+1 (X jθ 1 t+1 , P jθ 1 t+1 ) F j P iθ 2 t+1 (X jθ 2 t+1 , P jθ 2 t+1 ) ∀i, j, θ 1 , θ 2 (21)
This condition is similar to [START_REF] Borch | Equilibrium in a Reinssurance Market[END_REF] concerning agents' marginal rates of substitutions between two states of nature.

From the set of optimality conditions (eq. ( 16) to ( 20)), we obtain an implicit function Γ iθ between the number of permits allocated to each firm in a given state of nature and the total amounts of permits distributed for each of these states. We write the following reciprocity principle:

P iθ t+1 = Γ iθ ( P 1 , P 2 , ..., P θ , ..., P Θ ) (22)
The permits distributed by the agency depend on the aggregated sum of permits available in the economy over the two periods. If a change arises in firms' permits allocation rules, and without uncertainty on the total number of permits allocated during the second period, we obtain the following result:

For any given set of decisions of the regulator concerning firms' permits allocation criteria during the second period, the re-allocation of permits by the agency is Pareto-optimal for firms.

The expected firms' profits are similar to the case without uncertainty where the agency is in charge of redistributing the total number of permits available in the economy. If the agency only knows the total number of permits allocated during each period, Pt = i P i t and Pt+1 = i P i t+1 , it will be able to redistribute the total number of permits , Pt + Pt+1 , during each period for any change in permits allocation rules enforced by the regulator.

In this context, the agency is able to smooth changes in permits allocation between the two periods instead of firms. When Borch's condition is met, this allocation is also Pareto-optimal.

However, as we detail in the next section, if the uncertainty concerns the total amount of permits distributed during the second period, then the risk-sharing agency will only be able to enforce an optimal risk-sharing rule associated with permits holdings.

The Optimal Risk-Sharing Rule

For a given state of nature , θ, we may deduce from the optimality conditions the equality of permits marginal productivity between firms:

F i P iθ t+1 (X iθ t+1 , P iθ t+1 ) = F j P jθ t+1 (X jθ t+1 , P jθ t+1 ) (23)
Based on eq.( 17), the inputs X during the second period may be expressed as functions of second period permits allocations:

X iθ t+1 = Φ i (P iθ t+1 ) (24)
Plugging these functions in eq.( 23), we obtain a relationship between firms' final permits allocations taken pairwise:

F i P iθ t+1 (Φ i (P iθ t+1 ), P iθ t+1 ) -F j P jθ t+1 (Φ j (P jθ t+1 ), P jθ t+1 ) = 0 (25)
In order to derive the optimal risk-sharing rule, we study the variations of firms' permits allocations as a function of the variation of the second period permits bank. We consider the permits allocation constraint in eq.( 15) in two different states of nature θ 1 and θ 2 :

i

[ P i t + P iθ 1 t+1 ] = i P i t + i P iθ 1 t+1 (26)
and

i [ P i t + P iθ 2 t+1 ] = i P i t + i P iθ 2 t+1 ( 27
)
with

P θ 1 t+1 = i P iθ 1 t+1 and P θ 2 t+1 = i P iθ 2 t+1
, the total allocation for each state of nature, and we obtain:

P θ 1 t+1 -P θ 2 t+1 = i P iθ 1 t+1 - i P iθ 2 t+1 (28)
Using eq.( 25) and the implicit function theorem, we define, for each state of nature, a relationship, g(.), between firms' second period permits allocations taken pairwise. For each pair of firms i and j, we have:

P iθ t+1 = g θ ij (P jθ t+1 ) (29)
Using equation ( 28) and (29), we get:

dP jθ t+1 d P θ t+1 = 1 i g θ ij (P jθ t+1 ) (30)
with

g θ ij (P jθ t+1 ) = ∂F j P /∂P ∂F i P /∂P
Noting σ θ j the elasticity of the marginal productivity of the environmental variable as a function of firm's i production with respect to the variation of the number of permits:

σ θ j = P × ∂F j P /∂P ∂F j /∂P ,
we obtain the following proposition:

Proposition 3 If the total permits bank during the second period is random, any optimal risk-sharing rule between firms is such that:

dP jθ t+1 d P θ t+1 = σ θ j /P jθ t+1 i σ θ i /P iθ t+1
When the initial global permits allocation in the state of nature θ increases, the final second period permits allocation in this state increases proportionally to the elasticity of the marginal productivity with respect to the environmental variable (the stock of pollution). This sharing condition also takes into account the sensitivity of the firm's marginal productivity to the number of permits.

Having detailed in Section 2 the consequences of uncertainty on firms' banking behavior and in Section 3 the rationale for risk-sharing between firms, we examine in the next section whether these theoretical predictions meet firms' actual behavior on the European carbon market.

Empirical Evidence

This section hinges on the recent development of the EU ETS during 2005-2007 to discuss empirically the main theoretical findings. The early experi-ence of the world's largest greenhouse gases ETS to date indeed allows us to shed some light on (i) the banking behavior at the installation level as detailed in Section 2, and (ii) the permits pooling behavior at the group level as highlighted in Section 3.

We use Reuters Carbon Market Data11 to provide a qualitative discussion of these theoretical findings. From the 800 companies included in this database, we identify 7 companies that allow us to shed some light on the banking and pooling behaviors. Our case-studies are divided in three subsamples: firms with the highest pollution permits shortages at the group level, firms with the highest pollution permits surpluses at the group level and the highest emitter of CO 2 on the market. We focus our comments on the number of allowances banked forward at the installation level and the presumed pooling behavior at the group level as detailed respectively in Sections 2 and 3. All tables and figures may be found in the Appendix.

The Banking Behavior at the Installation Level

On the EU ETS, the political uncertainty regarding the second period permits allocation is linked to the negotiation of the second National Allocation Plans (NAPs) between Member States (MS) and the European Commission (EC). This situation fits well the theoretical framework developed in Section aspects, Phase I may be considered as a "warm-up" period for the EU ETS since several key provisions 13 of this newly created market were characterized by abrupt decision changes. By using this approach, we intend to provide an empirical discussion of the theoretical results regarding the banking of permits that corresponds to the early development of the European carbon market.

The risk underlying permits trading on the European carbon market is linked to increasing permits prices against which installations need to form hedging strategies 14 , and the firm's net short/long position that need to be carefully managed to save penalty costs (Buchner and [START_REF] Ellerman | Over-Allocation or Abatement? A Preliminary Analysis of the EU ETS Based on the 2005-06 Emissions Data[END_REF]). An installation is defined as net short (long) if the number of verified emissions is superior (inferior) to the number of allocated allowances during the com-pliance period. An installation with a short (long) position may buy (sell) allowances in order to be in compliance (make profits). If the installation does not match its mandatory emissions target with the corresponding number of allowances during the compliance period, it needs to pay back one permits plus a penalty 15 during the next period.

On the demand side, the volume of CO 89).

Based on the visual inspection of the data, we have highlighted in this section a wide variation in the amount of banked permits at the installation level during 2005-2007. Three kinds of arguments may explain these variations between firms. First, differentiated allocation rules were enforced by the regulator between EU ETS sectors that affect firms' permits supply.

Second, unforeseen economic activity events may impact firms' production levels and their permits demand. Third, these heterogeneous behaviors in terms of banked permits may come from the political uncertainty described earlier regarding banking provisions and NAPs II. The latter argument is in line with the theoretical framework regarding the effects of political decision changes on firms' banking behavior derived in Sections 2 and 3. This first step of the inspection of the data therefore brings us to a more detailed analysis analysis of the potential for permits pooling by the parent company in the next section.

The Pooling Behavior at the Group-Level

Let us detail first the rationale behind risks pooling, and second the actual behavior of the firms contained in our sample.

To our best knowledge, only Alberola et al. (2008) evoke the existence of pooling behavior in the EU ETS. In this article, we have detailed the economic intuitions behind it. Before investigating the pooling behavior empirically, it is worth emphasizing that the Community Independent Transaction Log (CITL) 16 , which oversees all national registries, displays extensive information at the installation level concerning allocation and verified emissions. However, not all registries have been connected to date, and the CITL contains raw data that needs to be reorganized between subsidiaries and parent companies 17 . Hence, we do not aim at an exhaustive discussion on this topic, but rather at introducing empirical perspectives that are relevant to our theoretical results.

On the EU ETS, pooling behaviors may emerge at the group level in order to save the cost of purchasing permits. The economic logic of such an argument unfolds as follows: if there exist both types of net short and net long subsidiaries, the parent company may transfer allowances internally between them so that the net position of the parent company is globally in compliance. Thus, the exposure to the risk of permits shortage during compliance periods may be reduced by this intra re-allocation of permits.

This type of behavior is close to the theoretical finding detailed in Section 3 with the role of the agency pooling risks. This logic holds only if there is

an alternate of net short and net long installations at the group level, that is why we detail several cases that may apply.

Among the three firms in our sample that are in a net short position, Union Fenosa exhibits in 2007. There appears to be little room for permits pooling within subsidiaries.

Only 2 out of 25 installations are in a slightly short position, which may 18 One EUA is equal to one ton of CO 2 emitted in the atmosphere.

easily be counterbalanced by permits reallocation from other installations within the group. This situation is confirmed by the visual inspection of the data in Figures 1 to 3. Overall, the parent company is a net seller on the permits market. In Table 4, Dalkia also exhibits a large surplus of 2.4M EUAs in 2007. 4 out of 125 installations are net short, which supposes similarly that their deficit may be compensated internally by the parent company, thereby covering the risk of permits shortage for its subsidiaries.

From Figures 4 to 6, one may remark that the distribution of installations is very heterogeneous with two installations above 1M of allocated allowances holding substantive surpluses. On a smaller scale, Eesti Energia displays in analysis has therefore confirmed the potential for permits pooling between subsidiaries by the parent company, which is conform to the theoretical finding on the optimal risk-sharing rule enforced by the agency derived in Section 3.

Conclusion

In this article, we have shown that, once permits trading between firms has occurred, the presence of uncertainty regarding political decision changes in permits allocation rules may provide incentives for firms to bank permits in order to hedge against this risk. The conditions under which risk-neutral firms hedge their risk by banking permits is linked to the third derivative of the production function. This condition is similar to the condition under which individuals adopt a prudent behavior 19 . Besides, we have characterized an optimal risk-sharing rule when the uncertainty is associated either to the repartition of permits between firms, or to the global permits allocation.

This rule depends on firms' technological characteristics, and more precisely on the concavity of the production function with respect to pollution.

As for the empirical evidence, we have selected three types of firms operating on the EU ETS characterized by the highest allowances surpluses, the highest allowances shortages and the highest CO 2 emissions level on the market. Thus, we obtain a sample of 7 firms out of 800 included in the Reuters Carbon Market Database. These case studies allow us to provide some qualitative insights of the theoretical results. First, the investigation of the banking behavior has revealed asymmetric banking (borrowing) patterns The latter result illustrates the economic logic derived in Section 3, i.e. the parent company acts as the agency to introduce an optimal risks-sharing rule between subsidiaries.

From a regulatory viewpoint, the management of the environment through the introduction of pollution permits implies that firms have the ability to bank permits in order to hedge the risks linked to political decision changes.

The banking of permits is not motivated here by adaptation concerns to environmental constraints, but by the need to counter-balance political risks attached with the introduction of pollution permits systems. Our analysis has therefore confirmed the key role played by banking provisions in order to cope with the potential political uncertainty related to the creation of pollution permits markets.

2 .

 2 The uncertainty concerns indeed the exact amount of permits being allocated from 2008 onwards, since the EC announced during Phase I its will to enforce stricter allocation rules for Phase II. The goal of the EC essen-tially consists in enacting stricter validation criteria for MS to create a real scarcity of pollution permits, and thus foster firms' investment in depolluting technologies and internal CO 2 emissions abatement efforts. The variation of banked allowances between Phase I (2005-2007) and Phase II (2008-2012) corresponds to the 2008 compliance result which will be disclosed by the EC by mid-May 2009 12 . Thus, we choose to focus our comments on the variation of banked permits by firms linked to the variation of the global level of regulatory uncertainty during 2005-2007. By many

  19 i.e. the study of the positive third derivative of the utility function at the installation level as a consequence of varying net long (short) positions during 2005-2007. Second, the investigation of the pooling behavior has confirmed the potential for internal permits transfers between subsidiaries at the group level. The former result is consistent with what was expected from Section 2, i.e. to observe a variation of the number of permits banked by firms in reaction to political uncertainty concerning Phase II of the EU ETS.

Table 2 :Table 3 :Table 4 :Table 5 :

 2345 Distributed Allowances, Verified Emissions and Net Short/Long Position for Dalkia, Installations #1-42 (2005-2007) from Reuters Carbon Market Data Distributed Allowances, Verified Emissions and Net Short/Long Position for Dalkia, Installations #43-84 (2005-2007) from Reuters Carbon Market Data Distributed Allowances, Verified Emissions and Net Short/Long Position for Dalkia, Installations #85-125 (2005-2007) from Reuters Carbon Market Data Distributed Allowances, Verified Emissions and Net Short/Long Position for Eesti Energia and Enel (2005-2007) from Reuters Carbon Market Data

Table 9 :

 9 Distributed Allowances, Verified Emissions and Net Short/Long Position for RWE, Installations #38-73 (2005-2007) from Reuters Carbon Market Data

Figure 1 :Figure 2 :

 12 Figure 1: Distributed Allowances and Verified Emissions for ArcelorMittal in 2005 from Reuters Carbon Market Data

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Distributed Allowances and Verified Emissions for ArcelorMittal in 2007 from Reuters Carbon Market Data

Figure 6 :

 6 Figure 6: Distributed Allowances and Verified Emissions for Dalkia in 2007 from Reuters Carbon Market Data

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Distributed Allowances and Verified Emissions for Eesti Energia in 2005 from Reuters Carbon Market Data

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Distributed Allowances and Verified Emissions for Enel in 2005 from Reuters Carbon Market Data

Figure 13 :Figure 14 :

 1314 Figure 13: Distributed Allowances and Verified Emissions for Eon in 2005 from Reuters Carbon Market Data

Figure 15 :Figure 17 :

 1517 Figure 15: Distributed Allowances and Verified Emissions for Eon in 2007 from Reuters Carbon Market Data

Figure 18 :Figure 20 :Figure 21 :

 182021 Figure 18: Distributed Allowances and Verified Emissions for RWE in 2007 from Reuters Carbon Market Data

Table 1

 1

	) and Dalkia (Figures 4 to 6, Tables 2-4), as well as for Eesti Energia's
	installation (Figures 7 to 9, Table 5). This comment applies especially for
	installations in the combustion sector which seem to have benefited from
	windfall profits during the NAPs I allocation process. In the sub-sample
	15 Equal to 40 = C per unit during 2005-2007 and 100 = C per unit during 2008-2012.

2 emissions at the installation level depends on economic activity, weather events and other energy market prices (Mansanet-Batallet et al. (2007), Alberola et al. (2008a), Alberola et al. (2008b),

[START_REF] Newbery | Climate Change Policy and Its Effect on Market Power in the Gas Market[END_REF]

). On the supply side, the allocation of permits in the EU ETS during Phase I followed the grandfathering principle, which allocates permits freely based on business-as-usual emissions. If a firm encounters an uncertainty as modelled in Section 2 and goes beyond its emissions forecasts during the current allocation period, then it is basically left with two choices: either use banked permits or go on the market to buy permits.

Given the underlying assumption of our model that permits trading has already occurred between firms, we investigate the changes in banked permits at the installation level that occured in a context of regulatory uncertainty on the EU ETS during 2005-2007. Banking behaviors greatly vary between the 7 companies that belong to our case-studies. In the sub-sample of firms which record the highest permits surpluses, we remark net banking patterns for the largest installations in terms of allocation for ArcelorMittal (Figures

1 to 3,

  Table 10 the largest shortage by 7.3M European Union Allowances (EUAs)18 in 2007. Out of 12 installations, 9 are in a short position which may only be compensated internally by 2M EUAs in surplus. Thus, the pooling of allowances by the parent company allows to reduce the risk of permits shortage by 25% for some, but not all, subsidiaries.The visual inspection of the data inFigures 19 to 21 reveals that three installations are especially short of permits over 2005-2007. Next, we turn our attention to E.ON which records in Tables6 and 7a net short position of 2.7M EUAs. 31 out of 89 installations encounter a permits shortage, and the potential for permits transfer at the group level amounts to 1.6M EUAs.Among the three firms in our sample that are in a net long position, ArcelorMittal stands out as holding the largest surplus of allowances. Indeed, as shown in Table1, it is net long by 18.9M EUAs during the compliance year

Hence, the risk pooling strategy by the parent company may save the costs of permits purchase on the market by 60%. The distribution of subsidiaries in Figure

13

to 15 also reveals a strong dispersion in terms of size, with one installation of 1M allocated allowances being consistently short over 2005-2007. In Table

5

, Enel records a net deficit of allowances by 1.5M in 2007. 5 out of 9 installations are net short, which may only be compensated by another subsidiary by 0.05M EUAs. Figures 10 to 12 confirms this analysis, with most installations being net short in 2007. From this sub-sample of firms with permits shortages, our analysis has confirmed the potential for risksharing and thereby the pooling behavior at the group level that constitutes one of the main finding in Section 3. Let us now examine another sub-sample of firms with permits surpluses.

Table 5

 5 Finally, we comment the case of RWE which is the largest CO 2 emitter in the current European system with 128M EUAs verified emissions in 2007.

	From Tables 8 and 9, we observe that RWE is in a net short position by 8.6
	M EUAs. 21 out of 73 installations encounter an permits shortage, which
	may be compensated internally by the parent company by 2.8M EUAs, i.e.
	33% of the total permits shortage. The distribution of installations displayed
	in Figures 16 to 18 reveal that RWE gathers very large installations, with
	4 installations being allocated above 1.5M EUAs in 2007. One installation

a net long position of 0.27M EUAs in 2007 for one installation being reported in the Reuters Carbon Market Database. Without commenting further the possibility of pooling risks, Figures 7 to 9 reveals that this permits surplus has been increasing from 2005 to 2007. This second sub-sample of firms has confirmed the liquidity of the permits market in terms of extraallowances available for trading during each compliance period. Given this high level of heterogeneity between firms, if parent companies are still in a net short position after pooling allowances internally, they may buy allowances on the market to be globally in compliance.

Table 1 :

 1 Distributed Allowances, Verified Emissions and Net Short/Long Position for ArcelorMittal (2005-2007) from Reuters Carbon Market Data

Table 6 :

 6 Distributed Allowances, Verified Emissions and Net Short/Long Position for E.ON, Installations #1-45 (2005-2007) from Reuters Carbon Market Data

Table 7 :

 7 Distributed Allowances, Verified Emissions and Net Short/Long Position for E.ON, Installations #46-89 (2005-2007) from Reuters Carbon Market Data

Table 10 :

 10 Distributed Allowances, Verified Emissions and Net Short/Long Position for Union Fenosa (2005-2007) from Reuters Carbon Market Data

i.e. less information is needed concerning firms' depolluting costs.

For instance, a tax or a lump sum.

Be it concerning the global permits allocation or its repartition between firms.

This behavior applies especially for firms with high abatement costs.

See Bosetti et al. (2008) for an empirical discussion.

Available at http://www.carbonmarketdata.com. To the purpose of our empirical section, we exploit from the Reuters Carbon Market Data the compilation of[2005][2006][2007] verified emissions between subsidiaries and parent companies that was accurate as of May 2008.

Moreover, due to institutional fungibility between the EU ETS and the Kyoto Protocol as of 2008, the possibility to transfer banked allowances from Phase I to Phase II has been restricted by all MS. For an extensive discussion of the inter-period ban on banking in the EU ETS, see Alberola and Chevallier (2007).

Such as allocation criteria or banking provisions.

See Ielpo et al. (2008) for an extensive discussion based on increasing futures prices over 2008-2012. We do not further comment this type of risk in the remainder of the article.

Available at http://ec.europa.eu/environment/ets/, cited May 2008.

For a detailed analysis on this compilation of emissions data at the group level, see[START_REF] Trotignon | Technical Memorandum on Analysis of the EU ETS Using the Community Independent Transaction Log[END_REF] and[START_REF] Trotignon | European CO 2 Market and the CITL : The Trial Period Under Scrutiny[END_REF]. As another preliminary remark, note it is not yet technically possible to track permits exchanges at the European level, although each permit is marked with an unique identifier, since this information will only be disclosed publicly after 5 years of permit trading in the EU ETS.


Christian Bidard

THE DYNAMICS OF INTENSIVE CULTIVATION

Keywords: JEL Classification: B12, D24, D33 intensive cultivation, land, rent, Ricardo

An increase in the demand for agricultural goods leads to the use of more intensive cultivation methods. Though Ricardo sees no difficulties in the intensification process, their existence is revealed by the possible occurrence of multiple equilibria. A general theory of intensive rent is based on a formal parallel with single-product systems without land.

INTRODUCTION

The2 recent rise in the prices of agricultural products and raw materials is generally thought to be a long-term phenomenon, which puts again to the fore the old Ricardian theory of rent: an increase in the demand for agricultural goods leads to a rise in their prices, which makes the exploitation of lands of lower quality profitable or allows for the introduction of more intensive cultivation methods. Some doubts on the validity of that classical piece of economic analysis arose when [START_REF] Montani | Scarce natural resources and income distribution[END_REF][START_REF] D'agata | The Existence and Unicity of Cost-Minimizing Systems in Intensive Rent Theory[END_REF] provided examples showing that different ways to meet a given demand basket may exist, each corresponding to different agricultural methods and being associated with different prices, wages and rents. Though established in a static framework, the phenomenon sets a challenge to Ricardo's dynamic analysis which suggests that a progressive increase in demand is met by introducing more productive methods which are land saving but were too costly to be used when corn was cheap. The sequence of substitutions determines a unique path and a unique solution is expected for each level of demand. The multiplicity puzzle has been analyzed in specific cases, assuming one agricultural good and no or restricted substitutions in the industrial sector, without convincing results. Let us isolate the intensification process by assuming that land is of uniform quality. A general theory of intensive rent proper, with one land of uniform quality, several agricultural and industrial goods and any number of methods in each sector, is deemed to be out of reach [START_REF] Kurz | Theory of Production[END_REF].

We show on the contrary that the general theory of intensive rent is simple. Under the hypothesis, faithful to Ricardo, that the given distribution variable is the real wage basket, the key argument is that the formalization of intensive cultivation with a uniform rent per acre is very close to that of single-product systems with a uniform wage per worker. This feature is used to transfer well known results relative to single production without scarce resources to the study of intensive cultivation.

3

Section 2 introduces the intensive cultivation model. Section 3 describes the Ricardian dynamics and analyzes the value side, then the physical side of the problem. Section 4 identifies the conditions of validity of those dynamics.

COUNTING EQUATIONS

The data of the general problem of intensive rent proper are the n reproducible commodities (their decomposition as industrial goods and agricultural goods does not matter, as soon as there is at least one agricultural good), the set of industrial and agricultural methods of production, which are of the single-product type with constant returns (land is not treated here as an output of agriculture), the total area l of the uniform grade land and the semipositive final demand basket d. After incorporation of the real wage basket into the physical inputs of production, labor does not appear explicitly: the jth method of production is described by the column-vector j a of its n input coefficients and the land area j l per unit produced (the land input of an industrial method is zero). On the physical side, the unknowns are the semipositive activity levels, represented by a column-vector y which satisfies n constraints stemming from the adjustment of the net product to demand

d Ay y  
(1) and one constraint relative to the full cultivation of land l ly 

(2) (taking into account the fallow method, with land as only input and no product, allows us to replace the inequality l ly  by an equality). Flukes apart, these conditions imply that at least n+1 methods are operated. On the value side, the unknowns of a long-term equilibrium are the uniform rate of profit r, the price row-vector p and the uniform rent  per acre (fallowing requires a zero rent). For a given numéraire, their number is reduced to n+1. Each method j producing commodity i satisfies an inequality with complementarity relationship

  j i j j y p l pa r     ) 1 ( (3)
Equalities hold for the operated methods ( 0  j y

). Flukes apart, at most n+1 equalities are met by the n+1 unknowns. On the whole, it turns out that the number of operated processes exceeds that of commodities by one, therefore two processes produce the same commodity.

That coexistence may well occur in industry: since land enters in the production of iron, the use of a land-saving iron method has the same effects as that of more intensive corn method [START_REF] Saucier | Le choix des techniques en situation de limitation de ressources[END_REF].

RICARDO'S ANALYSIS

With two methods producing the same good, the extension of the more productive method to the expense of the other allows for an increase in production. During this phase, the value equations are not affected and the prices remain constant. This continuous expansion comes to an end when the activity level of the less productive method vanishes. At this stage, a spasmodic rise in the price of the product in short supply (say, corn) occurs, the surge being sufficient for the introduction of a more productive method which was not used before because of its cost. The level of the uniform rent rises, the rate of profit falls and, since corn enters in the production of other commodities (and indirectly of all others, if the economy is basic), all prices change. To an observer, the new value conditions let appear a new method of production in some branch when another disappears: the stage is thus set for another phase of expansion followed by another jump in prices and distribution. This scheme is the core of the Ricardian dynamics of intensive cultivation.

The value side of these dynamics has a noticeable property: the value relationships (3) bear a striking formal similarity with those prevailing for the choice of methods in a purely industrial framework without land (Sraffa, 1960, chapter XII) if the land vector is identified with the labor vector and the rent per acre with the wage per worker. The value problem can therefore be solved by transposing the standard tools and results: the numéraire being chosen, consider an arbitrary n-set of methods, one for each commodity, and draw the r   curve obtained by eliminating the prices from the corresponding value equations. Repeat the procedure for all possible n-sets and consider the upper envelope of these curves. Each curve being decreasing, the same for their envelope, a property which expresses the trade-off between profits and rents. On the factor price frontier, each farmer or entrepreneur maximizes profits for the given prices and rent. The only significant difference with the industrial framework is that the number of equalities now exceeds that of commodities by one, but one more equality is met at a switch point between two consecutive n-sets along the frontier. That is, the switch points ) , ( r  , at which two methods producing the same good are equally costly and can operate simultaneously, are the noteworthy points for the theory of intensive cultivation. The Ricardian dynamics are made of two phases: (i) at each node, there are n+1 operated methods, with one 'old' method and one 'new' method producing the same commodity, and a progressive substitution of the new method for the old; and, (ii) when the old method disappears, the value system jumps from a node to the next on the factor price frontier, with one new method appearing in the same or another industry.

Consider now the quantity side. At a switch point, the coexistence of n+1 methods for n goods can be viewed as the coexistence of two neighboring square n-sets, with n-1 common methods and an alternative for the production of some commodity. Consider first a given n- two n-sets A and A' which differ by one method. At any given rate of profit, all prices in terms of wage are smaller in the selected system (Sraffa, 1960, section 94), therefore all labor values (which coincide formally with prices for zero profits) are also smaller in one system.

Transposed into the agricultural framework, the property means that one of the vectors V or V′ of land values is smaller than the other, therefore one of the simplices S or S′ is entirely above the other: without ambiguity, the corresponding n-set is more productive. Finally, consider a

Ricardian node: two neighboring n-sets work together and, depending on relative activity levels, can produce any basket which is a barycenter of baskets in S and S′, therefore any basket belonging to the 'slice' between S and S′.

THE RICARDIAN DREAM

It is worth noting that, in the above analysis, the dynamics result from the examination of the value conditions and that the posterior study of quantities has shown that the land value constitutes the relevant index to measure physical productivity. Ricardo's views on the dynamics of intensive cultivation can be criticized because they proceed by identifying profitability and productivity conditions. Let us rather quote Sraffa (1960, section 88): "The existence side by side of two methods can be regarded as a phase in the course of a progressive increase of production on the land. The increase takes place through the gradual extension of the method that produces more corn at a higher unit cost, at the expense of the method that produces less. As soon as the former method has extended to the whole area, the rent rises to the point where a third method which produces still more corn at a still higher cost can be introduced to take the place of the method that has just been superseded." The reference to a change in one method corresponds to the jump from a switch point to the next.

The implicit hypothesis is that the new method then introduced is indeed more productive, but the property is not ensured.

A 'Ricardian reversal' occurs when the newly introduced method is less productive than the one with which it is jointly operated. The phenomenon explains the occurrence of multiple solutions: if, after a series of jumps which allow to produce up to a certain level d,

the new method reduces it to a lesser level d', there are two possibilities to produce an intermediate basket. Suppose moreover that the next switch is of the normal type and allows to increase production up to some high level. In a static framework, one can find values and activity levels sustaining the production of a basket greater than d but, when demand is continuously increasing beyond d, a jump appears with the sudden introduction of new methods of cultivation at positive activity levels (on the factor price frontier, it corresponds to a jump over the next switch point). These discontinuities do not fit with smooth substitution on the physical side. From a theoretical point of view, the occurrence of a Ricardian reversal is linked with the so-called paradoxes of capital theory: 'capital reversal' occurs when more capital intensive methods are less productive.

The Ricardian story holds if the substitution along the frontier leads to the introduction of more productive methods. This condition can be characterized by an algebraic criterion which is some resemblance with [START_REF] Erreygers | On the Uniqueness of Square Cost-Minimizing Techniques[END_REF] uniqueness criterion:

Definition. Let us associate with a method of production and an arbitrary rate of profit r the (n+1)-column vector made of its output vector minus (1+r) times its material inputs, with the

scalar i l  in last position. ) (r 
is the determinant obtained by stacking n+1 vectors, with two methods competing to produce the last good.

We adopt the temporary convention that the coefficients of the old method are in the last but one column whereas those of the new method are in the last column. Since the next criterion derives from a property of single-product systems, we momentarily replace 'land' by 'labor' and 'rent' by 'wage'. When the rate of profit varies continuously, the determinant ) (r  vanishes at switch points because the value equalities (3) show that the product of price-and-wage vector by the corresponding matrix is zero. The determinant preserves a constant sign between switch points and this sign indicates which of the alternative methods is then cheaper. If it is positive, a fictitious increase in the labor coefficient of the method described in the last column lets decrease the determinant (by the Hawkins-Simon criterion) down to zero, the fictitious method is then as costly as the old, therefore the method before the fictitious change was cheaper. The argument shows that the determinant ) (r  is positive when the method ranked in last position is cheaper at the rate of profit r (Bidard, 2004). On the value side, the determinant vanishes at a switch point r and takes the sign of ) ( ' r   after it ('minus' because the rate of profit is decreasing), and since the introduction of the new method is justified by its lower costs, ) ( ' r   is positive. On the physical side, the change introduces a method with smaller labor values if ) 0 (  is positive. In the agricultural framework, and after abandoning the convention on the ranking of the columns of the determinant, the conclusion is:

Proposition. The substitution of methods along the frontier for decreasing rates of profit (or increasing rents) lets come in at each switch point a new method which operates jointly with an old method producing the same commodity. The Ricardian dynamics hold if the new method is more productive, which is the case if the derivative ) ( ' r  at the switch point and ) 0 (  have opposite signs.

Under the alternative hypothesis of a given rate of profit (Sraffa's hypothesis), similar lessons are obtained by showing that only the upper envelope of the wage-rent curves matters and that a jump from a switch to the next on the frontier correspond to the change in one method of production. Again, the question is whether the new method, which is selected for profitability reasons, is indeed more productive. Some differences regarding the conclusions are illustrated by the behavior of a simple corn-labor-land model with corn as the only produced commodity: for a given real wage, the model behaves like a neoclassical corn-land model with smooth substitution and the Ricardian dynamics hold; on the contrary, multiple solutions may coexist for a given rate of profit [START_REF] Freni | Capitale tecnico nei modelli dinamici ricardiani[END_REF].

CONCLUSION

Following an increase in the demand for agricultural goods, a rise in prices provides an incentive to use more costly methods on the already cultivated lands. The progressive substitution of a new method for an old one presumes that the new method, which has become profitable at the new prices and distribution levels, is indeed more productive, a coincidence which is not automatically ensured. The conditions for smooth Ricardian dynamics in a general intensive cultivation model have been identified. If they are not met, the occurrence of a reversal implies severe discontinuities on the physical side which make the dynamics of intensification obscure. From a theoretical standpoint, another implication of this study follows from a return to the traditional framework with homogeneous labor and no land. The construction then leads to an endogenous determination of the wage rate under a full employment hypothesis, for a given labor supply. The stage is thus set for a solid comparison between the classical approach and the conventional theory.

  set. The input matrix A has dimension n n  and the net baskets d which can be produced under the scarcity constraint (2) are those belonging to the simplex S = {d; d > 0 and Vd = l }, -vector representing the land values of the commodities. These land values are similar to the more usual labor values and the condition Vd = l means that the whole land enters directly or indirectly into the production of basket d. Consider now

With acknowledgements to Guido Erreygers for helpful comments.