Total Factor Productivity of Tunisia's manufacturing sectors: measurement, determinants and convergence towards OECD countries

Abstract:

The purpose of this paper is twofold. First, sector-based Total Factor Productivity (TFP) is calculated for six Tunisian manufacturing sectors over the period 1983-2002. Economic determinants of the productive performance are also investigated.

In doing so, we take care of the direction of the causality by using a panel data Granger type-test. The recent literature in international economics has placed a particular emphasis on the relation between TFP and variables reflecting the potential impact of both trade and financial openness. Sector-based TFPs proved to be sensitive to some of these variables, highlighting a causality that does not reject the stimulating impact of exports and foreign direct investments. Second, the paper implements some panel data unit root tests to investigate the statistical hypothesis of TFP catching up of Tunisia with OECD members. In benchmarking each of the six Tunisian sectors by those of the most developed countries, panel data unit root tests do not reject the hypothesis of an overall catching-up for five of them.

I. Introduction

In November 1995, the Barcelona Declaration established a global framework geared toward strengthening a multidimensional partnership between the European Union and twelve South and East Mediterranean (SEM) countries, including Tunisia. The main objective of this Declaration is to promote a shared prosperity on both sides of the Mediterranean Sea, mainly through the development of regional trade. The gradual creation of a Free Trade Zone, which is to be finalized before the end of 2010, will be the central instrument in achieving this result. This regional policy remains fully consistent with the principles of multilateral integration in the world economy, as required by WTO membership.

The pace of trade liberalization will depend on the ability of the SEM countries to promote deep structural reforms in their respective manufacturing industries. Indeed, until the beginning of the nineties, SEM manufactured goods benefited from significant unilateral trade advantages in entering the EU markets. The relative scope of these advantages has gradually faded with the extension of similar facilities to eastern European countries which recently joined the European community. In addition, the creation of the Free Trade Zone will rely on the reciprocity rule. This means that the SEM products will have to compete with European goods on their own national markets whereas until now, most of them were sheltered from European exports. Therefore, international competition will increase dramatically on both domestic and traditional export markets, necessitating a greater productive efficiency of manufacturing firms.

In the SEM countries, there is reason to believe that the ongoing process should particularly affect Tunisia. In 1999, the industrial sector accounted for 18% of the GDP, the highest percentage in North Africa and the Middle East, after Egypt. Moreover, not only is the EU a traditional selling market for Tunisian exporters, but most manufactured goods are also still highly protected on the national market. 1In this paper we focus on the long-run Total Factor Productivity (TFP) of six Tunisian manufacturing sectors over the period 1983-2002 and investigate the hypothesis of an international convergence process with the productive performance of OECD members. The paper is organised as follows. In section II, TFPs are calculated for food processing (FOOD); electrical and metal products (ELEMET); chemical activities (CHEM); textiles, clothing and leather (TCL); building materials and ceramics (BGC); other manufacturing products (OTHER).

Then, TFP levels are regressed on a vector of potential determinants including those reflecting the economic and financial impact of openness. Before running the regression we control for the direction of the causality. Section III is devoted to the analysis of international TFP convergence.

Tunisian sectors are then compared with those of OECD members'. The hypothesis of a stochastic convergence is not rejected for five sectors, but proved to be somewhat sensitive to the panel data unit root test as well as the OECD sample we refer to. In section IV, we summarise the results and discuss avenues for further research.

II-Tunisian TFP manufacturing sectors: levels and determinants II.1 The sector-based evolution of the Total Factor Productivity

For several decades after independence (1956), Tunisia supported an inward-oriented productive industry. In 1970, a first watershed occurred with the provision of fiscal incentives to export activities. Textiles, clothing and leather (TGL) benefited greatly from this policy change, which contributed to strengthening light industry beyond traditional food processing activities.

At the beginning of the eighties, the whole manufacturing sector accounted for 15% of the GDP.

Over four decades, the structure of the industry changed significantly with an increasing share taken by chemical and textile exports. The relative importance of exports continued to increase, first as a result of the structural adjustment policy (1986) and later, with the Barcelona Declaration (1995). With tariffs on imports being progressively eliminated, international competition gradually increases and forces Tunisian firms to raise the level of their productivity.

The performance of the whole industry can be appreciated both, in the long run and for sub-periods in light of trade policy changes. Over the 1983-1987 period, the effects of the second oil crisis were still being felt. Tunisia faced a severe economic and financial crisis resulting from the decline in oil export earnings and the slowdown in net worker's remittances (cf. Morrisson and Talbi, 1986). A more restrictive external trade policy was implemented to cope with the balance of payments crisis and the sharp fall in external reserves. The period 1987-1995 reflects the consequences of the structural adjustment policy and the move toward a more marketfriendly economy. During this period, Tunisia joined the GATT (1990), became a founding member of the World Trade Organization (WTO) and adopted current account convertibility (1993). Besides the multilateral approach to trade policy, the Euro-Mediterranean partnership also developed and was institutionalized through the Barcelona Declaration (1995). The last subperiod (1995)(1996)(1997)(1998)(1999)(2000) reflects the gradual preparation for the establishment of the Free Trade

Agreement and the strengthening of competition on both domestic and external markets.

For each of the six sectors, Total Factor Productivity, (TFP) has been measured by considering the value added at constant market prices. Production technology is defined assuming constant returns to scale with two primary inputs: the number of permanent employees (L) and the capital stock (K). With regard to the labour force, the data for the number of hours worked is not available. However, although the labour market became more flexible over the entire period, public regulations for hours worked did not change. Thus, we can reasonably assume that TFP is not biased when using the number of employees. The capital stock has been calculated, at constant prices, using the perpetual-inventory method for annual investment flows.

As we refer to aggregate data with no information about the composition of the capital stock, two average annual depreciation rates have been considered: 5% and 10%. 2 The stock has been constructed as follows, with (t) denoting the end of year.

K t = I t + (1-δ) K t-1 (1)
When measuring productivity, one assumes constant returns to scale and remuneration of production factors at their marginal product. As the relative contribution of the labour force (α) was calculated by considering the share of wages in the sector value added, we infer the capital share (1-α).

) 1

( α α - = t t t K L Q TFP t (2)
There is an extensive literature discussing the advantages and disadvantages of using growth accounting techniques to estimate TFP. The non-parametric procedure is preferred for it better serves our purpose of identifying TFP level determinants. Indeed, in adopting this method we do not have to face the simultaneous equation problem resulting from the endogeneity of inputs. We do not have to estimate the input elasticities by regression techniques, but derive them from the observed cost shares. Therefore the non-parametric procedure allows us to analyse the TFP effects of the variables reflecting openness in two steps. In the first step we generate the "Solow residuals" while in the second step, we use a regression analysis to evaluate TFP determinants.

In Table 1, we present the TFP growth rates over the entire period and relevant subperiods under the two above-mentioned hypotheses regarding the depreciation rate of the capital stock. TFP does not prove to be very sensitive to these rates, especially after 1987. Therefore, in the rest of the paper we will only use the 5% depreciation rate. For three sectors: electrical and mechanical (ELEMET), building and ceramics (BGC), textiles and leather (TCL), the long-run annual average rate of TFP growth, ranges between 1.5% for electrical and mechanical activities and 3.5% for textiles and leather over the period 1983-2002. The evolution is somewhat different for the two other sectors. After 1987, TFP strongly increases in the chemical industry (CHEM), by about 9% per year. In the food sector (FOOD), TFP falls sharply, with some deceleration over the 1995-2002 period. Understanding this empirical evolution would require a sub-sector-based analysis with disaggregated data, which unfortunately are unavailable 3 . A simple breakdown of TFP gains and losses according to the growth of the respective value added and input use shows that the change in inputs has been the prevailing determinant. 3 In an applied study of 21manufacturing industries from 16 OECD countries over the 1971-1994 period, [START_REF] Funk | Panel tests of stochastic convergence: TFP transmission within manufacturing industries[END_REF] have highlighted a high level of international heterogeneity in the sectoral productivities. The average growth rate of TFP is about 1.35% and 1.14% for iron and steel, and non-ferrous metals, respectively. For the same sectors, the maximum growth rate is 11.5% and 9.9% while the minimum stands at -0.07 and -0.15 Except for TCL and miscellaneous manufacturing industries, which exhibit a good productive performance, the 1983-1987 sub-period is characterized by negative sector growth rates. This poor performance results from difficulties in adjusting inputs in an adverse macroeconomic context. Cumbersome regulations governing the labour market hampered the rapid restructuring of enterprises whilst some of them also had to manage the overinvestment of the previous period. Thus, TFP suffers from the under-utilisation of productive capacities. The next sub-period, stretching from 1987 to 1995, was much more favourable. The macroeconomic recovery as well as the combination of trade openness and local deregulations contributed to a more efficient management of inputs. Firms benefited4 from the previous investments and were able to expand their production without a proportional increase in inputs.

In the last sub-period, the TFP growth rate slows down in the chemical industry and does not significantly differ from the growth rates of other manufacturing sectors. For textiles, clothing and leather, although the value added growth remained dynamic, about 5% a year, the growth of the sectoral TFP was far lower than the highest values reached in the eighties. The relative stability of the real exchange rate has contributed to the present difficulties of firms in managing the end of the multifiber agreement (2005). The long run productive performance of Tunisian industries turns to be heterogeneous across sectors and sub-periods. Some manufacturing activities such as food processing industries experience a decline while others strengthen their position and contribute to modify the national production structure.

II.2 The econometric determinants of TFP levels

In this overview of potential TFP determinants, a special interest is given to the variables reflecting trade and financial openness. In analyzing both the level and the growth of productivity, the recent empirical literature has given a great importance to this dimension. If empirical studies generally provide evidence of a positive correlation between trade/financial openness and productivity performance (see [START_REF] Dollar | Outward-Oriented Developing Economies Really Do Grow More Rapidly: Evidence from 95 LDCs, 1976-85[END_REF][START_REF] Bhagwati | Foreign Trade Regimes and Economic Development: Anatomy and Consequences of Exchange Control Regimes, Lexington, Ma Ballinger[END_REF][START_REF] Sachs | Economic Reform and the Process of Global Integration[END_REF][START_REF] Edwards | Openness, Productivity and Growth: What do we really know?[END_REF] there are some "prominent trade liberalisation sceptics" (cf., [START_REF] Edwards | Openness, Productivity and Growth: What do we really know?[END_REF] 5 . [START_REF] Rodriguez | Trade Policy and Economic Growth/ A skepic's Guide to The Cross-National Evidence[END_REF] point to the hotly debated methodological issue of identifying the direction of the causality6 . Do exports enhance productivity or is productivity a prerequisite for a high external trade performance? Indeed, TFP can result from a learning-byexporting effect, but also from a self-selection effect, exporting firms or sectors then being those having the best initial TFP performance (see [START_REF] Haddad | Morocco, 1984-89: Trade Liberalization, Exports, Industrial Performance[END_REF]. In view of the above comments, the subsection is divided as follows. First, we discuss the potential determinants of TFP levels. An analysis based on productivity growth rates would be more general, but it has the disadvantage of being much more demanding in terms of data quality.

Then we deal with the causality problem and explore the impact of the exogenous variables through a multiple regression analysis7 .

TFP and its potential determinants

The level of exports (EX) can be considered as a first relevant trade variable. Indeed, competition on foreign markets provides a strong stimulus for promoting economic efficiency. In the case of Tunisia, the integration of the country into the world economy entails a progressive loss of the preferential advantages that the European Union has granted for decades. Europe remains by far the main exporting market for Tunisian manufacturing goods. Avoiding the "trade diversion" effect that may result from the stretching of the EU 8 requires greater productive efforts by firms. The sector-based net exports (NEX) can also be considered as a proxy of the Tunisian comparative advantage. This other indicator is defined as the ratio of the difference between exports and imports to the sector-based value added. A causality problem can be reasonably suspected for those two trade variables. The risk also applies to the effective rates of protection (ERP). Until the beginning of the eighties, numerous authors agreed that developing countries had to protect their infant industry9 . This theoretical argument suffers from the detrimental effects of protectionism, such as the risk of perpetuating X-inefficiency with domestic costs keeping above internationally competitive levels 10 . Thus, high unit costs, and consequently low TFP levels, might be correlated with ERP. But if the Euro-Mediterranean Free Trade Agreement (FTA) stimulates productivity by providing a clear signal that there will be no protection behind which to shelter, the rhythm of the tariff removal can be predetermined by the ability to achieve higher productivity.

The economic environment also matters through the domestic demand in the manufacturing sectors. Here, domestic demand is defined as the current output augmented by imports minus exports (DEM). A slowdown in domestic demand represents a real challenge for producers, who have to redeploy their activities in order to compete on external markets. The impact of this variable is potentially difficult to differentiate from the previous one. Thus, the greater the level of trade protection is, the stronger will be the firm's vulnerability to the swings of domestic demand. Also, DEM can be seen as correlated with unobservable variables such as the under-utilisation of the capital stock, which cannot be rapidly adjusted downward, as well as with labour regulations, which slow down the speed of adjustment of the workforce to the optimal level. 8 Let us recall that the integration process of the new eastern European countries was completed in 2004.

How can we assess the impact of international price competition on TFP? Tunisia is a "world price-taker". This exogenous constraint bears on local costs and, then, on the behaviour of firms as regard TFP levels. Forty-eight real effective exchange rates, henceforth referred to as REERs, have been calculated over the 1983-2002 period, one for each of the three-digit industries (j) as defined by the Standard International Trade Classification (SITC). For a jindustry, the world price being given, the REER allows to take into account the differential of cost evolution across competing partners. Under the assumption that the law of one-price holds, a long run deviation in the REER (i.e. appreciation) constitutes a potential loss of the Tunisian competitiveness and an incentive to raise TFP levels. Six sector-based geometric averages of the REERs have been calculated from the forty-eight relevant 3-digit groups, one for each of the manufacturing sectors we refer to in this paper. Each of the three digit level industries (j) has been weighted by its respective contribution to the value added of the sector, q industries being considered. In measuring REERs, the consumption price index (CPI) has been used for the ten largest world exporters (i) of the j-industry. The weight given to each partner was based on its average contribution to exports over the period 1999-2002. In the formula that follows (3), NERs are the nominal exchange rates of the Dinar vis-à-vis the currencies of those ten exporters (i).

qj pi q ij i i j i CPI REER NER CPI = =                   = × ∏ ∏ (3)
For fifteen years, the Tunisian authorities have targeted a stable "macroeconomic" real effective exchange rate by periodically adjusting the nominal rate. The Dinar is pegged to a basket made up of the currencies of the main economic partners, with a prominent role given to the euro. This exchange rate policy has been quite successful in terms of promoting macroeconomic stability. The inflation rate has remained low, close to the rate of the European economies, with the changes in the nominal exchange rate offsetting price differentials.

However, this policy did not take into account the specific sectoral constraints of the trade liberalization program. Therefore, some of the manufacturing sectors had to compensate for the loss of price competitiveness by a higher TFP.

As regard financial openness, the common wisdom suggests that Foreign Direct Another variable has been introduced to capture reallocation effects within each of the six industries. Bernard and Jones (1996 b) have shown that these reallocations (or share effect) accounted for 4% of the aggregate TFP growth of OECD countries over the 1970-1987 period 11 .

As we work on aggregated data, similar redistributions of the sector-based value added, hereafter referred to as STRUC, may occur with a non-neutral impact on the evolution of TFPs. This composition effect is measured on a yearly basis by the sector-based average of the 3 digit level absolute differences in value added shares (3):

, , 1 , 1 , , 1 1 1 1 k k k J j t j t k t J J j k j t j t j j Q Q STRUC J Q Q - = - = = = - ∑ ∑ ∑ (4)
where t j Q , is the value added of the j-th-industry and k J the total number of industries in the k-th-manufacturing sector. STRUC values range between 0 (no structural change within the sector) and 1 (full change). In contrast with other approaches to the share effect, we consider that STRUC captures the impact of a short term or a spontaneous reallocation effect. STRUC reflects the aggregated dimension of the data set and not a specialisation or a long run supply reallocation effect. Therefore, the sign of the regression coefficient is not predetermined.

The dynamic panel and the issue of causality

The analysis of the potential determinants of openness has suggested that we should control for the direction of causality between variables. The Granger-causality test (1969) is commonly used. As long time series are not available here, we consider a panel causality test as follows. We suppose that X fails to Granger cause Y if in the regression of Y on its lagged variables and X, the coefficients j δ associated with X t-j in (5) are not significant 12 . The unobservable heterogeneity over the six manufacturing sectors will be controlled for through the introduction of fixed effects.

+ = = + + + = ∑ = - ∑ = - t i n j j t i j m j j t i j i t i X Y Y ε δ β β (5)
The lag pattern of the variables has to be identified through a specification test13 . Equation ( 5) is a general dynamic panel data model. Lagged dependent variables tend to generate biased and non consistent estimators when the within or the ordinary least squares methods are used. An appropriate way to overcome this problem consists in taking the first difference of the equation in order to remove the fixed effect and then resorting to the consistent Generalized Method of Moments estimator (GMM). The error terms

t i, ε ∆
in the differentiated equation are correlated with the independent variables j t i Y - ∆ , . Therefore, the latter need to be instrumented by a suitable set of lagged endogenous variables expressed in levels (

' j it Y -), or in differences ( ' , j t i Y - ∆
) 14 . The causality test from X to Y will consist in testing the null hypothesis of the coefficients in (4) : 0 ....

2 1 = = = = n δ δ δ
. The statistic of the test asymptotically follows a Chisquare distribution with n degrees of freedom. As the time series are short, the simplest version of the panel causality test has been performed. In equation ( 5), the slope coefficients are assumed 11 The Bernard and Jones(1996b) method consists in decomposing the productivity growth into a pure productivity growth effect and a share effect by using an accounting framework (RES,1996 p140) 12 We use the easiest presentation of the causality test on panel data. A more complex analysis, as in [START_REF] Nair-Reichert | Causality tests for cross-country panels: a new look at FDI and economic growth in developing countries[END_REF], would be to hypothesise that j β and j δ are specific to individuals in (3), here the sectors.

to be the same across the sectors. This hypothesis can potentially affect the validity of our results if there are disparities across sectors15 . The results of the empirical analysis are given in Table 2. (1,3) (3,3) The symbol " Y X ⇒ " indicates the direction of the Granger causality. The symbol "X ≠ Y", means that there is no causality from X to Y. ***, **,* denote significance levels 1%, 5% and 10%, respectively.

A unidirectional causality is evidenced for Foreign Direct Investment (FDI) and net export (NEX) with past values of these variables explaining the present sector-based TFP levels.

A bidirectional impact is found for both domestic demand (DEM) and structure (STRUC). This last result suggests a potential endogeneity risk while considering TFP determinants. For ERP, EX and REER, the Granger causality test did not prove to be statistically significant.

The TFP determinants: the regression analysis

Policy during the period 1990-2002 was more focused on trade openness than the previous one 16 . Tunisia joined the GATT in 1990, became a founding member of the World Trade Organisation (WTO) and initiated current account convertibility in 1993. The euro-Mediterranean partnership also took shape and was institutionalized through the Barcelona Declaration. For the aforementioned reasons, we have checked for the stability of the TFP model coefficients and rejected this hypothesis by comparing the 1983-89 and the post WTO integration period (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002), through a Chow test. The results with the within estimator are reported in table 3. On the left hand side of this table, the explanatory variables are considered as predetermined. On the right hand side, STRUC and DEM have been instrumented to control for the potential endogeneity bias that was detected by the causality test pertaining to the whole period (table 2). Lagged values for the composition (STRUC) as well as the level of the domestic demand (DEM), net exports (NEX), imports (IMP), and the effective rate of protection (ERP) have been used as instruments. The over-identification p-value of the Hansen J-statistic did no reject, at the 5% level, the relevance of these instruments. The explanatory power of the estimated models is quite good (table 3). The variance of the TFP level is explained from 90% for the first sub-period to 88% for the second one (1990-02), with the instrumental variable estimator. The fixed effects F-test do not reject the presence of a strong intersectoral heterogeneity. From the regression coefficients we observe that until 1990, the net export (NEX), the domestic demand (DEM) are the only variables statistically significant at the 99% level of confidence. The second sub-period is marked by a better econometric fit, all coefficients proved significant with a clear impact of the variables capturing the openness strategy in both trade and financial dimensions. It is worth mentioning that the FDI effect turned to be significant in the second sub period at a time when FDI inflows greatly increased as a result of the privatisation policy pursued in several sectors. This positive impact supports the recent Tunisian policy target to raise foreign capital inflows at a yearly rate of 10%.

As regard the relative price indicators, some significant changes occurred between the first and the second sub-period. The coefficient of the real exchange rates (REERs) is negative and suggests that price competitiveness played against sector-based TFP gains. The effective rates of protection display a negative impact on TFPs. This result is consistent with John Hicks' remark about protected firms enjoying a "quiet life" leading to X-inefficiency. Therefore the dynamic of the trade liberalisation policy should and proves to be a stimulating force for increasing productivity levels. Over the whole manufacturing sector, the average ERP decreased from 194% in 1983 to 85% in 1990 and 56% in 2002. Finally, STRUC is negatively correlated to TFP and mainly reflects the intra-sector based heterogeneity as discussed above.

III. Tunisian sectors and the international TFP convergence hypothesis

III. 1: The convergence hypothesis revisited

In this section we want to highlight the TFP performance of Tunisia vis-à-vis those of OECD countries. In other words, did Tunisian manufacturing firms converge toward the best international standards over the 1983-02 period? To answer this question panel data unit root tests are used to evaluate convergence between Tunisia and a sample of 11 OECD countries. In the Solow-type economic growth model, the international convergence of per capita GDPs is assumed to result from the diminishing returns to individual factors. The smaller the per capita capital stock is, the higher will be its marginal efficiency. In developing countries, economic and financial openness modifies the relative endowments of capital and labor, allowing the import of the most efficient technology available on the market. The long-run dynamics should therefore support the international productivity convergence hypothesis.

However, there are some restrictive assumptions underlying the international productivity convergence and, to some extent, the sector-based convergence arguments. Indeed, in a context of increased integration into the world economy, the process of trade specialization may exclude the possibility of a long-run international convergence of productivities for all sectors. In addition, the progress characterizing the technology is neither exogenous nor the same worldwide. OECD countries have a high rate of innovation. For manufacturing activities, they can be seen as leaders while developing countries are only followers seeking to replicate innovations based on their own human and institutional abilities. If FDI contributes to the international diffusion of the appropriate production technology, some differences inevitably remain, partly connected with non-transferable cognitive apprenticeships. Efficient static and dynamic routines are firms' core competencies in industrialized countries. In a situation where skills are idiosyncratic and non-transferable, TFP can be higher and production costs lower in OECD countries.

III.2 Unit root tests and convergence

The empirical literature on productivity convergence can be decomposed in two ) ln (ln ) ln (ln ) ln (ln

1 * 1 * 1 * t p j j t j t j t t t t u Y Y Y Y Y Y + - ∆ + - + = - ∆ ∑ = - - - - φ ρ µ (6)
If the difference in log TFPs is non stationary ( 0 = ρ ), the TFP series between the two countries diverge, suggesting a persistent TFP shock. On the other hand if the gap is stationary 0 < ρ , the technology shocks are not persistent, and we conclude for a convergence process. Equation ( 6) is the standard regression equation for the augmented Dickey Fuller test, (ADF). Any univariate unit-root test on this equation helps to test for stochastic convergence.

This methodology has been used by [START_REF] Harris | Manufacturing industry in Northern Ireland and Great Britain: Was there convergence during the 1949-92 period?[END_REF] to investigate regional convergence within UK over the 1963-92 period17 .

Although this methodology allows identifying the country pairs for which there is bilateral convergence, long time series are required in order to avoid the low power of univariate unit-root tests. To increase this power, Bernard and Jones (1996 b)

Y Y Y Y Y Y + - ∆ + - + = - ∆ ∑ = - - - - φ ρ µ (7) * it
Y represents the TFP of each of the 11 OECD countries, while t Y is the TFP of the Tunisian sector. Appendix 1 reports the TFP differentials by industry.

Over the last ten years, the empirical literature analyzing international TFP convergence has significantly stretched. To limit the references to some of the most influential works, let us mention Bernard and Jones (1996 b)'s pioneering paper on a sample of 14 OECD countries over the 1970-87 period. Stochastic convergence is tested by using a Levin and Lin (1992) type-test imposing the same autoregressive coefficient in equation ( 7): ρ ρ = i . Martin and [START_REF] Mitra | Productivity growth and convergence in agriculture versus manufacturing[END_REF] referred to a similar test to investigate the phenomenon in agriculture and manufacturing over the period 1967-92, based on sample of 50 countries. [START_REF] Freeman | interpreting cross-section and time series tests of convergence: the case of labor productivity in manufacturing[END_REF] applied the Maddala and Wu (MW)'s test to evaluate the labour productivity convergence in the manufacturing sector of 8 OECD countries from 1950 to 1998, while Mukherjee andKuroda [2002, 2003] have considered the productivity convergence of the agricultural sector in 14 Indian regions by adopting several panel data unit root tests18 . Finally, lets us mention the recent paper by [START_REF] Funk | Panel tests of stochastic convergence: TFP transmission within manufacturing industries[END_REF] who resorted to the IPS test to analyse the TFP convergence of 21 industries in 16 OECD countries (1971-94).

In this paper, we test for sector-based stochastic TFP convergence between the Tunisian sectors and a sample of 11 OECD countries over the period 1983-02. Using the aforementioned methodology, we test the stationarity of the log TFP gap. The OECD benchmark countries are: Austria, Belgium, Canada, Finland, France, Italy, the UK, Norway, Spain, Sweden and the USA.

On the basis of equation ( 7), two standard panel data unit root tests (IPS, MW) are performed.

For both tests, the null hypothesis means: 0 ... The IPS statistics used is the so-called t-bar statistics defined by:

) 1 , 0 ( ) ( )) ( ( N t Var t E t N IPS → - = (8)
where

∑ = i c i t N t ) / 1 ( , c
i t is the Dickey-Fuller unit root statistic which is specific to each country (i). The t -statistic follows a normal distribution with unknown mean, E ( t ), and variance, Var ( t ). These values can be either simulated or obtained from Im et al (1997,2003)'s tables. The MW statistic combines the observed significance levels associated to the individual ADF. Presuming cross-sectional independence, p i denoting the p-value from the Dickey Fuller test on the i th time series, MW statistic follows a Chi-square distribution:

∑ = → - = N i N Ln MW 1 i ) 2 ( ) (p 2 κ (9)
Simulation experiments are carried out to obtain the p-values derived from the empirical distribution of the test. Moreover, for both tests we assume that one lag length is sufficient to remove serial correlation for each country-pair of the panel 20 . Table 4 reports the IPS and the 19 In this particular case, we can employ the [START_REF] Levin | Unit root in panel data: new results[END_REF] test. 20 As the time series dimension is limited, we did not conduct the usual strategies (sequential method, AIC criterium…) to identify the appropriate lag length to save more degrees of freedom.

MW statistics with their respective p-values. According to the MW test, we can reject the null that sector-based TFP differentials have a unit root and conclude that there is an overall convergence for the six sectors. For two of them, Building, Electrical and Mechanical activities, the IPS test does not reject the null hypothesis that all the series are stationary, suggesting an overall divergence. Due to the limited dimension of the panel dataset, the heterogeneity of the autoregressive coefficients could affect these empirical results. Conclusions derived from the two tests be sensitive to a mixture of stationary and non stationary series. In other words, Tunisian sectors may experience a long-term convergence process with some OECD countries and in a divergence process with others. To appraise this potential sensitivity, the two tests have been rerun by dropping 3 countries from the 11 OECD-sample, one by one, according to the lowest individual ADF test values. That way, the benchmark countries to which Tunisian sectors are likely to converge have been dropped sequentially. The results of panel data unit root stationary tests have to be interpreted with caution, especially when the individual dimension is limited as it is the case in this work. When we reject the null hypothesis, it does not mean that Tunisian sectors are in a convergence process with the whole OECD country sample. A TFP convergence process may exist within a restricted subsample. The sensitivity analysis conducted by considering sub-panels has shed some light on this problem. Beyond this limitation and its potential implication for the statistical inference, the two standard unit root type tests have given some support to the existence of an overall TFP convergence process for five sectors.

IV Conclusion

In this paper, we addressed two issues regarding the productive performance of the Tunisian manufacturing industry. First, we focused on measuring sector-based TFP levels and analyzing their economic determinants over the 1983-02 period. Causality relationships between identified determinants sectoral TFPs were assessed using panel data Granger causality type test.

The GMM estimator has been used to conduct this test in order to take into account the potential endogeneity bias resulting from the dynamic specification of regressions. Over the first subperiod (1983)(1984)(1985)(1986)(1987)(1988)(1989), domestic demand and net exports were found to have a significant impact on TFP levels. Over the second sub-period (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002) all the coefficients of the reduced form proved to be statistically significant, including the variables reflecting trade and financial openness, as measured by the effective rate of protection and the Foreign Direct Investment flows, respectively.

The second empirical objective of this paper was to use the international TFP comparison for assessing the potential convergence of Tunisian sectors toward the productive performance of a sample of 11 OECD countries. Trade liberalisation goes along with increased specialization. In this context, international convergence is far to be a trivial process although developing economies have an economic interest in adopting the best OECD productive practice. The IPS and the MW panel data unit root tests were used by considering the country TFP differentials.

Empirical results suggest a stationary process for five sectors, therefore supporting the assumption of a long run convergence. On the contrary, a divergence process was found for building and ceramics industries.

Empirical results are subject to some refinements. Specifically, for improving the power of the panel data unit root tests but not only, it would worth reconsidering convergence by using more disaggregated data. The three-digit industry level would deserve a particular attention, allowing a better understanding of the intra-sectoral heterogeneity. Given the sector, in line with the trade specialization argument we referred to, Tunisia should improve its TFP level in manufacturing activities offering the best export opportunities while leaving others. Although detailed information is lacking for carrying out such an empirical analysis, it remains a stimulating question for further research. It would also be interesting to extend the sample to some large countries such as China or India. Both have a wide range of common specialisations with the Tunisian manufacturing industries and are becoming major rivals in world markets.

  Investment (FDI) facilitates productivity gains and the integration into the world economy by several channels. The enduring presence of foreign enterprises provides easier access to efficient technologies and organizational methods. In comparison with standard financial debt, FDI incorporates both human and organizational know-how. It allows saving on the fixed costs of producing technological innovations as well as on the marginal cost of their replication in the domestic environment. FDI also contributes to lowering the transaction costs of penetrating external markets. A capital stock ratio of FDI has been constructed for each of the six manufacturing sectors we are interested in. Inflows have been summed since 1980 with the same yearly depreciation rates as those used for the calculation of the sector-based capital stock (10%, 5%).

  categories. The first category is based on cross-sectional data and uses the Sigma or the Beta convergence tests. The second category explores the time series properties of the data and refers to stochastic convergence tests. According toBernard and Jones (1996b), stochastic convergence implies that sectoral TFP differentials between countries are stationary. The econometric methodology based on the unit root tests is resorted to in this study. For each of the six sectors, we test for stochastic convergence between Tunisian TFP and those of an OECD country sample. Let t Y be the TFP level of a specific Tunisian industry and * t Y the TFP level of one OECD country considered as the benchmark. Testing for convergence consists in conducting the following simple unit root test.

.

  Rejecting the null suggests a convergence process and implies the stationarity of the Tunisian TFP gap with a subset or at least one OECD country. The specification of the test does not impose the equality of is, a converging process at a common rate with respect to all these countries.

  Figure 2: TFP Evolution in Food Processing Industries, by Country

Figure 3 :

 3 Figure 3: TFP Evolution in Building Materials and Ceramic Industries, by Country

Figure

  Figure 5: TFP Evolution in Chemical Industries, by Country

Figure

  Figure 7: TFP Evolution in Other Industries by Country

Table 1 : Sector-based TFP growth and the relative share of the Tunisian manufacturing sectors 1983-2002 period (%)

 1 Annual average growth rate of TFP for each period. Figures refer to two different depreciation rates of the capital stock, 10% and 5%, respectively; II: Relative share of the sector-based value added over the period for all Tunisian manufactured goods.

	Periods	1983-1987		1987-1995		1995-2002		1983-2002
	Industries	I	II I	II I	II	I
	Food processing	-	28.7	-	20.9	-	18	-
	TFP (10%)	-5.7	-	-6.3	-	-1.6	-	-4.6
	TFP (5%)	-7.4	-	-6.3	-	-2.2	-	-5.2
	Building and ceramics	-	10.5	-	11	-	10.2	-
	TFP (10%)	-2.7	-	5.1	-	4.1	-	3.4
	TFP (5%)	-3.9	-	4.8	-	3.3	-	2.7
	Electrical, Mechanical	-	14.8	-	13.9	-	13.3	-
	TFP (10%)	-0.7	-	2.7	-	5.3	-	2.1
	TFP (5%)	-1.9	-	2.4	-	4.9	-	1.5
	Chemical	-	4.6	-	9.5	-	10.9	-
	TFP (10%)	-0.3	-	12.3	-	5.1	-	9.7
	TFP (5%)	-0.8	-	11.6	-	4.4	-	9.0
	Textiles and leather	-	28.3	-	31.8	-	34.7	-
	TFP (10%)	8.9	-	3.4	-	3.4	-	3.5
	TFP (5%)	6.8	-	3.3	-	2.8	-	2.8
	Miscellaneous indus.	-	12.2	-	12.6	-	12.9	-
	TFP (10%)	3.4	-	-0.2	-	1.2	-	0.4
	TFP (5%)	1.4	-	-0.3	-	0.5	-	-0.3

I:

Table 2 : Results of the panel data Granger causality test: GMM estimation method (1983-2002)

 2

		Variables (levels)	P-value	Number of lags
	Variables				(m,n)
	(TFP, DEM)	DEM ⇒ TFP PTF ⇒ DEM	K c =6.49*** K c =4.37**	0.01 0.04	(1,3) (1,1)
	(TFP, FDI)	FDI ⇒ TFP PTF ≠ FDI	K c =3.69** K c =0.32	0.05 0.57	(1,3) (1,1)
	(TFP, ERP)	ERP ≠ PTF	K c =0.08	0.77	(1,3)
		TFP ≠ ERP	K c =1.76	0.41	(2,1)
	(TFP, NEX)	NEX ⇒ TFP TFP ≠ NEX	K c =3.03* K c =0.76	0.08 0.86	(1,3) (3,3)
	(TFP, EX)	EX ≠ PTF	K c =0.69	0.40	(1,3)
		TFP ≠ EX	K c =2.55	0.47	(3,3)
	(TFP, REER)	REER ≠ TFP	K c =0.24	0.62	(1,3)
		TFP ≠ REER	K c =5.13	0.16	(3,3)
	(TFP, STRUC)	STRUC ⇒ TFP TFP ⇒ STRUC	K c =3.97** K c =29.4***	0.05 0.00

Table 3 : Tunisian manufacturing sectors and TFP level determinants (1983-2002)

 3

	Variables	Within	Variables	Instrumental Variable
							estimator
		1983-1989	1990-2002				1983-	1990-2002
							1989
	FDI	0.002	0.00028		FDI	0.0003	0.00024
		(1.19)	(4.67)***				(0.20)	(4.70)***
	DEM	0.00027	0.00008		DEM	0.0004	0.0001
		(2.42)***	(3.13)***				(2.97)***	(3.84)***
	NEX	0.104	0.046		NEX	0.165	0.036
		(3.43)***	(3.16)***				(2.76)***	(2.41)***
	REER	0.001	-0.0036		REER	0.0020	-0.003
		(1.00)	(-2.67)***				(2.57)	(-2.65)***
	STRUC	-1.066	-2.325	STRUC	-1.060	-3.25
		(-2.08)***	(-8.40)***				(-1.12)	(-5.28)***
	ERP	0.9 10 -6	-0.0006		ERP	-0.00001	-0.0006
		(0.15)	(-1.92)**				(-0.52)	(-2.15)***
	Chow Test	12.56***	Chow Test	25.03***
	R 2	0.87	0.90		R 2	0.90	0.88
	Fixed Effect	14.03***	83.59***	Fixed Effect	15.49***	63.74***
	F-Test	F(5,30)	F(5,66)		F-Test	F(5,24)	F(5,66)
		Hansen over-identification J-statistics	χ	) 3 (	5.54	6.886
	Numbers in brackets are the t-statistics at the following significant levels of confidence: (*) 90%;
	(**) 95 %; (***) 99 %. The sectoral fixed effects are not reported in this table.

  have suggested running panel unit root tests on a sample of OECD countries. The test is based on following the regression

	equation:
	(ln	* it	ln	t	)	i	i	(ln	* it	1	ln	t	1	)	j	1 p	j	(ln	* it	j	ln	t	j	)	u	it

Table 4 : Tunisia and the international TFP stochastic convergence tests

 4 Chemical activities. Therefore, not only a mixture of stationary and non-stationary TFP gaps does exist, but the IPS test also proves to be more sensitive to the sample heterogeneity of the TFP differentials than the MW test.

	Sectors	IPS	Test		MW	Test
		IPS Statistic P-value	Conclusion	MW Statistic P-value	Conclusion
	Food	-1.473*
	Building	2.40
	Electric
	Chemical
	Textile
	Miscellaneo
	us

Table 5 :

 5 Unit root test sensitivity to the exclusion of some OECD countries

	Sectors
		IPS Statistic P-value MW Statistic P-value	Sub-sample
	Food	-1.323	0.13	48.937***	0.00	10 countries
		-0.844	0.20	42.474***	0.00	9 countries
		-0.546	0.29	36.162***	0.00	8 countries
	Building	3.918	1.00	18.953	0.53	10 countries
		3.995	1.00	15.712	0.61	9 countries
		3.982	1.00	13.060	0.67	8 countries
	Electric	-0.241	0.41	47.741***	0.00	10 countries
		0.397	0.65	37.873***	0.00	9 countries
		1.023	0.85	28.890**	0.02	8 countries
	Chemical	-1.534*	0.06	53.975***	0.00	10 countries
		-1.281*	0.10	47.236***	0.00	9 countries
		-1.005	0.16	40.520***	0.00	8 countries
	Textile	-1.475*	0.07	62.887***	0.00	10 countries
		0.285	0.61	39.861***	0.00	9 countries
		1.486	0.93	26.045*	0.05	8 countries
	Miscellan	-2.050**	0.02	57.186***	0.00	10 countries
	eous	-1.212	0.11	45.568***	0.00	9 countries
		-0.795	0.21	37.907***	0.00	8 countries
	N.B: The exact denomination of sectors is: Food processing, Building and ceramics, electrical
	and mechanical, textile and leather, miscellaneous

In 1995, when the principles of the partnership were accepted, and ratified in early 1998 by the Association Agreement with the EU (AAEU),

28% of public revenue resulted from import tariffs. The removal of Tunisian trade restrictions is therefore a highly important measure for the government as well as the manufacturing producers. Its macroeconomic and social impacts explain why the public programme of "mise à niveau" was launched.

In his analysis of Chilean trade liberalization and its aftermath, James[START_REF] Tybout | Chile, 1979-1986: Trade Liberalization and Its Aftermath[END_REF] retains different depreciation rates: 5% for building, 10% for machinery and 20% for vehicles.

Textiles, clothing and leather account for nearly 35% of the manufacturing value added and the economic health of this sector is crucial for thousands of Tunisian workers. Its future will be narrowly conditioned by the ability of local producers to overcome the impact of foreign competition and its implications in terms of international prices and costs.

In his review of prominent trade liberalisation sceptics, Sebastian Edwards includes[START_REF] Krugman | The Myth of Asia's Miracle[END_REF] and[START_REF] Rodrik | Trade and Industrial Policy Reform[END_REF]

See Edwards (1993) for a survey

Other potential variables would deserve interest such as the level of human capital or the quality of institutions. They are not considered in this paper because of data limitations. We assume that this does not create an omitted variable bias.

This is the best known argument, reasoning in favour of temporary protection until the "infants" have learned to stand on their own feet, becoming internationally competitive without any state support.

We use the sequential method starting with 3 lag lengths for both variables.

In that particular case, it would be less restrictive to replace these slopes by sector specific coefficients, see[START_REF] Hurlin | Granger causality tests in panel data models with fixed coefficients[END_REF] for a good discussion of this problem of heterogeneity.

A predictive test for stability of the coefficients over the whole period, not reported here, has been carried out. The stability hypothesis was rejected. 1990 proved to be the relevant year for the structural break. Different subsets were consideredwith 1991, 1992, 1993 and 1994, as alternative cut-off years.

These authors also extend the unit root test by adding the trend component in equation (4) and by controlling for structural break.

Let us mention[START_REF] Levin | Unit root in panel data: new results[END_REF] Lin's, Im, Pesaran and[START_REF] Im | Testing for unit roots in heterogeneous panels[END_REF]'s (IPS),[START_REF] Harris | Manufacturing industry in Northern Ireland and Great Britain: Was there convergence during the 1949-92 period?[END_REF] Tzavalis (1999)'s, Hadri (2000)'s tests

Results of the sensitivity are reported in table 5. As regard the building industry and according to the MW test, the convergence evidenced in table 4 is now rejected with the elimination of only one country. Conclusions remain unchanged for the other five sectors.

Conversely, the IPS test shows a stronger sensitivity to the subtraction of countries. Let us observe that the convergence process vanishes when dropping one country for Food industry, two countries for Textile, garment and leather, but also for miscellaneous industries, and three


Jean-Louis Arcand

Pessimism, Optimism and Credit Rationing

Keywords: Pessimism, Optimism and Credit Rationing rank-dependent expected utility, credit rationing JEL: D81, D82

est destinée au dépôt

Introduction

In their celebrated article, [START_REF] Stiglitz | Credit Rationing in Markets with Imperfect Information[END_REF] (henceforth, SW) showed that rationing could obtain in competitive credit markets as a result of adverse selection stemming from the inability of lenders to observe the riskiness of projects undertaken by borrowers. In this paper, I prove two results. First, if borrowers, whose preferences can be described by the rank-dependent expected utility (RDEU) model [START_REF] Quiggin | A Theory of Anticipated Utility[END_REF]), are sufficiently pessimistic, then there may be no adverse selection in the market for loans because the Choquet expected return of a project to the borrower will not necessarily be an increasing function of its risk. Second, the Choquet average expected return of the loan to the lender can be a decreasing function of the interest rate, thereby rendering credit rationing possible, either (i) when borrowers are not overly pessimistic and lenders are not overly optimistic (an example being the standard expected utility (EU) case), or (ii) when borrowers are highly pessimistic and lenders are highly optimistic. In all other cases, the Choquet average expected return to the lender will be an unambiguously increasing function of the interest rate.

Both of these findings stand in contrast to two key aspects of the SW model of credit rationing, and show how its mechanics stem in part from the assumption that the preferences of agents can be adequately described by the EU model. Furthermore, the results presented here show that the manner in which risk is perceived by borrowers and lenders can potentially be an important, and hitherto neglected, determinant of the operation of the market for loans.

The standard result

Consider the standard SW model in which banks identify a pool of loan applicants who have projects with equal borrowing requirements D, and equal expected gross returns. While projects have identical mean returns, they differ in their riskiness. More precisely, a loan applicant in risk class ρ has a cumulative distribution function (cdf ) for gross returns θ ∈ [θ, θ] given by F (θ, ρ), where ρ is the [START_REF] Rothschild | Increasing Risk I: A Definition[END_REF] parameter of increasing risk and expected gross returns are denoted by μ = R θ θ f (θ, ρ) dθ, where f (θ, ρ) = dF (θ,ρ) dθ . An increase in risk, as parameterized by ρ, is defined by the usual integral conditions:

(i) Z θ θ F ρ (θ, ρ) dθ = 0, (ii) Z θ 0 θ F ρ (θ, ρ) dθ > 0, ∀θ 0 ∈ [θ, θ], (1)
where ∂ ∂ρ F (θ, ρ) = F ρ (θ, ρ). In the absence of collateral possibilities, under the usual limited liability assumption, and given a gross interest rate i, the return to the borrower is e π = max [θ -iD, 0]. The expected return to a loan applicant in risk class ρ is therefore:

Π(iD, ρ) = E [π] = Z θ iD (θ -iD) f (θ, ρ) dθ = θ -iD - Z θ iD F (θ, ρ) dθ, (2)
where the second equality follows by integration by parts. The key to the SW adverse selection result is that the expected return to the borrower is an increasing function of the risk ρ of the project, since:

∂Π(iD, ρ) ∂ρ = - Z θ iD F ρ (θ, ρ) dθ = Z iD θ F ρ (θ, ρ) dθ > 0, (3)
where the second equality follows from integral condition (1 (i)) and the inequality is a direct consequence of integral condition (1 (ii)). Mechanically, the result flows from e π = max [θ -iD, 0] being convex in θ. Let ρ * be such that Π(iD, ρ * ) = K, where K is the borrower's reservation level of return. Then, since ∂Π(iD,ρ) ∂ρ > 0, only projects such that ρ > ρ * will be undertaken. By implicit differentiation of (2) evaluated at ρ * :

dρ * di = - ∂Π(iD,ρ * ) ∂i ∂Π(iD,ρ * ) ∂ρ * = D (1 -F (iD, ρ * )) R iD θ F ρ (θ, ρ * ) dθ > 0. (4)
Equation ( 4) is the SW adverse selection result: as the gross interest rate i increases, borrowers with low risk projects drop out of the market for loans, thereby increasing the average riskiness of the remaining projects faced by the lender. For the latter, the gross return on a loan is given by y = min [iD, θ]. The expected return to the lender is therefore:

Y (iD, ρ) = E [y] = Z iD θ θf (θ, ρ) dθ + Z θ iD iDf (θ, ρ) dθ = iD - Z iD θ F (θ, ρ) dθ, (5)
where the second equality follows from integration by parts. Differentiating (5) with respect to ρ and applying integral condition (1 (ii)) allows one to show that the lender's expected return is decreasing in the riskiness of the project, ceteris paribus:

∂Y (iD, ρ) ∂ρ = - Z iD θ F ρ (θ, ρ) dθ 6 0. (6) Note also that ∂Y (iD,ρ) ∂i = D [1 -F (iD, ρ)] > 0.
Since the lender cannot directly observe which borrower undertakes the project, she must calculate an "average" expected return. Let ρ ∈ [ρ, ρ] be distributed in the population of borrowers according to the cdf H(ρ). Then the "average" expected return to the lender is given by:

Y (iD) = R ρ ρ * Y (iD, ρ)h(ρ)dρ 1 -H(ρ * ) , (7)
where h(ρ) = dH(ρ) dρ . Differentiating with respect to i and rearranging yields the standard SW expression:

dY (iD) di = R ρ ρ * D [1 -F (iD, ρ)] h(ρ)dρ 1 -H(ρ * ) + h(ρ * ) 1 -H(ρ * ) ¡ Y (iD) -Y (iD, ρ * ) ¢ dρ * di . (8)
The first term in ( 8) is positive. The second term is negative because dρ * di > 0 by ( 4), while Y (iD) < Y (iD, ρ * ) since the expected return to the lender is greater on the "least risky" than on the "average" loan. Since Y (iD), Y (iD, ρ * ) and ρ * do not depend upon h(ρ * ), the negative term can be made arbitrarily large in absolute value terms by choosing h(ρ * ) large. Thus, an increase in the gross interest rate reduces the lender's average expected return from loans.

RDEU preferences

Let S = {s 0 , s 1 , ..., s i , ..., s n } be the finite set of states of nature. Consider the set of subsets of S, denoted by E = 2 S , which we shall refer to as the set of events. Let X : S → R with s → X(s). Then

ν : A ∈ E → ν(A) ∈ [0, 1] is a capacity if ν(∅) = 0, ν(S) = 1, and A ⊆ B ⇒ ν(A) 6 ν(B); ν is convex if ν(A ∪ B) + ν(A ∩ B) > ν(A) + ν(B), ∀A, B ∈ E.
The [START_REF] Choquet | Théories des Capacités[END_REF] integral of X with respect to ν is given by:

E C [X] = Z 0 -∞ [ν ({X > x}) -1] dx + Z ∞ 0 ν ({X > x}) dx. (9)
By Theorem 7 in [START_REF] Wakker | Under Stochastic Dominance Choquet-Expected Utility and Anticipated Utility are Identical[END_REF], if the decisionmaker's preferences are consistent with first-order stochastic dominance with respect to probability measure P , the capacity takes the form ν

({X > x}) = ϕ (P ({X > x})) = ϕ (1 -F (x))
, where F (x) is the cumulative density function of X, and ϕ(.) is nondecreasing with ϕ (0) = 0 and ϕ (1) = 1; ϕ (.) is unique and plays the role of a probability transformation function. As shown by [START_REF] Roell | Risk Aversion in Quiggin and Yaari's Rank-Order Model of Choice under Uncertainty[END_REF], [START_REF] Demers | Price Uncertainty, the Competitive Firm and the Dual Theory of Choice under Risk[END_REF], and [START_REF] Guriev | On Microfoundations of the Dual Theory of Choice[END_REF], assuming that ϕ (.) is differentiable and integrating by parts then allows one to rewrite the Choquet expectation in (9) in terms of the Lebesgue-Stieltjes integral:

E C [X] = Z ∞ -∞ xϕ 0 (1 -F (x)) f (x)dx, (10)
where f (x) = dF (x) dx . The "distorted" expectation given in (10) corresponds to [START_REF] Yaari | The Dual Theory of Choice under Risk[END_REF] dual theory functional, and its differentiability has recently been studied by [START_REF] Carlier | Core of Convex Distortions of a Probability[END_REF]. While one can take ϕ(.) to be a component of the agents' risk preferences in a RDEU context, the results that follow can also be reinterpreted in terms of Choquet-Schmeidler expected utility (CEU, [START_REF] Gilboa | Expected Utility Without Purely Subjective Non-Additive Probabilities[END_REF], [START_REF] Schmeidler | Subjective Probability and Expected Utility Without Additivity[END_REF]), where a convex (concave) ϕ(.) corresponds to ambiguity aversion (preference).

For the case at hand, application of (10) yields the following expression for the Choquet expectation of the borrower's return, where ϕ (.) is understood to be the borrower's probability distortion function:

Π C (iD, ρ) = E C [e π] = Z θ iD (θ -iD) ϕ 0 (1 -F (θ, ρ)) f (θ, ρ) dθ = Z θ iD ϕ (1 -F (θ, ρ)) dθ, (11)
where the last equality follows from noting that Let ρ * * be such that Π C (iD, ρ * * ) = K. On the one hand, ∂Π C (iD,ρ * * ) ∂i = -Dϕ (1 -F (iD, ρ * * )). On the other, differentiating Π C (iD, ρ * * ) with respect to ρ * * , integrating by parts, applying integral condition (1 (i)), and assuming that ϕ 0 (0) is bounded from above, immediately yields the following

d dθ [-ϕ (1 -F (θ, ρ))] = ϕ 0 (1 -F (θ, ρ)) f (θ,

Proposition:

Proposition 1 Under RDEU preferences, the limit risk class ρ * * is not necessarily increasing in the interest rate i:

dρ * * di = - ∂ΠC (iD,ρ * * ) ∂i ∂Π C (iD,ρ * * ) ∂ρ = Dϕ (1 -F (iD, ρ * * )) ⎛ ⎝ ϕ 0 (1 -F (iD, ρ * * )) ³ R iD θ F ρ (z, ρ * * ) dz - R θ iD ϕ 00 (1 -F (θ, ρ * * )) f (θ, ρ * * ) ³ R θ θ F ρ (z, ρ * * ) dz ´dθ ⎞ ⎠ .
The key difference between the EU and the RDEU cases is thus apparent: for ϕ(.) convex, dρ * * di may be negative. The reason lies in the denominator of the expression given in the Proposition ( ∂Π C (iD,ρ * * ) ∂ρ ), which is not necessarily positive, as for the EU case given in (4). Formally, this is because:

ϕ 0 (1 -F (iD, ρ * * )) Ã Z iD θ F ρ (z, ρ * * ) dz ! > 0, (12)
since ϕ 0 (.) > 0 and R iD θ F ρ (z, ρ * * ) dz > 0 by integral condition (1 (ii)), whereas:

-

Z θ iD ϕ 00 (1 -F (θ, ρ * * )) f (θ, ρ * * ) Ã Z θ θ F ρ (z, ρ * * ) dz ! dθ 6 0, (13)
since ϕ 00 (.) > 0, f (θ, ρ * * ) > 0, and

R θ θ F ρ (z, ρ * * ) dz > 0.
The intuition for the result is straightforward: pessimism in the borrower's probability distortion function can induce sufficient concavity in her objective function to overcome the convexity of e π that lies at the heart of the SW result. In this case, it will be the high risk borrowers who will progressively drop out of the market for loans, and there will be no adverse selection. In the EU case, the term that includes ϕ 00 (.) vanishes and ϕ 0 (.) = 1, ensuring that the limit risk class ρ * is unambiguously increasing in the interest rate.

As an illustration, consider the parametric example in which ϕ (x) = x 2 and θ ∼ N (1, σ 2 ). For this illustration based on the normal density, ρ corresponds to the standard deviation σ. The right-hand panel of Figure 1 represents the RDEU case, whereas its EU counterpart (ϕ (x) = x) is represented on the left. Expected returns to the borrower are close to zero at iD ≈ 1 for low values of ρ because we have set in the left-hand panel (the standard SW result), while there are many configurations in the right-hand panel that yield a Π C (iD, ρ) that is decreasing in ρ. Figure 2 provides a further illustration of the RDEU case by plotting ∂Π C (iD,ρ) ∂ρ as a function of iD and ρ, under the same parametric assumptions. As should be clear, ∂ΠC (iD,ρ) ∂ρ becomes positive only once iD and ρ are sufficiently large. Consider now the lender's side. Denote the lender's probability distortion function by φ(.). Applying (10) and integrating by parts yields the Choquet expectation of the lender's return:

E[θ] = μ = 1, which implies that E C [θ] = μ -σ √ π . Π(iD, ρ) is unambiguously increasing in risk
Y C (iD, ρ) = θ + Z iD θ φ (1 -F (θ, ρ)) dθ. (14)
Differentiating with respect to ρ, integrating by parts, assuming that φ 0 (1) is bounded from above, and applying (1 (i)) yields the following Proposition:

Proposition 2 Under RDEU preferences, the expected return to the lender Y C (iD, ρ) is not necessarily a decreasing function of the risk ρ of the project:

∂Y C (iD, ρ) ∂ρ = -φ 0 (1 -F (iD, ρ)) Ã Z iD θ F ρ (z, ρ) dz ! - Z iD θ φ 00 (1 -F (θ, ρ)) f (θ, ρ) Ã Z θ θ F ρ (z, ρ) dz ! dθ.
Proposition 2 shows that, contrary to the EU case as given by ( 6), the Choquet expectation of the return to the lender is not necessarily a decreasing function of the riskiness of the project. Indeed, if the lender's probability distortion function is sufficiently concave (i.e., if the lender is sufficiently optimistic), then the lender's expected return to the project can be an increasing function of its riskiness. On the other hand, if the lender is pessimistic, one obtains the standard SW result since ∂Y C (iD,ρ) ∂ρ will be negative. Note also that ∂YC (iD,ρ) ∂i = Dφ (1 -F (iD, ρ)) > 0. Consider now the Choquet average expected return to the lender. If borrowers are not overly pessimistic, as set out in Proposition 1, and therefore dρ * * di > 0:

Y C (iD) = R ρ ρ * * Y C (iD, ρ)φ 0 (1 -H (ρ)) h(ρ)dρ R ρ ρ * * φ 0 (1 -H (ρ)) h(ρ)dρ . ( 15
)
Differentiating with respect to i yields:

dY C (iD) di = R ρ ρ * * Dφ (1 -F (iD, ρ)) φ 0 (1 -H (ρ)) h(ρ)dρ R ρ ρ * * φ 0 (1 -H (ρ)) h(ρ)dρ (16)
+ φ 0 (1 -H (ρ * * )) h(ρ * * ) R ρ ρ * * φ 0 (1 -H (ρ)) h(ρ)dρ ¡ Y C (iD) -Y C (iD, ρ * * ) ¢ dρ * * di .
As in the EU case considered in (8), the second term is negative, as long as lenders are not overly optimistic (as spelled out in Proposition 2, so that ∂Y C (iD,ρ)

∂ρ < 0) so that Y C (iD) < Y C (iD, ρ * *
).1 Credit rationing may therefore obtain. On the other hand, if borrowers are sufficiently pessimistic so that dρ * * di 6 0:

Y C (iD) = R ρ * * ρ Y C (iD, ρ)φ 0 (1 -H (ρ)) h(ρ)dρ R ρ * * ρ φ 0 (1 -H (ρ)) h(ρ)dρ , ( 17
)
where it is important to note that the limits of integration have changed, and:

dY C (iD) di = R ρ * * ρ Dφ (1 -F (iD, ρ)) φ 0 (1 -H (ρ)) h(ρ)dρ R ρ * * ρ φ 0 (1 -H (ρ)) h(ρ)dρ (18)
+ φ 0 (1 -H (ρ * * )) h(ρ * * ) R ρ * * ρ φ 0 (1 -H (ρ)) h(ρ)dρ ¡ Y C (iD, ρ * * ) -Y C (iD) ¢ dρ * * di .
Here, ρ * * corresponds to the riskiest project and Y C (iD, ρ * * ) < Y C (iD) if lenders are not overly optimistic (so that ∂Y C (iD,ρ) ∂ρ < 0). The second term in (18) will then be unambiguously positive, as will dY C (iD) di : no credit rationing can then obtain. The converse of the preceding two cases can be constructed for optimistic lenders for whom ∂YC (iD,ρ) ∂ρ > 0.

Concluding remarks

In this paper I have shown that the key adverse selection result of the SW model of credit rationing depends upon borrowers not being overly pessimistic, as this concept is defined in the RDEU model. If borrowers are particularly pessimistic, it will be the high risk group that will be the first to drop out of the market for loans, contrary to the usual SW result. The perception by lenders of the Choquet average expected return on their remaining pool of loan applicants determines whether credit rationing can obtain or not. If borrowers are sufficiently pessimistic so as to eliminate the adverse selection problem, pessimistic (or EU) lenders are sufficient to ensure that the Choquet average expected return to lenders will be unambiguously increasing in the interest rate. Conversely, if lenders are not pessimistic enough and adverse selection exists, sufficiently optimistic borrowers can also ensure that the Choquet average expected return to lenders is unambiguously increasing in the interest rate.

It is straightforward to show that all of the results presented here carry over to the case of collateral, as considered in [START_REF] Wette | Collateral in Credit Rationing in Markets with Imperfect Information: Note[END_REF]. An important extension would be to study the existence of adverse selection in the SW model under RDEU preferences using richer notions of mean-preserving increases in risk such as those proposed by [START_REF] Chateauneuf | Four Notions of Mean-Preserving Increase in Risk, Risk Attitudes and Applications to the Rank-Dependent Expected Utility Model[END_REF].

  ρ) and integrating by parts. When ϕ (x) = x, equation (11) boils down to the EU expression given in (2).

Figure 1 :

 1 Figure 1: The expected return and Choquet expected return to the borrower as functions of iD and ρ. The right-hand panel represents Π C (iD, ρ) for ϕ(x) = x 2 and θ ∼ N (1, σ 2 ). The left-hand panel represents the corresponding EU case (Π(iD, ρ)).

Figure 2 :

 2 Figure 2: ∂Π C (iD,ρ) ∂ρ

The termφ 0 (1-H(ρ * * ))h(ρ * * ) ρ ρ * * φ 0 (1-H(ρ))h(ρ)dρis unambiguously positive.