J D Byers
email: byersjd@cardiff.ac.uk

D A Peel

D A Thomas

AGGREGATION AND LONG MEMORY: EVIDENCE FROM TELEVISION AUDIENCE DATA

Keywords: Long Memory, Fractional Processes, Aggregation, Habit. JEL Classification: C22., D12

Many economic outcomes appear to be influenced by habit or commitment, giving rise to persistence. In cases where the decision is binary and persistent, the aggregation of individual time series can result in a fractionally integrated process for the aggregate data. Certain television programmes appear to engender commitment on the part of viewers and the decision to watch or not is clearly binary. We report an empirical analysis of television audience data and show that these series can be modelled as I(d) processes. We also investigate the proposition that temporal aggregation of a fractionally-integrated series leaves the value of d unchanged.

Introduction

Decisions which have to be repeated often display a considerable degree of persistence which appears to reflect habit or commitment. Examples include attendance at religious ceremonies, support for political parties, attendance at sporting and theatrical events and consumption of tobacco, alcohol and other drugs. Typically, we do not have data at the level of the individual but must work with aggregate data and the two may exhibit quite different properties. One possibility is 'short memory' at the individual level but 'long memory' at the aggregate level .

In this paper we report an analysis of several data series generated by a particular set of choices which are often believed to be strongly affected by habit -the viewing of television programmes. We present evidence that -consistent with a cross-sectional aggregation result which we discuss below -the temporal aggregate of individual choices exhibit long memory as represented by a fractional differencing parameter, d, greater than zero. We also investigate the proposition that temporal aggregation of a fractionally-integrated series leaves the value of d unchanged -an implication of the self-similar behaviour of long-memory processes.

A characteristic of the data is that, at the individual level, it consists of a sequence of values which are either 1 or 0 corresponding to the decision to partake in the activity or not. Since the standard choice problem in economics involves selecting from a continuum of possible outcomes, it may appear that the relevance of the result is limited. Consider, however, the mundane example of purchasing baked beans.

Although the rate of consumption can vary continuously, the purchaser also has to select a brand from the range of brands available. The decision to buy the same brand as last time or to try some other is a 0/1 decision, one that firms expend substantial sums to influence. It would be easy to multiply examples but a little thought suggests that this aspect of choice is widespread.

Theoretical Issues

At the individual level, it would appear plausible that the inertia resulting from habit will lead to an autoregressive component in behaviour. This may be justified by appeal to either 'myopic' or 'rational' theories of addiction (Becker andMurphy, 1988, Becker, Grossman and[START_REF] Becker | An Empirical Analysis of Cigarette Addiction[END_REF]. The intertemporal structure of aggregate outcomes, however, may not be so straightforward since, to take a well-known example, the sum of n independent AR(1) processes with different parameters will, in general, be ARMA(n, n-1). [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF] pointed out that the panel aggregate of a wide range of time series processes may display the long memory property of a fractionally integrated process. Specifically, suppose that the choice over the level of some variable, x , of the j th agent can be represented by the process,

x
x y W j t j j t j t j t j t , , , ,

= + + + 1 (1)
where y j t , , W t and t are independent processes with the latter white noise, then the aggregate series, x x

t j t j J = = , 1
, derived by summing these dynamic microrelationships over J agents will be integrated of order d if the j are drawn from a Beta distribution over the range (0, 1) with a probability mass of zero at unity. The size of d is affected by the order of integration of the y and W processes but x t is fractionally integrated even if these are I(0). [START_REF] Byers | Modelling Political Popularity: a Correction[END_REF][START_REF] Byers | The Dynamics of Aggregate Political Popularity: Evidence from Eight Countries[END_REF][START_REF] Byers | Modelling Political Popularity: an Analysis of Long Range Dependence in Opinion Poll Series[END_REF] apply this result to support for political parties in the UK and a number of other countries. An approach based on the Granger aggregation result is not practicable in the current context i but an alternative can be found in [START_REF] Willinger | Self-Similarity in High-Speed Internet Traffic Measurements[END_REF] and [START_REF] Taqqu | Proof of a Fundamental Result in Self-Similar Traffic Modelling[END_REF] who consider a stochastic mechanism which generates a sequence of ones -the source is switched on -and zeros -the source is switched off. An 'ON' period is a sequence of 1s, and an 'OFF' period is a sequence of 0s. Taqqu et al analyse the case where ON and OFF periods alternate with i.i.d. lengths, though with possibly different distributions, and show that the superposition of a large number of such series may be fractionally integrated. The crucial condition is that the upper tails of the distribution functions, F x i ON OFF i b g = , , of the ON and OFF periods decline in accordance with a 'power law', i.e. 1 F x i b g behaves like x i as x with 1 2 < < At the individual level, we have a nonlinear, 'regime-shifting', process but, at the aggregate level, the observed process has linear characteristics. As [START_REF] Granger | A Simple Nonlinear Time Series Model with Misleading Linear Properties[END_REF]

A Model of Television Viewing

Consider the following informal model of television viewing. On day (or week) t an individual has to decide whether to watch a programme or not. The Rational Addiction approach suggests that this decision is taken in the light of previous choices and expected future utility. In particular, by watching a programme an individual builds up a stock of 'consumption capital' which raises the current utility from watching the programme but which depreciates over time. Given the current capital stock, an individual will choose to watch a programme if the expected utility from doing so exceeds the utility of not watching. Viewing the programme increases the stock of consumption capital and makes it more likely that the programme will be watched again. The model allows for random shocks to utility with 'positive' shocks raising current utility and making it more likely that the programme will be watched again and 'negative' shocks reducing enjoyment and, perhaps, resulting in crossing a threshold which means that the next broadcast will not be watched. The effects of depreciation may then result in the programme not being watched again regularly until a sufficiently large positive shock occurs.

Suppose we record a value of 1 when the programme is watched and 0 when it is not watched. Habitual viewers will exhibit long sequences of 1s with occasional 0s and uncommitted viewers only occasional 1s among many 0s. For really committed viewers the ON period approaches infinity, i.e. they always watch. Other viewers, however, may become attached for a number of periods, then, for some reason, not watch for a time and later become a regular viewer once more. Some individuals may have extremely long sequences of zeros -in effect, they never watch. The implication of the model is, however, that for each individual there exists an alternating sequence

The Data

The data which we analyse consist of daily viewing figures for three programmes broadcast by Sianel 4 Cymru (S4C) -the UK's Welsh-language TV service ii . The

/ , / + = = 2 1 2 K (3)
In particular, s = 1 corresponds to the annual cycle of 260 days, s = 13 corresponds to the monthly cycle and s = 52 corresponds to the weekly cycle. [START_REF] Harvey | Forecasting, Structural Models and the Kalman Filter[END_REF] discusses the use of this method and points out that, in practice, most of the seasonal variation in a series be accounted for by the initial few sine/cosine pairs plus a small number of other frequencies. All three series were adjusted by removing the first eight trigonometric seasonals plus those corresponding to the weekly cycle and its first harmonic in addition to a constant and trend.

The adjusted series exhibit remaining stochastic seasonality at the weekly frequency.

We deal with this in two distinct ways: firstly by allowing seasonal autoregressive and moving average terms to enter the model for each adjusted series and, secondly, by analysing weekly series obtained from the original series by separate consideration of the different days of the week. This procedure has the advantage of allowing us to compare the estimates of the long memory parameter at two different levels of aggregation since theory suggests [START_REF] Chambers | Long Memory and Aggregation in Macroeconomic Time Series[END_REF] that the fractional differencing parameter will be the same both for an original series and for an 'aggregate' series obtained by selecting every n th value from the original series. The weekly series are adjusted for the annual cycle by removing the first two trigonometric seasonals

Empirical Results

For Range test [START_REF] Lo | Long-term memory in stock market prices[END_REF] provides a formal test of long memory. The test rejects the null hypothesis of d = 0 against 0 0 5 < < d . in the upper tail and involves a correction for short-term dependence in the data using the Newey-West variance estimator to scale the range of the process. We report results for lags of 0, 1, 3 and a value, q, determined by Andrew's (1984) data dependent rule for AR(1) processes. The latter would appear to be an appropriate simple alternative model for the aggregate data.

Table 1 reports the values of the test for the three daily series with the number of lags in the data-dependent test given in parentheses. The test clearly rejects d = 0 against d > 0 for Heno and Pobol Y Cwm but is less clear in the case of Newyddion.

We obtain a potential model for each series by comparing the values of the Schwartz Information Criterion (SIC) for a range of ARFIMA processes. Given the evidence of stochastic seasonality in the data, we conduct the search over a range which includes first order seasonal ARMA components. Specifically, we assume multiplicative weekly seasonality and compare models of the form ARFIMA p d q p q s s ( , , )( , , ) 0 for 2 0 p q , and where p s and q s can be 0 or 1.

Weeding out models with implausible parameter values or other obvious deficiencies we arrived at the models which are reported in Table 2. In each case the fractional differencing parameter is significantly different from zero. The cross-sectional aggregation result implies that each aggregate series is a linear, Guassian, I(d) process for d 0 . To test this we report statistics on skewness and kurtosis along with tests for serial correlation and ARCH effects. Notice firstly, that the ARFIMA models do a remarkable job of providing a clean set of residuals. Pobol-y-Cwm is something of an exception since the Ljung-Box Q statistics are significant from Q(10). Estimating a higher-order seasonal AR fails to solve this problem but also leaves the estimate of d largely unchanged. We look at this issue below when examining the weekly series.

All three series appear to be free of low-order ARCH effects.

The normality tests are less satisfactory. Skewness and kurtosis tests are notoriously sensitive but these three tell a consistent story of positive skewness and excess kurtosis. This is borne out by examination of the histograms. It is, however, noticeable that while the skewness and kurtosis statistics for Heno and Newyddion are similar to each other, the residuals for Pobol Y Cwm appear to be much closer to normality with an insignificant Sk statistic. One possible explanation is that the residual series are subject to outliers which, arguably, would often be positive because viewers would be much more likely to watch, say, the national or international news if some major event had taken place and much more likely to watch a soap opera if it were knownas it often is -that a major plot development was about to take place. is also insignificant at 0.828. There is, therefore, evidence that outliers are responsible for the rejections of normality in the daily series.

Table 3 gives the estimated models for the five weekly series for each programme.

Here the SIC procedure chose a pure fractional model in each of the 15 cases. In all but one case (the Wednesday series for Newyddion ) the estimated d is significantly different from zero, often at the 1% level. As with the daily data, the chosen models However, setting the 'large' (> 2.65 standard deviations) residuals equal to their expected value of zero renders all of the kurtosis coefficients and all but one of the skewness coefficients insignificant at 5%. The Jarque-Bera statistics for these trimmed residuals are all insignificant at 5%.

The lack of normality in the residuals for both daily and weekly models is the main inconsistency with the predictions of the Taqqu et al theorem. Removing this problem by removing 'large' residuals seems arbitrary, though one should bear in mind that the procedure removes both large negative as well as large positive values so that it will not obviously eliminate the positive skewness in the untrimmed residuals. One possible explanation lies in the sample size. The Taqqu et al theorem is a large sample result and it could simply be that the sample used here -420 -is not large enough and that viewers who watch only occasionally are overrepresented. In this context it is interesting to note that non-normality is less of a problem with Pobol Y Cwm, the series where one would expect habit to play the most important role, with fewer people watching just the occasional episode, as opposed to current affairs and news programmes which may occasionally attract surprisingly large audiences. The

Conclusion

Nothing in the basic theory requires that aggregate data on television viewing will be fractionally integrated. Both the Granger and

  non-negligible probabilities of very long ON and OFF periods. If all the sources are identical, the resulting aggregate time series will have a differencing parameter given by d = 1 2 min / . If the sources are heterogeneous the source with the smallest dominates as T . Notice that min = 2 implies an I 0 b g series.

  OFF periods in which the lengths of the ON and OFF periods are random variables.Aggregate data on viewing numbers is collected by various agencies for television companies. If individual households behave in accordance with the model outlined above then these aggregate series may exhibit long memory.

  programmes are Heno, a general interest programme covering topical issues, Pobol Y Cwm, a soap opera, and Newyddionddion, the evening news programme. Each lasts for half an hour and the three are broadcast in sequence between 6:30 pm and 8:00 pm from Monday to Friday. For Heno and Newyddionddion the series are nearly continuous over the period from 2 January 1995 to May 22, 2000, being broken only by occasional public holidays -especially around Christmas. The series for Pobol Y Cwm runs continuously only from 11 September 1995 since the programme was not broadcast during the summer months of that year. Breaks in the series after this time are mainly around Christmas. The occasional missing values in each series are interpolated by the simple expedient of taking the average of the same day in the previous and succeeding weeks iii .The viewing figures for each of the three programmes vary over the days of the week and over the weeks of the year. Heno and Pobol Y Cwm, in particular, display strong annual cycles. Since there is a danger of confusing the pattern of low-order autocorrelations generated by this cycle with any long memory in the data, we remove the former using a set of trigonometric seasonals in the form of S/2 pairs of sines and cosines where S is the number of seasons and the typical term is

  each of the adjusted daily series the sample autocorrelations decline slowly and are positive for a substantial number of lags. The autocorrelations of Heno are positive to lag 34, those of Newyddion are positive to lag 44 while the first negative autocorrelation of Pobol Y Cwm occurs as late as lag 100. Lo's Modified Rescaled

  are almost entirely free of serial correlation and ARCH effects. These results also show that the problem with the daily Pobol Y Cwm series can be attributed to the observations for Wednesday. Only three ARCH statistics are significant, indicating first order ARCH effects in two of the series. The skewness and kurtosis statistics are, again, often significantly different from zero but less so in the case of Pobol Y Cwm than the other two series. The Jarque-Bera test rejects normality at 5% for 10 of the 15 series, the exceptions being the Monday and Thursday series for Newyddion and the Tuesday, Wednesday and Friday series for Pobol Y Cwm.

  then that the aggregation effect is strong enough to generate long memory even in quite small panels. Finally, as mentioned above, the temporal aggregation involved in moving from daily to weekly series should leave the differencing parameter unchanged. The average d value for Heno is 0.145 compared to the estimated value for daily data of 0.207, the average d value for Newyddion is 0.138 which is very close to the estimated d for the daily data of 0.131 and the average d value for Pobol Y Cwm at 0.206 is close to the estimated d for daily data of 0.185. Thus in two of the three data series the temporal aggregation result appears to carry through. The exceptional case, Heno, is the only series which includes a non-seasonal AR coefficient. Estimating a pure fractional model for this series results in a d of 0.146 which is much closer to the average d derived from the weekly data, though the residuals fail the Q test. Hence, it would appear that the inclusion of an autoregressive component has had the effect of boosting the value of the differencing parameter for this series. It should also be noted that the estimated d's for purely fractional process for Newyddion and Pobol Y Cwm are 0.146 and 0.197 respectively.

  Taqqu et al approaches to aggregation require, in effect, an appropriate set of 'weights' and some restrictions on a behavioural parameter to deliver a d between zero and unity rather than a d of zero or unity. Nevertheless, such outcomes do appear to arise quite frequently in practice. In this paper we appealed to habit or commitment in a context where behaviour can be characterised in such a way as to apply the Taqqu et al result. We find that all three daily series of viewing figures, once adjusted for seasonality, behave broadly in line with the Taqqu et al result. In addition, all of the weekly series appear to be adequately modelled as purely fractional process with Gaussian noise after the removal of an annual cycle. The fractional differencing parameters are quite small, but significantly different from zero, and there is no strong evidence of further structure, linear or nonlinear. By considering both daily and weekly data we were able also to investigate the effects of temporal aggregation and show that the different frequencies share similar values of d.

	Submitted Manuscript Submitted Manuscript	Page 14 of 17
	F o r P e F o r P e F o r P e Table 1 MRR Tests Truncation Lag 0 1 2.247 2.114 Newyddiondd Heno ion 1.819 1.696 Pobol Y Cwm 5.305 4.754 Upper Tail Critical Values: 5%, 1.747, 10%, 1.620 3 Andrews' q 1.901 1.733(5) 1.564 1.454(5) 4.219 3.427(8) F o r P e e r R e v i e w
	e e e
	r r r
	R e R e R e
	v i v i v i
	e w e w e w

Page 12 of 17 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Notes: Standard Errors in parentheses, * (**) denotes significance at 5% (1%). Sk and Ku are the coefficients of skewness and excess kurtosis, SC(1) is the Lagrange Multiplier test for corrletion at lag 1, Q(5) and Q(70) are Ljung-Box portmanteau tests using 5 and 70 lags respectively, ARCH(1) and ARCH(5) are tests for autoregressive conditional heteroscedasticity at lags 1 and 5 respectively.

						Table 2
					ARFIMA Models: Daily Data
		d	AR1	SMA	Sk	Ku	SC(1)	Q(5)	Q(70)	ARCH(1)	ARCH(5)
	Heno	0.207**	-0.132**	0.059*	0.585**	1.364**	0.112	7.659	89.282	2.507	5.226
		(0.024)	(0.036)	(0.026)
	Newyddionddio	0.131**		0.093**	0.626**	1.155**	0.160	2.923	69.863	0.281	3.796
	n
		(0.020)		(0.026)
	Pobol Y Cwm	0.185**		0.099**	0.127	0.562**	0.267	6.435	131.174**	0.026	4.192
		(0.016)		(0.025)

Page 15 of 17 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript F o r P e e r R e v i e w

Table 3 ARFIMA Models: Weekly Data

 3

	Heno	d	s.e.	sk	ku	SC(1)	Q(5)	Q(30)	ARCH(1)	ARCH(5)
	Mon	0.143*	0.056	0.204	0.187	0.436	2.451	32.229	12.756*	15.739*
	Tue	0.141**	0.054	0.620**	1.353**	0.294	1.737	23.338	0.406	1.686
	Wed	0.166**	0.043	0.382**	-0.055	0.199	4.551	30.732	0.021	4.632
	Thu	0.136*	0.054	0.287*	0.222	0.369	2.435	19.730	2.004	7.881
	Fri	0.137*	0.060	0.898	3.056	1.350	10.001	38.774	0.031	1.841
	Mean	0.145
	Newyddion
	ddion
	Mon	0.144**	0.055	0.464**	0.573	0.104	6.166	35.224	0.013	18.548**
	Tue	0.125*	0.058	0.713**	0.954**	0.805	3.787	34.168	0.292	3.516
	Wed	0.089	0.065	0.742**	1.233**	0.054	7.590	30.619	0.018	1.045
	Thu	0.180**	0.043	1.240**	4.415**	0.034	3.183	36.589	0.112	4.137
	Fri	0.152*	0.069	0.422**	0.241	0.153	4.166	28.579	0.175	5.886
	Mean	0.138
	Pobol Y
	Cwm
	Mon	0.228**	0.053	-0.099	0.844*	0.183	1.448	31.141	0.169	1.974
	Tue	0.179**	0.046	0.016	-0.058	1.009	5.042	27.488	0.746	1.831
	Wed	0.189**	0.043	0.112	-0.064	1.759	7.049	100.946**	0.501	3.887
	Thu	0.185**	0.069	0.492**	0.881**	0.004	2.820	24.919	0.306	8.874
	Fri	0.250**	0.047	-0.012	0.544	0.021	3.188	26.261	0.189	1.345
	Mean	0.206
	Notes: See Notes to Table 2

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Submitted Manuscript

F o r P e e r R e v i e w i Although the individual decision to watch or not watch a given programme -like the decision to support or not support a given political party -is binary in nature, we do not have an estimate of the proportion of the relevant population engaging in the activity. Should one, for instance, use the population of all viewers at a given time, or all potential viewers? Since our data is for programmes in Welsh, we could also use the population of Welsh speakers or of Welsh-speakers watching television at the time.

ii We are very grateful to S4C for making these figures available to us. iii There are, in total, 28 missing values for Heno, 8 for Newyddion and 20 for Pobol Y Cwm Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Estimation of cost functions in a data poor environment: the case of capacity estimation in fisheries

Keywords: C61 -Optimization Techniques|Programming Models|Dynamic Analysis <, C81 -Methodology for Collecting, Estimating, and Organizing Microeconomic Data <, C21 -Cross-Sectional Models|Spatial Models translog cost function, Data Envelopment Analysis, capacity, fisheries translog cost function, Data Envelopment Analysis, capacity, fisheries

Estimation of cost functions in a data poor environment:

the case of capacity estimation in fisheries

Introduction

Applied economic analysis in fisheries is characterised by limited, and often poor quality, data. This is especially so for fisheries economic data. While price information is often readily available, largely due to the predominance of fish auction markets as a means of distributing the catch, information on the costs of fishing is extremely limited. Collection of this information is costly, as it is usually undertaken through personal interviews. Further, the number of participants in a given fishery 1 is also relatively small, resulting in small sample sizes for particular fleet segments operating in particular areas. Further, the industry is often subject to many other systems of data collection, primarily for scientific purposes and involving activity information (catch, fishing effort, vessel characteristics etc). These data are usually collected through some mandatory requirement, so fishers are reluctant to participate in "voluntary" surveys to provide detailed information on their individual financial situation. As a result, economic panel data series are often short and unbalanced, often consisting of a one period cross section rather than a true panel data set.

Despite this lack of reliable economic data, policy analysis is still required. This requires the creative use of the available information in order to derive robust policy conclusions.

The study that is the focus of this paper is one such example where combining techniques enables the estimation of reliable econometric results that can be used to inform policy advice. The particular issue in this case was the potential size and 4 substantial decline in many key whitefish stocks in EU fisheries. Reductions in total allowable catches (TACs) in excess of 50 per cent were imposed for many North Sea whitefish stocks in 2002, with stocks in other areas subject to TAC reductions of between 10 and 30 per cent [START_REF] Fish | Fisheries: "alarming state of fish stocks" requires further substantial cuts for 2002 fishing catches[END_REF]. Further cuts in quotas of the order of between 30 and 40 per cent were made in 2003, and these lower quotas were carried through to 2004 also.

Of particular concern to the UK industry is the state of the cod stocks. These are still considered to be outside safe scientific limits in the North Sea, English Channel and Irish Sea, with scientific advice being to close the cod fishery in the North Sea and Irish Sea and reduce fishing mortality by 90 per cent in the English Channel [START_REF] Ices | Report of the ICES Advisory Committee on Fishery Management[END_REF]. While such extreme measures have not been implemented, the low cod stock creates particular problems for the UK demersal trawl fleets operating in these areas. Cod is the major target species for these vessels, and the UK takes the largest share of total catch of cod. Restrictions on days fished and reduced quota results in many fishers operating at less than full capacity with resultant economic inefficiency. In 2001, 35 per cent of whitefish vessels were making a financial loss or no profit, and average financial profit (i.e. before depreciation costs were considered) was only 2 per cent of earnings (Rural Development Committee 2003). The most recent reductions in catch quotas are likely to result in a significant proportion of the fleet being economically unsustainable. The Prime Minister's Strategy Unit (2004) recommended urgent actions to reduce capacity in the whitefish sector through an additional decommissioning scheme to permanently remove fishing capacity and a series of short-term tie-ups to relieve pressure on the stocks. In addition, the Unit proposed that individual tradeable rights for resource access in the form of individual transferable quotas (ITQs) be introduced into all fleet segments by the end of 2006.

The latter proposal is aimed at providing incentives for continuing fleet rationalisation and enhancing long term economic performance. ITQs have been successful in many fisheries internationally in facilitating fleet adjustment and removing excess capacity. The theory and practice underlying the use of ITQs is well documented in the fisheries economics literature (e.g. [START_REF] Hannesson | From common fish to rights based fishing: Fisheries management and the evolution of exclusive rights to fish[END_REF].

Excess capacity is removed through quota trade, with some vessels exiting the fishery and the remaining vessels consolidating the available quota. Incentives are created to reduce the costs of capture, with fleet adjustment favouring the least cost producers and reinvestment occurring in vessels that will operate at minimum average cost.

The UK demersal trawl fleet

The UK demersal trawl fleet undertakes three main activities -otter trawling, danish seining and Nephrops trawl. Otter trawlers and danish seiners both target similar whitefish species, but using different types of trawl gear, while Nephrops trawlers target primarily Nephrops (also known as scampi, langoustine and Dublin Bay prawns). In 2002, there were 929 demersal trawlers over 10 metres in length 3 The demersal trawl fleet is currently regulated through a series of input and output controls. Licence limitations restrict entry to the fishery, while a unitisation system restrictions boat replacement. Aggregate total allowable catches are set at the European level for each stock of the key species and distributed to the individual Member States in relatively fixed proportion. In the UK, these are further distributed to individual vessels greater than 10m in length in the form of fixed quota allocations. Although termed "fixed", the quotas are transferable on an annual basis through quota leasing. Permanent quota transfers can also be arranged, although the restrictions associated with this have prevented wide-scale permanent transfer of quota.

The whitefish trawlers (otter trawlers and seiners) operate primarily in the North Sea, English Channel, Celtic Sea and Irish Sea targeting cod and other whitefish species. The catch composition varies in the different areas, with the English Channel trawlers being characterised by a relatively high proportion of non-quota species in the catch. In contrast, catch in the North Sea is dominated (i.e. in excess of 90 per cent) by quota species. The Nephrops trawlers are predominantly based in Scotland, and operate in the North Sea as well as off the west coast of Scotland. The focus of this study is the whitefish trawlers as these have been most adversely affected by quota cuts. In contrast, Nephrop fisheries have experienced increased stock sizes over recent years, largely as a result of the reduction in predation from whitefish.

Long run and short run measures of capacity

The concept of capacity has been well established in the economics literature. [START_REF] Johansen | Production functions and the concept of capacity. Recherches recentes sur la fonction de production[END_REF] defines capacity as "the maximum amount that can be produced per unit of time with existing plant and equipment, provided that the availability of variable factors of production is not restricted". This primal measure of capacity does not take into account the additional costs of increasing output, and is essentially an "engineering" or "technological-economic" definition [START_REF] Felthoven | Multi-output, non-frontier primal measures of capacity and capacity utilization[END_REF]. In contrast, [START_REF] Klein | Some theoretical issues in the measurement of capacity[END_REF] defined capacity output as the level of output corresponding to the minimum point of a short-run average cost curve, while Berndt and Morrison (1981), [START_REF] Morrison | Primal and dual capacity utilisation: an application to productivity measurement in the U.S. automobile industry[END_REF] and Seguerson and Squires (1990) defined capacity output as the level of output corresponding to the tangency between short-run and long-run average cost curves. These definitions relate to essentially short run measures of capacity, as the assumption is that capital remains fixed. Indeed, in most instances, capacity is considered a short term concept. Earlier studies, however, considered long run concepts of capacity. Cassel (1937) and [START_REF] Hickman | On a new method for capacity estimation[END_REF] suggested that capacity corresponded to the level of output at which the long-run total cost curve was at its minimum. This requires changes in the level of fixed inputs as well as variable inputs. Such a definition, however, has been found to have limited practical value, as empirical studies of the average cost function have tended to conclude that the long-run curve is either linear [START_REF] Klein | Some theoretical issues in the measurement of capacity[END_REF] or "L-shaped" [START_REF] Prior | Long-and short-run non-parametric cost frontier efficiency: and application to Spanish savings banks[END_REF] rather than the usually assumed "U-shaped", and hence defining a minimum is not feasible.

The difficulty in estimating long run cost curves is primarily a function of the available data. Most cost functions are estimated using either cross sectional or panel data. However, in any one time period, the data are short run by nature, with total cost and output being a function not only of the level of capital but also its utilisation. Further, different levels of efficiency exist that distort the relationship between capital and output. As the purpose of this study was to consider the fleet size and configuration that, if fully utilised, could harvest the available catch at the least cost, then it is desirable to remove these effects from the data in order to directly estimate a long run cost function.

To this end, a two stage approaches was adopted. First, the fully utilised level of output (i.e. the capacity output) of each vessel given its level of fixed inputs was estimated using Data Envelopment Analysis (DEA). The information produced using DEA was used to adjust the output as well as the variable costs associated with producing this level of output. Although DEA was initially used to overcome these data deficiencies, it also provided useful information on returns to scale and optimal vessel size. A translog cost function was then estimated using the adjusted data in order to determine the long run cost function and the cost-minimising level of production.

Data Envelopment Analysis

DEA is a linear programming (non-parametric) method for estimating a production possibility frontier, and where individual firms lie in relation to the frontier. The method is data driven, and, in the context of the fishery, the frontier is defined based on the actual output of boats in the fishery, thereby conforming to the assumption of normal fishing behaviour. [START_REF] Felthoven | Multi-output, non-frontier primal measures of capacity and capacity utilization[END_REF] describe these approaches as "technological-economic", as the derived relationships are based on observed data that implicitly reflect underlying economic decisions. If all fishers operating on the frontier maximised profit and faced similar cost functions to those operating at lower levels of utilisation, then this would be a reasonable approximation to an economically efficient measure of capacity. The measures are essentially short run in nature, as the assumption is that output is maximised given the existing set of fixed inputs.

Although the estimation of capacity in fisheries using DEA is relatively new, a number of studies have already emerged (e.g. Pascoe et al., 2001;[START_REF] Dupont | Capacity utilization measures and excess capacity in multi-product privatized fisheries[END_REF][START_REF] Felthoven | Effects of the American Fisheries Act on capacity, utilisation and technical efficiency[END_REF][START_REF] Vestergaard | Measuring capacity and capacity utilization in fisheries: the case of the Danish gillnet fleet[END_REF][START_REF] Tingley | Estimating capacity utilisation in multi-purpose, multi-metier fisheries[END_REF]Kirkley et al., 2003;[START_REF] Reid | An analysis of fishing capacity in the western and central Pacific Ocean tuna fishery and management implications[END_REF]Walden et al., 2003;[START_REF] Tingley | Factors affecting capacity utilisation in fisheries[END_REF].

= k k k i k i k k k m k k m z z i x x z k y z y (1)
where 1 is a scalar denoting how much the output of the target boat (i.e. k=0) can be increased, y k,m is the output m produced by boat k, x k,i is the amount of input i used by boat k and z k are the weights that relate the target boat to the set of peers (i.e. the vessels against which it is compared). The restriction k z = 1 allows for variable returns to scale (VRS), while excluding this constraint implicitly imposes constant returns to scale (CRS). The sum of the weights when CRS is imposed provides an indication of the returns to scale. are estimated), with the target boat's outputs and inputs being denoted by y 0,m and x 0,i respectively.

Capacity utilisation (CU) is defined as CU=1/ 1 . The measure of CU ranges from zero to 1, with 1 being full capacity utilisation (i.e. 100 per cent of capacity). The capacity output of each vessel is determined by

m k m k y y , 1 , =
.

A firm's outputs may not be produced efficiently and hence some of the apparent capacity under-utilisation may actually be due to technical inefficiency (i.e. not producing to the full potential given the level of both fixed and variable inputs). If all inputs (both fixed and variable) are not being used efficiently, then it would be expected that output could increase even without an increase in the level of variable inputs through the more efficient use of these inputs.

By comparing the capacity output to the technically efficient level of output, the effects of inefficiency can be separated from capacity under-utilisation. Further, the ratio of these measures has been found to be less susceptible to bias due to random error than the initial capacity utilisation and efficiency estimates [START_REF] Holland | Impacts of random noise and specification on estimates of capacity derived from Data Envelopment Analysis[END_REF].

The technically efficient level of output requires an estimate of technical efficiency of each firm, and requires both variable and fixed inputs to be considered. The DEA model for this is given by:

F o r P e e r R e v i e w 12 2 Max subject to 0 1 , 0 , , , 0 2 = k k k i k i k k k m k k m z z i x x z m y z y (2)
where 2 is a scalar outcome denoting how much the production of each firm can increase by using inputs (both fixed and variable) in a technically efficient configuration. In this case, both variable and fixed inputs are constrained to their current level and 2 represents the extent to which output can increase through using all inputs efficiently. The technically efficient level of output ( * , CU CU* 1. The difference between the measures reflects the degree to which random variation and technical inefficiency affect the output levels of the different firms.

Cost function approach

An implicit assumption of a primal approach such as implicit in the DEA model illustrated above is that output can increase to the full utilisation level. Under an ITQ system, economic efficiency is determined by cost minimisation given the quota allocation rather than output maximisation given the set of inputs available to the fisher. While the DEA model can be specified with an input orientation, and hence can provide a measure as to the extent to which input use can be reduced to achieve efficient production, this does not provide information on the capacity of the vessel. With ITQs, vessels can adjust output levels, but have incentives to produce this output at the lowest cost possible. For this reason, the estimation of the cost function can be considered a more appropriate means of assessing capacity under an ITQ system. Relatively few applications of the cost function approach have been made in fisheries (see [START_REF] Lipton | Effect of Stock Size and Regulations of Fishing Industry Cost and Structure: The Surf Clam Industry[END_REF][START_REF] Weninger | Assessing Efficiency Gains from Individual Transferable Quotas: An Application to the Mid-Atlantic Surf Clam and Ocean Quahog Fishery[END_REF], Bjørndal and Gordon 2000), largely due to difficulties in obtaining cost and revenue data for commercial fishing vessels.

The translog cost function for a single output industry 4 can be specified as where C is the total cost, w i is the price of input i and y is the (aggregated) level of output. By differentiating equation 5 with respect to the input prices and using Shephard's lemma, the set of cost-minimising factor cost shares can be derived, given by

+ + + = Q w w S iq n j j ij i i i ln ln ln (6)
where S i is the cost share of the ith input, given by w i x i /C.

The cost function and the associated set of share equations need to be estimated simultaneously. As the input shares sum to 1 (one), one of the share equations needs to be excluded in order to avoid problems of singularity. A number of restrictions also need to be imposed on the system to ensure consistency with economic theory. Homogeneity in input prices and output requires The set of coefficients from estimating the system provides additional information about the nature of the production system, including the propensity to respond to input price changes by changing input use or even substitute inputs, and the returns to scale associated with different production levels. The Allen partial elasticities of substitution between the factor inputs ( ij ) are given by

j i j i ij ij i i i ii ii S S S S S S S / ) ( , / ) ( 2 2 + = + = (7)
and the partial price elasticity of demand for input factor i ( i ) are given by

j ij ij i ii i S S = = , (8)
A positive elasticity of substitution and cross price elasticity indicates substitution possibilities exist, while negative values indicate a complementary relationship.

The returns to scale of an individual vessel can be given by

+ + = = i i iy yy y w Y R C RTS ) l n ln /( 1 ) / /( 1 (9)
The inclusion of both fixed and variable costs in the cost function implicitly assumes that the vessels are operating at their long run optimum level. However, where capital has a relatively long life, such as in the case of fishing vessels, capacity may not be fully utilised. In such a case, the fisher may be operating on Burney (2002) suggest that in such cases it is more appropriate to estimate the short term cost curve (i.e. excluding fixed and capital costs) and include a variable representing the level of capacity utilisation. Further, inefficiency may exist in the industry that could result in bias in the estimated coefficients if ignored (see Kumbhaker 2001). As adjustment in the fishery as a result of ITQs is likely to result in a more efficient fleet on average, assuming current efficiency levels may not be appropriate.

Given this, three separate cost functions were estimated. The first is the standard cost function presented in equations 5 and 6. Second, the output measure was adjusted using the results of the DEA analysis to reflect the full capacity output.

Costs and cost shares were similarly adjusted to represent the full capacity output.

Finally, output and costs were adjusted to represent the fully efficient, full capacity level of output. This latter model is assumed to be consistent with a long run cost function for a fishery not subject to excess capacity.

Data

Data on costs, revenues and physical characteristics for 67 UK demersal whitefish trawlers relating to the 2001 financial year were available, representing roughly 9 per cent of the total whitefish trawl fleet. These vessels were all above 10m in length. 5 A summary of the key characteristics of the data set is presented in Table 1. Although the data are relatively old, more recent data are subject to problems that limit their usefulness for such an analysis. Although the main stocks exploited by the fleet segments under consideration have been declining since the early 1980s, catch restrictions have generally not declined by the same degree. As a result, quotas were not fully binding in many years. Total Allowable Catch (TAC) reductions in excess of 50 per cent were imposed for many North Sea stocks in 2002, whilst most other stocks were subject to TAC reductions of between 10 and 30 per cent [START_REF] Fish | Fisheries: "alarming state of fish stocks" requires further substantial cuts for 2002 fishing catches[END_REF]. Further cuts in quotas of the order of between 30 and 50 per cent were made in both 2003 and 2004. By 2005, the UK cod quota was less than 28 per cent of its level in 1999, and only 18 per cent of that a decade ago. These quota cuts have had a substantial impact on capacity utilisation and have distorted the recorded output mix (i.e. overquota catch is discarded or, in some cases landed illegally, so that the recorded output is not representative of the actual production of the vessels). As the methods for assessing capacity utilisation are data driven and relative, comparing vessels in more recent years where all are substantially underutilised would result in the derived (relative) average utilisation measure being fairly high. Consequently, the average costs would appear substantially greater than they should if the fleet were fully utilised, even given the low stock levels.

The individual cost items were aggregated into four cost categories: crew costs, running costs, capital costs and 'other' costs. Crew costs were the payments to crew. Running costs consisted of fuel costs, ice, box charges and food.

Information on the capital value of the vessel was not provided by most skippers. Data on input prices were not available, but proxy measures of input prices were derived from the survey data. The crew price was derived from total crew payments divided by the number of crew. This is a potentially misleading measure, as crew are paid a share of the net revenue (i.e. revenue less running costs). As a result, a relatively high crew price may indicate a relatively high labour productivity, but may also be a consequence of 'luck' (i.e. higher than expected catches). Running costs are a function of both the amount of time fished and the size of the vessel. Information on fishing effort (e.g. days fished) was not available for most of the vessels. The input price associated with running costs was assumed to be the running cost of the vessel if it was operating at full capacity divided by the number of vessel capacity unit (VCUs). 6 An assumption was made that running costs were proportional to the level of capacity utilisation.

Hence the running cost if fully utilised was given by the observed running cost divided by the capacity utilisation rate. 7 The prices of capital and other inputs were also derived from the costs information and the physical boat characteristics.

Various combinations of measures were tried. The physical measures that resulted in the lowest variance in input prices were length for 'other costs' and the VCUs for capital costs. Input prices for other costs and capital costs were therefore taken as other costs per unit length and insurance cost per VCU. As mentioned in the methodology section, the analysis was also run assuming full capacity utilisation and full efficiency. The level of capacity utilisation and technical efficiency were derived using DEA. The revenue and running costs were scaled up by the appropriate factor for each analysis. Crew are currently paid a share of the revenue (and hence capture some of the rent). As a consequence, the price of labour and crew costs were also assumed to increase in proportion to the revenue increase.

The costs and revenue values were normalised (after appropriate adjustments to account for capacity utilisation and efficiency) such that the mean values of the normalised data were 1.

Empirical results

DEA: Capacity utilisation, efficiency and returns to scale

The DEA model was run with revenue as the output measure and length and engine power as the fixed inputs. Fuel costs, which were assumed to be proportional to days fished, were included as the variable input for the purposes of estimating technical efficiency and the 'unbiased' estimate of capacity utilisation.

Estimates of capacity utilisation were also obtained for the case of both constant returns to scale and variable returns to scale. The ratio of these measures provides a measure of the scale efficiency. 2. On average, the vessels were operating at around 87 per cent capacity and at around 69 per cent efficiency. If the vessels operated at both full capacity and efficiency, average output could potentially increase by 67 per cent (i.e. 1/0.6). In contrast, if the vessels were fully utilised but remained at their current (in)efficiency levels, potential output could increase by around 15 per cent on average. Scale efficiency was estimated relative to both capacity utilisation and technical efficiency. The seiners and North Sea otter trawlers were, on average, closer to the 'optimal' scale. The optimal scale in this case is defined where constant returns to scale exist. Both these boat groups were larger, on average, than the other two in terms of length and engine power as well as in terms of output.

A measure of returns to scale can be derived from the sum of the weights from the CRS technical efficiency model. Only four boats were found to be operating at the optimal scale, with three boats operating at above the optimal scale (and therefore subject to decreasing returns to scale). The remaining vessels were all found to be operating with increasing returns to scale. Of the four boats operating at the optimal scale, only 2 were both fully efficient and operating at full capacity.

These vessels where 26m and 30m in length with respective engine powers of 750kW and 500kW, and respective revenues of £1.16 and £0.97m (an average revenue of £1.06m). While they were at the top end of the vessels in the fleet (in terms of size), they were not the largest vessels.

Cost function

The cost function was estimated excluding the capital share equation in order to avoid singularity. As mentioned above, three variants of the model were run using different manipulations of the data. The first run was assuming the industry was in a long run equilibrium. The second run took into account capacity underutilisation and the revenue, crew and running costs were re-estimated. The third run took into account the existence of inefficiency as well as capacity underutilisation. In this run, revenues and crew costs were increased to take into account both of these factors while running costs were increased to take into account the increased utilisation only.

The parameter estimates from the three model runs are presented in Table 3. In all three models, most parameters were significant at the 1 per cent level. The adjusted R 2 values were also reasonably high for the cost function itself, but less so for the share equations. While the adjusted R 2 values varied for the different models, these cannot be compared as the values of the dependent variable also differed in each model run.

The estimated partial own and cross price elasticity for the demand for factor i are presented in Table 4. As would be expected, the own price elasticity was negative for each input and the cross price elasticities were generally positive indicating the potential for substitution. The exception to this was capital and running costs, The returns to scale derived at the mean prices and output levels for each model is given in Table 5. In all three models, increasing returns were found at the mean.

The optimal scale of fishing vessel can be found by solving equation ( 9) for the case where returns to scale are equal to 1 (one). In the base model, the optimal vessel is 17,020 times greater than the current average sized vessel, suggesting an optimal vessel length or around 254km -approximately half the southern UK coastline. Despite this magnitude, the scale factor is not significantly different to zero. In contrast, if considering fully efficient and fully utilised vessels, the optimal scale is about 2.8 times the current average sized vessel, with the value being statistically significant.

Optimal vessel size and profits

From the DEA analysis, the average of the 'optimum' level of output was £1.04m.

The vessels from which this average was obtained were both fully efficient and operating at full capacity. From the cost function analysis, the optimal vessel size (if fully efficient and fully utilised) was 2.793 times larger than the current average vessel. Given that the current average vessel if full efficient and fully utilised would produce revenue of £0.625m, the optimal vessel size would produce an output of around £1.74m. Although the cost function estimate of optimal yield is 67 per cent greater than the DEA estimate, the lower DEA estimate of optimal output is within the 95 per cent confidence interval of the corresponding cost function estimate. Hence, the two estimates are not statistically significantly different. The DEA estimate of optimal production, by the nature of its calculation, is restricted to be within the range of the available data. Also, the DEA estimate is based on a primal output oriented function with output maximisation the implicit objective. In contrast, the cost function derived estimate of optimal production is not restricted to fall within the range of observed output levels, and the dual function has the objective of minimising costs as well as maximising output in order to maximise profits. However, extending beyond the range of the data creates problems for obtaining reliable and robust estimates. The translog function underlying the cost function is lest robust the further the variable values deviate from 1. These difficulties in obtaining reliable estimates not withstanding, estimates of the profits associated with the "optimal" scale vessels are presented in Table 6. These are not true rents, as the non-cash capital costs (i.e. economic depreciation and opportunity cost of capital) have not been taken into account in the estimation of total costs. However, they provide an indication as to the potential increase in vessel profits that may occur through restructuring.

From Table 6, if the vessels tend over time to move to the optimal scale identified by the DEA, then the fleet would need to reduce by nearly two thirds in order to enable the vessels to operate at full capacity (assuming also full efficiency). In

Discussion and conclusions

The estimates of potential profits are based on current stock conditions, which are currently at a low level. If stocks recover, the unit cost of capture will decrease and the potential profit from a fully utilised vessel is likely to increase above that suggested in the above analysis. Further, assuming that the full capacity estimates of output are, in fact, representing the capacity of the vessel, higher stock levels would be able to support a larger number of vessels than the above analysis suggests.

Both the DEA and cost function approach provide useful information on the level of excess capacity in fisheries. The DEA approach is primarily a short run analysis as it assumes that fixed factors remain fixed and output is a function of their utilisation. From the DEA results, average capacity utilisation was 0.87, but average technical efficiency was 0.69. This suggests that inefficiency is a greater problem for the fleet than underutilisation. If all vessels were fully utilised and fully efficiency, then total output would be roughly two thirds greater than the current level. Given that output is currently restricted by quotas, this suggests that excess capacity is excessive in the whitefish fishery. The cost function approach provides a longer-term perspective in that it allows for all inputs to vary. Further, when output is restricted such as through ITQs, then incentives exist to minimise costs rather than maximise output. Hence, the cost function approach is theoretically more appropriate than the output oriented DEA approach. However, a difficulty arises if fixed inputs are underutilised in the short term. As seen from the econometric results, ignoring capacity under-utilisation in the estimation of the cost function results in unrealistic 'optimal' levels of output.

Combining the results of the DEA capacity utilisation analysis into the cost function analysis overcame this problem.

The cost function approach, however, requires detailed information on factor input prices. This is a particular problem when cross sectional data are used, such as in this study. When a time series of data are available, then industryindependent price indexes can be constructed for factors such as fuel, labour (e.g. average wage) and capital (e.g. interest rate) that vary from year to year. However, within a given time period, all firms face the same set of prices, so an industryindependent set of price indexes are not appropriate. Deriving proxy measures for input prices from the available data may result in measurement error that could affect the regression results. Further, apparent differences in 'prices' may reflect heterogeneity in input quality. For example, in the fleet segments examined, crew costs are based on a share of the revenue less running costs. While average crew earnings can be derived as a proxy for the price of labour, price differentials most likely reflect differences in skill of the crew and skipper. Labour in such a case is Capturing the full capital cost and appropriate cost of capital in an ITQ fishery is also problematic. While interest rates can be considered an appropriate price of capital, this is common to all vessels within a given time period. The approach adopted in this study was to use the average insurance cost per unit of physical capital. Again, this is subject to measurement errors as it assumes that the insurance costs are proportional to the value of capital invested.

These problems not withstanding, the results from the cost function conformed with a priori expectations with respect to the signs of the derived own and cross price elasticities of demand. Further, the derived scale elasticities were consistent with the returns to scale estimated using DEA, and the 'optimal' scale estimated using both DEA and the cost function were not significantly different.

The results of the study suggest that the whitefish demersal fleet is likely to adjust in both scale and size to consist of fewer, but larger vessels than currently exist. This consolidation of fishing activity into fewer, larger units has been commonly observed in other fisheries subject to ITQ management, and is often used as an argument by industry to prevent their implementation. In the UK, transferability is limited and involves high transactions costs. While this may slow the rate of adjustment, pressures exist for managers to reduce capacity in line with the reduction in the resource base. As a consequence, fleet sizes will, by necessity, decrease, and the social problems associated with fleet reduction (e.g. increased unemployment in rural areas), will have to be incurred. Freeing up quota transferability may facilitate this process at lower cost to the taxpayer and result in greater long run economic benefits (in terms of rent generation) than other capacity reduction management measures.

Acknowledgements

This project was undertaken as part of the EU funded project "Modelling fishermen behaviour under new regulatory regimes" (QLRT-2000-01535). Footnotes 1 A fishery is generally considered to consist of a set of vessels targeting a similar set of species in a particular geographical area. Usually, the size of the geographical area is defined by the distribution of the stock. For some species, this can be expansive, while for other species it may be a relatively small area. As several different types of fishing gear are often used, the fishery is broken down into a number of fleet segments.

Alam

2 Production functions and stochastic production frontiers have been employed to consider the implications of input controls in fisheries (e.g. Pascoe et al 2001[START_REF] Valle | Fishing effort validation and substitution possibilities among components: the case study of the VIII division European anchovy fishery[END_REF], Weninger and Strand 2003[START_REF] Kompas | Technical efficiency effects of input controls: evidence from Australia's banana prawn fishery[END_REF]. The assumption underlying these analyses is that capital is effectively fixed in the short term, and that incentives exist to maximise output or revenue (e.g. [START_REF] Kirkley | The technology and management of multispecies fisheries[END_REF]. When considering the adoption of individual transferable quotas, this assumption is not valid, as fishers would be expected to adjust their capital in order to minimise the costs given the output constraint. Hence, a cost or profit function may be considered more appropriate (see [START_REF] Lipton | Effect of Stock Size and Regulations of Fishing Industry Cost and Structure: The Surf Clam Industry[END_REF]

  ), of which around 230 were Nephrops trawlers. Between 1999 and 2002, the over 10m demersal trawl fleet decreased by almost 25 per cent as a result of decommissioning programmes and voluntary retirement as a result of the adverse economic conditions facing the industry.

  caught in the Irish Sea, a high proportion of which is caught by vessels moving down from the west coast of Scotland on a seasonal basis.

  k z < 1 implies the vessel is subject to increasing returns to scale while k z > 1 implies decreasing returns to scale. The ratio of the 1 's with VRS and CRS imposed provides a measure of the scale efficiency (i.e. scale efficiency = CRS VRS /).Inputs are divided into fixed factors, defined by the sub-set , and variable factors defined by the sub-set ˆ. For the purposes of estimating capacity, only considered. The value of 1 is estimated for each vessel separately (i.e. so effectively a set of k , 1

  cost curve rather that the long term cost curve. Al-Mutairi and

  information on capital values was provided, this was generally based on the insurance value of the vessel. The insurance cost was therefore used as a proxy measure for capital costs. All other costs were included in the 'other' cost category.

  the DEA results is presented in Table

  which were found to have a complementarity relationship. As running costs are a the level of capital and its utilisation, an increase in capital prices would lead to lower levels of capital and, consequently, also lower running costs.

  the vessels tend to increase in size over time to the optimal scale identified by the cost function, the fleet size would need to reduce by almost 80 per cent.

  input. Adjusting the crew costs and labour price for differences in efficiency overcomes this problem to an extent.

  , M.F., Omar, I.H, and Squires, D. (2002) Sustainable fisheries development in the tropics: trawlers and licence limitation in Malaysia, Applied Economics, 34, 325-337. Al-Mutairi, N. and Burney, N.A. (2002) Factor substitution, and economies of scale and utilisation in Kuwait's crude oil industry, Energy Economics, 24, 337-354. Berndt, E.R. and Morrison, C.J. (1981) Capacity utilisation measures: underlying economic economic theory and alternative approaches, American Economic Review, 71(2), 48-52. Bjørndal, T. and Gordon, D.V. (2000) The Economic Structure of Harvesting for Three Vessel Types in the Norwegian Spring-Spawning Herring Fishery, Marine Resource Economics, 15, 281-292.

  . and Strand, I. (2003) An empirical analysis of production distortions in the mid-Atlantic surf clam and ocean quahog fishery, Applied Economics, 35, 1191-1197. Walden J.B., Kirkley J.E., Kitts A.W. (2003) A limited economic assessment of the Northeast groundfish fishery buyout program. Land Economics 79

  and Alam et al 2002 for examples).

3

  The delineation of the fleet into "over 10m" and "10m and under" length categories has implications for management regulations both within the UK and also at the European level. The under 10m fleet segment dominate the industry in terms of vessel numbers (74 per cent in 2002 (DEFRA 2003)), but contribute less than 10 per cent of the value of the catch. The under 10m fleet are not subject to individual quota controls, but are generally subject to catch limits that vary month to month.

  be highly correlated with fishing capacity in trawl fisheries (see[START_REF] Pascoe | Physical versus harvest based measures of capacity: the case of the UK vessel capacity unit system[END_REF].7 This essentially assumes constant returns to fishing effort. Previous studies of revenue functions for the North Sea and English Channel demersal whitefish trawl fleet have found the production elasticity associated with days fished is around 1 (one) (see[START_REF] Tingley | Estimating capacity utilisation in multi-purpose, multi-metier fisheries[END_REF], suggesting that such an assumption is realistic.

F

Table 1 .

 1 Key characteristics of the sample, 2001

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Table 2 .

 2 Average capacity utilisation and technical efficiency

			Otter Trawlers		Seiners	All boats
		Irish Sea	North Sea	Channel
	Fully efficient CU (1/ 1 )	0.53	0.61	0.68	0.54	0.60
	Technical efficiency (1/ 2 )	0.59	0.68	0.79	0.67	0.69
	Capacity utilisation (CU*) ( 2 / 1 )	0.88	0.89	0.86	0.82	0.87
	Scale efficiency: CU ( 1,VRS / 1,CRS )	0.71	0.88	0.46	0.94	0.83
	Scale efficiency: TE ( 2,VRS / 2,CRS )	0.83	0.92	0.70	0.96	0.90

Table 3 .

 3 Results from econometric analysis

			Base Run	Full capacity utilisation	Technically efficient full CU
		Coeff	St. Err.	Coeff	St. Err.	Coeff	St. Err.
	Constant	-0.023	0.020	-0.018	0.028	-0.029	0.030
	Crew	0.343	0.009 ***	0.357	0.011 ***	0.429	0.014 ***
	Running	0.240	0.006 ***	0.243	0.004 ***	0.196	0.005 ***
	Other	0.349	0.006 ***	0.327	0.006 ***	0.278	0.005 ***
	Capital	0.068	0.006 ***	0.073	0.006 ***	0.097	0.014 ***
	Revenue	0.549	0.030 ***	0.630	0.050 ***	0.754	0.064 ***
	Crew 2	0.003	0.013	0.024	0.010 **	0.051	0.009 ***
	Running 2	0.065	0.013 ***	0.101	0.008 ***	0.096	0.008 ***
	Other 2	0.100	0.014 ***	0.046	0.009 ***	0.039	0.011 ***
	Capital 2	0.017	0.010 *	0.016	0.009 *	-0.003	0.019
	Revenue 2	0.023	0.017	0.069	0.023 ***	0.120	0.021 ***
	Crew*running	-0.019	0.022	-0.056	0.010 ***	-0.085	0.007 ***
	Crew*other	-0.061	0.019 ***	-0.057	0.012 ***	-0.082	0.007 ***
	Crew*capital	0.074	0.017 ***	0.066	0.010 ***	0.065	0.018 ***
	Crew*revenue	0.014	0.016	0.019	0.017	0.031	0.020
	Running*other	-0.070	0.018 ***	-0.042	0.012 ***	-0.021	0.014
	Running*capital	-0.040	0.017 **	-0.104	0.013 ***	-0.085	0.020 ***
	Running*revenue	0.053	0.013 ***	-0.002	0.010	0.001	0.011
	Other*capital	-0.069	0.020 ***	0.007	0.016	0.026	0.025
	Other*revenue	-0.100	0.016 ***	-0.041	0.013 ***	-0.030	0.014 **
	Capital*revenue	0.033	0.013 **	0.024	0.013 *	-0.002	0.025
	Irish	-0.008	0.062	-0.001	0.093	-0.092	0.087
	Channel	-0.206	0.053 ***	-0.162	0.080 **	-0.079	0.077
	Seine	0.042	0.034	0.001	0.050	-0.016	0.049
	Adjusted R 2
	Total costs		0.969		0.934		0.901
	Running share		0.573		0.723		0.505
	Other share		0.380		0.562		0.561
	Crew share		0.150		0.245		0.189
	*** significant at 1% level; ** significant at 5% level; * significant at 10% level

Table 4 .

 4 Own and cross price elasticities for demand for the factor inputs

		Crew	Running	Other	Capital
	Base run
	Crew	-0.650 ***	0.177 ***	0.177 ***	0.288 ***
	Running	0.260 ***	-0.489 ***	0.054	-0.104
	Other	0.170 ***	0.035	-0.364 ***	-0.123 **
	Capital	1.401 ***	-0.343	-0.621 **	-0.683 ***
	Full CU
	Crew	-0.569 ***	0.094 ***	0.168 ***	0.243 ***
	Running	0.139 ***	-0.344 ***	0.156 ***	-0.359 ***
	Other	0.190 ***	0.119 ***	-0.535 ***	0.084 *
	Capital	1.414 ***	-1.410 ***	0.434 *	-0.678 ***
	Full TE CU
	Crew	-0.425 ***	0.024	0.098 ***	0.192 ***
	Running	0.055	-0.330 ***	0.171 **	-0.359 ***
	Other	0.166 ***	0.128 **	-0.583 ***	0.146
	Capital	1.687 ***	-1.400 ***	0.761	-0.997 ***
	*** significant at 1% level; ** significant at 5% level; * significant at 10% level

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Table 5 .

 5 Estimated returns to scale

			Base Run	Full capacity utilisation	Technically efficient full CU
		Coeff	St. Err.	Coeff	St. Err.	Coeff	St. Err.
	Returns to scale	1.822	0.099 ***	1.588	0.126 ***	1.327	0.111 ***
	Scale factor	17020	125869	14.599	16.321	2.793	1.178 **
	*** significant at 1% level; ** significant at 5% level; * significant at 10% level

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Table 6 .

 6 Estimated revenues, costs and profits

				Page
		Current "average"	DEA "optimal"	Cost function
		vessel	vessel	"optimal" vessel
	Revenue (£m)	0.373	1.065	1.747
	Costs (£m)	0.311	0.633	1.182
	Profits (£m)	0.062	0.431	0.565
	Profits as proportion of revenue (%)	16.6	40.5	32.4
	Potential fleet reduction (%)	-	65	79

40 of 40 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The cost function can also be developed for a multi-output industry. The single output case is presented for the sake of simplification.

Data on a small number of vessels under 10m were also available. As these vessels are not subject to the same individual quota regulations as the larger vessels these data were not used. The data were collected through personal interview by the Seafish Industry Authority for the North Sea and Irish Sea, and by CEMARE for the English Channel.

In the UK, VCUs are defined by length*breadth+0.45*engine power. These

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript