APPENDIX B: PEAK-TO-PEAK ANALYSIS

Peak-to-peak analysis is a relatively simple method to assess the capacity utilization of an industry over time. An advantage of peak-to-peak analysis is that it requires information on only one output measure and one input measure, and hence is suited to estimating capacity utilization with only Level 1 data (see Table A.1). Peak-to-peak analysis has been applied in fisheries by Ballard and Roberts (1977), Ballard and Blomo (1978) and Hsu (2003).

The underlying theory

Peak-to-peak analysis is based on an underlying assumption that output is a function of the level of inputs and a technology trend, such that

(1)

where Y_t is the output in time, t; a₀ is a proportionality constant; V_t is a composite or aggregate index of inputs; and T_t is the technology trend that represents productivity change. An implicit assumption in the use of a composite index of inputs is that the technology displays constant returns to scale. That is, increasing all inputs will result in a proportional increase in output.

The level of technology is determined by the average rate of change in productivity between peak years, where productivity is given by Y_t/V_t (i.e. average output per unit of input). The technology in any one year is thus

(2)

where m is the length of time from the previous peak year, and n is the length of time to the following peak year, and T_t-m is the level of technology at the previous peak (i.e. year m) equivalent to the average productivity (e.g. catch per unit of effort) in that period. The other term on the right hand side (i.e. the term inside the brackets) represents the cumulative change in productivity between the two peaks. This is added to the average productivity in the previous peak year (i.e. year m) to give an estimate of the average productivity of capacity in subsequent years.

An alternative way of estimating the level of technology between peaks is given by

(3)

where Y_n/V_n is the average productivity in the upper peak and Y_m/V_m is the average productivity in the lower peak. The term in the brackets represents the average change in productivity between the two peaks. Both approaches produce identical results.

Assuming the proportionality constant has a value of 1, the estimate of the level of technology is equivalent to the capacity level of productivity (i.e. T_t = Y_t*/V_t, where Y_t* is the capacity level of output). From this, the capacity level of production can be estimated from the product of the inputs and the capacity level of productivity, such that

Y_t* = V_tT_t (4)

and capacity utilization can be estimated by

CU_t = Y_t*/Y_t. (5)

A particular difficulty in interpreting the results of a peak-to-peak analysis in fisheries is that no consideration is given to changes in the stock level. Apparent changes in productivity may be due to either changes in technology (the underlying assumption of the technique) or changes in the stock level.

This problem may be particularly pertinent in developing fisheries, where catch rates may increase rapidly initially, with the main peak occurring in the middle of the time series. Subsequent declines in catch rates may reflect falling stock levels. However, if the main peak is used as the last peak in the series (all other years showing a steady decline), it is likely that the technique will over-estimate capacity output and under-estimate capacity utilization.

The problem can be minimized by including lower peaks rather than successively higher peaks as is generally used in other industries that do not rely upon a biological resource base.

Example of use: Nigerian artisanal fishing sector

Data on the artisanal fishing sector in Nigeria were used as an example of how peak-to-peak analysis can be used to estimate capacity. The data were derived from Amire (2003), and are presented in Table B.1.

Table B.1 - Nigerian artisanal fisheries productivity, 1976-1994

Source: Amire (2003).

The choice of input may have an impact on the measure of capacity output and, consequently, capacity utilization. In the Nigerian artisanal fleet, the number of canoes active in the fishery had declined over time while the number of fishers remained relatively constant (a result of more fishers operating per canoe). Over the same period, motorization increased in the fishery from 8.7 percent in 1996 to 20.8 percent in 1994 (Amire, 2003). As a result, it would be expected that there was substantial technological change in the fishery. Developing a composite index of inputs in such a case is difficult without first estimating a production function and imposing constant returns to scale.

For purposes of illustration, capacity was assessed using both canoes and fishers (separately) for the input measure. From Table B.1, it can be seen that the peak productivity periods for both inputs were 1976, 1979, 1982, 1988 and 1992. These peaks also are apparent by graphing the catch per unit input series (Figure B.1).

Figure B.1 - Catch-per-unit input, Nigerian artisanal fleet

Table B.2 - Peak-to-peak analysis using canoes as input measure

Note: Peak years in bold, a) estimated by [(Y_n/V_n)-(Y_m/V_m)]/(n-m)

The analyses, undertaken in an Excel spreadsheet, are given in Tables B.2 and B.3 using canoes and fisher numbers respectively. Average technological change was estimated between the peak years (indicated in bold). For example, between 1976 and 1979, average productivity change was (2.669-2.438)/(4-1) = 0.0768.

Capacity CPUE is estimated by adding the average technological change to the preceding year’s value. Capacity output is estimated by multiplying the capacity CPUE by the input level. The utilization rate is estimated by dividing actual output by capacity output.

Table B.3 - Peak-to-peak analysis using number of fishers as input measure

Note: Peak years in bold, a) estimated by [(Y_n/V_n)-(Y_m/V_m)]/(n-m)

Despite differences in the input measure used, the estimated capacity output was fairly similar in both instances (Figure B.2a). The estimated capacity utilization in each year was also relatively similar (Figure B.2b).

Figure B.2 -a) estimated capacity and b) estimated capacity utilization