Demand equations for qualitatively different foods under fixed-ratio schedules: a comparison of three data conversions

Abstract

Concurrent schedules were used to establish 6 hens' preferences for three foods. The resulting biases suggested wheat was preferred over honey-puffed and puffed wheat, and puffed wheat was the least preferred food. The hens then responded under fixed-ratio schedules for each food in 40-min (excluding reinforcer time) sessions, with the response requirement doubling each session until no reinforcers were received. At the smaller ratios, the less preferred the food, the faster the hens' overall response rates (mainly as a result of shorter postreinforcement pauses) and the more reinforcers they received. The relations between the logarithms of the number of reinforcers obtained (consumption) and the response ratio (price) were well fitted by curvilinear demand functions. Wheat produced the smallest initial consumption (ln L), followed by honey-puffed and puffed wheat, respectively. The response requirement at which the demand functions predicted maximal responding (P(max)) were larger for wheat than for the other foods. Normalizing consumption and price, as suggested by Hursh and Winger (1995), moved the data for the three foods towards a single demand function; however, the P(max) values were generally largest for puffed wheat. The results of normalization, as suggested by Hursh and Silberberg (2008), depended on the k value used. The parameter k is related to the range of the data, and the same k value needs to be used for all data sets that are compared. A k value of 8.0 gave significantly higher essential values (smaller alpha values) for puffed wheat as compared to honey-puffed wheat and wheat, and the P(max) values, in normalized standard price units, were largest for puffed wheat. Normalizing demand by converting the puffed and honey-puffed wheat reinforcers to wheat equivalents (by applying the bias parameter from the concurrent-schedules procedure) maintained separate demand functions for the foods. Those for wheat had the smallest rates of change in elasticity (a) and, in contrast to the other analyses, the largest P(max) values. Normalizing demand in terms of concurrent-schedule preference appears to have some advantages and to merit further investigation.

Keywords: behavioral economics; concurrent schedules; demand functions; domestic hens; fixed-ratio schedules; key peck; magnitude-of-reinforcer; normalization; reinforcer quality.

Fig 1

The natural logarithms of the numbers of reinforcers per session plotted as functions of the natural logarithms of the FR value for each food and for each hen. The data are the averages across the two series of increasing FR values with each food. The solid lines were fitted using Equation 1, and their parameter values appear in Table 3. The dotted lines and dashed lines were fitted using Equation 5 with k values of 3.5 and 8.0, respectively, and their parameter values appear in Table 6.

Fig 2

The left panel shows the natural logarithm of the numbers of reinforcers per session plotted as functions of the natural logarithms of the FR value for each food and for each hen. The data are the averages across the two series of increasing ratios with each food. W data are indicated by plus marks, HPW data by crosses, and PW data by circles. The central panel shows the data after the normalization suggested by Hursh and Winger (1995), where consumption was normalized to a value of 100. The natural logarithm of 100 is indicated by the dashed horizontal line. Price was also modified (see text for details). The right panel shows the data after they were adjusted by the preference values. Here consumption was converted to W equivalents by dividing by the W preferences found in Part 1 of the experiment. The solid lines were fitted to the pooled data in each panel using Equation 1. Their parameter values appear in Table 4. The parameters of functions fitted to the unmodified data and preference adjusted data from each food separately appear in Table 3 and those for the normalized data in Table 5.

Fig 3

Both panels show the natural logarithm of the consumption normalized as suggested by Hursh and Silberberg (2008) and plotted as functions of the natural logarithms of the normalized standard price (see text for details) for each food and for each hen. The data are the averages across the two series of increasing ratios with each food. W data are indicated by plus marks, HPW data by crosses, and PW data by circles. The left and right panels show the functions found by fitting Equation 5 with a k value of 3.5 and 8.0, respectively. The solid lines show the functions for the W data, the dotted lines the functions for the HPW data, and the dashed lines the functions for PW data. The parameter values are in Table 6.

Fig 4

Means of the running response rates (responses per second) over the two series with W and PW for each hen plotted against the normal logarithm of the FR value. Open circles mark data from the W condition and filled diamonds from the PW condition.

Fig 5

Mean PRP durations (in seconds) over the two series using W and PW for each hen plotted against the normal logarithm of the FR values. Open circles mark data from the W condition and filled diamonds from the PW condition.

Fig 6

Examples of cumulative records from Hen 61 responding under FR 1, 2, 4, 8, and 64 schedules for W (left panels) and PW (right panels). The vertical dotted line indicates the end of the session.