Elsevier

Behavioural Processes

Volume 103, March 2014, Pages 199-210
Behavioural Processes

Reinforcer magnitude and demand under fixed-ratio schedules with domestic hens

https://doi.org/10.1016/j.beproc.2013.12.013Get rights and content

Highlights

  • Domestic hens’ demands for different durations of access to food were determined.

  • Three procedures for normalizing demands were compared.

  • A preference-based normalization procedure was in some regards best.

  • Further research examining normalization procedures is warranted.

Abstract

This study compared three methods of normalizing demand functions to allow comparison of demand for different commodities and examined how varying reinforcer magnitudes affected these analyses. Hens responded under fixed-ratio schedules in 40-min sessions with response requirement doubling each session and with 2-s, 8-s, and 12-s access to wheat. Over the smaller fixed ratios overall response rates generally increased and were higher the shorter the magazine duration. The logarithms of the number of reinforcers obtained (consumption) and the fixed ratio (price) were well fitted by curvilinear demand functions (Hursh et al., 1988. Journal of the Experimental Analysis of Behavior 50, 419–440) that were inelastic (b negative) over small fixed-ratios. The fixed ratio with maximal response rate (Pmax) increased, and the rate of change of elasticity (a) and initial consumption (L) decreased with increased magazine duration. Normalizing consumption using measures of preference for various magazine durations (3-s vs. 3-s, 2-s vs. 8-s, and 2-s vs. 12-s), obtained using concurrent schedules, gave useful results as it removed the differences in L. Normalizing consumption and price (Hursh and Winger, 1995. Journal of the Experimental Analysis of Behavior 64, 373–384) unified the data functions as intended by that analysis. The exponential function (Hursh and Silberberg, 2008. Psychological Review, 115, 186–198) gave an essential value that increased (i.e., α decreased significantly) as magazine duration decreased. This was not as predicted, since α should be constant over variations in magazine duration, but is similar to previous findings using a similar procedure with different food qualities (hens) and food quantities (rats).

Introduction

Knowing the aspects of an animal's world that are important to that animal is essential to maximize its welfare and to predict its future behavior. Several different methodologies can be used to gain information about the importance of a given commodity. For example, assessing the degree to which an animal selects one commodity over another indexes the relative value of the two commodities to that animal. Several such procedures, termed preference assessments, were described by Sumpter et al. (2002). Normally they involve the animal making a response to gain access to one of two or more commodities. The response may be simply moving from one location to another, or selecting one arm of a maze or operating a manipulandum, such as a key or lever. Preference is assessed by the degree to which an alternative is selected over the others, e.g., the proportion of choices of or relative time allocated to that alternative. Measures of preference obtained in this way are always relative to the commodities on offer and the results are taken to be the animals’ preferences between the commodities on offer at that time, that is, they are measures of the relative values of the commodities. Such procedures allow direct comparison of the commodities and it is possible to conclude which is of more importance to the animal in that context.

Another methodology that provides information on the importance of commodities to animals comes from applications of consumer demand theory (Dawkins, 1983). In one such procedure the effort (or price) required to gain access to a commodity is varied and the way consumption changes is examined. In this procedure, termed own-price demand (Green and Freed, 1998), the relation between the amount of the commodity consumed at each price and price is taken to be a description of animal's demand for that commodity and is known as a demand function (Hursh, 1984). In basic research with animals, price is typically operationalized as the number of responses required to produce a reinforcer (e.g., fixed-ratio (FR) size) and consumption as the number of reinforcers earned.

Hursh et al. (1988) proposed that demand functions could be described using the equation:lnQ=lnL+b(lnP)aP,where Q refers to total consumption, P denotes price, and L, a, and b are free parameters. The parameter L estimates the initial level of consumption obtained at the minimal price and reflects the height of the demand function above the origin. When consumption is measured on a common scale, the larger the L value the more is consumed at minimal price. The parameters a (the rate of change in the slope of the function across price increases) and b (the initial slope of the function) reflect aspects of the elasticity of the demand function. Both a and b are required to describe the elasticity of the function and if, for example, two demand functions have very different b values, then the a values cannot be sensibly compared. When a function is inelastic (i.e., with slope less steep than −1) over low prices but changes to being elastic (i.e., falling with a slope steeper than −1) as price increases, then a and b can be used to find the price associated with maximal response output. This is the price at which demand changes from inelastic to elastic and is termed Pmax (Hursh and Winger, 1995), which is calculated as:Pmax=(1+b)a.

The higher the price at which demand changes from inelastic to elastic, the larger the value of Pmax. Equations (1), (2) have proven to be useful in describing the data from many research studies (e.g., Foster et al., 1997, Foltin, 1992, Sumpter et al., 1999, Hursh and Winger, 1995).

Because a demand analysis encompasses the effects of changing price or effort, it can be viewed as a more general measure of the value of a commodity than preference estimates alone. For example, one commodity might be preferred over another or might be preferred similarly to another when little effort is required to obtain either but the relative preference might change when the amount of effort required to obtain the commodities changes. Such a relation is evident in a study by Williams and Woods (2000), in which monkeys preferred a 0% ethanol solution (tap water) to a 32% ethanol solution under an FR 4 schedule, but preferred the 32% ethanol solution at FR values of 32 and 64. To obtain the same total “value” in demand sessions with each of two commodities if one was relatively more preferred than the other in a preference assessment, the animal would need to obtain more of the less preferred than of the more preferred commodity. Therefore, at low prices the preference between the commodities may result in consumption of the less preferred commodity (i.e., the number of reinforcers earned) in a session being higher than consumption of the more preferred commodity. If price were increased, then the animal might maintain this difference or responding might reduce for the less preferred commodity more rapidly, resulting in a more elastic demand function. In the latter case, where b is similar for two commodities, a would be larger (i.e., a higher rate of change of elasticity) and Pmax smaller (i.e., it would maintain behavior to a lower price) for the less preferred commodity. Such an analysis involves comparisons of the demand functions from the different commodities, a point made by Williams and Woods (2000).

Comparison of demand functions requires that consumption of the various commodities be measured on a common scale. To do so, Hursh and Winger (1995) suggested that, when the aim was to compare demand for different commodities such as different drugs, the measure of consumption of the various drugs could be normalized. Their normalization involved converting the consumption measures to a percentage of consumption at the lowest price, thus giving all demand functions an initial consumption value of 100. They normalized the price, converting this to the price per unit of normalized consumption. Madden et al., 2007a, Madden et al., 2007b applied this normalization to data from prior studies (e.g., Ko et al., 2002, Winger et al., 2002) to compare the relative reinforcing efficacy of various drugs. The ranking of reinforcing efficacy that resulted was consistent with that predicted by other means.

The approach suggested by Hursh and Winger (1995) relies on normalizing using the initial level of consumption obtained in generating the demand function. Foster et al. (2009) offered another strategy for normalization. They suggested that it should be possible to use a preference measure to normalize consumption data, a strategy they called “preference-adjusted demand.” This strategy involved comparing commodities using a concurrent-schedule choice procedure (see Davison and McCarthy, 1988) and then applying the resulting preference measure to normalize the demand data. The suggested preference measure was based on the generalised matching equation (Baum, 1974, Baum, 1979). Matthews and Temple (1979) previously demonstrated that the following version of that equation could be used to assess bias or preference resulting from qualitatively different reinforcers:logP1P2=aslogR1R2+logbc+logqwhere P1 and P2 are the numbers of responses to the two concurrently available schedules, R1 and R2 are the number of reinforcers obtained under the two schedules, as reflects the sensitivity of behavior to changes in reinforcement rate, log bc quantifies the bias (i.e., the tendency to respond more under one schedule than under the other) resulting from factors other than reinforcer differences, and log q measures bias resulting from differences between the two reinforcers. Log q is taken as a measure of the preference for one reinforcer over the other. The total bias, log bc + log q, is often termed log c. When the two schedules deliver reinforcers equally often, then R1 will equal R2 and, as log (R1/R2) will equal 0, the equation reduces to:logP1P2=logbc+logq=logc

As Sumpter et al. (2002) pointed out, the value of log bc, or preference, can be found using the same reinforcer on both schedules (so that log q equals 0). When different reinforcers are arranged under the two schedules, with equal reinforcer rates, then a measure of log c (i.e., log bc + log q) is obtained. Subtracting log bc from this value gives the value of log q alone.

Foster et al. (2009) used this process to assess hens’ preference among three foods, wheat (W), puffed wheat (PW), and honey-puffed wheat (HPW), by pairing W with W, W with PW, and W with HPW. They found W was preferred to HPW and PW, and that PW was least preferred. They also used single FR schedules with each of the three foods and increased the number of responses required to gain access to a food over sessions (i.e., increasing FR schedules). This procedure assessed demand for each of the three foods when presented alone. The analyses proposed by Hursh et al. (1988) and Hursh and Winger (1995) were then compared with that from a preference-adjusted demand analysis based on the Hursh et al. equation (i.e., using the preference data from the concurrent schedule phase to normalize consumption).

While the functions generated by all three analyses fitted the data well, the relations between the various parameters and preference were not clear. The unmodified data (Hursh et al., 1988) resulted, paradoxically, in the lowest initial consumption (measured as number of reinforcers obtained) for the most preferred food. In line with the preference data, however, the most preferred food had the highest Pmax value. Hursh and Winger's (1995) analysis necessarily reduced the initial consumption differences between the foods, and this normalization resulted in the least preferred food having the highest Pmax value. Normalizing the data in this way reduced the impact of the preferences between the foods on the resulting demand functions. It unified the demand functions with the rates of change of elasticity (a values) across the functions for the three food being similar. The preference adjusted-demand analysis, based on the Hursh et al. (1988) equation, reduced the differences in initial consumption seen in the unmodified data, and resulted in the most preferred food having the highest Pmax value. Also the rates of change of elasticity were found to be greatest for the least preferred food, as might be expected if the preference interacted with demand, and so this analysis seemed promising. In addition, as price increased the hens reduced responding more quickly for the less-preferred than for the more-preferred food. A possible explanation of these findings, suggested by Foster et al., was that preference affected the demand so that the hens responded to gain more reinforcers with the less preferred foods at low prices, achieving equivalent total ‘value’ across the three foods at these prices.

In another approach to comparing demand functions for different commodities, Hursh and Silberberg (2008) suggested that comparisons would be simpler if the function used gave a single measure of the value of a commodity. They proposed an alternative to Eq. (1) that, they argued, provided such a measure. The exponential equation for this alternative is:logQ=logQ0+k(eαP1),where Q and P are as in Eq. (1), Q0 is equivalent to L in Eq. (1), k is the range of consumption in logarithmic units, and α is the rate constant and reflects the rate of decrease in consumption with increases in P. To deal with the scale on the consumption axis, they suggested that consumption be normalized using consumption at a price of zero (Q0), in a similar manner to Hursh and Winger (i.e., the number of reinforcers obtained at each price divided by Q0 and multiplied by 100) so that the normalized consumption at a price of zero will be 100. To compare across commodities, they standardized price as the total cost required to defend the consumption at a price of zero (Q0) for each schedule requirement (C), therefore, P = (Q0 × C). When k is the same for two commodities, then α can be compared directly and it is this parameter that Hursh and Silberberg term a measure of the “essential value” of a commodity. The essential value is inversely related to the value of α.

Standardizing cost means that when different amounts of the same commodity are evaluated the essential value should stay the same. Hursh and Silberberg (2008) and Hursh et al. (2013) argue that this was the case for the rats in Hursh et al.’s (1988) study, where demand for 1 and 2 food pellets was compared using this exponential analysis. When this analysis is used with demand for different commodities, the essential value should differ for commodities that differ in value. Studies have shown that α provides a meaningful index of the relative value of different drugs self-administered by nonhumans (Madden et al., 2007a, Madden et al., 2007b) and of different brands of products purchased by humans (Oliveira-Castro et al., 2011).

The maximum output, equivalent to Pmax, can be calculated from the parameter values in Eq. (3) although, as Hursh et al. (2013) point out, it is only possible to determine an exact Pmax value for this function based on an iterative solution. They suggest an intuitive approximation to Pmax, one that is inversely related to α, for this analysis. The approximation is calculated as:Pmax=0.65(αQ0k1.191),where the parameters are as in Eq. (3). As for the Hursh et al. (1988) analysis, this value reflects the point at which the elasticity of the demand curve is −1 and is the price at which maximal responding is achieved. The standardized price associated with maximal responding is this value multiplied by Q0.

Foster et al. (2009) fitted Hursh and Silberberg's (2008) equation to their data. The analysis proved problematic because the range of consumption (k) differed greatly across hens and foods and it was not clear what k value should be used for the analysis. A constant k value is required if α is to be compared sensibly across data sets. Foster et al. (2009) analysed the data using two different k values and both generally and unexpectedly resulted in highest essential values (smallest α values) for the least preferred food and gave estimates of Pmax (measured in normalized standard price units) that were largest for the least preferred food.

In summarising the findings from the three analyses aimed at comparing demand for different commodities covered here, Foster et al. (2009) reported that the Pmax values from the Hursh and Silberberg (2008) analysis suggested the least preferred food maintained behaviour to higher value than the more preferred food, as did the Pmax values from the Hursh and Winger (1995) analysis. However, the Hursh and Winger analysis gave similar rates of change of elasticity for the three foods. The preference adjustment resulted in the most preferred food having the smallest rates of change in elasticity (a) and the largest Pmax values, demonstrating that it had the greatest value according to these measures. Although the results obtained with the analysis proposed by Hursh and Silberberg appear to be counterintuitive when applied to these data, it was possible that, contrary to the results obtained under the concurrent-schedule arrangement, PW really did have a greater essential value than W, and that PW would have been preferred to W if the price of both was increased in the preference test.

Given that the different strategies for normalizing demand data that they compared did not support the same conclusions, Foster et al. (2009) suggested further exploration of these strategies was warranted. One suggestion they made was to systematically replicate their procedures using different amounts of the same commodity, rather than different commodities. As previously mentioned, different amounts of the same commodity should have the same essential value in Hursh and Silberberg's (2008) analysis, and so this analysis should result in different amounts of the same commodity having the same α values. Hursh and Winger's (1995) analysis should result in similar rates of change of elasticity regardless of amount of food and, if this normalization removes the effects of preference, the Pmax values, in terms of normalized price, should be the same. The preference-adjustment procedure should show the larger reinforcers giving lower rates of change of elasticity and higher Pmax value than smaller reinforcers.

The present study examined the results obtained under conditions where hens worked to produce access to different durations of food (wheat) delivery. In order to be able to apply the preference-adjusted analysis, concurrent schedules were used with the hens to obtain preference (i.e., bias) measures for different durations of access to wheat. The hens also responded for three food-access times under increasing FR schedules. In order to make the present data comparable to those reported by Foster et al. (2009), the same general procedures were used. This resulted in data that allowed further comparison of the different demand analyses.

Section snippets

Subjects

Six Shaver-Starcross hens (Gallus gallus domesticus) (41–46) served as subjects. Hen 45 was approximately 5 years old at the start of the experiment and the remaining hens were 3 years old. All the hens were housed individually in home cages measuring 450 mm long by 200 mm wide by 430 mm high and had free access to water. They were maintained at 80% (±5%) of their free-feeding body weights by daily weighing and the provision of supplementary commercial feed pellets when necessary. Grit and

Concurrent schedules

The logarithms of the ratios of the numbers of reinforcers obtained, with the right schedule data over the left schedule data, averaged over the last five sessions of Conditions 1, 2 and 6, are given in Table 2. In all but one case (Condition 1 for Hen 42), these values were within 0.08 of 0. The logarithms of the ratios of responses to the two schedules give measures of bias, log c (Eq. 5). These log c values, taken as the averages of the logarithms of the ratios of responses over the last five

Discussion

The purpose of the present study was to assess the demand for three different durations of access to wheat and to examine three different analyses of the data for comparison with data obtained previously using a similar procedure but different qualities of food. Foster et al. (2009) used the number of reinforcers obtained in the session as the consumption measure when they compared demand for qualitatively different foods; the same metric was used in the present study. In the present analyses

Author note

These data were colleted for a Master's thesis completed by the first author. Raw data are available in electronic form from the second author ([email protected]). We thank Jennifer Chandler for her technical assistance in conducting this research and the research cooperative of staff and students who helped to conduct these experiments.

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