Eight human subjects pressed a lever on a range of variable-interval schedules for 0.25¢ to 35.0¢ per reinforcement. Herrnstein's hyperbola described seven of the eight subjects' response-rate data well. For all subjects, the y-asymptote of the hyperbola increased with increasing reinforcer magnitude and its reciprocal was a linear function of the reciprocal of reinforcer magnitude. These results confirm predictions made by linear system theory; they contradict formal properties of Herrnstein's account and of six other mathematical accounts of single-alternative responding.Key words: linear system theory, Herrnstein's equation, quantitative law of effect, choice, reinforcer magnitude, variable-interval schedules, lever press, humansThe linear system theory is a set of mathematical techniques that can be used to calculate the response of a system to a known input provided the system can be described at least in principle by a linear differential equation (Aseltine, 1958). McDowell and Kessel (1979) used a modified version of the linear system theory to calculate the response of organisms to reinforcement inputs provided by variableinterval (VI) schedules. Their application of the theory entailed writing the reinforcement "input" to the system and the response "output" of the system as mathematical functions of time, calculating the Laplace transformations (McDowell, Bass, & Kessel, 1983) of these functions, and taking their quotient. For a system that can be described by a linear differential equation, the ratio of the transformed output function to the transformed input function must be constant (Aseltine, 1958 tion of reinforcement rate, reinforcer value, and response value. McDowell and Kessel (1979) showed that the rate equation accounted for response-rate versus reinforcement-rate data from VI schedules as accurately as Herrnstein's (1970) hyperbola.In a later paper, McDowell (1980) compared the rate equation with Herrnstein's hyperbola and demonstrated that at ordinary rates of reinforcement and responding the hyperbola was an approximation of the rate equation. When the exponential terms in the rate equation were replaced by their series expansion approximations, the equation assumed a hyperbolic form identical to that of Herrnstein's equation. However, McDowell also demonstrated that the identity of the two equations did not extend to the structure or interpretation of their parameters. In particular, Herrnstein's (1970Herrnstein's ( , 1979 derivations require the y-asymptote, k, of the hyperbola to remain invariant with respect to changes in reinforcement parameters like magnitude or immediacy (Herrnstein, 1974). McDowell (1980) showed that the y-asymptote of the hyperbolic form of the rate equation was required to vary directly with these reinforcement parameters and, specifically, that its reciprocal was required to vary as a linear function of the reciprocal of reinforcement parameters like magnitude or immediacy. Unlike the constant-k property of Herrnstein's hyperbola, the variable-k property of the...

Four human subjects worked on all combinations of five variable-interval schedules and five reinforcer magnitudes (¢/reinforcer) in each of two phases of the experiment. In one phase the force requirement on the operandum was low (1 or 11 N) and in the other it was high (25 or 146 N). Estimates of Herrnstein's k were obtained at each reinforcer magnitude. The results were: (1) response rate was more sensitive to changes in reinforcement rate at the high than at the low force requirement, (2) k increased from the beginning to the end of the magnitude range for all subjects at both force requirements, (3) the reciprocal of k was a linear function of the reciprocal of reinforcer magnitude for seven of the eight data sets, and (4) the rate of change of k was greater at the high than at the low force requirement by an order of magnitude or more. The second and third findings confirm predictions made by linear system theory, and replicate the results of an earlier experiment (McDowell & Wood, 1984 (1970, 1974)