This function allows the user to check whether a given data set is consistent with the Strong Axiom of Revealed Preference at efficiency level e (eSARP) and computes the number of eSARP violations. We say that a data set satisfies SARP at efficiency level e if q_t R_e q_s implies ep_s'q_s < p_s'q_t (see the definition of R_e below). It is clear that by setting e = 1, we obtain the standard version of SARP. While if e < 1, we allow for some optimization error in the choices to make the data set consistent with SARP. The smaller the e is, the larger will be the optimization error allowed in the test. It is well known that SARP is a necessary and sufficient condition for a data set to be rationalized by a continuous, strictly increasing, and strictly concave preference function (see Matzkin and Richter (1991)).
A T X N matrix of observed prices where each row corresponds to an observation and each column corresponds to a consumption category. T is the number of observations and N is the number of consumption categories.
A T X N matrix of observed quantities where each row corresponds to an observation and each column corresponds to a consumption category.T is the number of observations and N is the number of consumption categories.
The efficiency level e, is a real number between 0 and 1, which allows for a small margin of error when checking for consistency with the axiom. The default value is 1, which corresponds to the test of consistency with the exact SARP.
The function returns two elements. The first element (
passsarp) is a binary indicator telling us whether
the data set is consistent with SARP at a given efficiency level e. It takes a value 1 if the data set
is eSARP consistent and a value 0 if the data set is eSARP inconsistent.
The second element (
nviol) reports the number of eSARP violations. If the data is eSARP
nviol is 0. Note that the maximum number of violations in an eSARP inconsistent data is
For a given efficiency level 0 ≤ e ≤ 1, we say that:
bundle q_t is directly revealed preferred to bundle q_s at efficiency level e (denoted as q_t R^D_e q_s) if ep_t'q_t ≥ p_t'q_s.
bundle q_t is strictly directly revealed preferred to bundle q_s at efficiency level e (denoted as q_t P^D_e q_s) if ep_t'q_t > p_t'q_s.
bundle q_t is revealed preferred to bundle q_s at efficiency level e (denoted as q_t R_e q_s) if there exists a (possibly empty) sequence of observations (t,u,v,\cdots,w,s) such that q_t R^D_e q_u, q_u R^D_e q_v, \cdots, q_w R^D_e q_s.
Matzkin, Rosa L., and Marcel K. Richter. "Testing strictly concave rationality." Journal of Economic Theory 53, no. 2 (1991): 287-303.
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# define a price matrix p = matrix(c(4,4,4,1,9,3,2,8,3,1, 8,4,3,1,9,3,2,8,8,4, 1,4,1,8,9,3,1,8,3,2), nrow = 10, ncol = 3, byrow = TRUE) # define a quantity matrix q = matrix(c( 1.81,0.19,10.51,17.28,2.26,4.13,12.33,2.05,2.99,6.06, 5.19,0.62,11.34,10.33,0.63,4.33,8.08,2.61,4.36,1.34, 9.76,1.37,36.35, 1.02,3.21,4.97,6.20,0.32,8.53,10.92), nrow = 10, ncol = 3, byrow = TRUE) # Test consistency with SARP and compute the number of SARP violations sarp(p,q) # Test consistency with SARP and compute the number of SARP violations at e = 0.95 sarp(p,q, efficiency = 0.95)
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