Measuring information in the market: an application to physician services.

I. Introduction

Background

Ever since Stigler's |47~ seminal piece on the economics of information, there has been a great deal of research investigating equilibrium in markets with imperfect information. While most of this research has been concerned with theoretically establishing the conditions under which there exists a distribution of prices in equilibrium, the field now contains a small, but growing, body of empirical research.

This work has followed the suggestion of Stigler and utilized the dispersion of prices (usually the variance) as a measure of incomplete information about price, such as Stigler |47~, Stigler and Kindahl |49~, Pratt, Wise and Zeckhauser |44~, Carlson and Pescatrice |12~, Marvel |34~, Mathewson |35~, Cox, DeSerpa and Canby |13~, Dahlby and West |14~, Van Hoomissen |51~, and Lach and Tsiddon |29~. There are two disadvantages to using the variance (or another measure of dispersion, such as the range) of prices as a measure of incomplete information about price. The first disadvantage, recognized by Stigler and others, is that price can vary for many reasons other than incomplete information. Thus dispersion is not a pure measure of incomplete information about prices. The second disadvantage, which has not been commonly recognized in the empirical literature, is that price dispersion encompasses incomplete information both on the part of buyers and sellers as indicated by Rothschild |45~, Axell |4~, Butters |9~, and Telser |50~, and hence fails to distinguish between these two components.

In this paper we propose a method for measuring incomplete information about price which builds on Stigler's original suggestion to use dispersion as a measure of incomplete information, but which avoids the pitfalls mentioned above. We apply our approach to the physicians' services market, a market in which incomplete information is widely cited as a source of market failure. After we adjust for physician characteristics and other factors which can influence price we measure incomplete buyer information as the difference between the seller's (adjusted) reservation price (which is the lowest price at which the seller the product) and the (adjusted) price which the consumers pays. This is the money cost to the consumer of not knowing where to obtain the lowest price. Analogously, incomplete seller information is the difference between the (adjusted) buyer's reservation price (i.e., the highest price the buyer will pay) and the (adjusted) price the seller accepts for the product. This is the seller's cost of incomplete information about buyer reservation prices. Thus, these measures provide money metrics for the amount of buyer and seller information in a market. Obviously in a full information competitive market when a unique price evolves both measures are zero.

The estimation entails a generalized frontier estimation technique developed by Polachek and Yoon |43~ to separate observed price dispersion into purely random variation, variation due to incomplete buyer information, and variation due to incomplete seller information.(1) The application is to the physicians' services market. Measures of incomplete buyer and seller information are computed for eight different physician services(2) and by the proportion of patients a physician receives through referrals as well as by patient income. Hypotheses concerning expected differences in information across these groups correspond to well established hypotheses concerning differences in the costs and benefits of search.

The estimation results are strikingly consistent with the hypotheses regarding search. Incomplete patient information is approximately 50 percent greater than incompletely physician information. Incomplete patient information is larger than incomplete physician information for smaller ticket items, less frequently purchased items, more heavily insured items, and treatments associated with severe illness. Measured incomplete patient information is also higher relative to incomplete physician information both for high referral practices and those in high income areas.

In the rest of the paper we provide institutional detail on the market for physician services, describe the model and econometrics (sections II and III, respectively), generate hypotheses (section IV), discuss the data sources and the variables employed (section V) and present the empirical results (section VI). Section VII contains a summary and conclusions.

The Market For Physician Services

The health care market in general, but especially the physicians' services market, is often viewed as one in which market forces fail to work effectively, as Arrow |3~, Newhouse |36~, and Pauly |38; 41~ have indicated. Incomplete consumer price information (i.e., lack of consumer information about price) has been viewed as an important reason for this market failure |11; 18~.(3)

If consumer incomplete information is substantial, policies failing to address this problem could prove ineffective in strengthening the role of market forces |33; 52~.

Despite universal agreement that information deficiencies are important causes of failure in the physician services market, there are no direct estimates of the extent of consumer incomplete information in this market. This is not surprising, given that information is difficult to observe and quantify. Pauly |38; 41~ points out that, while consumer information is perhaps the most important feature of the health services marketplace, nearly absolutely incomplete consumer information has been accepted as fact with very little empirical evidence.

A considerable number of studies have documented the large dispersion in physician fees.(4) This is commonly interpreted as indicative of the extensive lack of information in this market, a la Stigler's suggestion, despite the fact that there are product differences among physicians and these product differences are reflected in price. As Phelps |42, 193-95~ contends, although quality and price are undoubtedly connected, "The dispersion of prices seems far too large to correspond to quality differences." Some evidence on this can be gleaned from data on the price distributions from several cities. The mean price for a standard office visit in Dayton, Ohio differed by only $1.50 between general practitioners and internists, while within each specialty the highest price was more than twice as high as the lowest price |42~. Presumably internists are perceived as being of higher quality. It is hard to believe that quality alone is responsible for such wide specialty price variations, when the variation across these specialties is so small. Newhouse and Sloan |37~ report coefficients of variation for selected physician fees in New York and Chicago. The coefficient of variation for all these services is higher by an order of magnitude than for the asking price of a Chevrolet in Chicago or for bids on federal contracts for delivery of anthracite coal. As both Phelps and Newhouse and Sloan indicate, it is difficult to believe that quality differences can account for these wide variations in price, after controlling for specialty, market area, and service performed.

Another body of empirical work examines the link between information and the price level.(5) These studies attribute higher prices to incomplete consumer information, using advertising or other variables as proxies for information. As Pauly |38; 41~ indicates, incomplete consumer information alone is not sufficient to reduce consumer welfare relative to the full information equilibrium. An informational asymmetry favoring the seller must also be present for incomplete information to present a problem. This study devises a way to measure incomplete price information as well as disentangle incomplete buyer and seller information.

II. The Model

The approach taken here is based on results from the extensive literature on search theory. Price dispersion is possible in an otherwise competitive market if informational imperfections are present. For instance, buyers are motivated to purchase the product for the lowest possible price. They know the distribution of prices in the market, but not the price charged by any given seller. Buyers choose an optimal amount of search based on a weighing of the expected savings from search versus its associated costs. Let a buyer's reservation price, i.e., the highest price a patient will pay, be a function of factors which affect the patient's search cost and benefits such as the extent of insurance coverage, the patient's education and income, the severity of the patient's illness, and the frequency with which the physician's services are needed. For transaction i this reservation price (|P.sub.b~) can be expressed as

|P.sub.bi~ = |X.sub.bi~||Beta~.sub.b~ + |u.sub.bi,~ (1)

where |X.sub.b~ are the exogenous factors just noted, |u.sub.b~ is a random error term and subscript i denotes transaction i.

Similarly, sellers seek to sell at the highest price possible. For physicians the reservation price is a function of factors that affect the lowest price a physician can charge, e.g., input prices, technology, age of equipment, and factors affecting efficiency. For transaction i this reservation price (|P.sub.s~) is

|P.sub.si~ = |X.sub.si~||Beta~.sub.s~ + |u.sub.si~, (2)

where |X.sub.s~ are the exogenous factors affecting the physician's reservation price, |u.sub.s~ is a random error term and i denotes transaction i.

In a market with imperfect information buyers on average pay more than the seller's reservation price and sellers will sell at a price lower than some consumers are willing to pay.(6) For a given transaction we define the gap between the seller's reservation price and the price the consumer actually pays as incomplete consumer information,

|W.sub.i~ |approximately equal to~ |P.sub.ci~ - |P.sub.si~, (3)

where |W.sub.i~ is consumer ignorance, |P.sub.ci~ is the price paid by the consumer, and |P.sub.si~ is the seller's reservation price. This is the monetary cost of paying more than "necessary" and as such is a measure of the pecuniary cost to the consumer of his incomplete information about price.(7)

Incomplete seller information is defined in the same manner. The seller seeks to sell his product at the highest possible price. The seller also knows the distribution of consumer reservation prices, but does not know what any specific buyer will pay.(8) Since sellers do not have perfect information they will, on average, receive less than the highest price that a consumer would offer. The difference between the price received and the buyer's reservation price is9)

|V.sub.i~ |approximately equal to ~ |P.sub.ri~ - |P.sub.bi~, (4)

where -|V.sub.i~ is seller incomplete information, |P.sub.ri~ is the price received by the seller, and |P.sub.bi~ is the consumer's reservation price. This is the pecuniary cost to the seller of incomplete price information.

In equilibrium for each transaction the price paid by the consumer (|P.sub.ci~) must equal the price received by the seller (|P.sub.ri~),

|P.sub.ci~ = |P.sub.ri~ = |P.sub.i~, (5)

where |P.sub.i~ denotes the transaction price.

An expression for the equilibrium price can be obtained by substituting equation (1) into equation (4), substituting (2) into (3), imposing the equilibrium condition (5), and adding (3) and (4). After simplifying we obtain

|P.sub.i~ = |X.sub.i~|Beta~ + |u.sub.i~ + |v.sub.i~ + |w.sub.i~, (6)

where

|X.sub.i~|Beta~ = |X.sub.si~(||Beta~.sub.s~/2) + |X.sub.bi~(||Beta~.sub.b~/2),

|u.sub.i~ = (|u.sub.si~/2) + (|u.sub.bi~/2), and

|v.sub.i~ = |V.sub.i~/2, |w.sub.i~ = |W.sub.i~/2.

Note that E(P/X) = (|P.sub.b~/2) + (|P.sub.s~/2) + (P/2) - (|P.sub.b~/2) + (P/2) - (|P.sub.s~/2) = P. In equilibrium there will be a distribution of prices due to variations in information across buyers and sellers.

III. Hypotheses

To complete the model we consider some established hypotheses from search theory. For example, it is well known that the net benefits of search are less for goods which are cheaper as indicated by Stigler |47; 48~ and Stigler and Kindahl |49~, for goods which are purchased infrequently Pratt, Wise, and Zeckhauser |44~, and (for health care) for more heavily insured services, and for treatments associated with more severe illnesses in which timely information is needed as discussed by Newhouse |36~, Dionne |15~, and Marquis |32~.

We therefore estimate the model using the prices of eight different physician services as a means of testing these hypotheses. The services are: office visits to any physician; office visits to general practitioners (G.P.'s); office visits to pediatricians; office visits to general surgeons; hospital follow-up visits, dilation and curettage (D&C) by obstetricians; blood counts; and chest x-rays. We expect the information gap between buyers and sellers to be smaller for office visits than for hospital follow-up visits, D&C's, blood counts, or chest x-rays. Hospital care and D&C's are purchased less frequently than office visits, are associated with more severe illness, and are more heavily insured. Blood counts and chest x-rays are not necessarily infrequent nor associated with severe illness, but both are generally fully covered by insurance. In addition, hospital follow-up visits and blood counts and chest x-rays are usually tied to the purchase of another service, e.g., a hospital inpatient stay in the case of hospital follow-up visits. Similarly, the incomplete information of buyers relative to sellers is expected to be larger for office visits to general surgeons than for office visits to G.P.'s or pediatricians, since the services of the former are purchased less frequently and are associated with more severe illness.

We also hypothesize that patients who are referred from another physician substitute the referral for search, and thus have less information.(10) To test this, we stratify physicians into those who have a high proportion of their practice from referrals and those who have a low proportion from referrals. The magnitude of buyer relative to seller incomplete information should be greater for the high referral stratum than for the low referral stratum.

Last, we hypothesize that the opportunity cost of search is higher for patients with higher incomes. Thus, the measure of incomplete buyer information should be higher in areas with high per capita income than with low per capita income.

One potential problem with these as well as Stigler type dispersion measures is that they are not based on actual measures of information, but are inferred from price data alone. Given that, it is useful to question the interpretation of the estimates as measuring information. In particular, there are undoubtedly some unmeasured quality differences which are captured in the total residual, |Epsilon~. Nevertheless, there are two reasons why our estimates should represent reasonable measures of relative buyer to seller incomplete information. First, some of the services examined, such as blood count or chest x-ray, can be considered homogenous. As reported in section VI, the estimates of patient relative to physician incomplete information for these homogeneous services are consistent with our hypotheses. Further, the results for the other services and strata follow the same pattern as for these homogeneous services. Second, to infer that quality differences drive our measure of incomplete information requires the dispersion in quality to be larger for those strata with greater measured incomplete patient information. For example, consider the patient referral strata. It could be argued that a larger measure of incomplete patient information for the high referral stratum actually reflects higher quality physicians receiving more referrals. However, since the measure of incomplete patient information is an estimate of dispersion, for our measure to be reflecting quality there must not only be greater quality among high referral physicians, but greater dispersion in quality. There is no obvious reason to believe this is true. We now turn to the econometric specification of the model.

IV. Econometrics

Consider equation (6). Let the price observed for transaction i, |P.sub.i~, be determined by a vector |X.sub.i~, a parameter vector |Beta~, incomplete buyer information, |w.sub.i~, incomplete seller information, |v.sub.i~, and random noise. Then |P.sub.i~ can be written as

|P.sub.i~ = |Beta~|prime~|X.sub.i~ + ||Epsilon~.sub.i~, i = 1, 2, . . . , n (7)

where ||Epsilon~.sub.i~ is an error term such that

||Epsilon~.sub.i~ = |u.sub.i~ + |v.sub.i~ + |w.sub.i~ (8)

and where the density of |u.sub.i~ has support (-|infinity~, |infinity~), the density of |v.sub.i~ has support (-|infinity~, 0~, and the density of |w.sub.i~ has support |0, |infinity~).

Equations (7) and (8) represent a three-error component model which allows for both systematically positive and negative error components in addition to the usual random error component. As such |u.sub.i~ is a symmetric, purely random, error, and |v.sub.i~ and |w.sub.i~ are systematically negative and positive errors, representing seller and buyer information, respectively.

The interpretation of v and w as measures of information implies one-sided errors of differing signs such that E(v) = ||Mu~.sub.v~ |is less than~ 0 and E(w) = ||Mu~.sub.w~ |is greater than~ 0. As explained previously, ||Mu~.sub.v~ represents the average difference between the price received by sellers and the highest price buyers will pay, thus ||Mu~.sub.v~ is negative. The term ||Mu~.sub.v~ can then be interpreted as incomplete seller information. The term ||Mu~.sub.w~ represents the average difference between the actual sale price and the lowest price at which sellers will sell, and hence can be interpreted as buyer incomplete information. Thus, ||Mu~.sub.w~ is positive.

To obtain a form suitable for estimation which allows the identification of the one-sided error components |v.sub.i~ and |w.sub.i~, assume that |u.sub.i~ has a normal distribution with mean and variance (0, |Mathematical Expression Omitted~); that |v.sub.i~ is exponentially distributed with mean and variance (||Mu~.sub.v~, |Mathematical Expression Omitted~), and that |w.sub.i~ has an exponential distribution with mean and variance (||Mu~.sub.w~, |Mathematical Expression Omitted~).(11) In addition, for tractability, assume |u.sub.i~, |v.sub.i~ and |w.sub.i~ are independent.

Following Yoon and Polachek |55~, the marginal density of ||Epsilon~.sub.i~ is (suppressing the subscript i)

|Mathematical Expression Omitted~

where |Phi~ denotes the standard normal distribution function.

The parameters of the model (|Beta~, ||Sigma~.sub.u~, ||Mu~.sub.v~ and ||Mu~.sub.w~) can be estimated by maximizing the likelihood function:

|Mathematical Expression Omitted~

where the density g(|center dot~) is as given in (9).

The log-likelihood function is

|Mathematical Expression Omitted~

where

||Theta~.sub.u~ = 1/||Sigma~.sub.u~,

||Theta~.sub.v~ = ||Sigma~.sub.u~/||Mu~.sub.v~, and

||Theta~.sub.w~ = ||Sigma~.sub.u~/||Mu~.sub.w~.

The parameter ||Theta~.sub.u~ is the inverse of the dispersion (||Sigma~.sub.u~), i.e., the precision, of the two-sided error component u, and the parameters ||Theta~.sub.v~ and ||Theta~.sub.w~ measure the inverses of the relative magnitudes (with respect to ||Sigma~.sub.u~) of the mean negative-sided error component v and the mean positive-sided error component w.

V. Data

Sources

The data utilized for this study are from a national sample of physicians practicing in medical group practices.(12) The data were assembled by Mathematica Policy Research, Inc., under contract to the National Center for Health Services Research, Department of Health and Human Services, U.S. Government. The bulk of the data set is composed of surveys conducted by Mathematica from March to June of 1978, although some secondary data sources have been included. The final sample included 957 groups and 6353 physicians practicing in those groups. Five medical practice specialties were sampled: general practice, internal medicine, pediatrics, general surgery, and obstetrics/gynecology. Approximately 60 percent of all office-based physicians practice in these specialties.

This data set also includes data measuring characteristics of the area in which the group practiced as well as data on the hospital with which the group is affiliated. The area characteristics data were obtained from many sources, including the American Medical Association, and The County and City Data Book. For a full listing of all these data sources see Boldin, Carcagno, Held, Jamieson, and Woolridge |7~. The hospital data were obtained from the American Hospital Association Guide for 1978.

Variables

We estimated the three-error component model of the price function (equation (6)). Eight different prices were used as dependent variables: the usual fee for an office visit for all physicians, the usual fee for an office visit for general practitioners, the usual fee for an office visit for pediatricians, the usual fee for an office visit for general surgeons, the fee for a hospital follow-up visit for all physicians, the fee for a dilation and curettage (D&C) for obstetrician/gynecologists, the fee for a complete blood count, and the fee for a chest x-ray.(13)

The independent variables represent exogenous factors affecting the demand and supply of physician services: measures of insurance coverage, fee collection, market area characteristics, input prices, and physician characteristics.(14) There are three measures of insurance coverage: the proportion of the physician's patients who are covered by Medicare, the proportion covered by TABULAR DATA OMITTED Medicaid, and the proportion without insurance coverage. The proportion covered by Medicare is expected to have a positive impact on price, since Medicare beneficiaries are hypothesized to have less elastic demand and since Medicare was relatively generous in reimbursement to physicians in 1978. The proportion on Medicaid is expected to have a negative effect on price, since Medicaid has been penurious in its reimbursement to physicians. The proportion without insurance should also be negatively related to price, since those without insurance pay the full cost of services provided. Fee collection is measured by the proportion of bills collected by the physician's practice. Using this variable enables adjustment for the proportion of fees physicians receive, thus providing a more accurate measure of the average transaction price. It may also proxy for price discrimination, to the extent that it is practiced by forgiving fees.(15)

Market area characteristics are per capita income, percent of the area that is urban, percent of the population on AFDC, and median gross rent. The first three variables are demand shifters controlling for income distribution, population density, and poverty. Median gross rent is a proxy for the price of office space. The wage paid to registered nurses by the physician's practice is a measure of the input price of labor.

Physician characteristics are: the experience of the physician, whether they practice in a subspecialty, whether they are female, whether they graduated from a foreign medical school, and the physician's specialty. The physician's experience, whether they practice in a subspecialty, and whether they are a foreign medical graduate are characteristics which may be associated with quality or productivity differences. They may either determine quality, and thus be valued indirectly, or they may be valued directly by patients as in an hedonic model, e.g., Goldman and Grossman |20~. The physician's gender may also be directly valued by patients, or may be related to productivity. Dummies for the physician's specialty were included in regressions which pooled physicians across specialties. Table I lists the definitions of these variables and Table II provides descriptive statistics.

VI. Estimation Results

Table III reports the complete set of parameter estimates for office visit fees. Tables IV and V report the estimates of the incomplete information indices by the various services and strata. The estimation results are striking. They reflect the common wisdom concerning information asymmetries. For all the specifications, the patient incomplete information estimates are greater than the estimates of physician incomplete information.(16) For example, in Table III, the estimate of incomplete patient information is 4.08 and of incomplete physician information is -- 2.78.(17) Thus, on average, patients pay a premium of almost 30 percent over the average of $13.94 due to their incomplete information. This is approximately 50 percent more than the estimate of physician incomplete information.

The parameter estimates are consistent with prior expectations. We report the parameter estimates for office visit fees pooling all physicians, since they did not differ much for the various strata. Table III contains two columns: one for the maximum likelihood estimates, and one containing least squares estimates for purposes of comparison.(18) Since the estimates are very similar, we only discuss the maximum likelihood estimates.

Table II. Variable Means and Standard Errors

Variable                                Mean   Standard Error

Office Visit Fee: All Physicians       13.94       3.88
Office Visit Fee: G.P.'s               12.61       2.94
Office Visit Fee: Pediatricians        13.76       3.14
Office Visit Fee: General Surgeons     14.23       4.46
Hospital Follow-Up Visit Fee           16.25       5.20
D & C Fee                             191.45      71.02
Blood Count Fee                         9.10       4.23
Chest X-Ray Fee                        21.70       8.09
Collection Percent                     91.86       9.09
Percent Medicare                       21.96      14.82
Percent Medicaid                       10.46      10.86
Percent No Insurance                   10.39      10.84
Female Physician                        0.05       0.21
Foreign Medical Graduate                0.06       0.24
Experience                             20.29      10.85
Subspecialty                            0.28       0.45
Registered Nurse Wage                   4.87       0.99
Internist                               0.35       0.48
General Practitioner                    0.33       0.47
Pediatrician                            0.14       0.35
Obstetrician                            0.15       0.36
Per Capita Income                    4752.99    1162.91
Percent Urban                          72.88      28.19
Median Gross Rent                     104.77      22.77
Percent AFDC                            3.94       2.52
Percent of Practice from Referrals     20.12      26.52

The insurance variables all have the expected effects. The proportion of patients with Medicare coverage is positively related to price, and the proportion with Medicaid or no insurance have negative effects on price, although the proportion with no insurance is not statistically significant. These results may be due to less elastic demand by Medicare beneficiaries and to Medicaid's below market reimbursement, respectively. The percent of bills collected by the practice has a negative, but insignificant, effect on the price charged. Thus the negative sign implies that price increases as the collection percent falls, a way of compensating for lost revenue. This is also consistent with price discrimination by means of bill forgiveness.

The market area variables are mostly consistent with expectations. Per capita income and percent urban both have positive and significant coefficients. Higher median gross rent, which increases costs, has an estimated positive effect on price, as hypothesized but is not statistically TABULAR DATA OMITTED significant. Registered nurse wages have a positive effect on price. This is as expected, since an increase in input prices will increase the marginal cost of production. The percent of the population on AFDC has a positive and significant effect on price.

Table III. Estimates of the Three-Error Term Price Function (a,b)

Dependent Variable: Office Visit Fee: All Physicians
Independent Variables              MLE                        OLS

Constant                         12.18(***)                   10.48
                                 (2.15)                       (1.62)
Collection Percent               -3.58                        -4.41(***)
                                 (2.89)                       (1.35)
Percent Medicare                  0.03(***)                    0.0275(***)
                                 (0.0085)                     (0.008)
Percent Medicaid                 -0.061(***)                  -0.056(***)
                                 (0.012)                      (0.012)
Percent No Insurance             -0.0057                      -0.0017
                                 (0.012)                      (0.010)
Female Physician                  0.0311                       0.139
                                 (0.135)                      (0.61)
Foreign Medical Graduate         -0.055                       -0.306
                                 (0.197)                      (0.33)
Experience                       -0.071(**)                   -0.093(***)
                                 (0.03)                       (0.04)
Subspecialty                      0.7797(***)                  0.726(***)
                                 (0.203)                      (0.21)
Wage of a Registered Nurse        0.915(***)                   0.795(***)
                                 (0.171)                      (0.122)
Internist                         1.158(***)                   1.065(***)
                                 (0.357)                      (0.362)
General Practitioner             -1.785(***)                  -1.796(***)
                                 (0.273)                      (0.285)
Pediatrician                     -2.276(**)                   -2.58(*)
                                 (1.14)                       (1.52)
Obstetrician                      3.113(***)                   3.636(***)
                                 (0.585)                      (0.50)
Per Capita Income                 0.447 x |10.sup.-3~(***)     0.0003(***)
                                 (0.12 x |10.sup.-3~)         (0.00015)
Percent Urban                     0.015(**)                    0.0134(**)
                                 (0.0047)                     (0.005)
Median Gross Rent                 0.0016                       0.002
                                 (0.003)                      (0.004)
Percent AFDC                      0.325(***)                   0.414(***)
                                 (0.049)                      (0.04)
||Theta~.sub.u~                   0.574(***)
                                 (0.067)
||Theta~.sub.v~                 1.255(***)
                                 (0.29)
||Theta~.sub.w~                   0.853(***)
                                 (0.166)
||Sigma~.sub.u~                   1.74
||Mu~.sub.v~                      1.39
||Mu~.sub.w~                      2.04
Number of Observations                                         1,046

a. Standard errors reported in parentheses below the parameter estimates.
b. *: significant at the 10% confidence level, **: significant at the 5%
confidence level, ***: significant at the 1% confidence level.

We hypothesized that physician experience would have a positive effect on price through patients valuing experience or through greater efficiency in producing quality. However, we find that experience of the physician has a negative and significant impact on price. This is not consistent with the hypothesis that experience represents quality, however it is consistent with a hypothesis that experience is associated with productivity. Physicians who practice in a subspecialty charge higher fees, consistent with the story that this represents quality. Neither the physician's gender (whether they are female) nor whether they are a foreign medical graduate has a significant effect on price. Last, the physician specialty dummies are consistent with observed pricing patterns: G.P.'s and pediatricians charge less than general surgeons, and internists and obstetricians charge more.

Table IV contains the maximum likelihood estimates for each of the three components of the error term for the seven other services: office visits by G.P.'s, office visits by pediatricians, office visits by general surgeons, hospital follow-up visits, D&C's by obstetricians, blood counts, and chest x-rays. Patients are found to have less information than doctors. The measure of incomplete patient information ranges from 10.424 for D&C's to 3.918 for pediatric office visits, and the difference between patient and physician incomplete information varies from 8.168 for D&C's to 2.48 for G.P. office visits. In addition, the measures of incomplete patient information and the differences between patient and physician incomplete information are greater for those services which are purchased less frequently, associated with greater severity of illness, or are more heavily insured.(19) The largest measures of incomplete patient information, and the largest differences between the two information measures, are for D&C's and hospital follow-up visits. These are both services which are purchased infrequently, are associated with greater severity of illness, and are heavily insured. The smallest estimates of incomplete patient information and the smallest differences between patient and physician information are for office visits to G.P.'s and pediatricians. These services are used relatively more frequently, are associated with lesser severity of illness, and are lightly insured. Intermediate values of these measures are found for blood counts and chest x-rays, services which are heavily insured, but are not purchased frequently and are not associated with severe illness. These findings, combined with the findings from Table III, provide strong empirical support for the notion that information, and especially asymmetry in information, plays an important role in price determination in this market.

To link this with previous work, we attempted to replicate the regression of consumer incomplete information on price and frequency of purchase contained in Pratt, Wise, and Zeckhauser |44~. They regress their measure of incomplete information for a product, log standard deviation of price, on mean price and the frequency with which goods are purchased. They find that incomplete information is positively related to mean price and negatively related to frequency of purchase.(20) We regressed our measure of consumer incomplete information (2||Mu~.sub.w~) on mean price and dummy variables for very frequently purchased goods and moderately frequently purchased goods. The observations were the seven types of physician services in our data. Office visits to G.P.'s and pediatricians were classified as very frequently purchased, blood counts, chest x-rays and office visits to surgeons were classified as moderately frequently purchased, and hospital follow-up visits and D&Cs were classified as infrequently purchased. The estimated regression is

ln(2||Mu~.sub.w~) = 1.93 + 0.078 lnP - 0.68|D.sub.1~ - 0.42|D.sub.2~

(0.16) (0.037) (0.087) (0.081) Adjusted |R.sup.2~ = 0.96 |F.sub.3,3~ = 48.72,

where |D.sub.1~ and |D.sub.2~ are dummies for very and moderately frequently purchased services, respectively (the omitted category is infrequently purchased services), and P is mean price (standard errors are in parentheses beneath the coefficients).(21)

Incomplete information rises with price, but by a small amount. Doubling the mean price leads to a 5.6 percent increase in consumer incomplete information (|2.sup.0.078~ |is approximately equal to~ 1.056). This is consistent with Pratt, Wise, and Zeckhauser's proposition that, holding frequency of purchase constant, the expected gains to search increase with price, so that in equilibrium incomplete information rises much less rapidly than price. Incomplete information is smaller for moderately frequently purchased goods than for infrequently purchased goods and is smallest for very frequently purchased goods. These results are consistent with theory and with the empirical work of Pratt, Wise and Zeckhauser and others, but they should be regarded as suggestive due to the extremely small number of observations.(22)

We also hypothesized that the proportion of a physician's practice drawn from referrals and per capita income in the market area are related to the cost of search. Referrals substitute for patient search. Since patients are searching less, they will pay a higher price. Physicians also likely know more about referred patients. Consequently, we expect the estimate of incomplete patient information to be higher and the estimate of incomplete physician information to be lower for physicians in the subsample with high referral practices. This is precisely the case. Examining TABULAR DATA OMITTED the estimates in Table V, the measure of incomplete patient information is 4.722 when the percent of practice from referrals is 0-3 percent, and 5.71 when referrals make up 20 percent or more of the physician's practice. Incomplete physician information is estimated to be 3.53 for the low referral subsample, and 2.338 for the high referral physicians. Even more telling is the difference between incomplete patient and physician information estimates for the two subsamples. The gap equals 1.192 for the low referral subsample and is equal to 3.372, or almost triple that, for the high referral subsample.

Per capita income is positively related to the opportunity cost of time, therefore search should be more costly (on average) in high than in low per capita income areas. Thus, the estimate of patient incomplete information should be higher in areas where the per capita income is higher. This is indeed the case. Again, using the estimates for office visit price for all physicians, we see that the estimate of patient incomplete information is 5.678 where per capita income is $5,567 or greater, and 4.062 where per capita income is $4,455 or lower.

Both of these results are consistent with hypotheses about how these variables affect search costs. They provide further support for interpreting the one-sided residual estimates components as measures of incomplete information. Last, the overwhelming number of estimates following the same pattern of asymmetry between patient and physician and the consistency between the estimates for homogeneous products such as blood count and chest x-ray and all the other estimates provide evidence in favor of interpreting these estimates as measures of information.

VII. Summary and Conclusions

In this paper a new method for measuring information is proposed which extends the Stiglerian measure of price dispersion. This method controls for the effects of supply and demand on price and uses a new econometric method to separate price dispersion into measures of incomplete buyer information, incomplete seller information, and random noise.

The method is applied to data from the market for physician services, a market which is commonly thought of as being characterized by incomplete information about price. The empirical results accord with intuition: the measure of incomplete patient information exceeds the measure of incomplete physician information by approximately a factor of 1.5. Incomplete patient information is also measured as being relatively larger than incomplete physician information for smaller ticket items, less frequently purchased items, more heavily insured items, and treatments associated with severe illness, as has been hypothesized. Last, measured incomplete information is higher when proxies for search costs are higher. These results are striking and consistent across strata. They suggest that this method has promise in obtaining estimates of money metrics for price information.

1. Single tiered frontier models have been used by Hofler and Polachek |23; 24~, Hofler, Schlottman, and Herzog |25~, Warren |53~ and Hofler and Murphy |22~ to estimate measures of worker (seller) information in the labor market. Our approach is generalized to a two-tiered frontier model, and allows estimation of measures of both seller and buyer information in a market.

2. The services are: pooled office visits, G.P. office visits, pediatrician office visits, surgeon office visits, hospital follow-up visits, blood counts, chest x-rays, and D&C's. Note that some of these are arguably homogeneous, e.g., blood count and chest x-ray.

3. The Supreme Court viewed this as of enough importance to uphold an FTC ruling forcing the AMA to drop member restrictions on advertising |16~.

4. Newhouse and Sloan |37~, Hsiao |26~, Marquis |32~, Marquis et al. |33~, and White-Means |54~ have documented the dispersion in physician fees. Beazoglou and Heffley |5~ have done the same for dentists.

5. E.g., studies relating advertising to eyeglass and retail drug prices such as Benham |6~, Cady |10~, Feldman and Begun |17~, Kwoka |28~ and Gaynor and Mullahy |19~ and a study by Pauly and Satterthwaite |40~, relating proxies for search costs to physician fees.

6. Some uninformed buyers may serendipitously stumble on the lowest priced seller and will thus not incur any cost due to their incomplete information, and similarly for sellers.

7. Again, we emphasize that this incomplete information is optimal from the individual's viewpoint, given the expected costs and benefits of search.

8. Telser |50~ explicitly models search on the part of sellers. Rothschild |45~, Axell |4~, Butters |9~, and Burdett and Judd |8~ assume that sellers know their demand functions in only a probabilistic sense.

9. This incomplete information is optimal, as it is for buyers, given the expected benefits and costs of search.

10. As the patient's agent, it is possible that the referring physician may have searched optimally for the patient. This is an unlikely scenario, given the well known difficulties with physician agency, as pointed out in Pauly |39~.

11. These specific distributional assumptions are not required, but |v.sub.i~ and |w.sub.i~ must be distributed differently from |u.sub.i~ over their ranges in order to achieve identification. Experimentation with the more general gamma distribution as in Greene |21~ or Stevenson |46~ is possible. However, Aigner, Amemiya and Poirier |1~ and Aigner, Lovell, and Schmidt |2~ provide some evidence indicating that frontier estimators are robust to distributional assumptions. Our estimator exploits information about the shape of the distribution of prices as well as the location and spread of the distribution. This is somewhat related to the empirical work by Pratt, Wise, and Zeckhauser |44~ and Lach and Tsiddon |29~, which they report estimates of shape parameters for price distributions.

12. Since not all physicians practice in groups, the data are not necessarily representative of the physician services market, consequently the empirical results in this paper should be regarded as illustrative.

13. All money variables are deflated by the consumer price index (four person family, intermediate budget) for the county in which the physician is located.

14. We considered employing a number of other variables which could be hypothesized to determine price: patient waiting time, length of a follow-up office visit, physicians per capita, number of physicians in the group, and physician compensation method, but specification tests rejected the exogeneity of these variables.

15. For an interesting discussion of this point, see Lachs, Sindelar, and Horwitz |30~.

16. As a preliminary test we examined the distribution of the OLS residuals (|Epsilon~) to the price function. Since |Epsilon~ is the sum of a normally distributed random variable and two exponentially distributed random variables, it is not normally distributed. A Kolmogorov-Smirnov test and a Jarque-Bera |27~ test on the OLS residuals both rejected normality at the 1% confidence level.

17. We estimate ||Mu~.sub.w~ and ||Mu~.sub.v~. Since buyer incomplete information is W = 2w and seller incomplete information is -V = 2v, the measures of incomplete information are equal to twice the parameter estimates.

18. Since theory provides no guide to the functional form of the price equation, we tested the validity of the two most common functional forms: linear and logarithmic, using the |P.sub.E~ test of MacKinnon, White, and Davidson |31~. The test failed to reject linearity, while the logarithmic form was rejected at the 1% level, therefore we employed the linear functional form.

19. We compare the absolute levels of the incomplete information measures since the cost of incomplete information is foregone expenditure on all other goods.

20. Carlson and Pescatrice |12~ also find that empirical price dispersion is positively related to a good's expense and negatively related to frequency of purchase. As Stigler |48~ argues, since the marginal benefits of search rise faster for more expensive goods, in equilibrium (with marginal benefits equated across all goods) buyers will be less informed about more expensive goods, although the amount of incomplete information will increase at a decreasing rate as price rises.

21. Correcting for heteroskedasticity produces very small changes in the standard errors. None of these changes affect the significant tests.

22. It is not possible to duplicate this regression for physician in incomplete information, since physicians do not sell all these goods, whereas patients are (potential) buyers of all the goods.

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