Batter up! moral hazard and the effects of the designated hitter rule on hit batsmen.

Goff, Brian L. ; Shughart, William F., II ; Tollison, Robert D. 等

I. INTRODUCTION

Despite its relatively recent appearance in the literature, the theory of moral hazard has proved to be rich in empirical implications. It has been applied fruitfully in a wide variety of settings, including medical insurance, property and casualty insurance, labor contracts, savings and loan failures, and credit rationing.(1) Perhaps the most frequently cited application of this idea is Sam Peltzman's [1975] study of mandatory seat belt laws. He reported evidence suggesting that drivers respond rationally to such laws, which reduce the risk of injury or death in automobile accidents, by driving more recklessly. Hence, while mandatory seat belt laws have resulted in fewer automobile fatalities, they have increased the number of non-fatal accidents and accidents involving damage to property and pedestrians.

In this paper we argue that the introduction of the designated hitter (DH) rule in the American League but not the National League of Major League Baseball in 1973 created the potential for a classic moral hazard problem. Because they are not required to appear at the plate, American League pitchers can throw at opposing hitters with greater impunity (i.e., at a lower cost) than National League pitchers, who must take their turn at bat. The economic theory of moral hazard clearly predicts that more batters will be hit in the American League under such circumstances, an implication which is supported by the empirical evidence reported herein. After controlling for other relevant factors, we find that American League batters have been hit by pitches at rates 10% to 15% higher than their National League counterparts in the typical post-DH rule season.

II. THE SIMPLE ECONOMICS OF HITTING BATTERS

We model pitchers as the ultimate decision makers with regard to behavior that leads to hit batters. In making this assumption, we ignore the principal-agent relation between pitchers and managers as well as any externalities arising among players that may influence the number of batters hit by pitches. Managers sometimes instruct pitchers to throw at a batter, presumably to cool off opponent's bats or to retaliate for opposing pitchers throwing at his team. On occasion, a pitcher may view his own personal interests., in terms of runs and income, as diverging from the manager's interests, so that he would not carry out such instructions to the letter.(2) In any case, a batter may be hit when the pitcher makes a deliberate attempt to do so or when the pitcher has not expended much effort to avoid hitting him. While the distinction between the two causes of hit batsmen may be important in its own right, we treat them as being observationally equivalent. Both amount to the pitcher altering the probability of hitting a batter by changing the intended location and speed of his pitches.

In our elementary approach, a pitcher's decision to raise or lower this probability hinges on a marginal decision-making calculus of weighing the expected benefits against the expected costs of hitting a batter. These benefits and costs would ultimately affect the pitcher's expected utility by influencing his lifetime income and consumption possibilities, his utility from the indulgence of personal preferences, or both.

Increasing the probability of hitting a batter can potentially benefit the pitcher in several ways. Hit batsmen may become more reluctant now and in future games to lean over the plate, thereby reducing their effectiveness against pitches low and outside; they may become more susceptible to being fooled by curve balls that give the illusion of being inside; or they may "dig in" less and thereby reduce their power. All of these effects lower the probability of runs being scored against the pitcher in present and future appearances. Additionally, the pitcher may alter the probability of hitting batters in order to retaliate against similar tactics being used by opposing pitchers. This may be a means of neutralizing an opponent's strategy of throwing at batters, and, hence, help increase run production by the pitcher's own team. Aside from these benefits related to run scoring, pitchers may gain satisfaction from hitting a batter (or at least making him dive out of the way) as one way of securing revenge for earlier home runs or other hitting success by that batter or his team.(3)

On the other hand, pitching in ways that raise the probability of hitting batters entails higher expected costs. Hitting a batter automatically places him on base and may advance runners already on base. Opposing pitchers may retaliate by throwing at the pitcher's teammates. Finally, and what is more important for our analysis, the pitcher may suffer a loss in utility, including the possibility of physical harm due to the retaliation by opposing pitchers against him personally.

Pitchers strive to equate these benefits and costs at the margin, which vary from game to game and which depend on closeness of the score, the number of runners on base, the number of remaining innings, the importance of the game's outcome, and so on. The fact that such a rational choice calculus is integral to pitchers' decisions to alter the probability of hitting batters would seem obvious from even casual observation. For instance, batters are hardly ever deliberately hit in very close games, where the marginal costs of another base runner may be very high. Similarly, batters are rarely hit by pitches when the bases are loaded and the expected cost of hitting a batter is the certainty of giving up another run. On the other hand, batters are hit most frequently when the score differential is large and the marginal costs, especially to the trailing team's pitcher, are consequently low.

The effect of adopting the DH rule within this framework is straightforward. Introducing the DH rule creates a classic moral hazard problem: The pitcher no longer bears the full costs of his own actions. Because the DH rule lowers the marginal cost of throwing at the opposing team's batters, the expected number of batters hit by pitches increases relative to the situation in which the pitcher must take his turn at bat. If the marginal benefits and marginal costs of hitting batters were similar across baseball's two major leagues beforehand, we would expect the American League's adoption of the DH rule in 1973 to cause an increase in the number of hit batters there relative to the National League where pitchers are still required to appear at the plate.

III. STATISTICAL ANALYSIS OF THE DH RULE

We next investigate whether the adoption of the DH rule in 1973 actually led to an increase in the number of batters hit in the American League relative to the National League. The data on hit batsmen, HB, are the number of American (A) and National (N) League batters hit by pitches each year from 1901 through 1990. The two time series - normalized by at-bats - are plotted in Figure 1.

Basic Model

To test whether the observed divergence in the two hit batsmen series remains significant while controlling for other relevant factors, we specify the following regression model:

[Mathematical Expression Omitted],

where Z represents a vector of explanatory variables that determine the position and shape of the marginal benefit and cost curves in each league in the absence of the DH rule. The variable DH is a dummy equal to zero before 1973 and equal to one from 1973 through 1990.

One advantage of using inter-league differences in hit batters as the unit of observation is that it allows us to hold constant factors that help explain variations over time in this variable that are common to both leagues. As discussed in section II above, game-specific and batter-specific situations are important determinants of the marginal benefits and costs of throwing at batters. But once the observations have been aggregated across teams and over games within seasons, pitchers in both leagues in all likelihood will on average have faced similar situations in terms of closeness of score, early versus late innings, home runs allowed, and so on. Whatever other differences may exist are assumed to be random and captured by the error term, [Epsilon].

TABLE I

OLS Estimates of Hit Batter Differences Between American and
National Leagues

Variable 1901-1990 1920-1990 1947-1990

Intercept 11.54 9.31 14.13
 (1.34) (0.91) (1.48)

DH 49.74 47.52 43.37
 (2.28)(**) (2.15)(**) (2.17)(**)

At-Bats .006 .006 .007
 (3.55)(***) (3.89)(***) (4.06)(***)

[Rho] .52 .54 .25
 (5.51)(***) (5.16)(***) (1.65)(*)

[R.sup.2] .68 .74 .74

D-W 2.07 2.05 2.02

Note: t-statistics are in parentheses; [Rho] is the estimated
first-order autocorrelation coefficient and D-W is the Durbin-Watson
d-statistic. The dependent variable is American minus National
League hit batters per season. DH equals zero until 1972 and one
from 1973 onward. At-Bats is the difference in total at-bats
(American minus National). Asterisks indicate significance at the 1%
(***), 5% (**), and 10% (*) levels.

A key remaining difference between leagues concerns the total number of batters faced by pitchers in a given season. Total at-bats differ over time because of differences in the number of clubs in each league. For our basic model, we therefore include the inter-league difference in total at bats, [Mathematical Expression Omitted], as the sole explanatory variable.

Table I presents the results of estimating our regression equation by OLS for three time intervals: 1901-1990, 1920 - 1990, and 19471990. We disaggregated the data into these three subperiods in order to take account of changes over time in official baseball rules, especially those occurring during Major League Baseball's early, formative years. These time intervals were chosen on the basis of the following considerations: the American League began play in 1901, the use of "spit-ball" pitches was outlawed in 1920, and the game entered a new era following World War II when many important changes were introduced, including racial integration. Each of the equations is corrected for first-order serial correlation and the correction factor is reported.(4) Inclusive of the autocorrelation correction, the equations explain between 57% and 73% of the variation in inter-league differences in hit batsmen, with the explanatory power increasing with the use of more recent data.

In each time frame, the estimated coefficient on the designated hitter variable is positive and statistically significant at the 5% level. Depending on the sample, the DH coefficients indicate that after the rule's introduction in 1973, between 44 and 50 more American League batters were hit by pitches in a typical season after controlling for differences in at-bats between the leagues. Given that both leagues were averaging between 300 and 400 hit batters per year in the late 1960s and early 1970s, these figures translate into a 10% to 15% increase in hit batters in the American League relative to the National League.

TABLE II

OLS Estimates of Hit Batter Differences Between American and
National Leagues, 1921-1989

Variable

Intercept 13.43 7.62
 (1.11) (.77)

DH 60.76 62.89
 (2.47)(**) (2.93)(***)

At-Bats .003 .003
 (1.05) (1.66)(*)

Slugging Average .16
 (.33)

Home Runs .006
 (.10)

Bases on Balls -.01
 (1.19)

Strikeouts .005
 (.51)

Saves .33 .275
 (1.95)(*) (1.91)(*)

Std. Dev. of Winning Percentage -.004
 (.98)

Attendance 5.3E-6 4.7E-6
 (1.92)(*) (2.01)(**)

[Rho] .53 .51
 (4.53)(***) (4.63)(***)

[R.sup.2] .77 .77

D-W 2.09 2.08

Note: t-statistics are in parentheses; [Rho] is the estimated
first-order autocorrelation coefficient and D-W is the Durbin-Watson
d-statistic. See Table I note. Except for DH, all explanatory
variables are computed as American League minus National League.

To explore the validity of our assumption that the other elements of a vector of explanatory variables (Z) in each league tend to cancel out at the level of data aggregation employed, we collected observations on the differences between American League and National League season totals for several additional explanatory variables. These additional variables include measures of pitcher control/ability, hitter ability, degree of competitiveness of games, the amount of reliance on relief pitching, and the financial rewards of winning. In particular we include inter-league differences in bases on balls and strikeouts for pitcher control/ability; home runs and slugging averages for hitter ability; yearly standard deviation of league winning percentages for competitiveness of games; number of saves for amount of relief pitching; and game attendance as a proxy for the pecuniary returns to winning.(5) The results of estimating this regression specification by OLS for 1921-1989 are reported in Table II.(6)

TABLE III

Cointegration Tests

Sample Period Johansen Likelihood Ratio 5% Critical Value

1901-1972 18.0 15.4
1901-1990 10.9 15.4

Note: The Johansen [ 1991] likelihood ratio, statistic tests the
null hypothesis that no cointegrating vectors exist. The version
employed here includes a constant and no time trend.
TABLE IV

Dickey-Fuller Tests for Differences in Hit Batters

Sample Period Dickey-Fuller t-statistic

1901-1972 -3.73(***)
1901-1990 -2.04

Note: For each sample period, the Dickey-Fuller t-statistic is from
the augmented Dickey-Fuller equation, [Delta][DHB.sub.t] = [c.sub.0]
+ [c.sub.1][DHB.sub.t-1] + [c.sub.2][([Delta]DHB).sub.t-1], where
DHB is the difference in hit batters (American League minus National
League). The null hypothesis of non-stationary residuals is rejected
at the 1% level for the 1901-1972 period.

In this revised version of our regression equation, the estimated coefficient on the DH rule is considerably larger than before. Of the additional explanatory variables included, only saves and attendance are significantly different from zero (at the 10% level). The second column of Table II shows the results of estimating the same equation with only those two additional explanatory variables included. More relief pitching appears to increase the number of hit batsmen in the American relative to the National League - possibly a moral hazard problem one step removed. Greater fan interest, which increases the pitcher's expected returns to winning, also increases the relative number of hit batters.

Cointegration Tests

In addition to these OLS estimates, we searched for additional evidence of the DH rule's impact by investigating the stationarity properties of the time series. Since both [HB.sup.N] and [HB.sup.A] are non-stationary according to Dickey-Fuller methods, we first posed the question, are the series cointegrated?(7) That is, do the two series share a common underlying trend? For the period prior to the DH rule's introduction, we would expect this to be the case based on our earlier assumption that the other factors driving both series are nearly identical once the data are aggregated over games and teams. On the other hand, if the DH rule has had an important impact, the post-DH American League hit batsmen series should diverge both from its earlier time path and, ceteris paribus, from the National League series.

Table III presents the results of Johansen [1991] cointegration tests for the time intervals 1901-1972 and 1901-1990. The null hypothesis of no cointegration can be rejected at the 5% level for the pre- 1973 sample, but cannot be rejected for 1901-1990.

Next, and in a closely related test, we considered whether the null hypothesis of non-stationarity for the differenced series of hit batters, DHB, could be rejected exclusive and inclusive of the rule change. This is really just an alternative cointegration test where we look at a particular linear combination of H[B.sup.A] and H[B.sup.N]. Specifying the test in this way implicitly restricts the cointegrating vector to be equal to one. Table IV presents these results, and they are consistent with those reported in Table III. The null hypothesis of non-stationarity for the difference between H[B.sup.A] and H[B.sup.N], the two leagues' hit batters, can be rejected at the 1% level for the time frame prior to the DH rule (1901-1972). The calculated Dickey-Fuller test statistic for the time frame inclusive of the DH rule (1901-1990) cannot reject the null hypothesis at standard levels of statistical significance.(8)

IV. CONCLUDING REMARKS

In recent years, growing media attention has been brought to bear on bench-clearing brawls prompted by hit batters and "knock downs" of batters. Some observers of the game have speculated that this violence may be in part due to the fact that American League pitchers do not have to bear the full costs of their pitching tactics. Whether this has caused more brawls or whether brawls have become more prevalent in the American League than in the National League remains an open question. Our results lend support to the idea that American League pitchers became much more willing to throw at batters after the DH rule went into effect.

We thank Mike Belongia of the University of Mississippi and Keith Carlson of the St. Louis Fed for help in gaining access to the hit batsmen data that are the paper's centerpiece and Pete Palmer, co-editor of Total Baseball, for making it available to us. Valuable comments were provided by Robert McCormick, Keith Womer, Luke Froeb, Fred McChesney, and seminar participants at the University of Kansas and the University of Mississippi. The suggestions of Mark Zupan and two anonymous referees were especially helpful in improving the paper. Tim Greer and Robert Trimm supplied able research assistance. We are also grateful to James M. Buchanan for earlier discussions which led to the development of this paper. As is customary, however, we accept full responsibility for any remaining errors.

1. Some of the well-known works in these areas in-elude Pauly [1968] and Arrow [1971] on medical markets; Holmstrom [1979] on general principal-agent problems; Cheung [1969], Stiglitz [1974], and Newberry [1977] on labor contracts; and Stiglitz and Weiss [1981] on credit rationing.

2. We also ignore the possibility that even with the DH rule in effect, the pitcher's teammates may continue to bear the retaliation costs of throwing at opposing batters. By bringing their own influence to bear on the pitcher (in the form of locker room threats or deliberate fielding errors) they might be able to make the pitcher behave as if the costs of hitting batters has not changed.

3. This source of additional utility may not be completely separable from the run-reducing effect of raising the probability of hiring batters. In many public interviews, pitchers who gloat about their control of the inside part of home plate usually explain that this is necessary for success.

4. The results are not sensitive to omitting this correction for first-order autocorrelation. Observations on these variables were collected from the Baseball Encyclopedia [1990] and Thorn and Palmer with Gershman [1995].

5. Attendance is included to test the hypothesis suggested by one of the referees that National League pitchers have grown more fearful of being retaliated against because their salaries have increased over time relative to their American League counterparts. A consistent pitcher salary time series, which takes bonuses, incentive pay, and deferred compensation into account, is unfortunately not available. As a proxy, we assume that the returns to winning are a monotonic transformation of game attendance. We estimated two other specifications of our basic model to respond to issues raised by the journal's referees and other commenters. First, we interacted DH and at-bats to allow the difference in hit batsmen to be proportional to the difference in at-bats across leagues. It is not. Second, we added an additional dummy variable, equal to one for the years 1973-1976 and zero otherwise, to distinguish an "expansion effect" (the American League added two weak teams in 1977) from the DH effect. The estimated coefficient on this dummy variable was not significantly different from zero while the coefficient on the DH variable was unaffected.

6. Our original sample is restricted to the 1921-1989 subperiod because observations on league attendance are not available prior to 1921, and we lose the last observation in calculating the standard deviation of league winning percentages.

7. We do not report these Dickey-Fuller tests in tabular form but would be happy to supply them on request.

8. See McCallum [1993] for a good discussion of the limitations of cointegration tests, especially where such tests do not support a finding of cointegration.

REFERENCES

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Baseball Encyclopedia, 8th ed. New York: Macmillan, 1990.

Cheung, Steven N. S. "Transactions Costs, Risk Aversion, and the Choice of Contractual Arrangements." Journal of Law and Economics, April 1969, 23-42.

Holmstrom, Bengt. "Moral Hazard and Observability." Bell Journal of Economics and Management Science, Spring 1979, 74-91.

Johansen, Soreu. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models." Econometrica, November 1991, 1551-80.

McCallum, Bennett T. "Unit Roots in Macroeconomic Time Series: Some Critical Issues." Federal Reserve Bank of Richmond Quarterly Review, Spring 1993, 13-44.

Newberry, David. "Risk Sharing, Sharecropping and Uncertain Labour Markets." Review of Economic Studies, October 1977, 585-94.

Pauly, Mark V. "The Economics of Moral Hazard." American Economic Review, June 1968, 531-37.

Peltzman, Sam. "The Effects of Automobile Safety Regulation." Journal of Political Economy, August 1975, 677-725.

Stiglitz, Joseph. "Incentives and Risk Sharing in Share-cropping." Review of Economic Studies, April 1974, 219-55.

Stiglitz, Joseph, and Andrew Weiss. "Credit Rationing in Markets with Imperfect Information."American Economic Review, June 1981, 393-410.

Thorn, John, and Pete Palmer with Michael Gershman, eds. Total Baseball: The Official Encyclopedia of Major League Baseball, 4th ed. New York: Viking, 1995.

Brian L. Goff: Professor of Economics, Department of Economics Western Kentucky University, Bowling Green Phone 1-502-745-3855, Fax 1-502-745-3893 E-mail brian.goff@wku.edu

William F. Shughart: Professor of Economics and Self Free Enterprise Chairholder, Department of Economics & Finance, University of Mississippi, Oxford Phone 1-601-232-7579, Fax 1-601-232-5238 E-mail shughart@bus.olemiss.edu

Robert D. Tollison: Duncan Black Professor of Economics, Center for Study of Public Choice, George Mason University Fairfax, Va., Phone 1-703-993-2315 Fax 1-703-993-2323