The theory of incentives is concerned with the problem that a planner (alternatively called a designer, principal, or government, depending on context) faces when his own objectives do not coincide with those of the members of society (whom we shall call agents). This lack of coincidence of goals distinguishes incentives theory from the theory of teams (Marschak and Radner, 1972), which postulates identical objectives, but which otherwise shares many features with our subject. In turn, the assumption that the planner, often the surrogate for society itself, has well-defined objectives separates incentives theory from most of social choice theory, which, since Arrow (1951), examines the possibility of deriving social objectives from those of individual preferences.

For an incentive problem to arise, noncoincidence of goals is not enough; the planner must care about either what agents know or what they do. That is, his objective function must depend either on agents' information or on their behavior.

An example of pure informational dependence is provided by the literature on resource allocation mechanisms. There, the planner's objective - social welfare - is a function of consumers' (agents') preferences and endowments. The incentive problem is, typically, that of eliciting this information.

Pure behavioral dependence is exhibited by an employee-employer relationship in which the employer is interested only in the employee's output. In this case, incentives pertain not to revealing what the employee knows but to inducing him to work hard. Of course, incentive problems typically involve both kinds of dependence.

Introduction

Three general principles employed for hypothesis testing in econometrics are the likelihood-ration (LR), Wald (W), and Lagrange-multiplier (LM) criteria. The W test was introduced by Wald (1943) and the LM test by Aitchison and Silvey (1958) and Silvey (1959). The LM test, which is also the same as the score test of Rao (1947), has been the subject of recent reports by Breusch (1979), Godfrey (1978a, 1978b, 1978c), and Breusch and Pagan (1979, 1980).

Savin (1976) and Berndt and Savin (1977) showed that a systematic numerical inequality exists for the test statistics for testing linear restrictions on the coefficients of certain linear models. The inequality relationship for the values of the test statistics is W≥LR≥LM. Breusch (1979) established that the inequality relationship holds for a general linear model with normal disturbances, provided that the unknown elements of the covariance matrix can be estimated by maximum likelihood (ML) and that the ML estimates of the coefficient parameters are asymptotically uncorrelated with those of the covariance matrix parameters.

The exact sampling distributions of the three test statistics can be complicated, so that in practice the critical regions of the tests commonly are based on asymptotic approximations. In regular problems the test statistics have the same asymptotic chi-square distribution under the null hypothesis. When the tests use the asymptotic chi-square value, they have the same critical region, and they are asymptotically equivalent. The inequality relationship among the test statistics implies that there are samples for which the large-sample tests will produce conflicting inferences.

In this chapter, a new method of approximating the probability density functions (pdf's) of econometric estimators and test statistics is developed. It is shown that best uniform approximants to a general class of pdf's exist in the form of rational functions. A procedure for extracting the approximants is devised, based on modifying multiple-point Padé approximants to the distribution. The new approximation technique is very general and should be widely applicable in mathematical statistics and econometrics. It has the advantage, unlike the Edgeworth and saddlepoint approximations, of readily incorporating extraneous information on the distribution, even qualitative information. The new procedure is applied to a simple simultaneous-equations estimator, and it gives exceptionally accurate results even for tiny values of the concentration parameter.

Introduction

The idea of approximating small sample distributions, rather than extracting their exact mathematical forms, has a long history in statistics, and a number of different techniques have been explored. Kendall and Stuart (1969) gave an introductory survey of some of these techniques in their Chapters 6, 12, and 13. Approximations are clearly of importance in those cases in which mathematical difficulties have prevented the development of an exact theory. An example is provided by regression models with lagged endogenous variables as regressors, models that are of particular relevance in econometrics. Approximations to distributions are also useful in those cases in which the exact mathematical expressions are too complicated for numerical computations. Some examples of the latter have been discussed previously (Phillips, 1980a, 1980b).

Introduction

The issue of incentives is one of the grand themes of economics; it is a main thread on which the rich tapestry of economic theory is woven. Notions of self-interested behavior are at the foundation of all microeconomics or theories of individual agents' behavior. Until quite recently, however, most investigations have been concerned with analyzing self-interested behavior under market institutions, as for example, in consumer theory or the theory of the firm.

An early exception to the focus on incentives in a market context was the famous debate in the early years of this century over the feasibility of market socialism - known as the socialist controversy. Here, the issue of incentives arose in a new form - namely, given a specified procedure or rules for allocating resources and making economic decisions in a socialist state, will the agents, in fact, have any incentive to behave as specified by the procedure or in accordance with the rules?

More recently, this question has been posed and analyzed in a variety of models arising from different literature - some models have been very general while others have been very specific. Two main branches in this new, growing field are the theory of economic mechanisms - an outgrowth and development of the theory of socialist and other planning - and social choice theory. However, in both these branches, interestingly enough, the problem of incentives was not one of the first issues explored - although it was raised early by some. In the social choice literature, although the question of honest voting was raised by Vickrey (1960), nearly twenty years passed after Arrow's original work, c.f. Arrow (1963, 2nd ed.), before the contributions of Farquharson (1969), Gibbard (1973), and Satterthwaite (1975) on strategic voting and straightforward or nonmanipulability of social choice rules.

The problem

Econometrics is concerned with understanding and predicting the behaviors of economic agents. Some behavioral responses are structurally or observationally qualitative (categorical, discrete), rather than continuous. This is a survey of models and methods that have been developed for the analysis of qualitative responses.

Table 1.1 gives examples of economic qualitative responses. Empirical studies have concentrated on a constellation of problems in labor supply, such as occupational choice and employment status, and on travel behavior. The models that have been developed are strongly influenced by the specific features of these applications.

The examples in Table 1.1 are classified by economic agent: household, firm, or an interaction of agents. There are several other aspects of qualitative responses that may affect the choice of an appropriate method for analysis. First, the categorical response may be binomial (yes/no) or multinomial, and multinomial responses may be either naturally ordered or unordered. For example, the number of children is naturally ordered, whereas the brand of automobile purchased is not.

Second, the primary purpose of examining qualitative responses may be an intrinsic interest in explaining and forecasting the observed categorical behavior, or it may be the correction of biases induced by self-selection into a target population. For example, occupational choice is of direct interest because of its impact on labor supply, but it is also a potential source of bias in a study of labor hours supplied by a self-selected population of independent professional workers.

Econometric models for the analysis of duration data have recently come into widespread use in economics. Recent studies of employment and nonemployment (Flinn & Heckman, 1982a; Heckman & Borjas, 1980), unemployment (Flinn & Heckman, 1982b; Kiefer & Neumann, 1981; Lancaster & Nickell, 1980; Toikka, 1976), fertility (Gomez, 1980), strike durations (Kennan, 1980; Lancaster, 1972), and infant mortality (Harris, 1980) have estimated econometric models for durations of events. All of these models have two features in common: they are nonlinear in an essential way, and the methods used to secure estimates of structural models require strong a priori assumptions about the functional forms of estimating equations. Most of these studies have also assumed that the distributions of the unobserved variables in these models are of a simple parametric form. The choice of these distributions usually is justified on the basis of familiarity, ease of manipulation, and considerations of computational cost.

Because of the novelty of the new methods, it is not yet widely appreciated that empirical estimates obtained from these models are extremely sensitive to the choice of a priori identifying assumptions. This chapter will demonstrate this point and present an analysis of identification problems in models for the analysis of duration data. We shall demonstrate that current practice overparameterizes econometric duration models.

The literature about the so-called equilibria with rationing is one attempt (among many) to tackle questions that arise at levels that can be categorized for analytical convenience.

On one hand, there has been some discomfort with the way in which competitive Walrasian equilibrium is formulated classically. The classical definition does not depict any relations among economic units and, consequently, there is an unknown of the system (the equilibrium price vector) that is not chosen by any agent. Alternatively, one of the first proofs of the existence of a competitive Walrasian Equilibrium (Arrow and Debreu, 1954) reduces an economy with m agents to a game in normal form with m + 1 agents, the last one of which is the auctioneer whose role is to choose the price vector. This provides a centralized version of Walrasian competitive equilibria (WCE). Clearly, the theorem regarding the equivalence of the core and the set of Walrasian equilibria in large economies gives a decentralized version of Walras's notion.

On the other hand, there has been a gap between macroeconomics and microeconomics. Whereas, given the assumption of competition, it was assumed in microeconomic theory that individuals act only on the basis of price, it is supposed in macroeconomic Keynesian theory that agents take into account quantity constraints.

More precisely, it is possible to connect the recent development of the formalized theory of equilibrium with rationing with four anterior trends of analysis.

The first connection is well known and goes back to Keynes. In a nutshell, Clower (1965) and Leijohnufvud (1977) emphasize the necessity of taking into account the way in which transactions are made and the difficulty of getting an equilibrium price vector in a decentralized way.

The second one seems to be largely unknown although it is a natural development of Keynesian thinking. I refer to the works of economists who tried to give foundations to French Planning (see, for instance, Masse, 1965). French Planning - a kind of decentralized planning - is usually defined as a “generalized market study.” It is argued that the price mechanism is often ineffective even when markets exist and that there is no spontaneous way of coordinating agents' plans when there is no market, for instance, in the case of almost all future goods. Note that the method consists first of all in providing a set of compatible trades between sectors that they are, in principle, interested to take into account as a basis for their computations. It is not known whether the trades are not too much aggregated to be useful, even for big firms.

These are the expanded notes of a course intended to introduce mathematicians to some of the central ideas of traditional economics. They are just notes; they lack the “corroborative detail, intended to give artistic verisimilitude to an otherwise bald and unconvincing narrative”.

There appears to be no book doing what the course attempts to do. Perhaps the nearest approach is E. Malinvaud, Lectures in microeconomic theory (North Holland and American Elsevier, 1972), which is particularly relevant to the first four chapters. There is also a useful account of the subject matter of Chapters 1–4 in D. Dewey's Microeconomics (O.U.P. paperback, 1975). The topics of Chapter 5 are those of the last chapter of David Gale's admirable Theory of linear economic models (McGraw Hill, 1960), but we take them somewhat further. Finally, the simple models of Chapter 6 are discussed at exhaustive length in P.A. Samuelson's Economics (McGraw Hill, Kogakusha, 10th ed., 1976) (especially Chapters 12, 13, 18). This is a massive text intended for the mathematically underdeveloped, but it can be read without a shudder. Indeed it seems to be the best general introduction to the background of the whole course and explains the buzzwords, without which no discussion in economics is complete. Adam Smith's Wealth of Nations still makes interesting reading.

There is a fair number of books on the market with the title “Mathematical economics” or something similar. Those I have sampled have been disappointing. They devote considerable space to expounding standard mathematics.