INTERACTION EFFECTS

John Mullahy

Univ. of Wisconsin-Madison and NBER jmullahy@wisc.edu

ROUGH, PRELIMINARY, AND INCOMPLETE DRAFT

December 2, 2008

Comments Welcome; Please Do Not Quote/Circulate Acknowledgments Thanks are owed to seminar participants at Yale and Wisconsin and poster session participants at the 2008 ASHE Conference at Duke for helpful comments and discussions. Josh Angrist, Jon Gruber, Don Kenkel, Gene Laska, Will Manning, Mike Morrisey, Edward Norton, Paul Rathouz, Jon Skinner, Doug Staiger, and Aaron Stinnett provided helpful comments on the much earlier working paper (Mullahy, 1999) out of which grew the ideas for the present analysis. Some of this work was done while the author was a visiting scholar at University College Dublin's Geary Institute, whose hospitality was brilliant. Partial financial support has been provided by the RWJ Health & Society Scholars Program at UW-Madison.

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There is another curious Question I will just venture to touch upon, viz. Whence arises the sudden extraordinary Degree of Cold, perceptible on mixing some Chymical Liquors, and even on mixing Salt and Snow, where the Composition appears colder than the coldest of the Ingredients? I have never seen the chymical Mixtures made, but Salt and Snow I have often mixed myself, and am fully satisfied that the Composition feels much colder to the Touch, and lowers the Mercury in the Thermometer more than either Ingredient would do separately.

Benjamin Franklin, to John Lining, April 14, 1757

0. Prologue: Two Examples

The clinical and epidemiological literature on neonatal health outcomes suggests that

infant health is influenced by birthweight and gestational age, and also that there may be

interactions effects involving birthweight and gestational age (e.g. Alexander et al., 2003;

Kierans et al., 2007; McIntire et al., 1999). To explore this hypothesis in the context of

non-accident-related neonatal mortality outcomes, a random 10% sample was drawn from

the Linked Birth & Infant Death Data for the 1991 U.S. Birth Cohort, whose overall sample

(population) size is 4,115,494. The analytical sample comprises 395,996 observations on

singleton births for which the sample neonatal mortality rate .0071 (2,817 deaths; accident-

related deaths were omitted from the analytical sample). Figure 1 displays the estimation

sample marginal distributions of the birthweight (in pounds) and gestational age (in weeks)

variables. Table 1 summarizes OLS and logit model estimates of survival outcomes (1

minus death) in specifications that also control for the gender of the neonate, the age of the

mother, and the infant's birth order. In two specifications quadratic terms in birthweight

and gestational age are included.

Motivating the second example is the risk-adjustment literature in which

comorbidities between and among different health condition indicators are analyzed as

potential influences on health care expenditures (e.g. Stukenborg et al. 2001; Robinson,

2008). To study this, data on health care expenditures for non-elderly (<65) individuals are

drawn from the U.S. Medical Expenditure Panel Surveys combined over the years 1996-

2005 (N=115,637). The outcome data are individual-level two-year total expenditures.

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Subpopulations are defined by binary indicators of whether or not individuals do or do not

report having any activity of daily living (ADL) or instrumental activity of daily living (IADL)

limitations at the baseline of the two-year data window. Individual age and sex are also

included as covariates. Figure 2 displays the subsample means for the unadjusted (left

panel) and age-sex-adjusted (right panel) expenditure measures by ADL/IADL status.

In both the neonatal mortality and the risk-adjustment examples, is the evidence

provided consistent with the reader's sense of an "interaction effect," and -- if so -- what

are the sign and magnitude of that effect so identified?

Plan for the Paper

The plan of the remainder of this paper is as follows. Section 1 provides some basic

concepts and background and describes briefly some of the historical and cross-science

contexts in which issues of interaction effects arise. Section 2 builds the analytical

foundations for the subsequent discussion. Section 3 reviews various features of the "cross-

derivative" approach to interaction effects that is prominent in applied economics. Section 4

contrasts the approaches of section 3 with alternative perspectives that build on concepts

from pharmacology and production economics. Section 5 characterizes the concepts from

the previous two sections in the context of three specific functional forms. Section 6

considers issues of statistical inference. Section 7 revisits the examples described above in

the Prologue in light of the intervening discussion. Section 8 concludes.

1. Interaction Effects: Concepts, Background, Scientific Context

What is an "interaction effect"? While this would appear to be a simple concept to

define, a survey of the economic literature as well as of scientific literatures outside of

economics suggests that the answer to this question is anything but simple and is certainly

not unambiguous. Occasional queries of seminar or classroom participants have shown as

well that individuals often have roughly the same notion in mind at an abstract level yet

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when differ (or balk) when pushed to provide algebraic or other quantitative representations

of their intuitions. Related terminology that is often associated with discussions of

interaction effects -- e.g. "synergy", "complementarity", etc. -- is no less ambiguous in its

meaning.

A not-unreasonable conjecture is that scholars working across scientific disciplines

share a fairly common intuition about what they intend or interpret when they express or

encounter the term "interaction" or the term "interaction effect" (henceforth "IE") in

scientific discourse. Yet it is not always obvious how such intuitions map into specific

parameters or quantities of first-order scientific or policy concern. And while many common

conceptualizations of what constitutes an interaction effect tend to point in the same

direction, as will be demonstrated formally later on, it turns out that there are other

competing and important characterizations of IEs that have structurally different meanings

that do not necessarily lead to the same answer as to whether an interaction effect is

present and, if present, its sign and magnitude. Moreover, while textbooks in epidemiology

and other disciplines provide rigorous characterizations of concepts like additive and

multiplicative IEs, such characterizations will not necessarily be informative regarding the

scientific question at hand. Alternatively, while it is straightforward to point to a regression

model parameter (say jkβ ) that multiplies a particular summand in a linear index function

(say j kx x× ) as an "interaction parameter," it is not obvious that such a parameter will on

its own will characterize any quantity of fundamental scientific or policy interest, as will be

demonstrated below.

Furthermore, it will be suggested here that as a general matter -- and regardless of

one's algebraic conceptualization of the IE of concern -- the phenomenon of there being no

or zero interaction effect in a particular circumstance is a razor's edge situation. As such,

considerations of the magnitudes of the IE -- and, even more fundamentally, the decision or

policy question informed by knowledge of or inference about an IE quantity -- will be more

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prominent in this discussion than the issue of whether or not a particular IE differs from

zero. (See Ziliak and McCloskey, 2008, for discussion of the importance of magnitude in

statistical investigations.)

The main contributions of the paper are to suggest a unified framework within which

the various conceptualizations of IEs can simultaneously assessed and to provide some

novel approaches to measurement/quantification of one of these conceptualizations. While

the measurement of IEs corresponding to some of these conceptualizations has been well

established, the measurement or quantification for others has not, and the paper offers

several suggestions to redress this imbalance.

"Interaction Effects" in Earlier Literature

In a recent JSTOR search, the earliest hit in the economics category on "interaction

effect" in full text search was Klein, 1951. In that paper, Klein effectively characterizes an

IE in the sense of a non-zero cross-partial derivative; quoting from Klein's paper:

In an equation of the form (12) ( )0 1 2 1 3 1 1 4 4 1S / Y logY /N L / Y Y Y / Y a L u,− − − −= β + β + β + β − + β + β +

the marginal effect of liquid assets on savings is

(13) 2 51

SY.

L−

∂= β + β

∂

Equation (13) states that the marginal asset effect depends on the income level, which is analogous to the previous results which claimed that the marginal asset effect depends on income change. Either case may be considered as a representation of an interaction effect.

As will be discussed below, this nonzero cross partial derivative relationship (or quantities

base on it) is perhaps the most standard interpretation in the economics literature.

In his seminal paper on the use of categorical dummy variables in regression models,

Suits, 1957, extends his basic framework to introduce what he called "an interaction term

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involving X and the dummy variables" to accommodate slope variations across the dummy

categories. The model so specified was

( )= + + + + + + + +1 1 2 2 3 3 1 1 2 2 3 3Y a d R d R d R X b R b R b R c u .

Finally, with this bold pronouncement Morgen and Sonquist, 1963, set the stage for over 45

years of research on various ways to explore quantitatively so-called interaction effects:

Most of the problems of analyzing survey data have been reasonably well handled, except those revolving around the existence of interaction effects.... Where interaction effects exist, the concept of a main effect is meaningless, and it is our belief that in human behavior there are so many interaction effects that we must change our approach to the problems of analysis.

"Interaction" and IEs across Scientific Fields

Concepts of "interaction" and IEs arise broadly across the sciences. Perhaps most

obvious is statistics, where different characterizations of and tests for interactions are

prominent in the literature (e.g. Cox, 1984; Friedman et al., 2000). In genetics,

explorations of gene-gene and gene-environment interactions as features of disease

phenotypes are highly visible (e.g. Hoffjan et al., 2005). Considerable effort has been

devoted to developing quantitative methodologies to understand such interactions (e.g. Lou

et al., 2008), while -- consistent with a theme of this paper -- some recent work has

suggested the importance of actually defining "genetic interaction" (Mani et al., 2008).

In the clinical literature, too, one finds frequent use of terminology relating to IEs.

One prominent domain is the consideration of the clinical efficacy or outcomes of so-called

combination therapies relative to those arising from monotherapies (e.g. Clavel and Hance,

2004; Wadden et al., 2005). Related considerations arise in behavioral, clinical, and social

psychology, as well as psychoneuroimmunology (Friedman et al., 2005; Gruenwald et al.,

2006; Schnittker, 2002). Sociologists and related social scientists frequently examine

empirically IEs (e.g. Mirowsky and Ross, 2007). In psychology, sociology, and related fields

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considerations of interaction and IEs often entail considerations of the interacting variables

in roles of mediators or moderators (see Baron and Kenny, 1986). Even engineers

sometimes get into the interaction act (e.g. Box, 1990).

The field of epidemiology has a long and broad history involving considerations of

interactions and IEs (see Rothman and Greenland, 1998; Ben-Shlomo and Kuh, 2002).

Many of the conceptual, definitional, and testing issues involving IEs have been addressed

in this literature. For present purposes, two important epidemiological definitions are those

of additive and multiplicative interactions. In a 2x2 context (with, e.g., "treatment" and

"nontreatment" options for two variables and standard outcome notation of y00, y01, y10, and

y11), additive interaction is defined in sign and magnitude by the difference-in-differences

( ) ( )− − −11 10 01 00y y y y ,

whereas multiplicative interaction is defined correspondingly by the ratio-of-ratios:

( ) ( )11 10 01 00y / y / y / y .

It might be noted that while these concepts are oftentimes applied in contexts of particular

functional forms (i.e. linear models and log-linear models, respectively) there is no a priori

reason to make such a one-to-one linkage, e.g. an additive interaction may be a meaningful

or decision-relevant quantity in the case of a log-linear statistical model (Thompson, 1994).

Finally, the field of pharmacology has a deep tradition of assessing interaction

effects, often involving the efficacy of (or side effects arising from) two or more

pharmacotherapies delivered individually or jointly (Berenbaum, 1977, 1985; Greco et al.,

1995; Laska et al., 1997; Michel et al., 2008). This pharmacological tradition will be

discussed in considerable detail below since it turns out that the standard characterizations

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of interaction and IEs in this literature accord more-or-less exactly with concepts familiar in

many microeconomic contexts yet differ in important ways from many of the other

characterizations discussed above.

To gauge the breadth of scholarly effort dedicated to studying IEs, a JSTOR search

was conducted on 10/20/07 with the term "interaction effect" in the full-text search option.

The search resulted in the following rates of hits in the indicated subject areas:

JSTOR Subject Area # Hits # Journals

Anthropology 31 23 Biological Sciences 1082 89 Business 1498 75 Economics 412 52 Education 1034 50 Health Sciences 57 7 Mathematics 91 22 Political Science 465 43 Psychology 696 7 Sociology 1810 46 Statistics 390 23 Total 6329 437

In addition, the search on "interaction term" resulted in an even greater total of 8453 hits.

Many of these have little or nothing to do with what will be discussed below, yet it is

apparent that discussions of IEs are prominent in the scientific literature as well as in more

popular contexts, e.g. wine-music interactions (Gray, 2007).

Interactions in Current-Day Economic Analysis

For clarification, the discussion here does not entail social interactions (Becker,

1974), itself an important area of academic inquiry. Much of the discussion of interactions

and IEs in the modern economic literature arises implicitly from the evaluation literature in

which difference-in-differences (D-I-D) identification and estimation strategies are

prominent (e.g. Athey and Imbens, 2006). In the simplest such structures, a linear

statistical model is specified to have terms in (often binary) xj, xk, and "xj times xk" (as well

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as other covariates), with the parameter on the "xj times xk" term capturing the D-I-D

treatment effect (e.g. Currie and Hyson, 1999). When the D-I-D paradigm is extented

broadly to nonlinear model contexts, a variety of conceptual and empirical complications

arise (see Athey and Imbens, 2006; Ai and Norton, 2003; Norton et al., 2004; Puhani,

2008). Some of these issues are at the core of this paper's central discussion, below.

Obviously, considerations of interactions arise beyond the D-I-D context in

economics as well. Some economic models have as their core consideration issues involving

interactions of variables, e.g. the analysis of quantity-quality interaction in the classic work

by Becker and Lewis, 1973. In important recent work, for instance, Heckman, 2007,

discusses the interactions between early life and later childhood investments as they

operate on human capital outcomes. Finally, the econometric literature has also treated

extensively interactions and IEs, often implicitly under the rubric of flexible functional forms

or 2nd-order approximations (e.g. Chambers, 1988; Diewert, 1971)

2. Interaction Effects: Conceptual and Empirical Foundations

When studying an "effect" in an quantitative empirical exercise like an application of

regression analysis, it is logical that the question of "effect" entails a consideration of "an

effect of (something) on (something else)." Applying similar logic for an "interaction effect",

one is led to consider "an interaction effect between (something) and (something else) on

(yet something else)." To lay the groundwork for the analysis, it is thus useful to provide

some characterizations of the somethings and something elses.

Let f(y|x) be the conditional distribution of a scalar outcome y given conditioning

covariates x. For our purposes, x=[xj,xk,xA], where xj and xk are scalars whose

"interaction" and/or "interactions effects" will be of interest, while xA is a vector of other

covariates that play a secondary (yet in most cases more than a nuisance) role in what

follows. Importantly, while the population covariance matrix of x may be entirely non-zero,

it is assumed here that it is logically possible to vary (e.g. through policy interventions)

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elements of xj and/or xk without varying xA. (That is, if w(x) is some functional of f(y|x),

one can ignore terms involving A jd / dxx or A kd / dxx in taking derivatives or differentials of

w(x).)

The main considerations of the analysis are assumed to entail the IE -- however

characterized -- between xj and xk on some functional v(x) of f(y|x). v(x) could be a

conditional expectation ⎡ ⎤⎣ ⎦

mE y x , conditional quantile ( )αfQ x , conditional set probability

( )∈Pr y S x , or any other x-dependent quantity that might be of interest to the decision

maker. The analysis will work with a particular, albeit quite general specification of v(x),

i.e.

( ) ( )( )= φ +j k Av v x ,xx x β

where ( )φ j kx ,x can be thought of as an "interaction function" in a sense to be made clearer

below. Because its role is secondary in what follows, the remaining discussion will use the

shorthand notation Ω to denote Ax β , thus ( ) ( )( )= φ + Ωj kv v x ,xx . It will be assumed that

v(.) is monotone in its arguments, at least locally or over some relevant range of the x

measures. In general v(x) and ( )φ j kx ,x will be nonlinear functions of the respective

arguments. In what follows it will be helpful to think of both of these functions as being at

least twice continuously differentiable; implicitly, then, the focus here will be on continuous

measures of xj and xk although many of the main arguments may apply more generally.1

In what follows, it will be useful in many contexts to view v(x) as a production

function While its formal character as a production function is not essential, the notion that

1 It is important to note that the assumption that jx and kx are continuous moves the

analysis away from the many standard difference-in-difference environments in which jx

and kx would typically (though not necessarily) be dummy variables.

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v(.) is a function that describes conversion of multiple inputs into an output is rather

encompassing. Moreover appealing to the particular properties of production functions will

turn out to be quite useful in section 4 and elsewhere. To further this interpretation, it can

be helpful to think of xj and xk as variable inputs and Ω as summarizing quasi-fixed inputs.

It will also be useful in specific cases to assume ( )( )φ + Ωkv 0,x and ( )( )φ + Ωjv x ,0 are

nonzero (thus, e.g., ruling out Cobb-Douglas functional forms).

Many empirical studies of IEs across various scientific areas focus attention on the

specification

( )φ = β + β + β ×j k j j k k jk j kx ,x x x x x .

(This can be viewed as a "restricted quadratic" specification, in the sense that it is a second-

order Taylor expansion in which the squared terms are omitted.) In this specification the

×j kx x is often considered to be the "interaction term." Yet on reflection there is nothing

particularly special about using multiplication to "interact" xj and xk. One could equally well

specify an "interaction term" to be j kx / x or xkj

x or.... Indeed, there needn't be any

product or quotient or power relationship between xj and xk in an "interaction term" for

there to be meaningful interaction effects in any of the contexts described below; e.g. a CES

specification ( ) ( ) ρρ ρφ = β + β1 /

j k j kj kx ,x x x is considered below. As such, one issue that may be

dispensed with immediately is that "interaction terms" are essential elements of

considerations of IEs. So, more generally, it is sensible to focus generally on an interaction

subfunction or sub-aggregator function ( )φ j kx ,x than on any particular multiplicative

"interaction term."

To motivate the discussion in sections 3 and 4, and beyond, it is useful to note that

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the different characterizations of IEs considered therein can all be described in terms of the

functional form ( ) ( )( )= φ + Ωj kv v x ,xx . Of greater importance, however, is the recognition

that the two main characterizations of IE (IE1 and IE2) -- while related -- ultimately speak to

different concepts. For now, let it suffice to note that neither measure generally implies the

other. For instance, it will be demonstrated that the functional form

( )β + β + Ωj j k kv x x

with v(.) nonlinear will imply nonzero IE1 but IE2=0, whereas the functional form

( )β + β + β + Ω2j j jj j k kv x x x

with v(.) linear will imply nonzero IE2 but IE1=0.

A Brief Digression: Why Does the Analyst or Decision Maker Want to Know

Something about IEs?

From the decision maker's perspective, is there any meaningful difference between

the equation

= + + +j ky 10 2x 3x u

and the equation

( )= + + − × +j k j ky 10 2x 3x .0000005 x x u ?

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In many instances, the obvious answer is "no". Yet as often defined there is an interaction

effect in the latter equation but not one in former owing to the presence of a nonzero

parameter multiplying the "interaction term" in the latter equation. With a sufficiently large

sample, of course, it would be possible to reject a null hypothesis of "zero interaction

parameter" if the latter equation represents the true data-generating process.

But is this useful? It is certainly common to encounter statements in the literature

akin to: "We also found a statistically significant interaction effect between (___) and

(___)." In an important sense, though: So what? Such an empirical finding should

generally be the beginning of the discussion of the IE at hand, not the end, although more

often than not this is where the discussion terminates.

Instead it will be argued here that in empirical investigations in which considerations

of IEs are prominent, it will generally be important to ask why such an interaction effect is

of decision or policy relevance and how the understanding the magnitude and/or sign of the

purported IE informs the decision or policy. Simply learning -- or drawing an inference --

that an IE is nonzero seems generally less interesting scientifically, in part since an exactly

zero IE (as will be suggested below) is a razor's edge situation. If bjk is known to be

-.0000005 in the above equation, would that change the policy or other decision relative to

a finding that a null of bjk=0 cannot be rejected? Perhaps so, but as Ziliak and McCloskey,

2008, suggest in more general contexts, resolving the existence question is typically far less

useful than resolving the magnitude (or what they call the "oomph") question. So, however

the IE is defined operationally (this is part of the task that follows) it will be presumed

generally to be more informative to policy or decision makers to estimate and interpret its

magnitude (and, perhaps, α − level confidence intervals for the point estimate) than merely

to report the p-value of the null that bjk=0.

3. Interaction Terms, Cross-Derivatives, and IE1 Measures

It is useful to begin this discussion with the simple specification (e.g. Currie and

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Hyson, 1999; Ghuman et al., 2005)

( )φ = β + β + β ×j k j j k k jk j kx ,x x x x x .

As the literature has demonstrated (Ai and Norton, 2003; Mullahy, 1999; Norton et al.,

2004), the "interaction parameter" jkβ is only partially informative about the sign and

magnitude of one version of IE when v(.) is nonlinear. As these arguments are now

relatively familiar in the applied econometrics literature, only a relatively brief discussion will

be undertaken.

Specifically, this characterization of IE, which we will term 1IE , is defined by the

cross-partial derivative or difference of v(x) with respect to xj and xk. This characterization

has become a standard one in economics and in the related evaluation literatures, in part

since it is a generalization of D-I-D approaches. The central issue that has occupied the

attention of analysts of late is that when v(.) is nonlinear, the cross derivatives or

differences will in general entail all the covariates and will, in particular, involve much more

than just the "interaction parameter" jkβ in determining the sign and magnitude of the IE.

Moreover, nonlinearity of v(.) also has implications for interpreting the interaction in

"treatment effect" contexts (see, e.g., Athey and Imbens, 2006; Puhani, 2008).

To fix ideas, it is useful to review the standard difference-in-differences ("D-I-D")

regression estimator and the role played therewithin by the interaction parameter jkβ .

Suppose ( )xv is linear, ( )x j j k k jk j kv x x x x= β + β + β + Ω . Then the D-I-D estimator is the

familiar cross-partial difference of ( )xv with respect to ( )j kx , x , resulting in the interaction

parameter jkβ :

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( ) ( ) ( )x x2

j jk k jkj k k j k

v v"D-I-D" x

x x x x x

⎛ ⎞Δ ΔΔ Δ= ≡ = β + β = β⎜ ⎟⎜ ⎟Δ Δ Δ Δ Δ⎝ ⎠

.

In the case of continuous ( )j kx ,x , the obvious translation is the cross-partial derivative

( )x2

jkj k

v

x x

∂= β

∂ ∂.

The cross-partial derivative ( )x2

j k

v

x x

∂

∂ ∂ with nonlinear v(.) is at the core of much recent

econometric work on interaction effects (e.g. Ai and Norton, 2003). Indeed, it has become

relatively common in some circles to consider this cross derivative as the interaction effect.

For now, however, we will define this as one version of an IE, i.e.

( )x2

1j k

vIE

x x

∂=∂ ∂

Such a quantity could, for instance, provide an answer to a question like "By how much

does the marginal influence of jx (resp. kx ) on v(.) change in response to a marginal

change in kx (resp. jx )?" For some questions of interest, the sign and magnitude of this

cross-partial may provide precisely the answer that is of interest. To presage later

discussion, however, note that this definition of an IE does not hold constant the level of the

outcome v(x) in the computation of the cross derivative.

For a brief glimpse at the issues pertaining to IE1 in the nonlinear v(.) context,

consider

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( ) ( ) ( )x

m jk pm

vv ' . x

x

∂= × β + β

∂, { } m,p j,k , m p∈ ≠

and

( ) ( ) ( ) ( ) ( )x

x x2

jk j jk k k jk jj k

vv ' v " x x

x x

∂= × β + β + β × β + β

∂ ∂.

So, for example, in a logistic regression with v(x)=Prob(y=1|x), one has

IE1 = ( ) ( )( ) ( )( ) ( )( ) ( )⎡ ⎤× × β + × × × ×⎣ ⎦jk j kv 1 - v 1 - 2v 1- v v T Tx x x x x

where

= β + βj j jk kT x ,

and

= β + βk k jk jT x .

With v(x) in (0,1), it is apparent that while the sign and magnitude of IE1 indeed depends

on the parameter jkβ , it also depends on a lot of other things. Even assuming the terms

defining the "marginal products" Tj and Tk to be positive, both the sign and magnitude of IE1

will depend on the term (1-2v(x)), which may be positive or negative depending on the

particular point of evaluation of the logit distribution function.

4. Pharmacology, Production Economics, and the IE2 Measures

As hinted in section 1, there turns out to be considerable kinship between

pharmacology and production economics regarding concepts relating to interaction. The

tradition in pharmacology is to consider how two or more agents "interact" in producing

"good" (e.g. health) or "bad" (e.g. side effects) outcomes. Importantly, the focus in much

of the pharmacological literature is on what it terms "response surface" analysis, or what

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production economists might think of as level-curve or output-constant analysis. Ultimately

this is where the measures that we will term IE2 differ fundamentally from their IE1 cousins

that do not hold "output" constant in taking the cross derivatives.

While the field of pharmacology has its own terminology, with a bit of translation it is

easy to see why there are important parallels on which one might draw:

Pharmacology Production Economics Isobol Isoquant

Isobologram Isoquant Field Additivity Perfect Substitutability Synergy Quasiconcavity

Perfect Synergy Perf. Complementarity Antagonism Concave Isoquant

As with output-constant production concepts, the analyses in pharmacology are "local" in

the sense of pertaining to specific level curves or surfaces of v(x) in input space, holding

constant other factors (e.g. Ω ) as appropriate. Formally, a level curve in ( )j kx , x space−

(isoquant, isobole, etc.) for output level κ given covariates Ω is defined in the standard

manner as

( ) ( ) ( )( ){ }κ Ω = φ + Ω = κ ≥ ≥j k j k j kL , x , x v x ,x , x 0, x 0 .

For some functional forms, it will be possible to solve analytically for ( )= λ κ Ωk jx , , x and

thus trace out level curves explicitly in ( )j kx , x space− holding constant ( )κ Ω, . Interestingly,

the basic nature of the IE defined in this context evidently does not depend crucially on the

particular functional form of v(.), i.e. ( ) ( )−φ = κ − Ω1j kx ,x v =Δ .

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Defining IE2

Given a level curve -- or, more generally, a family of level curves (isobologram or

isoquant field) -- it become meaningful to speak (locally) of characterizations of IEs like

synergy or antagonism or additivity. Such a characterization of this alternative form of IE is

not novel -- obviously it is drawn directly from pharmacology and production economics.

What appears novel, however, is the linkage between such a characterization and the

quantification of the magnitude of the IE, which will be the definition of IE2. Quantification

is important to the extent that questions like "Is the IE between ( )j kx ,x larger (in some

metric) than that between ( )j mx ,x or ( )p qx ,x ?" can be tackled. The strategies detailed

below permit such questions to be addressed.

Two scalar characterizations of IE2 are offered here. Both have as their basis the

observation that when, e.g., synergy or quasi-concavity describes the production surface,

there are "advantages" (whose definition is situation-specific) to combining ( )j kx , x --

letting them "interact" -- relative to utilizing either xj or xk in isolation. This is why, for

instance, a seemingly simple specification like ( ) = β + β + β + Ω2j j jj j k kv x x xx results in

meaningful interaction in the present context even though there is no interaction in the

sense of IE1. (Note that in some instances, there will be alternative ways to characterize

the IE2, e.g. using the elasticity of substitution in a CES framework.)

The first measure draws on the literature on Farrell-type production efficiency

measurement and measures the "advantage" in terms of input use. The essential idea is

depicted in figure 3. For a given level curve one identifies the "line of additivity" (in

pharmacological terms) and then determines the ray from the origin for which the ratio

OU/OL is greatest. (If the production surface is characterized by antagonism instead of

synergy, then one determines the greatest ratio OL/OU.) In this case, IE2 can be defined by

the OU/OL ratio or meaningful transformations thereof.

18

The second measure, which has the most directly interesting economic

interpretation, is based on the notion that in the presence of synergy or quasi-concavity,

combination of inputs is cost-saving relative to production that relies on the use of xj or xk in

isolation. This approach is available when there are meaningful unit prices or shadow prices

for xj and xk. The IE2 measure so-defined is the monetary difference between the minimum

cost of producing output level κ (holding Ω constant) with xj and xk allowed to vary versus

the smaller of the minimum costs of producing output level κ (holding Ω constant) using

either xj or xk in isolation. This situation is depicted in figure 4. (If there is antagonism, flip

the discussion around and figure out what is the most costly combination of xj and xk that

could be used to produce κ and define correspondingly a negative IE2.)

5. Three Functional Forms (INCOMPLETE)

This section considers the implementation of the IE1 and IE2 constructs for three

particular parametric specifications of the interaction function ( )φ j kx ,x , specifically:

Constant-elasticity of substitution (CES),

( ) ( ) ρρ ρφ = β + β1 /

j k j kj kx ,x x x ;

Diewert or Generalized Leontief,

( ) ( )φ = β + β + β ×j k j j k k jk j kx ,x x x x x ;

and restricted quadratic (or linear with a multiplicative interaction term)

( ) ( )φ = β + β + β ×j k j j k k jk j kx ,x x x x x

19

CES Specification

This specification provides an attractive analytically tractable approach for examining

IEs. Heckman, 2007, uses the CES specification in his analysis of interactions between

early- and later-childhood investments on human capital outcomes.

For the CES specification the input-based measure (that corresponds to figure 3) is

given (after some tedious algebra and calculus) by

( ) ρρ

ρ ρ

β + β

β + β

1 /

k j

1 / 1 /jk

s

s,

where

ρβ⎛ ⎞

= ⎜ ⎟⎜ ⎟β⎝ ⎠

1 /j

k

s .

The cost-based version of the IE2 (corresponding to figure 4) is determined in this case as

follows. The minimum cost of producing output level κ (net of constant Ω ) with input

prices p=(pj, pk) is:

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )ρ− ρ ρ− ρ− ρ ρ−

ρρ ρ− ρ ρ− ρ ρ− ρ ρ−

⎧ ⎫⎪ ⎪β + β⎪ ⎪κ Ω = Δ ⎨ ⎬⎪ ⎪β β + β β⎪ ⎪⎩ ⎭

1 / 1 / 1 1 / 1 / 1j jk k

1 // 1 / 1 / 1 / 1k jj j k k

p pc , ,

p pp

Minimum cost of producing output level kappa net of OMEGA using either only xj or xk:

( ) { }− ρ − ρκ Ω = = = Δ × β β1 / 1 /j k j kj k

c , , ;x 0 or x 0 min p , pp

Thus, the cost-based measure of IE-2 for the CES case is

20

( ) ( )κ Ω = = − κ Ωj kc , , ;x 0 or x 0 c , ,p p =

{ }( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )ρ− ρ ρ− ρ− ρ ρ−

− ρ − ρρρ ρ− ρ ρ− ρ ρ− ρ ρ−

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥β + β⎪ ⎪Δ β β − ⎢ ⎥⎨ ⎬

⎢ ⎥⎪ ⎪β β + β β⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

1 / 1 / 1 1 / 1 / 1j jk k1 / 1 /

j k 1 /j k / 1 / 1 / 1 / 1k jj j k k

p pmin p , p

p p

Given homotheticity in this case, this could also reasonably be given a unitless character as

( ) ( )= = −j kc ;x 0 or x 0 cp p =

{ }( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )ρ− ρ ρ− ρ− ρ ρ−

− ρ − ρρρ ρ− ρ ρ− ρ ρ− ρ ρ−

⎡ ⎤⎢ ⎥β + β

β β − ⎢ ⎥⎢ ⎥β β + β β⎢ ⎥⎣ ⎦

1 / 1 / 1 1 / 1 / 1j jk k1 / 1 /

j k 1 /j k / 1 / 1 / 1 / 1k jj j k k

p pmin p , p

p p

[Discussion of other functional forms forthcoming]

6. Empirical Implementation and Inference

[Discussion forthcoming]

7. Revisiting the Empirical Examples

[Discussion forthcoming]

8. Conclusions

[Discussion forthcoming]

21

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26

Figure 1 Sample Marginal Distributions: Birthweight, Gestational Age, Birthweight * Gestational Age

1991 U.S. Singleton Births (Table 1 Estimation Sample, N=395,996)

0

.1

.2

.3

.4

Den

sity

0 5 10 15 20Birthweight (lbs.)

0

.05

.1

.15

.2

.25

Den

sity

10 20 30 40 50Gestational Age

0

.002

.004

.006

.008

Den

sity

0 200 400 600 800Birthweight * Gestational Age

27

Figure 2 Two-Year Health Care Expenditures, Non-Elderly Population, By ADL/IADL Status

1996-2005 MEPS Samples (N=115,637)

0

10,000

20,000

30,000

Unadjusted

No ADL/IADL ProblemADL Problem OnlyIADL Problem OnlyADL & IADL Problem

0

10,000

20,000

30,000

Age and Sex Adjusted

No ADL/IADL ProblemADL Problem OnlyIADL Problem OnlyADL & IADL Problem

Subsample Means

Unadjusted Adjusted

No ADL/IADL Problem 3,291 3,318

ADL Problem Only 14,464 14,409

IADL Problem Only 16,585 14,837

ADL & IADL Problem 28,863 27,557

.

28

Figure 3 IE2 Input-Based Measure

29

Figure 4 IE2 Cost-Based Measure

30

Table 1 Estimates of Survival Probability, 1991 U.S. Singleton Births (N=395,996)

(Robust t-statistics in parentheses)

OLS Logit BIRTHWEIGHT 0.1736 0.1312 1.9296 0.9850

(50.8) (37.7) (23.6) (7.9) GESTATIONAL AGE 0.0315 0.0547 0.2014 0.5605

(49.9) (27.3) (19.2) (10.1) BW * GEST -0.0044 -0.0025 -0.0338 0.0189

(50.0) (28.0) (16.1) (4.6) BW ^ 2 -0.0020 -0.0871

(14.5) (15.7) GEST ^ 2 -0.0005 -0.0090

(18.4) (9.0) GIRL BABY 0.0017 0.0016 0.3967 0.3833

(6.9) (6.4) (9.3) (8.9) MOM'S AGE 0.0001 0.0001 0.0272 0.0281

(5.6) (3.3) (7.1) (7.2) BIRTH ORDER -0.0009 -0.0007 -0.1266 -0.1234

(9.0) (7.0) (10.8) (10.4) CONSTANT -0.2548 -0.5422 -7.1339 -10.8453

(10.4) (13.8) (23.7) (14.5)

31

Table 2 Estimates of Two-Year U.S. Health Care Expenditures, MEPS 1996-2005 (N=115,637)

(Robust t-statistics in parentheses)

OLS GLM, Log-Link ADL 11,101.7 1.722

(8.30) (13.85) IADL 11,670.2 1.294

(16.77) (27.91) ADL * IADL 1,581.5 -0.992

(0.67) (6.08) AGE 106.8 0.030

(56.87) (62.13) FEMALE -907.5 -0.289

(15.60) (16.58) CONSTANT 654.8 7.191

(11.83) (351.75)

.