IN HIS THEORY OF the normative structure of science, sociologist Robert K. Merton (1973) argues that the institutional goal of science is the extension of certified knowledge. In particular, the emphasis is on research originality, since it is originality that most advances scientific knowledge. Scientific inquiry is characterized by a market process in which scientists publish the results of their research using the prior research of others to which they give and receive recognition.
Merton's view on the structure of science assumes that the reward structure in science is congruent with the norms of science: common ownership of scientific knowledge, objective evaluation of scientific knowledge, and dissemination of scientific knowledge for verification and extension. The reward structure in science compensates scientists principally in the coin of recognition accorded research by fellow scientists. Rewards to scientists are distributed in accordance with the degree of accomplishment, primarily by some form of visible professional recognition (e.g., Nobel Prize, awards, memberships in honorary academies of science). In this specific way, the reward structure of science reinforces and perpetuates the institutional emphasis on research quality and cognitive content. The reward system provides greater incentives for scientists to undertake research that makes the largest contributions to scientific knowledge.
Published research offers many positive benefits to academic economists. First, publications provide evidence that an economist has done work that is relevant to the current research frontier. Second, publications are used by most economics departments to ascertain productivity differences. Most economics departments link tenure, promotion, and salary increases to publication performance. Third, published research is one of the most important means for communicating one's findings, while protecting a researcher's property rights for creative ideas and original contributions.
The second half of the 20th century has seen a fundamental change in the means used to produce economic knowledge: there has been a substantial increase in the incidence of co-authorship in economics (Durden and Perri 1995). Sole authorship of research publications in economics has become less commonplace. This notable change in the method used to produce economic research may have an effect on the advancement of scientific knowledge. The collaborative arrangement may affect the productivity of collaborators and concomitantly the extent of scientific progress. Understanding the input relationship between co-authors in economics is of interest not only to economists but also to sociologists, historians of science, and physical and natural scientists because the incentive structure in academia encourages and rewards collaboration. Many universities and grant-giving agencies promote and reward collaboration because of the belief that collaboration has a positive impact on research productivity (Laband and Tollison 2000).
Empirical evidence on the input relationship between co-authors in economics is a topic of dispute. Almost all of the empirical research has focused on the impact of collaboration on article production by economists. The empirical evidence is mixed. McDowell and Smith (1992), using cross-sectional data from 89 men and 89 women who received their doctorates in economics between 1968 and 1975, found that cumulative article production is not increased by an individual's propensity to co-author. Durden and Perri (1995), using time-series data on total economics article production between 1969 and 1992, found that the percentage of articles co-authored was positively related to the total and per capita economics article production. Hollis (2001) examined the 1981 publication records of 339 members of the American Economic Association whose last names begin with the letters a, b, c, d, s, t, u, and v. He found that more co-authorship was associated with fewer quality-weighted article pages per author.
Other researchers have used indirect empirical evidence to impute the input relationship between co-authors in economics. Diamond (1985) found that the monetary returns from a citation to a co-authored article are worth more to its author than a citation to a sole-authored article. Liebowitz and Palmer (1983) report that, in a survey of economics department chairs, faculty who co-author receive 70 percent of the merit pay of a writer of a sole-authored paper. McDowell and Smith (1992) find that co-authored articles have the same weight as sole-authored articles in determining promotion to a higher academic rank.
The common methodological problem of the aforementioned studies is that they all use a hedonic function approach to impute the input relationship between co-authors in economics rather than directly estimating an explicit production function. The answer to the question of what the input relationship is between co-authors in economics will provide insight into both the collaborative research process and the progress of scientific knowledge. The purpose of this article is to empirically estimate the input relationship between co-authors in economics using an explicit production function model.
THERE ARE TWO SCHOOLS OF THOUGHT as to why economists collaborate. The first explanation is based on Adam Smith's observation that increases in the size of a market permits greater division of labor. The growth in economic knowledge has made efficiency gains from division of labor more pronounced (McDowell and Melvin 1983). Further, the increasingly complex quantitative nature of economics has made it almost a technical necessity to specialize (Barnett, Ault, and Kaserman 1988). Researchers have greater opportunities for efficiency gains in the production of economic knowledge from specialization and division of labor. Collaboration allows each researcher the opportunity to focus on various narrowly defined complementary skills in producing scientific knowledge. For example, a researcher proficient in econometric techniques may find it more efficient to collaborate with a researcher skilled in modeling. Collaboration involves researchers who combine complementary skills that enhance economic knowledge. If co-authorship reflects researchers who combine complementary inputs in the production of economic knowledge, then one would expect to find that the partial elasticity of substitution between co-authors is negative.
Another explanation is that the typical collaborative combination involves the merger of equally skilled researchers. The rationale for this arrangement is the increasing emphasis on research in the academic labor market (Barnett, Ault, and Kaserman 1988). Research has become the primary method used by academic institutions to assess an academic's qualifications for tenure and promotion. Co-authorship provides a means for equally skilled researchers to reduce their research efforts without reducing their individual measured research output by producing papers with little addition to scientific knowledge. The incentive to compromise on potential research quality is reinforced even more if academic institutions do not discount co-authored articles by the number of authors in tenure and promotion decisions. If co-authorship reflects the merger of equally smiled researchers, then one would expect to find that the partial elasticity of substitution between co-authors is positive.
Direct Addilog Production Function
A PRODUCTION FUNCTION MUST be explicitly specified to empirically investigate the input relationship between co-authors. The production function delineates the technical substitutability between inputs as summarized by the partial elasticities of substitution. Economists have always been interested in the substitution properties of inputs because these properties describe the production technology. (1) Most multifactor production functions, however, are specified in such a way that they can handle only certain restrictive assumptions about the substitutability of inputs. For example, the Cobb-Douglas and the constant elasticity of substitution production functions have the property that the partial elasticities of substitution between any input pair is constant (in the Cobb-Douglas case, the constant is unity). The Uzawa function is a two-tiered constant elasticity of substitution production function in which the partial elasticity of substitution between any pair of inputs is the same constant within each tier and unity between tiers.
Ideally what is needed is a production model that places no a priori restrictions on the elasticities of substitution. Exactly such a model is provided by the direct addilog production function (Mukerji 1963). The direct addilog production function is attractive because it places no a priori restrictions on the values that the partial elasticities of substitution can assume, it permits direct estimation of the elasticities, and it has pair-wise distinct partial elasticities of substitution. The direct addilog production function is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
The function is homogeneous only when all [rho]s are equal; and when [[rho].sub.i] = [[rho].sub.j] = [rho], all i, j, the function degenerates to the CES case. From Allen (1968: 342) the partial elasticities of substitution are given by
[[delta].sub.ij] = [F.sub.ij] [summation over (i)] [X.sub.i][f.sub.i]/[Fx.sub.i][x.sub.j] (2)
[f.sub.i] = [partial derivative]q/[partial derivative][x.sub.i], [f.sub.ij] = [[partial derivative].sup.2]q/[partial derivative][x.sub.i][partial derivative][x.sub.j]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and [F.sub.ij] is the co-factor of [f.sub.ij] in F. Now from Equation (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[f.sub.ij] = 1 + [rho]/q [f.sub.i][f.sub.j] i [not equal to] j, (4)
[f.sub.ii] = [f.sub.i](1 + [rho]/q [f.sub.i] - 1 + [[rho].sub.i]/[x.sub.i]). (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Hence, substituting Equation (6) and (7) into Equation (2), Allen's elasticity formula gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
From Equation (8), the partial elasticities of substitution are variable and depend on the parameters from the direct addilog production function. Thus, an estimate of the direct addilog function permits one to calculate the partial elasticities of substitution.
THE DIRECT ADDILOG PRODUCTION FUNCTION IS
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where q is the research output, the [x.sub.i]s are the productivity inputs, and [[beta].sub.i], [[rho].sub.i], and p are parameters to be estimated.
The critical question is how to measure the research output q. Merton views a publication as an input in the production of scientific knowledge. The output of a research publication (the quality of the article) is the impact that publication has on scientific thought, as measured by the quantity of citations received by that publication. (2) Merton argues that citation counts directly measure a publication's contribution to scientific knowledge as reflected by the degree to which others have found that work relevant to their own research. Citations stimulate further thought and add to the stock of scientific knowledge. A publication that has been cited 20 times is likely to have contributed more to the production of scientific knowledge than a publication that has been cited only once. (3) In economics, the total number of citations an article has accumulated over time is the most widely used measure to assess the research productivity of economists. (4)
Thirty-one top economics journals were surveyed for 1990. (5) This core journal set includes 20 of the 24 journals used by Graves et al. (1982), 26 of the 27 journals used by Laband (1985), 20 of the 24 journals used by Hirsch et al. (1984), and all eight of the journals used by Conroy et al. (1995) to rank economics departments. This sample journal set overlaps those used in other studies and is more inclusive since it draws from a wider variety of journals. While it is possible that a larger journal sample would produce different empirical results from those reported in this article, it seems unlikely since the core journal sample includes virtually every top general, specialty (business, finance, law, labor, urban, demography), theoretical, applied, and econometric journal in economics. In 1990, there were 1,737 papers published in the core journal set, of which 671 were co-authored (i.e., had two authors). Only articles written by two authors are examined because the overwhelming majority of collaborative research in economics consists of this form and the number of papers with multiple authors is too small for estimation.
The dependent variable [q.sub.i] in Equation (9) is the total number of citations (excluding self-citations) received by each co-authored article i published in the 31 journal set, compiled from the Social Sciences Citation Index, for the 10 years following publication (1990-1999). (6)
The input variables [x.sub.i]s are author- or article-specific productivity variables. The labor variables (L1, L2), author experience, are calculated as the number of years (as of 1990) since the receipt of each author's Ph.D. degree. (7) Besides author experience, there may be other labor variables, such as research and teaching assistance, time available for research, research support services, and teaching load, that influence publication output q. Unfortunately, such individual-specific microdata are unavailable, and as a consequence it is assumed that the joint influences of all these unobservable labor research inputs are random such that their combined effects can be subsumed into the stochastic error term.
The capital variables (K1, K2) are the cumulative stock of citations (excluding serf-citations) received by each co-author to all previous research over the period 1970-1989. The total number of citations an economist has accumulated over a period of time is a measure of the flow from a stock of past publications. This flow from prior publications measures the value of the research capital stock an economist can draw upon.
The remaining three inputs are article-specific characteristics. The relative prestige of a journal may account for some of the difference in the scientific contribution and value of articles. Leibowitz and Palmer (1984) ranked journals "based on their relative influence on the writings of academics." They argue that their 1980 impact-adjusted weights come closest to an ideal measure of the impact on the economics profession of papers published in 108 journals. Laband and Piette (1994), using the same methodology, updated the results and provide the 1990 impact-adjusted journal quality weights used in this study.
A widely held view is that longer articles are more substantive than shorter articles. Substantive scientific contributions, it is argued, generally require greater exposition than less substantive papers. However, 10 pages in one journal may be 20 pages in another journal. In order to account for the variation in journal page sizes, the length of each co-authored article is standardized to American Economic Review equivalent sized pages.
Articles written in certain fields may have a greater impact on the economics profession. In particular, articles written in the Journal of Economic Literature field codes A-F (General Economics; Economic Thought; Quantitative Methods; Microeconomics; Macroeconomics; International) tend to be more theoretical and/or quantitatively oriented and consequently are presumed to have a higher expected scientific value than those in the applied and interdisciplinary Journal of Economic Literature field codes G-R (Financial and Public Economics; Health, Education, Labor; Law; Industrial Organization; Business Administration; Economic History, Development and Systems; Agriculture; Urban/Regional). Some indirect support for this hypothesis is found in the work of Hawkins et al. (1973). Chairs were surveyed and asked to rank a list of economic journals that included two fictious entries. The fictious theoretical journal ranked 24th out of 87 journals, while the applied journal ranked 59th. To account for the differences in content, a binary variable equal to one for articles written in the Journal of Economic Literature fields A-F and zero otherwise is included. (8)
THE DIRECT ADDILOG PRODUCTION FUNCTION (Equation (9)) is estimated using iterative Gauss-Newton nonlinear least squares. (9) The empirical results are shown in Table 1, Column 1. As might be expected a priori, all the productivity variables are positively related to publication output with the exception of the co-author labor experience coefficients, which are negative. This latter result is consistent with the findings of other studies that as faculty age they tend to engage in more integrative (less time-intensive) research rather than basic (more time-intensive) research (McDowell 1982). Using the estimated parameters from Column I and Equation (3), we can compute the Allen partial elasticities of substitution between the co-authors. Since the partial elasticities are variable, the estimates are calculated at the sample means.
The partial elasticities of substitution between co-authors, [[sigma].sub.K1 K2], [[sigma].sub.L1 L2], [[sigma].sub.L1 K2], [[sigma].sub.L2 K1], are shown in Table 2, Row 1. (10) All the point estimates are positive, suggesting that the capital and labor of author 1 are equally facile substitutes in production with the capital and labor of author 2. It is possible that the finding that co-authored economic research involves researchers who are substitutes in production is because the production function model (Equation (9)) does not control for whether an article was theoretical or empirically oriented. Greater specialization and division of labor may be possible on empirically oriented articles than on pure theoretical papers. Empirical papers may require more complementary research skills and tasks that are easier to specify, separate, and assign than purely theoretical papers. In order to test whether the empirical results are sensitive to the inclusion of article type, Equation (9) was reestimated including a dummy variable equal to one if article i was purely theoretical (had absolutely no empirical estimation in the paper) and zero otherwise. The empirical results found that the coefficients of all the other variables previously reported in Table 1, Column 1 remained virtually unchanged and that the coefficient of the theoretical article dummy variable was equal to 0.00001 and statistically insignificant (t-statistic equal to 0.23). This is not surprising, since the impact of a theoretical article is already accounted for by both the journal quality variable (theoretical journals are consistently ranked higher than empirically oriented journals) and the page length variable (theoretical articles tend to be longer than empirically oriented articles).
In the previous estimation, no allowance is made for the possibility that a researcher's capital stock may depreciate over time. Lovell (1973) estimated that the citation value of articles depreciate at a rate of 8.2 percent over time. Using this figure implies that the half-life of citations from an article is approximately eight years. In order to account for depreciable research capital, the direct addilog production function is reestimated using as research capital stock (K1, K2) the cumulative number of citations received by each co-author in the 10 years prior to 1990. The empirical results appear in Table 1, Column 2, and the partial elasticities of substitution between co-authors with depreciable research capital stock is given in Table 2, Row 2. The point partial elasticities of substitution estimates are all positive and virtually identical to those when undepreciated research capital stock is used. The positive partial elasticities once again suggest that co-authorship in economics is characterized by a merger of substitute skilled researchers who combine substitute inputs.
The previous analysis implicitly assumed that the research capital stock of each co-author is homogeneous. That is, all previously cited articles were treated as having the same relative productive worth. However, all citations are not necessarily equal in value. A citation that appears in highly ranked journals may generate an increase in scientific thought by fostering additional citations--which in turn translates into an increase in an author's research capital stock. To allow for this possibility, a weighted measure of each co-author's research capital stock is computed. This is done by weighting each citation in an author's research capital stock (1970-1989) by the Laband-Piette impact-adjusted journal ranking. (11) This procedure gives greater weight to citations received in higher-ranked journals.
The direct addilog production function is reestimated using the impact-adjusted research capital stock measure for each co-author. The empirical results appear in Table 1, Column 3. The partial elasticities of substitution between co-authors (when each co-author's research capital stock is impact-adjusted) appear in Table 2, Row 3. All the partial elasticities of substitution are positive and virtually identical to those when underpreciated research capital stock is used (Table 2, Row 1)--the differences are less than 1.8 percent.
As a final test of the input relationship between co-authors, the direct addilog production function is reestimated after allowing for an author's impact-adjusted research capital stock to depreciate over time. This is accomplished by utilizing the impact-adjusted research capital stock for only 10 years prior to 1990 for each co-author. The estimated parameters appear in Table 1, Column 4, and the resultant partial elasticities of substitution between co-authors are reported in Table 2, Row 4. (12) The positive elasticities of substitution once again show that collaborative research in economics involves researchers who are equivalent substitutes in the production of an article of given quality.
THE ROBUST FINDING in all the empirical results presented in the previous section is that co-authors in economics are equivalent substitutes in the production of an article of a given quality. A logical question is why economists collaborate if they are equivalent substitutes in production. There are five possible explanations.
A. Measurement Errors
It is possible that the dependent variable in Equation (9), article citations, measures article quality incorrectly. Researchers may need to collaborate to produce articles of higher quality that require the specialization skills of co-authors. Complementary researchers may collaborate in order to produce higher quality articles than they could produce on their own. To test for this possibility, each citation to article i was weighted by the citing journal, using the Laband-Piette (1994) normalized weights (0 to 100) of the relative impact that articles published in a particular journal have on the economics profession. This procedure gives greater weight to citations received from high-quality journals. Equation (9) was then reestimated, and the partial elasticities of substitution recomputed. The partial elasticities of substitution between co-authors, when output is citation quality, are virtually identical to those reported in Table 2 (the differences were less than 1.2 percent). Co-authors in economics are found to be equivalent substitutes in the production of high-quality articles.
B. Endogenous Collaboration
A second explanation is that the decision by researchers to collaborate may be endogenous. The collaborative matchings may be endogenously based on the quality of the research. An economist may choose to produce high-quality research alone and collaborate with equally capable researchers on lesser quality articles. The problem with this explanation is that it is not supported by prior studies that have found the quality of co-authored articles (as measured by the number of article citations) is not less than sole-authored articles (Laband and Tollison 2000).
C. Self-Selection Bias
A third possible explanation for the finding that co-authors in economics are equivalent substitutes in production is that there may be self-selection bias in the decision to collaborate. Substitute skilled economists may choose to collaborate for consumption reasons. Research is a particularly isolated endeavor. Collaboration offers an opportunity for kinship and social interaction. Working with other researchers based on camaraderie may not be the optimal mechanism for selecting a collaborator if the desired outcome is an article of high quality. This is particularly true if economics departments reward collegiality in the determination of salaries (Diamond 1985).
D. Market Returns and Output Effects
The decision to collaborate by substitute skilled economists may reflect the academic market returns from a co-authored article. If economics departments assign a monetary return to co-authored articles (regardless of their quality) with n authors greater than 1/n times the monetary return of a sole-authored article, then similar smiled researchers have an incentive to collaborate. The incentive to collaborate is enhanced even more if equivalent substitute researchers can produce more articles (regardless of their quality) by collaborating than they could produce individually. Prior research tends to support the conclusion that economics departments do not fully discount co-authored articles (regardless of their quality). Moore, Newman, and Turnbull (2001) find that there is no difference in the monetary returns from co-authored versus sole-authored economics articles. McDowell and Smith (1992) report that economics departments make no distinction between co-authored and sole-authored articles in promotion decisions.
E. Division of Labor Inefficiencies
The magnitude of any conjectured efficiency gains from collaboration by researchers with complementary skills may be overstated or exaggerated. This may be due to an increase in the transaction costs of collaboration because of shirking by a co-author, duplication of effort, difficulty in coordinating tasks and monitoring individual performance, differential intensity levels of work product, and different individual levels of commitment to the research project. To determine if the efficiencies from skill complementarities are overstated, the partial elasticities of substitution were recomputed, assuming that the labor and capital input skill levels for one of the co-authors was of significantly greater numerical magnitude than that of the other co-author. The recomputed partial elasticities of substitution, assuming the labor and capital values for one of the co-authors was 150 percent, 200 percent, 300 percent, 400 percent, 500 percent of their sample mean, appear in Table 3. All the recomputed partial elasticities of substitution are still positive--collaboration in economics involves researchers with substitutable inputs. This suggests that any efficiency gains from collaborating by complementary skilled researchers may be overstated, possibly because the division of labor arrangements is not as clearly delineated as it is in the physical sciences.
THE LAST HALF of the 20th century has seen a substantial increase in the incidence of co-authorship in economics. This change in the organizational structure used to produce scientific research may have an effect on scientists' productivity. Understanding the input relationship between co-authors in economics is important because it may impact the evolution of economic knowledge. Previous research all used a hedonic function approach to impute the input relationship between co-authors in economics.
This article directly estimates the partial elasticities of substitution between co-authors in economics using the direct addilog production function. The direct addilog production function is empirically attractive because it places no a priori restrictions on the values that the partial elasticities of substitution can assume and because it has pair-wise distinct partial elasticities of substitution. The empirical results showed that the partial elasticities of substitution are positive--co-authors in economics are equivalent substitutes in production. This finding is robust whether the direct addilog production function is estimated using each co-author's research capital stock, depreciated research capital stock, impact-adjusted research capital stock, or depreciated impact-adjusted capital stock.
The empirical results that co-authors in economics are equivalent substitutes in production are also consistent with the presence of Merton's (1973) "Matthew Effect" in science. The Matthew Effect refers to the phenomenon that the allocation of recognition of a co-authored article is disproportionately skewed in favor of the author with the more established reputation. A necessary condition for the existence of the Matthew Effect is that the nature and the quality of the contributions by the collaborators be equivalent. This raises the obvious question: Why do economists collaborate if they are equally smiled and capable of producing sole-authored articles that protect a researcher's intellectual property rights for an innovative or significant scientific contribution?
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(1.) In order to estimate the input relationship between co-authors, it is important to distinguish between the production function relationship between co-authors at a particular moment in time and changes in the production function relationship over time. It is the former static production function framework between co-authors that is of interest in this article.
(2.) Anecdotal evidence suggests that economists do believe the scientific contribution of a publication is measured by the number of citations generated. Many of the leading economists, discussing classic articles (which were initially rejected), asserted that the impact these classic articles had on subsequent literature and empirical work was evident from the number of citations these articles had received (Gans and Shepherd 1994).
(3.) The limitations in the use of citations is discussed and dismissed by Leibowitz and Palmer (1983). They ask rhetorically if an article (or an economist) with few citations is considered to be a significant scientific contribution, then why has it not generated more citations?
(4.) Citations have been used to rank the quality of economics journals (Liebowitz and Palmer 1984), economics departments (Laband 1985), and economists (Medoff 1989).
(5.) The 31 journal set used was comprised of: American Economic Review, Demography, Econometrica, Economic Inquiry, Economic Journal, Economica, Industrial and Labor Relations Review, International Economic Journal, International Economic Review, Journal of Business, Journal of Econometrics, Journal of Economic Theory, Journal of Finance, Journal of Financial Economics, Journal of Human Resources, Journal of Labor Economics, Journal of Law and Economics, Journal of Mathematical Economics, Journal of Monetary Economics, Journal of Money, Credit and Banking, Journal of Political Economy, Journal of Post-Keynesian Economics, Journal of Public Economic, Journal of Urban Economics, National Tax Journal, Public Choice, Quarterly Journal of Economics, Rand Journal of Economics, Review of Economic Studies, Review of Economics and Statistics, and Southern Economic Journal.
(6.) Self-citations are excluded because they may not truly be productive and may be more likely to be "salesmanship or advertising" citations.
(7.) The data on author experience were obtained from the American Economic Association's Directory of Members.
(8.) Because the field code independent variable ([x.sub.i]) has only two values, 1 or 0, this means its term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [x.sup.-[rho]i.sub.i] equals either [[beta].sub.i] or 0. Thus its exponent [[rho].sub.i] was set equal to one in the estimation of the direct addilog function.
(9.) The iterative Gauss-Newton nonlinear least squares procedure is described in Greene (1990).
(10.) It is not feasible to perform statistical tests of significance on the partial elasticities of substitution. From Equation (2), each partial elasticity of substitution is a nonlinear function of all the estimated [[beta].sub.i], [[rho].sub.i], and [rho] coefficients. The explicit standard errors of the partial elasticities of substitution is a complex nonlinear expression involving over 150 terms.
(11.) Author Impact Adjusted Research Capital Stock = [summation over (i)] [w.sub.i] [c.sub.i], where [w.sub.i] is the 1990 Laband-Piette impact-adjusted weight for journal i and c, is the number of citations that appear in journal i between 1970 and 1989. 12. The partial elasticities of substitution between all the inputs are available upon request.
MARSHALL H. MEDOFF, The author is at the Department of Economics, California State University, Long Beach, 1250 Bellflower Boulevard, Long Beach, CA 90840-4607; e-mail: email@example.com.
Table 1 Gauss-Newton Nonlinear Least Squares Estimates of Direct Addilog Production Function Independent Variables (Coefficient/Exponent) (1) (2) (3) (4) L1 [[beta].sub.1] -0.3716 -0.3687 -0.1691 -0.1621 (2.72) (2.72) (1.62) (1.55) [[rho].sub.1] 0.3943 0.3961 0.3982 0.3995 (1.21) (1.18) (1.31) (1.37) L2 [[beta].sub.2] -0.1018 -0.1224 -0.1373 -0.1539 (1.89) (1.48) (1.47) (1.63) [[rho].sub.2] 0.3019 0.3022 0.3012 0.3021 (1.51) (1.61) (1.69) (1.76) K1 [[beta].sub.3] 0.2588 0.2616 0.1451 0.1462 (3.04) (3.04) (2.53) (2.47) [[rho].sub.3] 0.2885 0.2841 0.2735 0.2728 (1.52) (1.56) (1.04) (1.05) K2 [[beta].sub.4] 0.0235 0.0305 0.0321 0.0420 (1.35) (1.44) (1.57) (1.41) [[rho].sub.4] 0.2982 0.2971 0.2937 0.2927 (1.17) (1.23) (1.25) (1.32) Journal Quality [[beta].sub.5] 0.3990 0.4096 0.4074 0.4049 (4.24) (4.35) (4.29) (4.17) [[rho].sub.5] 0.2831 0.2761 0.2815 0.2786 (2.11) (2.18) (2.20) (2.18) AER Pages [[beta].sub.6] 0.3589 0.3308 0.3450 0.3415 (2.16) (1.99) (2.08) (2.05) [[rho].sub.6] 0.4953 0.4942 0.4942 0.4926 (1.15) (1.16) (1.17) (1.16) JEL Fields [[beta].sub.7] 0.0551 0.0581 0.0539 0.0575 (1.99) (2.11) (1.92) (2.04) [rho] 1.0296 1.0279 1.0040 1.0105 (2.37) (2.42) (1.80) (2.01) Note. Absolute value of t-statistics in parentheses. K1, K2 = Research Capital Stock (1970-1989) in Equation (1). K1, K2 = Depreciated Research Capital Stock (1980-1989) in Equation (2). K1, K2 = Impact-Adjusted Research Capital Stock (1970-1989) in Equation (3). K1, K2 = Depreciated Impact-Adjusted Research Capital Stock (1980-1989) in Equation (4). Table 2 Partial Elasticities of Substitution Between Co-Authors Variables K1, K2 [[sigma].sub.K1 K2] [[sigma].sub.L1 L2] (1) Research Capital 1.076 0.992 Stock (1970-1989) (2) Depreciated 1.085 0.994 Research Capital Stock (1980-1989) (3) Impact-Adjusted 1.076 0.974 Research Capital Stock (1970-1989) (4) Depreciated 1.099 0.992 Impact-Adjusted Research Capital Stock (1980-1989) Variables K1, K2 [[sigma].sub.L1 K2] [[sigma].sub.L2 K1] (1) Research Capital 0.995 1.074 Stock (1970-1989) (2) Depreciated 0.998 1.081 Research Capital Stock (1980-1989) (3) Impact-Adjusted 0.981 1.071 Research Capital Stock (1970-1989) (4) Depreciated 1.004 1.091 Impact-Adjusted Research Capital Stock (1980-1989) Table 3 Partial Elasticities of Substitution Between Co-Authors Evaluated at Different Percentages of the Sample Means Percent of the Sample Mean 150% 200% 300% 400% 500% [[sigma].sub.K1 K2] 1.064 1.054 1.050 1.046 1.043 [[sigma].sub.L1 L2] 0.981 0.971 0.967 0.964 0.961 [[sigma].sub.L1 K2] 0.983 0.974 0.971 0.966 0.962 [[sigma].sub.L2 K1] 1.061 1.051 1.047 1.043 1.040 Note. K1,K2 = Author Research Capital Stock (1970-1989). L1,L2 = Author Experience (number of years since receipt of Ph.D. degree).…