14.662 Spring 2015 Lecture Note: Market Structure, Organizational Structure and Wage Structure Revised 3/1/2015 MIT and NBER 1 1 Capital-skill complementarity In 1969, Zvi Griliches advanced the hypothesis that capital and skill are relative complements. [Be sure you are clear what it means for two goods to be complements, and that you can distinguish between p-complementarity and q-complementarity.] By this he meant that although capital is likely to be complementary to both skilled and unskilled labor, it tends to be more complementary to skilled labor. If this hypothesis is correct, capital deepening—that is the process of capital accumulation— will tend to increase the relative demand for skilled labor. Under this hypothesis, explicit technical change per se is not needed to explain rising skill demands (though one might think that there is a deeper model of technical change underlying this). We simply need capital accumulation. This hypothesis is clearly wrong if one takes it to apply to all times and places in even recent human history. The capital deepening the 19th century (i.e., the factory system) was probably quite complementary to unskilled labor. But it could potentially be a valid hypothesis for the post-WWII era. 1.1 The Declining Price of Equipment Capital: Krussell, Ohanian, Rios-Rull and Violante 2001 (Econometrica) The ﬁrst ’modern’ paper that takes this idea most seriously is Krussell, Ohanian, Rios-Rull and Violante (2001 Econometrica). They build on an interesting fact observed by Robert Gordon that the relative price of capital equipment has been falling steadily in the postwar period. Moreover, this rate of decline of equipment prices may have accelerated sometime during the mid to late 1970s. This observation, combined with the assumption of capital-skill complementarity, could potentially give rise to an increase in the relative demand for skilled labor. KORV consider the following aggregate production function � � �μ/λ �(1−α)/μ Y = Ksα b1 Lμ + (1 − b1 ) b2 Keλ + (1 − b2 ) H λ where Ks is structures capital (such as buildings), and Ke is equipment capital (such as machines), and b1 , b2 ∈ [0, 1]. • Notice that structures capital Ks enters this production function Cobb-Douglas style—so there is no capital-skill complementarity between structures and skilled labor. • The parameter σes = 1/ (1 − λ) is the elasticity of substitution between equipment and skilled workers • The parameter σus = 1/ (1 − μ) is the elasticity of substitution between unskilled workers and the equipment-skilled worker aggregate. • [Note that a similar theoretical model is used by Autor, Acemoglu and Lyle 2004. You could consult that paper for an alternative exposition if you don’t ﬁnd KORV clear.] 1 • If σes < σus (i.e., μ > λ), equipment capital is more complementary (less substitutable) to skilled workers than unskilled workers, and as a result, an increase in Ke will increase the wages of skilled workers more than the wages of unskilled workers. The skill premium in this model is �(μ−λ)/λ � (1 − b2 ) (1 − b1 ) H λ−1 b2 Keλ + (1 − b2 ) H λ wH ω= = b1 Lμ−1 wL Diﬀerentiation shows that as long as μ > λ, ∂ω/∂Ke > 0. So provided that equipment capital is more complementary to skilled workers than unskilled workers, an increase in the quantity of equip ment capital will increase the demand for skills. Since the post-war period has been characterized by a decline in the relative price of equipment goods, there will be an associated increase in the of equipment capital, Ke , increasing the demand for skills steadily. KORV estimate this model and ﬁnd that declining price of equipment capital can explain a large share of the rise in relative demand for skilled workers in the United States. But there are a number of problems with this explanation that limit its plausibility: 1. First, it is hard to develop much conﬁdence in measures of the real or even relative price of capital. Due to issues of simple measurement, quality change and inﬂation, one needs to be quite skeptical about making too much of a potential change in the rate of change (i.e., the second derivative) of a price series. While no economist would dispute that the price of capital equipment declined dramatically in recent decades (computers being the best example), whether that rate has increased or decreased in a given period is much harder to prove. 2. One would generally expect a decline in the price of capital to yield faster productivity growth and higher living standards, etc. • As we discussed in the ﬁrst lecture in 14.999, incomes stagnated in exactly the period when relative equipment prices were ostensibly declining. It is hard to reconcile vast technological improvements with declining or stagnating living standards (although you can be sure that many authors have tried). Moreover, it is hard to see how this hypothesis can explain falling real (in addition to relative) wages of less-skilled workers. Since the equipment-high-skill CES aggregate is a q-complement of low-skilled workers, capital deepening should also raise real wages of low-skilled workers, even if it raises wages of high-skilled workers by more. • [Note this point above should be distinguished from the idea that technical change is skill-biased. A skill-biased technological shift does not have to raise living standards by a great deal to substantially aﬀect relative earnings (a point made by Krugman in his 2000 Journal of International Economics article). But a rapid decline in the real price of capital should directly raise real living standards.] 2 3. Probably the most damaging evidence against the simple capital-skill complementarity expla nation is empirical. • In discussing KORV’s paper, Katz-Autor (1999) write, “Their measure of the capitalskill complementarity eﬀect on relative wages evolves similarly to a linear time trend. Thus the aggregate time series model of Krussell et al. (1997) [when their article was just a working paper] attributes variations in changes in the skill premium around trend (such as a sharp decline in the skill premium in the 1970s and sharp rise in the 1980s) to variations in the rate of growth of the relative skill supplies and to unobserved demand shocks (the residual).” • Translation: the KORV time series on capital equipment prices is not the variable pro viding the explanatory power in their model; it is primarily the supply measure that is aﬀecting the skill premium as in the Katz-Murphy model. • Acemoglu (2002) makes this point more forcefully. He estimates the Katz-Murphy re gression augmented with the KORV series for relative capital prices and a simple linear time trend. As Table 2 of Acemoglu shows, the KORV measure performs less well than a linear time trend. In fact, it is never signiﬁcant when a time trend is included. • Bottom line: the KORV analysis is elegant but not especially persuasive. It is also widely misunderstood. Most readers (or citers) of KORV appear to believe that it is the equipment-price variable that is doing the work in their model, when in fact, KORV primarily rediscovered the Katz-Murphy 1992 ﬁndings: relative supplies plus a linear time trend do a pretty good job of explaining relative wages of skilled versus unskilled in the U.S. for the last 40 years. 1.2 Competing Organizational Forms and Skill Supply: Acemoglu 1999 AER The 1999 AER paper by Acemoglu considers a model that is conceptually related to B&G but does not make use of the crucial capital-eﬃciency assumption. Acemoglu considers a search model in which ﬁrms can use two types of technology: one that uses only high skill workers and the second that uses both high and low skill. The former technology has higher TFP conditional on having a high skill workforce. But if high skill workers are relatively rate, they will have a low arrival rate in the search model. This makes it costly to open vacancies using the strongly skill-dependent technology. Hence, ﬁrms will primarily use the mixed technology. However, an exogenous increase in skill supply may induce ﬁrms to open skill-dependent jobs. This will raise wages of high skill workers (since they are more productive in the skill-dependent sector) but lower wages of low-skill workers (under the assumption that high and low skill workers are q-complements under the mixed technology). This model is analytically complex, which may be a virtue if you are a search theorist, otherwise probably not. Acemoglu’s 2002 JEL paper oﬀers a much simpler alternative that has the same ﬂavor. There is a scarce supply of some factor K, which could be capital or entrepreneurial talent, 3 for example. Skilled workers work with the production function Yh = Aαh Kh1−α H α , and unskilled workers work with the production function Yl = Aαl Kl1−α Lα , where Yh and Yl are perfect substitutes and Kl + Kh = K. Observe: if both production functions are in use, the return to capital K must be equated across H and L sectors. Now consider a factor-augmenting technological change, speciﬁcally an increase in αh . This raises productivity in the H sector, which increases the return to K in the H sector (since K and H are q-complements). To equilibrate the return to capital across sectors, K must ﬂow from the L sector to the H sector. Given the q-complementarity between K and L, wages of L workers must fall. Hence, this model generates falling wages of low-skilled workers. This model also implies a rise in the return to K. So, if K is elastically supplied—that is, the stock of K rises until returns fall back to the prior level—the real wages of low-skill workers do not fall. The closely related paper by Beaudry and Green in AER in 2003 (“Wages and Employment in the United States and Germany: What Explains the Diﬀerences?”) features one critical diﬀerence with the model above. In B&R the rising supply of skilled workers causes work to reallocate from the low-skill to high-skill sector, ultimately leading to capital starvation in the low-skill sector. Thus, while in the model above, traditional factor-biased technical change leads to rising demand for high skilled workers whereas in B&G, supply itself is suﬃcient. In both models, it is the inelastic supply of capital that allows low skilled wages to fall. [B&G is actually a traditional Hecksher-Ohlin two-skill, two-sector model, with one unusual wrinkle: both sectors produce an identical good; thus, changes in factor supplies or prices aﬀect which ’technique’ is used to produce the ﬁnal good, and this in turn aﬀects sectoral allocation, wages and capital demands.] 1.3 Causal evidence on capital-skill complementarity One of the long sought holy grails of the SBTC literature has been a compelling causal test of the hypotheses that newer technologies (new capital, computers, etc.) are relatively complementary to skilled labor. The 2011 QJE paper by Ethan Lewis (“Immigration, Skill Mix, and CapitalSkill Complementarity”) provides one useful example. While the empirical test implemented by this paper is quite indirect, it is nevertheless insightful and reasonably compelling. You will work through this paper in Problem Set #1. The 2015 paper by Akerman, Gaarder and Mogstad (“The Skill Complementarity of Broadband Internet”) uses the staggered rollout of broadband availability across Norway to estimate the impact of broadband on both the productivity and wages of high, medium and low skill workers. The results are so strongly consistent with the economic prior that it almost leaves one skeptical (though 4 I’m not suggesting skepticism is warranted). In brief, AGM ﬁnd that broadband increases the skilled labor output elasticity in production (the exponent on skilled labor in a Cobb-Douglas production function), raises high skill wages and lowers medium and low skill wages, and causes wage polarization among high, medium and low skill occupations (even consistent on worker education). It actually adds to the paper’s plausibility that the estimated magnitudes are actually quite modest; that is, this paper does not suggest that broadband dramatically changed the Norwegian wage structure. 2 Microfoundations for Technologies and Tasks (and a Test of Sorts): A Framework (ALM 2003 QJE ) • There have been a variety of eﬀorts to formalize the observations from these closely observed studies. One is given by Autor, Levy and Murnane (2003). They take a ‘task-based’ ap proach to understanding the ‘skill-content’ of technical change: modeling the human tasks that computers complement and those for which they substitute. • ALM argue that the ‘skill bias’ of recent technological change is non-monotonic, in contrast to what the simplest capital-skill complementarity stories would suggest. Their argument, focusing on the role of computerization, has three papers: 1. Computer technology has two intrinsic characteristics. One, it can process symbols, meaning that it can execute any well speciﬁed information processing task. Two, its capability is circumscribed by what programmers, engineers and scientists know how to describe using computer code. This is more limiting than it might seem. Many tasks involving vision, locomotion, problem solving, pattern recognition, language interpreta tion and communication cannot current be described with computer code—that is, we do not know ‘the rules’—even though we accomplish these tasks almost eﬀortlessly. 2. Because of these properties, computers are primarily used to substitute for ‘routine’ tasks—those that are readily formalized and ‘routinized’ in computer code—while com plementing ‘non-routine’ cognitive tasks such as problem solving. 3. Although computer technology’s core attributes are static (that is, it hasn’t fundamen tally changed in 200 years), the price of computer capital has fallen considerably. Nord haus 2001 (“The Progress of Computing”) writes, “There has been a phenomenal in crease in computer power over the twentieth century. Performance in constant dollars or in terms of labor units has improved since 1900 by a factor in the order of 1 trillion to 5 trillion, which represent compound growth rates of over 30 percent per year for a century.” It is this (assumed) exogenous price decline that is the driving force in the ALM model. • The conceptual building blocks of the ALM framework are given in Table 1 of their paper. 5 • The ALM model makes three assumptions that appear reasonably well motivated by the qualitative evidence above. 1. Computer capital is more substitutable for humans in carrying out routine tasks than non-routine tasks (this is similar to Krussell et al. in Econometrica). 2. Routine and non-routine tasks are themselves imperfect substitutes. 3. Third, at least for cognitive tasks, greater intensity of routine inputs increases the marginal productivity of non-routine inputs (Q-complements). • The ALM model is derived below. I probably will not spend much time on this in class. You are not responsible for the speciﬁcs of this model. You may or may not ﬁnd the derivation interesting. 2.1 Production • ALM embed these three assumptions in a framework that nests a Roy (1951) model—whereby workers self-select the sector where they hold comparative advantage—inside a simple CobbDouglas production framework. (See also the 2006 Autor-Katz-Kearney NBER Paper #11986 which presents a slightly richer—and probably simpler—version of this model.) • Speciﬁcally, the ALM model is built on production function with ‘two tasks:’ Q = (Lr + C)1−β Lβn (1) where N, R are eﬃciency units of Routine and Non-Routine labor inputs and C is computer capital. (note, they are combined Cobb-Douglas, meaning that σrn = 1). • Computer capital is supplied perfectly elastically at market price ρ per eﬃciency unit ALM assume that computer capital is a perfect substitute for routine tasks (this assumption can be weakened of course), where ρ is falling exogenously with time due to technical advances. The declining price of computer capital is the causal force in the model. • Three observations on this prod’n function: 1. C and Lr are perfect substitutes 2. Routine and nonroutine labor inputs are Q-complements: doing more of one raises the marginal productivity of the other 3. The elasticity of substitution σ between Lr and Ln is 1 due to the Cobb-Douglas produc tion structure. This means that computer capital is more substitutable for routine than nonroutine labor. By implication, computer capital and nonroutine labor are relative complements. 6 2.2 Labor supply • Assume a large number of income-maximizing workers, each of whom inelastically supplies one unit of labor. • Workers have heterogeneous productivity endowments in both routine and non routine tasks, with Ei = [ri , ni ] and 1 > ri , ni > 0 ∀ i. A given worker can choose to supply ri eﬃciency units of routine task input, ni eﬃciency units of nonroutine task input, or any convex combination of the two. Hence, Li = [λi ri , (1 − λi ) ni ] where 0 ≤ λ ≤ 1. [In equilibrium, each worker will choose to specialize in one task or the other. Choices will diﬀer among workers.] • These assumptions imply that workers will choose tasks according to comparative advantage as in Roy [1951]. Hence, relative task supply will respond elastically to relative wage levels. [If instead workers were bound to given tasks, the implications of the model for task productivity would be unchanged, but technical progress, reﬂected by a decline in ρ, would not generate re-sorting of workers across jobs.] 2.3 Equilibrium conditions Two conditions govern equilibrium in this model: 1. Given the perfect substitutability of routine tasks and computer capital, the wage per eﬃciency unit of routine task input is pinned down by the price of computer capital: wr = ρ. (2) Computer capital in this model is a directly skill-replacing technology and advances in com puter technology lower the wages of workers for whom computers are a substitute. 2. Worker self-selection among tasks—routine versus nonroutine—must clear the labor market. • Deﬁne the relative eﬃciency of individual at nonroutine versus routine tasks as ηi = ni /ri . (3) Our assumptions above imply that ηi ∈ (0, ∞). • At the labor market equilibrium, the marginal worker with relative eﬃciency units η ∗ is indiﬀerent between performing routine and nonroutine tasks when η∗ = wr . wn (4) • Individual i supplies routine labor (λi = 1) iﬀ ηi < η ∗ , and supplies nonroutine labor otherwise (λi = 0). (Note that λ equals either zero or one—a result of the model not an assumption.) 7 So, if wr wn falls (the relative wage of nonroutine tasks rises), labor supply to nonroutine tasks rises. • To quantify labor supply, write the functions g (η), h (η), which sum population endowments in eﬃciency units of routine and nonroutine tasks respectively at each value of η. So, g (η) = ri · 1 {ηi < η} and h (η) = i ri · 1 {ηi ≥ η} where 1 {·} is the indicator function. We i further assume that η has non-zero support at all η ∈ (0, ∞), so that g (η) is continuously upward sloping in η, and h (η) is continuously downward sloping. • Assuming that the economy operates on the demand curve, productive eﬃciency requires: wr = ∂Q ∂Q = (1 − β) θ−β and wn = = βθ1−β , ∂Ln ∂Lr (5) where θ = (C + g (η ∗ ))/h (η ∗ ) , is the ratio of routine to nonroutine task input in production. • It is now an easy step to solve for the model’s 5 endogenous variables: (wr , wn , θ, C, η). 2.4 Solution • We know that a decline in the price of C must reduce wr one-for-one: ∂ ln wr /∂ ln ρ = 1. Using equation (5), this implies that ∂ ln θ 1 =− . ∂ ln ρ β (b/c ln wr = ln (1 − β) − β ln θ, hence ∂ ln wr /∂ ln θ = −β (∂ ln θ/∂ ln ρ). So, if ∂ ln wr /∂ ln ρ = 1, this implies ∂ ln θ/∂ ln ρ = −1/β.) • By implication—and not surprisingly—the economy becomes more ‘routine-task-intensive’ when the price of computer capital falls. • From the perspective of producers, the rise in routine task demand could be met by either an increase in C or an increase in Lr (or both). Only the ﬁrst of these will occur. Because routine and nonroutine tasks are q-complements, the relative wage paid to nonroutine tasks rises as ρ declines: ∂ ln (wn /wr ) 1 ∂ ln η 1 = − and = . ∂ ln ρ β ∂ ln ρ β Consequently, marginal workers reallocate labor input from routine to nonroutine tasks. This means that increased demand for routine tasks is met by an inﬂux of computer capital not labor. • So, an exogenous decline in the price of computer capital raises the marginal productivity of nonroutine tasks, causing workers to reallocate labor supply from routine to nonroutine task input. Although routine labor input declines, an inﬂow of computer capital more than compensates, yielding a net increase in the intensity of routine task input in production.. 8 2.5 Industry level implications • Does this model have testable microeconomic implications? So far, no. Because the causal force in the model is the price of computer capital, we have only a single macroeconomic time series available. • ALM argue that additional leverage may be gained by considering equation (1) as representing the production function of a single industry, with distinct industries j producing outputs qj that demand diﬀerent mixes of routine and nonroutine tasks. • Write the production function for industry j as, 1−βj βj nj , qj = rj where βj is the industry-speciﬁc factor share of nonroutine tasks, and rj , nj denote the industry’s task inputs. All industries use Cobb-Douglas technology, but industries with βj smaller are more routine task intensive. • Assume that consumer preferences in this economy can be represented with a Dixit-Stiglitz (AER, 1977), utility function, ⎛ U (q1 , q2 , ..., qJ ) = ⎝ ⎞ X qj1−ν ⎠ 1 1−ν , j where 0 < ν < 1. The elasticity of demand for each good is −1/ν, with the market clearing price inversely proportional to the quantity produced: pj (qj ) ∝ qj−ν . The key feature of this utility function is that all goods are gross-substitutes and demand for each is elastic. With many goods, one approaches constant elasticity of demand for each good. (This follows because the expenditure share for any one good is negligible so one can eﬀectively ignore the cross-price eﬀects.) • Industry proﬁt maximization yields the following ﬁrst order conditions for wages: � � β −β −ν (1 − βj ) (1 − ν) nj j rj j , −ν � � β −β = nβj −1 r1−βj βj (1 − ν) nj j rj j . β −βj w r = nj j r j wn • Rearranging to obtain factor demands gives ) (1−βj )(1−ν) ν ) (1 − β w j n = wn (βj (1 − ν))1/ν , · ρ βj ( ) βj (1−ν) ν wn (1 − βj ) 1/ν = wr ((1 − βj ) (1 − ν)) , · ρ βj ( nj rj 9 which are kind of messy. • Two or three propositions follow from these equations: 1. Although all industries face the same price of computer capital, the degree to which industries adopt computer capital as its price declines depends upon βj . For a given price decline, the proportionate increase in demand for routine task input is larger in routine task intensive (βj small) industries, as may be seen by taking the cross-partial derivative of routine task demand with respect to ρ and βj : βj (1 − ν) − 1 ∂ 2 ln rj ∂ ln rj 1−ν = < 0 and = > 0. ∂ρ νρ ∂ρ∂βj νρ Although we cannot observe βj , a logical proxy for it is the observed industry level of routine task input in the pre-computerization era. We can therefore test whether industries that were historically (i.e., pre-computer era) intensive in routine tasks adopted computer capital to a greater extent than industries that were not. 2. Due to the complementarity between routine and nonroutine inputs, a decline in the price of computer capital also raises demand for nonroutine task input. This demand increase is proportionately larger in routine task intensive industries: ∂ ln nj (βj − 1) (1 − ν) ∂ 2 ln nj 1−ν = < 0 and = > 0. ∂ρ νρ ∂ρ∂βj νρ Recall, however, that labor supply to routine tasks declines with ρ. Rising routine task demand must therefore be satisﬁed with computer capital. Hence, sectors that invest relatively more in computer capital will show a larger rise in nonroutine labor input and a larger decline in routine labor input. 3. Analogously (by an informal argument), ALM suggest that occupations that make rel atively larger investments in computer capital will, for the same reasons, show larger increases in labor input of nonroutine tasks and larger decreases in labor input of routine tasks. 2.6 ALM and related empirical evidence Using the Dictionary of Occupational Titles, which provides observational measures of job content in representative occupations in the U.S. over several decades, ALM provide a number of pieces of evidence indicating this ‘task based’ story has empirical traction: 1. The skill content measures move as predicted over 1960 - 1998 (Figure I). 2. ALM ﬁnd task movement in the predicted direction along 3 margins: (a) Within industries: Industries that computerize change task input as predicted, and this relationship grows much more pronounced with time. (Table III) 10 (b) Within education groups within industries: Industries that computerize relative more change task input even among workers with identical education. So, industry-level com puterization is not simply mechanically reﬂecting educational upgrading but is correlated with changes in the structure of task changes among workers at all education levels. (Ta ble V) (c) Within occupations: Using data from successive editions of the DOT, ALM ﬁnd that occupations that intensively computerized also experienced large declines in routine cog nitive tasks and increases in cognitive analytic content. (Table VI) 3. The ALM framework also predicts where computers should have been adopted: Industries with high levels of routine task content should have become computer intensive as the price of computing fell. They ﬁnd this Computer adoptionj,1960−1997 = −24.56 +1.85×Routine Task Sharej,1960 (19.18) (0.48) R2 = 0.10, n = 140 The 2006 JOLE paper by Alexandra Spitz uses much better data German to explore a similar set of task changes over 1979 - 1999. Surprisingly, her evidence is quite consistent with ALM. Her paper also potentially opens a new chapter in the debate over why wage structure changes occurred in the U.S. but not continental Europe. As noted in class, one of the huge puzzles of the SBTC literature is that almost all of the evidence linking technical change to wage structure changes comes from the U.S. and the U.K. By contrast, continental Europe—which presumably has access to the same technology—has not had the same surge of inequality. And the two biggest ‘slackers’ are France and Germany. This puzzle is discussed by Piketty and Saez in the QJE, by Krueger and Pischke in their widely cited book chapter on “Observations and Conjectures on the U.S. Employment Miracle,” by Card, Kramarz and Lemieux (1999), etc. This pattern has caused Piketty and Saez to advance “Social norms” rather than demand shifts as an explanation for rising inequality in the Anglo countries relative to continental Europe. Spitz’s paper oﬀers a new angle of attack on this question. Rather than focusing on wage struc ture changes—which are a long run outcome of changes in skill demand, possibly strongly mediated by labor market institutions—Spitz instead analyzes the link between changing technology and changing job tasks—a primitive in any skill-bias story. Here, Spitz demonstrates that in Germany, as in the United States: 1) job skill requirements have been strongly rising along the lines predicted by ALM; 2) these changing skill requirements are all strikingly concentrated in computer-using occupations. So, the Spitz paper possibly moves the debate forward by reframing the puzzle of the U.S. versus continental Europe (at least for Germany): rather than asking why SBTC has changed skill requirements in the US/UK but not continental Europe, it may be fruitful to ask why similar changes in skill requirements in Anglo countries and continental Europe have not lead to similar 11 changes in wage structures. An interesting paper by Goos and Manning (2003) takes another line of attack on the question of changing skill requirements. G&M note that the ALM task framework implies a ‘polarization’ of the distribution of jobs, with growth of very low and very high skill positions (non-routine manual and non-routine cognitive) and a culling of the ranks of ‘medium skill’ jobs such as accountants, clerical workers, insurance adjusters, etc. G&M take this idea to the data and ﬁnd evidence for “Lousy and Lovely Jobs: The Rising Polarization of Work in Britain.” (See the paper for details.) 3 Why is Information Technology Associated with Greater Worker Discretion and Teamwork (i.e., less ‘Taylorism’)? The models discussed so far operate at a very high level. They posit a dramatic change in the organization of work but do not tell a detailed story about how speciﬁcally work organization has changed nor why this change has been especially demanding of skilled labor. It would be helpful to have a substantive theory of the nature of recent technological or organizational change that didn’t ultimately boil down to the assumed sign on a derivative or the magnitude of some (ultimately unmeasurable) elasticity. Caroli and Van Reenan study—in a reduced form manner—the impact of work reorganization on skill demands. They provide evidence from British and French establishments that measures of organizational change—such as decentralization of authority—have a strong predictive power for the demand for skills at the establishment level, even after controlling for other determinants of the demand for skills, such as computers. Bresnahan, Brynjolfsson and Hitt (2000) provide U.S. based evidence in a similar vein. Neither of these studies get a great deal further than documenting correlations between reorganization and skill input, but the correlations are striking. There is also a variety of theoretical and empirical work that oﬀers explicit hypotheses on the link between computerization and changes in the content of work and organization of production. Some recent careful case studies are in my view among the richest material in this literature (which may suggest that economic theory would beneﬁt if economic theorists would leave their desks more often). Ann Bartel, Casey Ichniowski, and Kathryn Shaw have performed detailed case studies of changing production technology in three industries: steel, medical devices, and valves. The 2007 QJE paper on your syllabus is speciﬁcally about the valve industry. Many authors have noted that problem solving teams and other ‘lean production techniques’ are often paired with information technologies, suggesting that newer Human Resource Practices (HRM) and IT are complements. Why would this be so? Some recent careful case studies are in my view among the richest material in this literature (which suggests to me that theory would beneﬁt if theorists would leave their desks more often). Ann Bartel, Casey Ichniowski, and Kathryn Shaw have performed detailed case studies of changing production technology in three industries: steel, medical devices, and valves. The 2007 QJE paper on your reading list is speciﬁcally about the valve industry. 12 My interpretation of their main ﬁndings is as follows: 1. Information technology has removed much of the mechanized, repetitive, rote components of production. Many repetitive tasks are now performed by machines that coordinate and monitor production. In the BIS valve study, computer-guided industrial lasers inspect com pleted valves for speciﬁcation compliance with precision measured in microns. Previously, this time-consuming inspection step was done by hand. Similarly, the process of machine setup for new runs and the coordination of production as products move between machining phases have also been extensively computerized. This automation is feasible because computerized machining tools are far more ﬂexible than older forms of capital. Much of the cumbersome reconﬁguration of the assembly line needed for diﬀerent products and batches is now handled directly by the machinery. All of these production changes are process improvements. 2. One consequence of the increased ﬂexibility of the process is a change in the set of products produced. The valve ﬁrms studied by BIS, which are those that have continued producing in the U.S., have increasingly moved out of commodity production and into ‘mass customiza tion.’ They exploit the ﬂexibility of the new capital to do shorter runs of more complex products. This in turn requires greater ﬂexibility on the part of the workforce. But notice that commodity valve manufacturing, which is increasingly performed overseas, may not be undergoing similar changes in skill demand. [A 2005 paper by Chong Xiang of Purdue in the ReStat (“New Goods and Rising Skill Premium: An Empirical Investigation”) presents detailed evidence that new goods are increasingly skill intensive. This suggests that product as well as process innovations may contribute to the evolution of skill demands. 3. Workers are increasingly required to use abstract reasoning to manage production. Whereas workers used to spend much of their time in contact with physical materials, much of the ‘work’ is now performed from the ‘control pulpit’ of highly automated factories where the key task is to monitor multiple assembly lines via large displays. The challenge is to be alert to problems as they arise and make subtle adjustments that improve eﬃciency, reduce error, enhance quality. In this setting, a good decision can have enormous value added and a bad decision can destroy large amounts of capital. 4. Production work is frequently reorganized into teams where workers have responsibility for quality control and are required to solve problems and develop better ways to organize pro duction. As observed by BIS, and Bresnahan, Brynjolfsson and Hitt (2002), and Caroli and van Reenan (2001) problem solving teams and other ‘lean production techniques’ are often paired with information technologies, suggesting that these Human newer Resource Practices (HRM) and IT are complements. (More on why this might be below...) These observations are drawn from the manufacturing sector, which provides a declining share of employment in almost all advanced countries. How relevant are they to the service sector? Similar case studies for the service sector are not in abundance. One by Autor, Levy, Murnane (2002, ILRR) 13 on the computerization of the back oﬃce check-processing function of a bank provides descriptive evidence that is consistent with the observations in Bartel, Ichniowski and Shaw (though the ex amples are not nearly as dramatic as those in manufacturing). In the bank studied by ALM, the introduction of Optical Character Recognition and electronic imaging of paper checks reduced the rote component of check processing for the ‘exceptions processors,’ reducing time spent on routine ‘paper chase’ activities and increasing time spent on problem solving and account management. Notably, for the ‘check preparers’ who perform the physical handling of the checks, there was little net change in skill demands—but there was a dramatic reduction in employment. BIS speciﬁcally investigate four hypotheses: 1. New IT-enhanced machines improve production process eﬃciency. Setup time, run time, and inspection time fall after new IT-enhanced equipment in these stages is adopted. 2. New IT promotes product customization and innovation. New 3D-CAD technologies should directly aﬀect the plant’s capabilities of designing more customized valves, while other tech nologies that reduce setup time would also promote customization. 3. IT adoption may increase (or decrease) skill demand. 4. IT adoption may require new HRM practices. The theoretical foundation for these hypotheses is BIS’ observation that IT reduces setup time, which is otherwise particularly costly for customized, small batch jobs. This cost reduction diﬀer entially reduces the price of producing customized relative to commodity products. What happens when IT prices fall: 1. Firms purchase more IT 2. Production eﬃciency rises, setup time, run time and inspection time time fall. 3. Firms make a strategic move towards producing more customized products. This is due to fall in setup times. 4. Optimal skill demand changes, but the direction is ambiguous. If setup is the most skillintensive task (as seems likely), when setup time falls, skill demand falls. But the move to more setup-intensive products exerts a countervailing force. Third, IT-based machinery increasingly displaces routine tasks, thereby raising skill content of labor input. This goes in the direction of increasing skill requirements. 5. Finally, new HRM practices may complement use of higher skill levels or new machinery, though a micro-foundation for this idea is not given. 14 4 Technology and the organization of production A 2007 JPE paper by Wouter Dessein and Tanos Santos (“Adaptive Organizations”) oﬀers an ingenious hypothesis on the subtle linkages between information (or communication) technology and work organization. Quoting Dessein-Santos (p. 17): Adam Smith’s famous observation that the “the division of labor is limited by the extent of the market,” has been challenged by both the management literature as well as economists such as Becker and Murphy (1992). These two strands of the literature have emphasized that perhaps, more importantly, specialization is mainly constrained by the need to coordinate activities. In particular, a straightforward comparative static prediction in Becker and Murphy (1992 [QJE]) is that as coordination costs increase, one should see less specialization.” The ‘Smithian’ view would suggest that the expansion of markets via population growth, inter national trade, etc. would increase specialization by enlarging the extent of the market. The Becker-Murphy view would also suggest that the falling price of information technology would in crease specialization by lowering communications/coordination costs. Yet, we seem to see a world where jobs are becoming broader and workers are increasingly asked to exercise discretion. What is going on? One possibility is that casual empiricism is simply wrong: work is becoming more compartmen talized, perhaps not in advanced countries but in the world at large (though why not in advanced countries?). A second possibility is that the theory is wrong. The Dessein-Santos’ paper suggests the latter. 4.1 Adaptation versus Coordination Dessein-Santos consider a model of organizational design in which there are a number of tasks, n, to be performed. Performance of each task is optimized through two actions: (1) adaptation and (2) coordination. Adaptation is the adjustment that should be made to a particular task in response to current conditions (which are assumed not entirely forecastable). For example, when cooking a Japanese dinner, the optimal quantity of water to add to the rice cooker depends in part on the current humidity and temperature as well as the season when the rice was harvested relative to the cooking date. Coordination is the adjustment of other complementary tasks to the adaptations made on the primary task. For example, adding water to the rice cooker might mean that the meal will be delayed, in which case the vegetables should be sautéed several minutes later so they do not turn to mush. One way to accomplish both objectives simultaneously (adaptation and coordination) is to assign one person to perform both tasks (bundle). The authors assume that coordination is costless (or very low cost) if one person performs all tasks (that is, there are no within-person coordination 15 costs). In the limit, one can push coordination costs to zero by having all tasks performed by a single worker. But the downside to bundling of tasks is the loss of the gains to specialization. Productive eﬃciency at each task is probably greatest if one chef cooks the rice and the other the vegetables (picture a large restaurant). The authors therefore assume that the more tasks done by one worker, the lower her productivity at each. One can maximize the gains to specialization by having a separate person do each task. But coordination suﬀers, leading to lower quality or higher coordi nation costs to achieve the same quality. If tasks are highly specialized, there is a signiﬁcant risk of coordination failures—information may be lost in coordinating among the workers performing complementary tasks. If the rice chef adjusts the water perfectly but neglects to inform the veg etable chef (or the information is lost with some probability) then we’ll end up with better rice but perhaps a worse meal. There is therefore a trade-oﬀ between allowing discretion to maximize adaptation (which comes at the cost of greater coordination failures) and reducing discretion to improve coordination (which comes at the cost of reduced adaptation). One can also bundle more tasks into a single job, which improves both adaptation and coordination. But this reduces gains from specialization. Dessein-Santos map out two general overarching organizational strategies: 1. ‘Ex ante coordination’—The goal here is to reduce coordination failures by limiting discretion. So, the rice chef does not adapt the recipe by much to current conditions. This means the rice is perhaps of lower quality but it is well timed with the vegetables. In this case, it is clear that a ﬁrm will also want to subdivide tasks ﬁnely to take advantage of gains to specialization since there is little need to coordinate (i.e., the coordination is built-in since there is no discretion). This the so-called ‘rigid’ organization. In this type of organization, there will be little investment in communication (meetings, teams, etc.) because such communication is not needed. One could call this production mode ‘Fordism.’ 2. ‘Ex post coordination’—Here, the ﬁrm allows workers discretion to adapt to local conditions but then must rely on high quality ex post communications to coordinate among adapted tasks. The ﬁrm will therefore want to invest in high quality communications to make ex post coordination work. Notice also that ﬁrms may want to use broad job assignments in this model. The reason is that if communication is costly but workers can still coordinate costlessly within their groups of tasks, it is valuable to both have high quality communications and to reduce the degree of specialization so that these communications are not too costly in practice (e.g., not too many meetings). One might label this organizational model as ‘ﬂexible production.’ Now consider what happens when communications costs fall (i.e., due to computerization). Firms will ﬁnd ex post coordination cheaper (i.e., communications among adapted tasks). This may increase the value of giving workers discretion to adapt their tasks (less coordination failure ex post). But if so, it will also increase the value of bundling tasks so that workers can costlessly 16 coordinate within tasks they are adapting. Thus, broad task assignments and intensive ‘horizontal communications’ are complements under some circumstances. 4.2 Task structure of production Production can be thought of as taking place on an n × n grid of tasks: a11 a12 a13 ... a1n a21 a22 a13 ... a2n a31 a12 a33 ... a3n ... ... ... ... a4n an1 an2 an3 ... ann The diagonal elements of this grid are the primary actions. The oﬀ-diagonal actions are the coordinating complementary actions. Workers are assigned one or more rows of tasks, with the number of rows t assigned to each worker determined in equilibrium. For each row that a worker is assigned, it is her responsibility to select both the primary and coordinating actions for the tasks in that row. For simplicity, assume that task assignments are symmetric, so that all workers perform the same number of tasks. Although workers are responsible for rows, output is a function of the elements of a column. In particular, the complementary actions in each row must be coordinated with the actions in their respective columns not the actions in their row. This means that workers responsible for diﬀerent rows must coordinate with one another to choose tasks optimally within a column. Before discussing how tasks are chosen, it’s useful to understand the sequencing of the model, which goes as follows: 1. The ﬁrm determines the number of task per agent, t. 2. Local circumstances θ i for i = 1, 2, ..., n are realized and observed by employee(s) in charge of task i. 3. Workers communicate the realizations of local information, and each attempt at communi cation is successful with probability p. Workers conveying this information cannot determine whether or not communication was successful. 4. For each row i, the employee in charge of i chooses actions aij , where j ∈ {1, 2, ..., n}, to maximize the objective function given his information. 5. Proﬁts are realized given the realization of local information, the success of communication, and the chosen values of all tasks. Now let’s consider how actions are chosen. 17 4.3 Adaptation and coordination Task i consists of undertaking a primary action aii , whose eﬀectiveness depends on how well it is adapted to the local environment. Adaptation calls for use of local information, which pertains only to this task and can only be observed by the worker assigned to the task. The local information is a random variable θi with mean θ̂i and common variance σθ2 . To achieve perfect adaptation, the primary action aii should be equal to θi . The realization of θi is independent across tasks. To ensure that task i is coordinated with all tasks j = 6 i, the employee in charge of task i must { i1 i2 } perform a sequence of n − 1 actions a , a , ..., ain that are complementary to the primary actions ˙ To achieve perfect coordination between tasks i and j, action aij of task i should 6 i. of tasks j = be set equal to the primary action ajj . Notice (as above) that worker in charge of task i must coordinate the complementary actions with the tasks in rows j 6= i. So, if the organization consists of two tasks, then adaptation and coordination losses amount to: φ �� a11 − θ1 �2 �� 2 � �2 � �2 � + a22 − θ2 + β a12 − a22 + a21 − a11 , where the parameters φ ≥ 0 and β ≥ 0 determine the importance of adaptation and coordination respectively. Here, a11 and a22 are the primary tasks, and actions a12 and a21 are the subordinate tasks for each primary action respectively. 4.4 Specialization Let T (i) equal the set of tasks bundled with worker i. For simplicity, assume that organizational design is symmetric, so that all workers in an organization have an identical number of tasks t assigned to them, so that T (i) = t for all workers (and the total number of workers is n/t). Task variety is costly in the Smithian sense in that forgoes the gains from specialization. Specif ically, the labor cost of carrying out task i, denoted by h (t, α) , is increasing in the level of task bundling t and in α. Thus h (·) is a per-task cost, with: ¯ − hα (t) ≥ 0, and t¯ > t. h (t̄) − h (t) ≥ 0, and hα (t) (Put more simply h (0) ≥ 0 and ht (·) > 0.) Bundling tasks raises per unit costs of execution by sacriﬁcing gains from specialization. (The parameter α reﬂects the returns to specialization. When α is larger, the gains from specialization are larger because the costs of non-specialization are higher.) 4.5 Communication To improve coordination, workers can communicate the realization of the local information θi prior to the actual implementation of the actions. If tasks i and j are assigned to diﬀerent employees, then with probability p the value of θi will be communicated successfully and with probability (1 − p), it 18 will be pure noise. Thus, p is a measure of the quality of the communication channel. It is assumed that agents who receive communication know whether communication was successful or pure noise. Agents sending this information do not know whether their communication was successful. 4.6 Proﬁts Proﬁts of a ﬁrm are given by: π=− =− n X � � C i a1i , a2i , ..., ani , t|θi i=1  ⎡ n X � � ⎣φ aii − θi 2 + n X  ⎤ � �2 β aji − aii + h (t, α)⎦ j ∈T / (i) i=1 You can think of this proﬁt function as a loss function that the ﬁrm is attempting to minimize. Hence it might be more intuitive to write π ∗ = p×(y + π) , where p is the market price of a perfectly produced unit of output and π is the reduction in quality incurred by imperfections in adaptation and coordination during production. 4.7 Optimal choice of actions One can show that employees optimally choose the following primary actions: � � � φ θi − θ̂i . a (τ ) = θ̂ + φ + β (n − t) (1 − p) ii i � Notice that the degree of adaption is increasing in φ, decreasing in β, increasing in the quality of communication p, and increasing in task bundling t. Complementary actions are chosen as: aji (t) = aii(t) when task j learns θi θ̂i when task j does not learn θi . The covariance between local circumstances and the primary action are: � � σaθ (t) = Cov aii (t) , θi = � � φ σ2. φ + β (n − t) (1 − p) θ Thus, σaθ (t) characterizes how strictly employees adhere to the ex ante strategy (θ̂i ) versus tailoring their actions to local circumstances. Notice that σaθ (t) is increasing in the variability of local circumstances (in addition to the other comparative statics above). A key observation is that σaθ (t) is increasing in task bundling: σaθ (t̄) > σaθ (t) 19 Given t, expected proﬁts are: � � Π (t) = nφ σaθ (t) − σθ2 − nh (t, α) , t∗ = arg max Π (t), t∈T as a function of φ, α, σθ2 , β and p. 4.8 Some results 4.8.1 Specialization Task specialization is decreasing in the importance of adaption, φ, and the variance of local circum stances σθ2 , but increasing in the returns to specialization, α. 4.8.2 The relationship between coordination costs and specialization The Becker-Murphy 1992 QJE article would suggest that as coordination costs fall, specialization increases. In the current paper, coordination costs operate through β (holding communication technology ﬁxed). When β is higher, tasks are more interdependent, so coordination costs are higher. How does a rise in β aﬀect task bundling? Becker-Murphy suggests that increase bundling (reduce specialization). In this model, the answer depends on the importance of adaptation φ. An increase in task interdependence aﬀects two margins: 1. Holding worker ﬂexibility/adaptation constant, a rise in β makes it more important to improve coordination, which leads to increased bundling. 2. But a rise in β may also spur the organization to reduce employee ﬂexibility and become less adaptive. This reduces the need for task bundling. If β is very large, an employer may eliminate all ﬂexibility, in which case it is optimal to fully specialize t∗ = 1. Conversely, if β is very small so that tasks are virtually independent, then coordination can be ignored and so it is also optimal to specialize. Accordingly, the relationship between t∗ and β is non-monotone. For h (n, α) < ∞ and given α, there exists a unique φ̂ such that: ( lim t∗ β→∞ = n if φ > φ̂ 1 if φ < φ̂ Moreover, when φ < φ̂, t∗ is non-monotone in β, increasing and then decreasing. Initially, a rise in β leads to more bundling to improve coordination. As β gets larger, ﬁrms simply reduce adaption and make jobs very narrow. This resolves coordination problems, though also makes ﬁrms less adaptive. 20 4.8.3 Quality of communication Intuition would suggest that one would see more specialization as communication costs (1 − p) fall since coordination gets easier. But organizations may also take advantage of improved communica tions to become more adaptive. In this case, by bundling, the organization increases its adaptiveness to local environmental variables. This increased adaptation is complemented by improved commu nication because it becomes more important to coordinate the other tasks when the primary action is allowed to vary more. So, there is again a trade-oﬀ: 1. For given level of employee ﬂexibility, σaθ (t) , an improvement in communications makes it easier to coordinate, reducing the need for bundling. 2. But as coordination through communication improves, ﬁrms will want to increase employee adaptability. This favors additional task bundling. When communication channels are very poor, task bundling often increases as communication technology improves. Firms go from being extremely inﬂexible to somewhat more adaptive. In this range, communications and bundling are complements. But when communication technology gets suﬃciently good, bundling looks less and less attrac tive since it forgoes Smithian gains from specialization but saves little in coordination errors (since the quality of communications is good). In this range, communications and bundling are substitutes. 4.8.4 Other comparative statics There are many interesting extensions in the paper, which I leave for your further reading. An important choice variable that the paper studies is the optimal level of p when improvements in p can be purchased at a positive price. 5 The Economics of Superstars The 1981 AER article by Sherwin Rosen, “The Economics of Superstars,” is often cited as a prescient explanation for the rise in returns to skills that was to occur in the years immediately following its publication. Whether or not you believe that the Superstars hypothesis helps to explain the phenomenon of rising inequality, the paper oﬀers a fascinating set of insights that has considerable currency as an explanation for why wages of CEOs, entertainers and athletes are incomparably higher than for other occupations. (That is, the Superstars model certainly oﬀers a deep and “truthy” explanation for why the distribution of wages is so skewed in some occupations. Addition ally, the hypothesis may or may not explain why the overall distribution of wages has become more skewed (more skewered?)) Certain services appear to be characterized by a demand structure that places considerable weight on quality versus quantity. For example, a patient would rather have one really good heart 21 operation than two mediocre ones; an opera fan would rather see one Pavarotti concert than ten mediocre opera performances; no amount of Applebee’s food amounts to a meal at Oleana. Hence, if there is widespread agreement on who is talented among the providers of such services, one can imagine that talented service providers could earn considerable rents. There is also a natural check on the possibility that one ‘talent’ will control an entire market: congestion. A single heart surgeon cannot perform all of the surgeries; a great cook can only make so many dinners in an evening. It may be these congestion externalities that prevent single talents from controlling entire markets. Now, imagine a technological improvement that reduces congestion, allowing holders of talent to more eﬃciently serve larger markets. Examples: television, recording technology, the Internet, etc. These types of technological advances may increase the rents accruing to the most talented individuals by increasing their market share (possibly to a global scale). Some professions in which this is likely to be important: all performing artists and entertainers; managers of large corporations (CEOs); athletes; fund managers; possibly some academics. Is is actually hard to make a very long list. The Rosen model almost surely helps to explain why athletes and movie-stars earn such enor mous salaries, and why these salaries have risen as mass communications and entertainment have improved. The 2008 AER paper by Terviö explores this idea rigorously. 5.1 What are CEO’s Worth (and Why)? CEO pay is a divisive topic on which there has been an enormous amount of research. While there is room for debate about whether this topic receives more attention than it merits, it still oﬀers ‘teachable moments.’ One such moment is the 2008 AER by Terviö. This paper studies CEO pay and productivity through the lens of an assignment model. Assignment models were introduced by Nobel Laureate Jan Tinbergen in the 1950s, and they were ‘popularized’ (okay, made less obscure) by Sattinger’s widely cited 1993 JEL survey. The deﬁning feature of assignment models is the presence of indivisibilities among factors of production. In a situation where the amounts of two matching factors cannot be shifted across diﬀerent units of production, factors are not paid their marginal products in the standard sense. As we will see, it’s possible (indeed) likely for the wage paid to a given worker to rise or fall due to changes in the distribution of ability of other workers without any change in the productivity of that worker or in the value of his output. Assignment models have deﬁed widespread use, perhaps in part because they are typically rather non-intuitive and analytically diﬃcult. The paper by Tervio presents a beautiful exposition of the intellectual foundation of assignment models by reformulating the basic assignment model using distributional ranks rather than actual distribution functions. This formulation is completely natu ral because assignment models are inherently ordinal rather than cardinal. The paper’s substantive conclusions on CEO pay and its relationship to the distribution of CEO talent, are both insightful and intuitive. 22 5.1.1 Basic setup There are three assumptions that simplify the model: 1. One-dimensional quality of factors 2. Continuity of quality distributions 3. Complementarity between the qualities of factors The ﬁrst two assumptions are simply for analytical convenience. The third assumption is essential to the conceptual framework. The paper further assumes, risk neutrality and symmetric information (which implies an absence of agency problems). Worker quality is denoted as a. Firm quality is denoted by b, which will be referred to as ﬁrm size. There is a unit mass of workers and ﬁrms with continuos, ﬁnite support without gaps. (The unit mass should be thought of as a normalization of all units that are active in equilibrium.) The production function is continuous and strictly increasing in both its arguments. The key assumption is that this function has a positive cross-partial between a and b, so these inputs are complements. Accordingly, eﬃciency requires positive assortative matching. As is explained in the paper, one can write the production function without loss of generality as: Y (a, b) = a · b. The notation in the paper is actually an important contribution. Order abilities by quantile so that a [i] is the ability of the the ith quantile individual, and at [i] > 0. Denoting the distribution function by Fa , the proﬁle of a is deﬁned by a [i] = a ↔ Fa (a) = i. 5.1.2 Equilibrium conditions There are two types of constraints on the problem: a sorting constraint, which requires that no worker/ﬁrm pair wish to rematch; and an incentive compatibility constraint, which requires that all workers and ﬁrms earn at least their outside income: Y (a [i] , b [i]) − w [i] ≥ Y (a [j] , b [i]) − w [j] ∀i, j ∈ [0, 1] Y (a [i] , b [i]) − w [i] ≥ π 0 ∀i ∈ [0, 1] P C b [i] w [i] > w0 ∀i ∈ [0, 1] P C a [i] SC (i, j) It is assumed that π 0 and w0 are the same for all units, though it would be suﬃcient to assume that outside opportunities increase slower along the proﬁle than do equilibrium incomes. The lowest active pair breaks even: Y (a [0] , b [0]) = π 0 + w0 . 23 While there are in theory 2n! sorting constraints, where n is the number of units (if there were a ﬁnite number, which formally there are not), most constraints are redundant since for i ≥ j ≥ k, SC (i, j)+SC (j, k) implies SC (i, k) . The binding constraints are therefore: (1) the marginal sorting constraints that keep ﬁrms from wanting to hire the next best individual; and (2) the participation constraints of the lowest types. Ordering the binding sorting constraints, one can write: Y (a [i] , b [i]) − Y (a [i − ε] , b [i]) w [i] − w [i − ε] ≥ . ε ε (6) This equation becomes an equality as ε → 0, and yields the slope of the wage proﬁle (since the limit of the above expression is the partial derivative): wt [i] = Ya (a [i] , b [i]) at [i] , where Ya is the partial derivative. This gives us the slope. To get the full proﬁle, we need to integrate over the production function and add in the binding participation constraint: w [i] = w0 + ˆ 0 i Ya (a [j] , b [j]) at [j] d [j] . Similarly, the proﬁle of proﬁts must satisfy: π t [i] = Yb (a [i] , b [i]) bt [i] ˆ i 0 Yb (a [j] , b [j]) bt [j] d [j] . π [i] = π + 0 These conditions also imply that y = π + w at each ﬁrm. Note that this set of sorting conditions could equivalently be written in terms of workers choosing ﬁrms rather than ﬁrms choosing workers; these cases are isomorphic. This is an extremely tight set of constraints on the problem. It implies that the wages and proﬁts of each factor depend at quantile i depend on the full proﬁle of factors from quantile 0 to i − ε (but not on the proﬁle above i). A number of useful observations follow (and you should read the paper, which is chock full of astute insights, and even more so in the footnotes): 1. There is no bargaining in this model because of the continuity of the distribution of both factors. If there was a jump at some point in the proﬁle of one factor, then all of the surplus would go to the factor with the jump because the other side is perfectly competitive. 2. Payments to factors are only aﬀected by the quality of those below them in the ranking. This is because the binding constraint on each worker or ﬁrm is that the quality/price of the worker/ﬁrm just below it in the distribution. 3. The unobserved productivity characteristics a and b are essentially ordinal. Any increasing 24 transformation of the scale of measurement for a factor’s quality, combined with the inverse change in the functional form of the production function changes nothing of substance in the model. In this sense Y (a [i] , b [i]) = a [i] × b [i] is a general functional form (so long as we are assuming complementarity). 4. Quoting Tervio, “It would be incorrect to say that factors earn their marginal productivity by the usual deﬁnition of marginal productivity, because the increase in output if the individual of ability a[i] were to increase in ability is proportional to b[i]. But if she were to increase in ability, then, in equilibrium, she would also move up in the ranking and be matched with a higher b— and other individuals would have to move down and experience a decrease in productivity. This means that individuals in fact do receive their marginal product, once the margin is deﬁned correctly. As ability cannot conceivably be extracted from one individual and poured into another, the relevant margin here is whether an individual will participate in the industry or not—and if not, then the eﬀect of the resulting rearrangement of remaining individuals is part of the marginal product.” 5.1.3 Comparative statics Consider a change by which the production function Y is multiplied by some constant G with no change in distributions of a and b. Uniform productivity growth is mathematically equivalent to changing the units of measurement for output. This gives rise to a useful lemma: Scaling lemma: If Yt (a, b) = GY (a, b) , wt0 = Gw0 and πt0 = Gπ, then wt [i] = Gwt [i] and πt [i] = Gπt [i] for all i ∈ [0, 1]. Figure 1 summarizes much of the intuition of the model. In the strictly multiplicative case, the output accruing from matching a worker of ability a and a ﬁrm of ability b is a × b, which is simply a rectangle in a Cartesian graph. This graph is drawn as follows. Let a = ϕ (b) , deﬁned by a [Fb (b) = {(a, b) st.Fa (a) = Fb (b)}] with slope ϕt (b) = at [Fb (b)] fb (b) = at [i] bt [i] . i=Fb (b) ϕ (b) is strictly increasing in b, and the slope is given by the relative steepness of a and b at each quantile i. Be sure to understand Figure 1, since essentially all of the intuition of the assignment model is contained in this ﬁgure. Here’s an interesting exercise to build intuition. Let’s check Tervio’s assertion that “the relevant margin here is whether an individual will participate in the industry or not—and if not, then the eﬀect of the resulting rearrangement of remaining individuals is part of the marginal product.” Consider a hypothetical case where the highest ability worker (index a [1]) falls in ability to that of the lowest ability worker a [0]. Can we demonstrate that the reduction in total output is equal to this worker’s wage? 25 • The “demotion” of the highest ranked worker means that each ﬁrm other than b [0] will have to match with a slightly lower ranked worker. So, previously, total output was equal to: ˆ 1 Y = Y (a [j] , b [j]) d [j] 0 • Now, each ﬁrm except for the lowest ranked ﬁrm b [0] will have to pair with a slightly lower quality worker. Thus, the fall in output at each ﬁrm Y [j] is ΔY [j] ≡ Y [j] − Ŷ [j] = Y (a [j] , b [j]) − Y (a [j − ε] , b [j]) . (7) The output of the lowest ranked b is unaﬀected since it still pairs with the lowest ranked a. But for all units b above rank 0, they will now be pairing with a units that are ε below them in the prior ranking. Note that a [·] continues to refer to the values of a in the original distribution, not the new distribution. • Dividing equation (7) by ε and letting ε → 0, we take the limit of ΔY [j] Y (a [j] , b [j]) − Y (a [j − ε] , b [j]) = , ε ε to get Y t [j] = Ya (a [j] , b [j]) at [j] . • We integrate over the full distribution of units to obtain the total loss in output: ˆ 1 ΔY = 0 Ya (a [j] , b [j]) at [j] d [j] This is the net reduction in output caused by worker a [1]t s demotion. • Having solved for the reduction in output, let’s compare this to the change in the wage bill. The wage of the previously highest ability worker falls from ˆ 0 1 w [1] = w + 0 Ya (a [j] , b [j]) at [j] d [j] to ŵ [1] = w0 . Thus the change in the wage for worker a [1] is 0 ˆ 1 w [1] − ŵ [1] = w + Ya (a [j] , b [j]) at [j] d [j] − w0 0 ˆ 1 = Ya (a [j] , b [j]) at [j] d [j] , 0 26 which is identical to the fall in total output, ΔY . This conﬁrms Tervio’s assertion. 5.1.4 Applying this model to CEOs There are numerous conceptual and practical issues to confront before bringing this model to the data. I will discuss these assumptions and results only brieﬂy. The issues: 1. The surplus created by the CEO-ﬁrm interaction is unobserved. The market value of the ﬁrm is aﬀected by the current CEO and by expectations of future productivity (which also depend on future CEOs)—thus ﬁrm size, an outcome, cannot be treated as the b variable. 2. A second issue is that part of the market value of the ﬁrm will surely reﬂect the value of capital that can be readily transferred/resold among ﬁrms. This capital is not indivisible and so is not part of the surplus represented in Y (·) . 3. A third issue is that the current market value of a ﬁrm depends on the quality of the current CEO and the quality of past and (in expectation) future CEOs. Thus, one cannot necessarily infer the current CEO’s quality purely from contemporaneous data. 4. Productivity tends to grow over time, and the expectation of growth further aﬀects current market value. 5. The distribution of CEO ability and latent (exogenously determined) ﬁrm size (not market value) can change over time. 6. Outside options may shift. Thus, the subsequent analysis rests on a number of strong assumptions that necessarily make the exercise speculative (though perhaps still informative if one believes the foundational assumptions of the assignment model): 1. The distribution of a [i] and b [i] are time invariant. 2. Productivity grows deterministically at rate g at all ﬁrms (that’s why the scaling lemma is needed) 3. The value of outside options grow at rate g. Tervio uses values between 0.2 and 0.025. 4. The discount rate is constant. Tervio uses values between 0.08 and 0.05. 5. The impact of past and future CEO quality on current ﬁrm performance decays at a constant rate ατ +1 = ατ λ/ (1 + λ) . λ determines the decay rate. With λ → ∞, only the current CEO aﬀects contemporaneous earnings. Tervio uses values between ∞ and 0.1 27 6. Since adjustable capital must earn the market rate of return, it must be subtracted from Y. To determine the contribution of adjustable capital, Tervio assume’s that the gross surplus has constant elasticity θ with respect to adjustable capital. Tervio sets the share of adjustable capital in Y at values between 0 and 0.8. These assumptions are suﬃcient to pin down the contribution of ﬁrm size and ability to output up to an additive constant (one constant each for the two factors). The reason the constant is indeterminate is that one cannot infer the contribution of ﬁrm size and ability to surplus at the smallest ﬁrm in the sample since this depends on the full unobserved distributions of size and ability below this ﬁrm. However, these constants drop out when estimating counterfactuals of the form ‘how much would surplus rise or fall relative to the current baseline if the distribution of CEO ability were modiﬁed in various ways?’ Tervio explores three counterfactuals for CEO ability: all ﬁrms are paired with a CEO of the ability of the lowest ranked CEO; all ﬁrms are paired with the median CEO; all ﬁrms are paired with the highest ranked CEO. One can also perform a similar counterfactual for ﬁrm size. See Tables 1 and 2. Several points: 1. The contribution of ﬁrm size to CEO rents is much larger than the contribution of CEO ability. That is, in the logic of this model, the high levels of CEO pay at the top are mostly due to the exogenous component of ﬁrm scale. 2. The distribution of CEO ability relative to the lowest ability CEO can be inferred from the diﬀerential equations that describe the equilibrium outcomes. See Figure 5. A key conclusion, and one that should be intuitive, is that the distribution of CEO ability is not wide. It is estimated that the CEO at the top ranked ﬁrm is only 5 to 8 percent more able than the CEO at the bottom ranked ﬁrm. 3. To see the role of ﬁrm size in CEO pay, consider ﬁgure 4 where all ﬁrms are replaced with the lowest, median or highest ranked ﬁrm while the CEO ability distribution is held constant. Panel B is perhaps most informative. If all ﬁrms were the median size, the top CEO would earn 12.5 million less than the 500th CEO than is currently the case and the bottom CEO would earn an additional 2.5 million less relative to the 500th CEO than is currently the case. (Why greater pay dispersion at the bottom? Because the ﬂat proﬁle of ﬁrm size, all surplus is allocated to workers–unlike in the continuous ﬁrm distribution case. This surplus redistribution eﬀect can oﬀset the leveling eﬀect of greater equality of ﬁrm size. The relevant comparison here is not really among CEOs in these various scenarios but rather between scenarios–highlighting how changes in ﬁrm size hugely aﬀect the level and dispersion of CEO pay, holding CEO talent constant.) 5.1.5 How the heck does Terviö calculate what part of the capitalized market value of ﬁrms is due to CEO vs. ﬁrm ’quality?’ This is pretty tricky. 28 • Under the assumptions of (1) deterministic, constant growth; (2) proportionate growth in the value of outside options; and (3) no change in the quality distribution of factors over time, one obtains the following relationship w [i] a [i] b [i] + ν [i] = , 1−B 1−B where B = (1 + g) / (1 + r) is the growth adjusted interest rate. The ﬁrst term on the left is the present value of pay to all CEOs that ﬁrm i will ever employ and ν [i] is the market value of the ﬁrm. As shorthand, Terviö refers to 1/ (1 − B ) as the price to earnings ratio. • Adjustable capital can be partialled out from ν (i) using a Cobb-Douglas assumption. Only the part of ν that excludes adjustable capital is subject to the assignment process. • The diﬀerential equations (24) and (25) in the paper permit inference on the quality of CEOs and ﬁrms at each percentile in the distribution relative to the lowest quality units (a [0] , b [0]) participating in the market: a [i] = exp a [0] λ λ+1−B b [i] = exp b [0] ˆ i 0 ˆ i 0 wt [j ] dj , w [j ] + v [j ] (1 − B ) v t [j ] − wt [j ] /λ dj , w [j ] + v [j ] (1 − B ) where λ is the decay rate of present CEO contributions to future ﬁrm value. • I think a large part of the intuition of Figure 5 is that CEO pay increases very, very little from the smallest to largest ﬁrm, whereas market value increases by hundreds of billions (say from 19 billion at rank 1,000 to 750 billion at rank 1,000). This loosely implies that the slope of CEO talent must be relatively shallow. 5.1.6 Conclusions You do not have to be convinced that this model or empirical exercise fully (or even partly) explains the level or trends of CEO pay to ﬁnd much of value in this paper. As a labor economist, you should add the basic workings of assignment models to your intellectual toolkit. And this is best paper I’ve seen for acquiring those tools. As I will discuss in class, the assignment model can be used to interpret the Rosen 1981 AER superstar’s model, though Rosen’s original model is not constructed in the form of an assignment model. I will not spend class time on the related paper by Gerbil and Lanier (QJE 2008) and the critique of Gabaix-Landier by Gordon and Dew-Becker (“Unresolved Issues in the Rise of American Inequality,”Brookings). This is an important debate. 29 6 Terviö: Mediocrity in Talent Markets A closely related paper (one could even say a ’companion paper’) by Terviö proposes an alternative model that casts the same facts in quite a diﬀerent light. The point of departure between the Rosen and Terviö articles is in the mechanism by which talent is discovered. Rosen takes it as given that talent is known to the market. Given full information, the wages paid to talent are likely to be eﬃcient. Terviö’s paper takes a step back by asking: how does talent become known? Terviö observes that to be ‘discovered,’ one must ﬁrst ‘audition.’ This observation seems plausible for many talent-occupations. In the Terviö model, knowledge about worker quality is a joint output of production. (That is, to ﬁnd out if a worker would make a good actor, he needs to appear in a movie.) Economist Richard Caves refers to this ex ante uncertainty about talent as the “Nobody Knows” property – one cannot evaluate talent without putting it to use. Now, add a second key assumption: there are positive production costs. That is, workers cannot audition ‘for free’ because some other scarce is used for the audition. This resource could be capital or, more realistically, it could be the time of skilled evaluators (talent agents, moviegoers, academics, passengers on an airplane, etc.). More generally, consumers cannot become informed about the entire array of talents potentially available in the market because time and attention devoted to discovery is scarce. To evaluate and disseminate information about talent is costly. If these assumptions are correct, there is a positive opportunity cost to auditioning a worker’s talent by using him for production. The cost is that someone else better could ﬁll the slot (therefore making better use of the same scarce resource). Now assume: 1) learning is symmetric (i.e,. my talent is revealed to the public as the same time as it is revealed to the employer – this assumption can be weakened); 2) workers cannot commit to binding wage contracts (i.e., no indentured servitude); 3) workers are capital constrained so that they cannot simply ‘buy’ a job to audition their talents. 6.1 Setup • Firms need one worker and one unit of capital at cost c > 0 to produce output in each period. • Output of a ﬁrm is equal to the worker’s talent: Y = θ. • Talent is drawn from a distribution with a continuous and strictly increasing cumulative distribution function, F , with positive support [θmin , θmax ]. • There is an unlimited supply of potential workers, all of who have an outside wage of w0 . Hence, talent is not scarce. • Talent is industry-speciﬁc and becomes public knowledge after one period of work; the worker may then work in the industry up to T more periods, after which he ceases to be productive. • Both workers and ﬁrms are risk neutral, and there is no discounting. 30 • Industry output faces a downward-sloping demand curve pd (q). We think of the “quantity” of talent as referring to the quality of output times the number of workers I. Thus, all else equal, the equilibrium price of output is lower when talent in the industry is higher (i.e., less scarce) • The number of ﬁrms is “large” so that individual ﬁrms have no impact on total output, and there is no uncertainty about the realization of the distribution of talent. • Let the number of ﬁrms (and jobs) be a continuous variable I equal to the mass of the industry workforce, normalize so that I = 1 • Long-term wage contracts are not enforceable; workers cannot commit to decline higher oﬀers from other ﬁrms in the future. Some key mechanics • There will be a talent threshold ψ above which workers will be retained in the industry and below which they will not. • The social planner would like to maximize average talent in the industry. This is because the opportunity cost of talent in the outside market is uncorrelated with talent. Thus, the higher the level of talent in the industry, the higher is aggregate social welfare. (Note that output is linear in talent, so the average is a suﬃcient statistic for the maximization problem.) • In steady state, inﬂows of talent (new workers, that is) must equal outﬂows of talent. Some key economics In an eﬃcient market allocation 1. The veteran worker of the lowest type ψ retained in the industry must be indiﬀerent between exiting and remaining in the industry, and thus must earn exactly the outside wage w0 2. Novices must be indiﬀerent between entering the industry or pursuing the outside career, taking into account their option to exit for the outside career later on (that is, before T + 1 periods have elapsed) 3. Firms must expect zero proﬁts from hiring any talent 4. The output price must be market clearing 6.2 Talent threshold • In both market equilibrium and in the social planner’s optimal solution, individual careers will proceed in a simple manner: 31 – After one period of work, those whose talent is revealed to be below a certain threshold level ψ will exit the industry, while those above the threshold ψ will stay on for T more periods. This exit threshold will be the key variable in the model. • In steady state, inﬂows must equal outﬂows. Denote the fraction of novices in the work force by i; a fraction F (ψ) of them exit. The remaining fraction of jobs 1 − i are held by veterans; a fraction 1/T of these, the oldest cohort, retires each period. Equating the ﬂows of exits and entries yields: iF (ψ) + 1 (1 − i) = i T i (ψ) = 1 1 + T (1 − F (ψ)) • Average talent in the industry is therefore A (ψ) = i (ψ) θ̄ + (1 − i (ψ)) E [θ|θ ≥ ψ] . One can substitute the equation above for i (ψ) into this expression to obtain A (ψ) = 1 T (1 − F (ψ)) θ¯ + E [θ|θ ≥ ψ] . 1 + T (1 − F (ψ)) 1 + T (1 − F (ψ)) Notice that a higher ψ means that a larger fraction of workers are novices (which lowers average quality) but the average quality of retained veterans is higher, which raises average quality. • This expression will be maximized when the average talent in the industry (averaging over new entrants and veterans) is equal to the marginal talent retained. (When ψ is too low, raising it increases both the marginal and the average; when ψ is too high, raising it raises the marginal but reduces the average because there are too few veterans.) • The equilibrium condition for ψ can be written as ψ − θ̄ = T (1 − F (ψ)) (E [θ|θ ≥ ψ] − ψ) , (8) the solution to which is denoted as A∗ . The LHS of this expression is the foregone output of hiring a novice rather than a veteran (invoking that average output is equal to marginal output ψ at A∗ ). The RHS is the expected future gain from hiring a novice, equal to the probability that the novice is retained for T additional periods times expected output (conditional on being retained) in each subsequent period (invoking that ψ is constant across future periods). • The paper demonstrates that the solution to this condition is unique with no distributional assumptions on θ required (except continuity and boundedness). Here’s a numerical example of the optimal choice of ψ with θ ∼ N (0, 1) and T = 15: 32 E[TalentIPsi] Pr[VeteranIPsi] 1.10 1.08 1.00 0. 0 Notes: 0.80 Talent distributon is Theta - N(0,1) 0. 0 T is 15 periods 0. 0 0.50 0.40 0.30 0.20 0.10 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Psi (Cutof Value for Talent) 6.3 Indiﬀerence condition for entrants and veterans • Veterans must choose to exit if w (θ) < w0 , therefore the marginal veteran is indiﬀerent between exiting and remaining in the industry w (ψ) = w0 . • Similarly, novices must be indiﬀerent about entering the industry, accounting for the option to exit if θ < ψ: � � w θ¯ = T × E [max {w (θ) , w0 }] = (1 + T ) wo . � � This expression may imply that w θ¯ < 0, that is entrants will receive a negative wage. The optimal solution may therefore require an absence of binding credit constraints. • If constrained individuals cannot accept a wage below w0 − b where b > 0, then the wage of novices will be given by � � w θ¯ = w0 − b. This will yield substantial market-level ineﬃciencies as we shall see. 33 6.4 Zero proﬁt condition • Firms must expect zero proﬁt from hiring any talent. Denoting the output price as P , this implies that P θ − c − w (θ) = 0. • Since there is no risk aversion and information is symmetric, novices can be treated as a known ¯ quantity with talent θ. 6.5 Market clearing • The output price must clear the market. We’ve stipulated downward sloping demand, and we know that average talent is rising in ψ up to the optimum A∗ . Therefore, there will be a ﬁxed point for the market price equation where P = pd (I × A (ψ)) , where I × A (ψ) is total talent supplied. • Since talent in the industry is a function of ψ, we can rewrite the wage function in terms of individual talent and the market talent cutoﬀ w (·) = w (θ|ψ) . • We can use the zero proﬁt condition to get an expression for P in terms of the wage. – A ﬁrm employing the threshold obtains revenue P ψ and pays costs w (ψ|ψ) = w0 + c. – Thus, the equilibrium output price must satisfy P (ψ) = w0 + c . ψ (9) • Since ﬁrms must be indiﬀerent about hiring any level of talent, the surplus must go to the worker, implying that w (θ|ψ) = w0 + {P θ − (w0 + c)}. Rearranging: � w (θ|ψ) = w0 + � θ − 1 (w0 + c) . ψ (10) Thus, a worker at threshold talent of ψ earns w0 , which is her opportunity cost. A worker with θ > ψ earns a wage in excess of w0 while a worker with θ < ψ would exit the market and take w0 instead. 6.6 Solution with unconstrained workers • Competitive equilibrium with unconstrained individuals is socially eﬃcient, so the social plan ner’s solution already tells us that the exit threshold must be ψ = A∗ − θ̄. 34 • Inspection of the wage equation (10) makes clear that novices must accept less than the outside wage w0 since they have a positive probability of earning talent rents in the future while in the worst case they get the outside wage. If the novice wage were not below w0 , novices would earn rents. (Concretely, the fact that the novice wage is below ω0 is evident from the fact � � ¯ that θ/ψ < 1.) • Market equilibrium pins down the wage function as w (θ|A∗ ) and P ∗ = (w0 + c) /A∗ . Intu itively, note that unconstrained individuals bid for the chance to enter the industry up to the expected value of lifetime talent rents. Since veterans of the threshold type are available at � � the outside wage, novices have to pay P × θ¯ − ψ for this ﬁrst period job, which exactly compensates a novice-hiring ﬁrm for its expected revenue loss (compared to what it would get by hiring a threshold type). • In equilibrium, this novice payment must equal the expected lifetime rents: with threshold ψ , a novice has a probability 1 − F (ψ) of being retained, in which case he gets the excess revenue P × (θ − ψ) as a rent on each of the T remaining periods of his career. So, the market equilibrium condition is P × (θ − ψ) = P × T (1 − F (ψ)) (E [θ|θ ≥ ψ] − ψ) , where the P cancels out and hence this is identical to (8) above. • Payments by unconstrained novices raise the exit threshold to the eﬃcient level. Thus, the inability of workers to commit to long-term contracts does not cause problems if they are able to buy their jobs upfront. • The unconstrained payment (the price of a job) reﬂects the economic cost of the eﬃcient level of experimentation. ∗ b =P ∗ � � ( θ̄ A − θ̄ = (w0 + c) 1 − ∗ A ∗ ) As Terviö notes, the fraction of the total costs of production, w0 + c, that should be ﬁnanced by the novice is increasing in A∗ /θ̄ , which is a measure of the upside potential of novices. (For small values of b∗ the novice payment would merely be a wage discount below the outside wage.) 6.7 Equilibrium with constrained workers • Suppose now that the ability of individuals to pay for their ﬁrst period job is constrained at some b < b∗ due to an exogenous liquidity constraint. Now the novice payment is w (θ|ψ t ) = w0 − b, where ψ t denotes the threshold in the constrained case. 35 • Then ψ t (b) = ⎧� ⎨ w0 +c w0 +c−b � θ¯ for b < b∗ ⎩ A∗ for b ≥ b∗ (11) • Clearly the exit threshold is increasing in b. It follows that the average talent in the industry is also increasing in b. When novices cannot “subsidize” their employers, then the price of output must adjust upwards to induce the hiring of novices into the industry. • This implies that when there are liquidity constraints faced by novices: 1. threshold quality will be too low 2. turnover will be too low and average career lengths will be too long 3. average quality will be too low 4. output price will be too high 5. those who enter the industry earn rents in expectation 6. top wages will be ’too high’ because the top talents will still be drawn from the same distribution as in the unconstrained case, but they’ll face a higher output price. • I won’t go through the speciﬁcs of the solution in this setting, but they are straightforward provided that you are clear on the above. Key point: Credit constraints yield, in equilibrium, mediocre veterans in the market 6.8 � � ¯ A∗ ψ t ∈ θ, Two examples Let’s work with two simple examples. Both will have the same parameter values, but in one case we’ll assume that workers are not credit constrained and in the other case, we’ll assume that they are. For concreteness, let’s say that this industry is the movie business. The parameter c refers to the cost of making a movie. Everything is measured in 1, 000t s of dollars. So let’s say that c = 4, 000, that is, the cost of making a movie is $4 million. Parameter values • T = 15 (where the novice period does not count in T ) • Fixed cost of production is c = 4, 000 • Workers’ outside option is w0 = 0 • The talent distribution is uniform on 0 to 100. θ ∼ U [0, 100], which implies that θ¯ = 50 and E [θ|θ ≥ ψ] = 50 + ψ/2 36 • Output is equal to talent Y (θ) = θ • Surprisingly, this is all the info needed to solve the model. 6.8.1 No credit constraints case Start with the case where there are no credit constraints, meaning that the ﬁrst period wage in the industry can be negative. Solving for equilibrium: • Let’s start with the talent threshold (equation 8) ( )( ) ψ ψ ψ − 50 = 16 1 − 50 − . 100 2 This generates a quadratic equation with two roots, one of which is ψ = 80 and the other of which is out of bounds (430/3). Hence, ψ = 80. • We can now obtain the wage function � � θ w (θ|ψ) = w0 + − 1 (w0 + c) ψ � � θ w (θ|80) = − 1 4, 000 80 w (θ|80) = (θ − 80) 50 This function implies that P = 50. We know this because the worker obtains all of the surplus in excess of the threshold. So if the wage is increasing by 50 per unit θ, then output must be priced at 50. • Another way to obtain P is to invoke the zero proﬁt for hiring a marginal incumbent as in equation (9): P (ψ = 80) = w0 + c 0 + 4000 = = 50 ψ 80 • Now we need to solve for b, the cost that a novice pays to enter. From above, and using the fact that the marginal worker quality equals average worker quality, so A∗ = ψ = 80. ( ) � θ̄ ¯ b = P A − θ = (w0 − c) 1 − ∗ A ( ) 50 b∗ = 50 (80 − 50) = −4000 1 − = 1, 500 80 ∗ ∗ � ∗ Novices pay $1.5 million for the their ﬁrst job. Why is the worker paying to enter? Because if it turns out that θi > ψ, the worker will earn talent rents in the subsequent T periods. Those 37 rents must be zero in expectation, or else entry is ineﬃcient: novices are not indiﬀerent about entering, meaning they are paid too much, so other distortions must result. • What is the probability of a novice becoming a veteran? It’s Pr [θ ≥ ψ] = 0.20 • What fraction of active workers are novices? This is i (ψ) = 1 1 = = 0.25 1 + T (1 − F (ψ)) 1 + 15 (1 − 0.8) • What are the expected career earnings of a worker who exceeds the talent threshold? 15 × (50 × {E [θ|θ ≥ 80] − 80}) = 7, 500 • Expected lifetime earnings (rents) for a novice are therefore Pr [θ ≥ ψ] × 7, 500 − b = 0.20 × 7, 500 − 1, 500 = 0. • And ﬁnally, the top wage in the industry is w (100|ψ = 80) = 50 × (100 − ψ) = 1, 000. Thus, the top earner receives one million dollars per movie (yes, Austin Powers joke). 6.8.2 Equilibrium with constrained workers • Now assume that b = 0. Workers cannot pay to enter the industry. Optimality no longer applies. • One can calculate ψ t by using (11) t ψ (b) = ⎧� ⎨ w0 +c w0 +c−b ⎩ A∗ � θ̄ for b < b∗ for b ≥ b∗ With b = 0, ψ t (0) = 50. • Why is the talent threshold equal to mean talent in this example? Because novices cannot bid for jobs, veterans with θ ≥ 50 will always generate (in expectation) weakly more revenue for ﬁrms than will novices. This means that vets with θ ≥ 50 will always out-compete novices for a job. Thus, the talent threshold falls to the mean of novices. 38 • We can solve the wage equation as above � θ w (θ|ψ) = w0 + − 1 (w0 + c) ψt � � θ w (θ|ψ) = − 1 × 4, 000 50 � = [θ − 50] × 80 This implies immediately that P t = 80 • We can also invoke the zero proﬁt condition to obtain P (though the equation above already implies that P = 80): P t × θ̄ = 4, 000. Since θ¯ = 50, this implies that P t = 80. • What is the expected probability of being retained after the ﬁrst period? It’s Pr [θ ≥ ψ t ] = 0.50 • The average output of veterans is E [θ|θ ≥ 50] = 75, since veterans are retained if their talent exceeds the mean. • The fraction of novices in the industry is it (ψ) = 1 1 = = 0.118 t 1 + 15 (1 − 0.5) 1 + T (1 − F (ψ )) • The expected career earnings of a worker who exceeds the talent threshold is: 15 × (80 × {E [θ|θ ≥ 50] − 50}) = 30, 000 • Expected lifetime earnings (rents) for a novice are therefore ( 50 1− 100 ) × 30, 000 = 15, 000. • And ﬁnally, the top wage in the industry is � � � � w 100|ψ t = 50 = 80 × 100 − ψ t = 4, 000. Thus, the top earner receives four million dollars per movie (no Austin Powers joke). 6.8.3 Comparison Let’s compare the characteristics of these two equilibria 39 Constrained Equilibrium Eﬃcient Benchmark Talent Threshold ψ 50 80 Proportion Novices i 12% 25% E [Talent| Veteran] = E [θ|θ ≥ ψ] 75 90 72 80 Output price P $80K $50K Top Wage P × (100 − ψ) $4 million $1 million E[Career Earnings| Veteran] $30 million $7.5 million Wage of Novice $0 -$1.5 million E[Rents| Novice] $15 million $0 Average talent A∗ What is quite striking here is that the original equilibrium has lower talent and higher wages than the eﬃcient benchmark. The reason is that the original equilibrium has artiﬁcial talent scarcity. Because talent rents accrue to workers not ﬁrms, ﬁrms have insuﬃcient incentive to create new opportunities for talent revelation (in fact, they have none). So, there is under-experimentation, excess retention of mediocre talent, and ineﬃciently high rents. This problem could be solved in either of two ways: (1) create binding indentured servitude contracts such that a worker’s wage is set independent of output and the worker cannot quit once hired (but can be ﬁred in the 1st period) – this used to happen in European Soccer through the transfer fees system; (2) eliminate credit constraints so that workers can bid for jobs. This bidding will generate an eﬃcient market in talent revelation. You might ask yourself whether the markets for CEOs, athletes, movie stars and other celebrity professionals appears to satisfy (1) or (2). An interesting point on which to speculate is whether this model is, in some sense, more general than the Rosen model; could one could potentially nest the Rosen model as a special case where talent revelation was costless? Almost surely the answer is yes. 6.9 Mediocrities: Conclusion The key insight of this model is that talent can receive large rents even if that talent mediocre. The reason is that the (ex-post) realization of known talent may have artiﬁcial scarcity. It will be interesting to see whether this paper has an impact on the economic presumption that the market for superstars is in some intrinsic sense eﬃcient. 40

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