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Working Paper Series The Replacement Problem in Frictional Economies: A Near-Equivalence Result WP 05-01 Andreas Hornstein Federal Reserve Bank of Richmond Per Krusell Princeton University Giovanni L. Violante New York University This paper can be downloaded without charge from: http://www.richmondfed.org/publications/ The Replacement Problem in Frictional Economies: A Near-Equivalence Result Andreas Hornsteiny Per Krusellz Giovanni L. Violantex Federal Reserve Bank of Richmond WP 05-01 April 2005 Abstract We examine how technological change a¤ects wage inequality and unemployment in a calibrated model of matching frictions in the labor market. We distinguish between two polar cases studied in the literature: a “creative destruction” economy where new machines enter chie‡y through new matches and an “upgrading” economy where machines in existing matches are replaced by new machines. Our main results are: (i) these two economies produce very similar quantitative outcomes, and (ii) the total amount of wage inequality generated by frictions is very small. We explain these …ndings in light of the fact that, in the model calibrated to the U.S. economy, both unemployment and vacancy durations are very short, i.e., the matching frictions are quantitatively minor. Hence, the equilibrium allocations of the model are remarkably close to those of a frictionless version of our economy where …rms are indi¤erent between upgrading and creative destruction, and where every worker is paid the same marketclearing wage. These results are robust to the inclusion of machine-speci…c or matchspeci…c heterogeneity into the benchmark model. Keywords: Creative Destruction, Inequality, Technical Change, Unemployment, Upgrading. JEL Classi…cation: J41, J64, O33 We would like to thank two anonymous referees, Lars Ljungqvist, Joel Shapiro, Winfried Koeniger, and seminar participants at the ISOM Barcelona 2003, the 2003 SED meetings, and the 2003 IZA Workshop on Search and Matching for helpful comments. Krusell thanks the NSF for research support. Violante thanks the CV Starr Center for research support. Any opinions expressed are those of the authors and do not necessarily re‡ect those of the Federal Reserve Bank of Richmond or the Federal Reserve System. Our e-mail addresses are andreas:hornstein@rich:frb:org, pkrusell@princeton:edu, and gianluca:violante@nyu:edu. y Federal Reserve Bank of Richmond. z Princeton University, Institute for International Economic Studies, CAERP, CEPR, and NBER. x New York University and CEPR. 1 Introduction Technological progress is gradual and a large part of technological innovations are embodied in capital equipment. New machines tend to be much more productive than existing machines, but typically not every worker is matched with the most advanced equipment. Given that capital-embodied technical change creates substantial productivity di¤erences among machines, what are its e¤ects on wage inequality among labor that uses that equipment? Also, in light of the fact that new and more productive capital can make existing capital obsolete, what are the e¤ects of technological change on job creation, job destruction and, ultimately, unemployment? These are key questions for a macroeconomic analysis of labor markets that takes on a long-run perspective. If we view the labor market as a frictionless environment where workers are paid according to their marginal product, then the answer to these questions is relatively straightforward and well known. Here, the impact of technological change on inequality is limited to the extent that workers di¤er in their ability to use capital. Workers who use di¤erent vintages of capital but are otherwise identical will be paid the same wage. Moreover, the competitive equilibrium features no unemployment of labor.1 On the other hand, if frictions that prevent the free and timeless reallocation of labor among alternative uses are an essential part of the labor market, the situation is far more complex and none of the questions is settled even in the simple case of homogeneous workers. First, in frictional models wages of ex-ante equal workers re‡ect the relative productivity di¤erence of the technology they are working with. That is, wages are not equalized for identical workers due to a “luck”factor: workers are more or less fortunate in the matching process. But how important is this purely frictional source of inequality in terms of measured total wage dispersion for observationally equivalent workers? To put our analysis in perspective, consider that Katz and Autor (1999) report that the 90-10 wage ratio of the residual wage inequality in a typical wage regression, that is the inequality that remains after controlling for observable characteristics of the workers (gender, race, education, experience) 1 For the case of worker heterogeneity, there are also precise predictions in the literature. Akerlof (1969) showed that in a frictionless model with heterogeneous labor, unemployment of the worst type of labor can arise in equilibrium only if the aggregate capital stock is …xed and if there is some …nite limit to the marginal product of capital. Jovanovic (1998) showed that when capital embodies technological progress and machines are indivisible, faster growth raises wage inequality as long as skills complement capital in the production function. In this case, the most skilled workers will be the ones who are e¢ ciently assigned to work on the most productive machines. 1 and for individual …xed e¤ects to capture “unobserved individual ability,”is around 1.5. Second, since technological change may require the reallocation of labor towards new and more productive machines, but reallocation requires time with frictions, not all workers will be employed at any time. How does equilibrium unemployment react to changes in the rate of technological change? To this important question, two distinct qualitative answers emerge from the literature. Aghion and Howitt (1994) argue that when replacing an old obsolete machine with new capital requires destroying the existing match (the Schumpeterian “creative-destruction” model), then unemployment tends to go up as growth accelerates, due to a higher jobseparation rate. The models of Caballero and Hammour (1998) and Cohen and Saint-Paul (1994) bear similarities to the Aghion-Howitt approach. In contrast, Mortensen and Pissarides (1998) propose a mechanism whereby the new technologies can replace old machines at some cost, but without destroying the existing match (the costly “upgrading” model).2 The separation rate is una¤ected by faster growth and all the e¤ects work through job creation. For small values of the upgrading cost, unemployment falls with faster growth, thanks to the familiar “capitalization e¤ect”: investors are encouraged to create more vacancies, knowing that they will be able to incorporate and bene…t from future technological advances.3 Though we rely heavily on theory in our analysis— i.e., we explore mechanisms that are made explicit using dynamic general equilibrium theory— our …nal aim is to o¤er quantitative answers to these two questions. We wish to evaluate how much frictional wage inequality might be created by technological change, and how di¤erent quantitatively the implications for frictional unemployment of the two alternative replacement models really are. How do we accomplish these tasks? We study a calibrated version of a frictional model à la Diamond-Mortensen-Pissarides (DMP) with random matching and wages determined by cooperative Nash bargaining, augmented with vintage capital à la Solow (1960). The DMP model over the years has established itself as a standard framework of analysis of the labor market (see Pissarides, 2000, for an overview of the approach). Since capital-embodied 2 In this latter case, but not in the former, one could say that technology is “match-augmenting,”because it augments the value of existing matches. In the extreme case where upgrading can proceed at no cost, we have the Solowian model of disembodied technological change, even though the carrier of technology is equipment. 3 An interesting quali…cation to this result is provided by King and Welling (1995): if, unlike what is customarily assumed in this family of models, workers bear the full …xed search cost, then the capitalization e¤ect leads to an increase in the number of searchers and to longer unemployment durations. 2 technical change is, arguably, the key driving force of productivity growth in developed economies in the past three decades (see, e.g., Jorgenson, 2001), we model technological advancement through the introduction of new capital goods. Technically, our framework is a generalization of Hornstein, Krusell, and Violante (2003) where …rms cannot upgrade existing machines without separating from their employed worker. In particular, we are able to nest the two ways in which new equipment can enter the economy and replace the old one by assuming that (i) new entrant …rms can buy new equipment at price I0 and (ii) existing …rms, whether matched or not, can upgrade their equipment to the latest vintage at price Iu . Thus, if Iu = 0, technological change is fully match-augmenting and disembodied, whereas if Iu = 1, it has no match-augmenting or disembodied feature at all. In equilibrium, depending on parameter values, new equipment may enter through either channel. We parameterize our model economies tightly in order to match some micro-estimates and certain long-run facts. In particular, in a way consistent with our model formulation, we can use data on equipment prices, adjusted for quality, to measure the speed at which the new equipment investment carries technology improvements.4 In other words, in our framework the percentage productivity gaps across machines are restricted by data on relative prices of capital. Our main …ndings are quite striking. We …nd that the unemployment e¤ects of a rise in the rate of capital-embodied technological change are, qualitatively, those emphasized by Aghion and Howitt (1994): unemployment rises in response to higher rates of capitalembodied technological change. The di¤erence between the two ways of thinking about how technology is introduced is qualitatively the expected one— the Mortensen-Pissarides, or upgrading, perspective delivers a smaller increase in unemployment than does the AghionHowitt perspective— but quantitatively the two models are almost equivalent. Moreover, we …nd that an acceleration in the speed of technological progress raises wage dispersion, but that the amount of residual wage inequality generated by the calibrated DMP model of technological change is extremely small: the model delivers a 90-10 wage ratio between 1.06 and 1.08, or, at most, 1/6th of that measured. The intuition for these two results can be found in two key quantitative restrictions 4 Hornstein and Krusell (1996), Greenwood, Hercowitz, and Krusell (1997), and Cummins and Violante (2002) provide theoretical justi…cations and implementations of the price-based approach to the measurement of capital-embodied technical change. 3 that the equilibrium of our calibrated model has to obey. First, the average duration of unemployment in the United States is just above 8 weeks. In search models the duration of unemployment is proportional to the option value of search and the latter, for risk-neutral workers, depends positively on wage dispersion. A duration of unemployment as short as it is in the data can therefore only be consistent with a very small amount of frictional inequality.5 Second, the average duration of a vacant job with idle capital in the model (and in the U.S. data) is roughly 4 weeks, i.e., a very small amount compared to the average life of a new machine, which is over 11 years. When the meeting rate is so high, the …rm becomes roughly indi¤erent between scrapping and upgrading: intuitively, for a given cost of a new machine, the upgrading option is substantially better than the scrapping option only if the matching process is long and costly. Another way to look at these two conclusions is that, empirically, the labor market frictions are small. When the meeting frictions for both workers and …rms disappear, our model converges to the frictionless environment. In the frictionless model: 1) …rms are indi¤erent between upgrading and destroying a job since they face an in…nite arrival rate of idle workers, and 2) wage inequality is zero. Incidentally, our results are consistent with the conclusions of Ridder and van den Berg (2003) who estimate the severity of search frictions in the United States, United Kingdom, France, Germany and Netherlands and …nd that the United States has, by a signi…cant amount, the least frictional labor market. In the second part of the paper, we explore various extensions of the basic framework. The organizing principle we have chosen here is to maintain the assumption that workers are all ex-ante identical.6 Thus, we look at di¤erent forms of heterogeneity on the plant level, each of which can in‡uence wages in this form of labor market. There are two cases of interest. First, we consider heterogeneity in equipment/entrepreneurs that goes beyond vintage di¤erences: machine-speci…c heterogeneity. Second, we consider the kind of matchspeci…c productivity heterogeneity considered by Mortensen and Pissarides (1994), but now combined with the model of vintage capital. Thus, in the …rst extension, entrepreneurs that 5 Note that in our model, unlike in the more traditional Aghion-Howitt/Mortensen-Pissarides frameworks, there is vacancy heterogeneity in equilibrium, thus unemployed workers face a non-degenerate match distribution upon search. 6 Ex-ante worker heterogeneity in productivity levels raises new issues. First, such a model implies wage inequality and unemployment rates that depend nontrivially on worker type, thus making the output of the exercise less comparable to data, since a worker’s productivity cannot be measured directly. Second, it is well known (Sattinger, 1995) that random matching models with two-sided heterogeneity have multiple steadystates. In particular, identical (but identi…able) workers can have di¤erent combinations of equilibrium wage and employment rates. These issues are best left for future exploration. 4 enter the market are diverse and remain diverse as long as they are in the economy, whereas in the second extension, the heterogeneity appears in each new match and remains only so long as each match lasts. Our …ndings are that these two additional forms of productivity heterogeneity lead to very similar conclusions regarding the replacement debate: in response to a higher rate of capital-embodied technological change unemployment increases, whether or not upgrading is possible, but unemployment increases somewhat less if upgrading is possible. Wage inequality remains limited. Wage inequality is the highest, and becomes particularly visible through the emergence of a group of high-wage earners, in the economy with match-speci…c heterogeneity, because in this economy the breaking up of a match implies a larger loss of surplus than in an economy with machine-speci…c heterogeneity. The outline of the rest of the paper is as follows. In Section 2 we describe the frictionless benchmark of the vintage-capital economy. Section 3 contains the two basic versions of the replacement problem in the model with frictions: one where new machines enter in new matches, and one where they mainly enter through the upgrading of machines operated in existing matches. Section 4 contains the calibration to the U.S. economy and quantitative analysis, and also provides more detailed intuition for our two main results. Finally, Section 5 considers extensions to two additional sources of heterogeneity among matches: one due to permanent di¤erences among entrepreneurs/their equipment and one due to permanent di¤erences in match quality. Section 6 concludes. 2 The frictionless economy We begin by presenting a version of the Solow (1960) frictionless vintage capital model where production is decentralized into worker-machine pairs operating Leontief technologies: this decentralized production structure is typical in frictional economies. The competitive economy displays neither wage inequality nor unemployment; however, it is a useful benchmark for interpreting the main results of the richer frictional model. Demographics–Time is continuous. The economy is populated by a stationary measure one of ex-ante equal, in…nitely lived workers who supply one unit of labor inelastically, and there is also a set of entrepreneurs whose only role it is to pair up with workers and run the …rm. Workers and entrepreneurs are risk-neutral and discount the future at rate r. Production– Production requires pairing one machine and one worker and machines 5 are characterized by the amount of e¢ ciency units of capital k they embody. A matched worker-machine pair produces a homogeneous output good. Technological change– There is embodied and disembodied technical change. The economy-wide disembodied productivity level, ez(t) , grows at a constant rate > 0. Tech- nological progress is also embodied in capital and the amount of e¢ ciency units embodied in new machines grows at the rate to physical depreciation at the rate > 0. Once capital is installed in a machine it is subject > 0. A production unit that at time t has age a and is paired with a worker has output y (t; a) = ez(t) k (t; a)! = ez(0)+ t k0 e (t a) e a ! (1) ; where ! > 0. In what follows we set, without loss of generality, z0 = 0 and k0 = 1. How new machines enter– At any time t …rms can freely enter the market upon payment of the initial installation cost I0 (t) for a machine of vintage t. The cost of new vintages grows at the rate g. Existing …rms with older machines also have the opportunity to upgrade their machine and bring its productivity up to par with the newest vintage. The cost of upgrading Iu (t; a) 0 grows at the rate g but it is independent of age for a a ^. We assume that once a machine is too old, it becomes in…nitely costly to upgrade: Iu (t; a) = 1 for a > a ^. Rendering the growth model stationary–We will focus on the steady state of the normalized economy; this corresponds to a balanced growth path of the actual economy. It is immediate that for a balanced growth path to exist, we need g = + ! . In order to make the model stationary we normalize all variables by dividing by the growth factor egt . The normalized cost of a new production unit is then constant at I0 and Iu , and the normalized output of a production unit of age a which is paired with a worker is e a , where ! ( + ); thus, output is de…ned relative to the newest production unit. Note that the parameter represents the e¤ective depreciation rate of capital obtained as the sum of physical depreciation and technological obsolescence :7 Embodied technical change and the relative price of new capital–Since the cost of new vintage machines in terms of the output good, I0 and Iu , is growing at the rate g but the number of e¢ ciency units embodied in new vintages is growing at the rate , the 7 In Hornstein, Krusell, and Violante (2003), we describe the normalization procedure in detail. 6 price of quality-adjusted capital (e¢ ciency units) in terms of output is changing at the rate g = (1 !) . In the quantitative analysis of our model we will use this relationship to obtain a measure of the rate of embodied technical change from the rate at which the observed relative price of equipment capital changes. 2.1 Competitive Equilibrium Assume that the labor market is frictionless and competitive so that there is a unique marketclearing wage. In the steady state the wage rate w, now measured relative to the output of the newest vintage, is constant. Consider a price-taker …rm with the newest vintage machine. The …rm optimally chooses the age a that maximizes the present value of the current machine lifetime pro…ts V0 = max a Z a e (r g)a e a w da + e (r g)a max [V0 Iu ; 0] ; (2) 0 where r g is the e¤ective discount factor. While the wage w is the same for all …rms, a …rm’s output falls compared to new machines because of depreciation and obsolescence. Thus ‡ow pro…ts are monotonically declining in age and eventually become negative, and there is a unique age at which the machine will be discarded or upgraded. Pro…t maximization leads to the condition (r g) max [V0 Iu ; 0] = e a w: (3) The …rm upgrades/exits when the gain from upgrading/exit (left-hand side) is the same as the return from a marginal delay of the investment decision (right-hand side). If the value of a new machine is lower (higher) than the cost of upgrading, the …rm will scrap (upgrade) the machine. If the …rm scraps the machine, the wage is equal to the relative productivity of the oldest machine, which is also the marginal productivity of labor. The higher the wage, the shorter the life-length of capital since (normalized) pro…ts per period fall and thus reach zero sooner. Free entry of …rms with new machines requires that the value of a new machine does not exceed the cost of a new machine: V0 I0 . Obviously if the cost of upgrading a machine is greater than the cost of buying a new machine, I0 < Iu , then machines will never be upgraded. On the other hand, when the cost of upgrading is lower than the cost of buying a new machine, machines will never exit. Using the pro…t-maximization condition (3) in the 7 …rm value equation (2), the optimal investment condition can be written as Z a min fI0 ; Iu g = e (r g+ )a 1 e (a a) da: (4) 0 This is the key condition that determines the exit/upgrading age a, and hence wages. Existence and uniqueness–Equation (4) allows us to discuss existence and uniqueness of the equilibrium. The right-hand side of this equilibrium condition is strictly increasing in the upgrading/exit age a for two reasons. First, in an equilibrium with older …rms, the relative productivity of the marginal operating …rm is lower and therefore wages have to be lower and pro…ts higher. Second, a longer machine life increases the time-span for which pro…ts are accumulated. De…ne r r g+ = r + ! : The right-hand side of (4) increases from 0 to 1=r as a goes from 0 to in…nity. Taken together, these facts mean that there exists a unique steady state age aCE whenever min fI0 ; Iu g < 1=r. This condition is natural: unless you can recover the initial capital investment at zero wages using an in…nite R1 lifetime ( 0 e ra da = 1=r being the net pro…t from such an operation), it is not pro…table to start any …rm. Finally, with a unit mass of workers, all employed, the …rm distribution is uniform with density 1=aCE , which is also the measure of entering/upgrading …rms. The e¤ect of growth–Clearly, in this economy neither wage inequality nor unemployment is a¤ected by changes in the rate of technology growth: they are both zero. 3 The economy with matching frictions Consider now an economy with same preferences, demographics, and technology, but where the labor market is frictional in the sense of Pissarides (2000): an aggregate matching function governs job creation. The nature of the …rm’s decision process remains the same as in the frictionless economy: there is free entry of …rms which buy a new piece of capital, participate in the search process, start producing upon matching with a worker, and …nally either upgrade their capital once it becomes too old or exit if upgrading is too costly. Searching is costless: it only takes time. Existing matches dissolve exogenously at the rate : upon dissolution, the worker and the …rm are thrown into the pool of searchers.8 In this environment vacant …rms are heterogeneous in the vintage of their capital for two reasons: …rst, newly created …rms do not match instantaneously, so they remain idle until 8 We omitted separations from the description of the competitive equilibrium because, without frictions, it is immaterial whether the match dissolves exogenously or not as the worker can be replaced instantaneously by the …rm at no cost. 8 luck makes them meet a worker; second, …rms hit by exogenous separation will also become idle. Note here a key di¤erence with the traditional search-matching framework (Mortensen and Pissarides, 1998): the traditional models assume that a new machine can be purchased at no cost and that only posting a vacancy entails a ‡ow cost. This assumption implies that the pool of vacancies consists of the newest machines only, and that only matched machines age over time.9 Our setup is built on the opposite assumption: purchasing and installing capital is costly— an expense which is sunk when the vacant …rm start searching— whereas posting a vacancy is costless.10 As a result, it can be optimal even for …rms with old capital to remain idle. This class of economic environments is a hybrid between vintage models and matching models. The traditional assumption emphasizes the matching features of the environment, while the explicit distinction between a “large” purchase/setup cost for the machine and a “smaller”search/recruiting ‡ow cost (zero in our model) …ts more naturally with its vintagecapital aspects, whose emphasis is on capital investment expenditures as a way of improving productivity. In actual economies, new and old vacancies coexist, as in our setup. 3.1 The environment Matching– The number of matches in any moment is determined by a constant returns R1 to scale matching function m(v; u), where v (a)da is the total number of vacancies 0 of all ages v (a) and u is the total number of unemployed workers. We assume that m(v; u) is strictly increasing in both arguments and satis…es some standard regularity conditions.11 The rate at which a worker meets a …rm with capital of age a is w (a) and the rate at which R1 she meets any …rm is w w (a)da, where we assume that the integral is …nite. A …rm 0 meets a worker at the rate f. Using the notation = v=u to denote labor market tightness, 9 Aghion and Howitt (1994) also describe a vintage capital model with initial setup costs for capital, but they assume that matching is “deterministic”: at the time a new machine is set up, a worker queues up for the machine, and after a …xed amount of time the worker and …rm start operations. Hence, in the matching process, all vacant …rms are equal (although they do not embody the leading-edge technology). 10 Vacancy heterogeneity will survive the addition of a ‡ow search cost, as long as this cost is strictly less than the initial set-up cost I0 . 11 In particular, m(0; u) lim mu (v; u) = m(v; 0) = 0; = lim mv (v; u) = 0; lim mu (v; u) = u!1 u!0 v!1 lim mv (v; u) = +1: v!0 9 we then have that f = w (a) = m ( ; 1) (5) ; (a) m( ; 1): v (6) The expression for the meeting probability in (5) provides a one-to-one (strictly decreasing) mapping between f and . Thereafter, when we discuss changes in f, we imagine changes in . The measure of worker-…rm matches with an a machine is denoted R1 employment is (a)da. 0 (a) and total Scrapping versus upgrading–Values for the market participants are J(a) and W (a) for matched …rms and workers, respectively, V (a) for vacant …rms, and U for unemployed workers. We consider two forms of capital replacement and the value functions will di¤er in the two cases. We proceed immediately by guessing that there is a cuto¤ age for capital, a; such that either scrapping or upgrading takes place. The values of matched workers and …rms–Under both replacement decisions, the ‡ow value of a job for …rm and worker for a (r g)J(a) = maxfe a (r (r g)W (a) = maxfw(a) a is given, respectively, by w(a) [J(a) V (a)] + J 0 (a); (7) g)V (a)g [W (a) U ] + W 0 (a); (r g)U g: (8) The return of a matched …rm with an age a machine is equal to pro…t, i.e., production less wages w(a) paid to the worker, minus the ‡ow rate of capital losses from separation plus the ‡ow losses/gains due to the aging of machines.12 Analogously for a worker, the return on being in a match with an age a machine is the wage minus the ‡ow rate of capital losses from separation plus the ‡ow losses/gains due to the aging of machines. For the match to be maintained, the ‡ow return from staying in the match must be at least as high as the ‡ow return from leaving the match, i.e., from the …rm becoming vacant and the worker becoming unemployed. Note that the capital value equations for matched workers and …rms are de…ned only for matches with machines not older than a, since all machines either exit or are upgraded at age a. 12 In Appendix A we describe a typical derivation of the di¤erential equations above. 10 Values of idle workers and …rms– In the economy where capital is replaced by scrapping, the values of idle workers and …rms are (r g)V (a) = max f f [J(a) V (a)] + V 0 (a); 0g Z a (r g)U = b + U ] da: w (a) [W (a) (9) (10) 0 The return on a vacant …rm is equal to the capital gain rate from meeting a worker plus the ‡ow losses/gains due to the aging of capital. Vacant machines that are older than the critical age a exit.13 The return for unemployed workers is equal to their bene…ts b plus the capital gain rate from meeting vacant …rms. When capital is upgraded, we have instead 8 a a; < max f f [J(a) V (a)] + V 0 (a); 0g ; (r g)V (a) = : max f J(0) IuJ (a) V (a) + V 0 (a); 0 ; a < a Z a (r g)U = b + U ] da w (a) [W (a) 0 Z a^ + IuW (a) U da: w (a) W (0) (11) a ^; (12) a Vacant machines that are older than the critical age a do not exit, but wait until they meet a worker and then upgrade. At the time the machine is upgraded, the …rm pays its share of the upgrading cost IuJ (a). When age a ^ is reached, upgrading becomes in…nitely costly, so the vacant …rm exits with value equal to zero. The return for unemployed workers contains an additional term that gives the value of meeting vacant machines older than a: These …rms will upgrade upon meeting a worker, and the worker contributes IuW (a) to the upgrading cost and starts working with brand-new capital. Wage determination–In the presence of frictions, a bilateral monopoly problem between the …rm and the worker arises and, thus, wages are not competitive. As is standard in the literature, we choose a cooperative Nash bargaining solution for wages. In particular, we assume that the parameter de…ning the relative bargaining power is the same when the pair negotiates over how to split output and over how to share the upgrading cost Iu . With outside options as in the above equations, the wage is such that at every instant a fraction 13 This also means that a matched machine of age a or higher would be scrapped, since there is no cost for a vacant machine of waiting for a worker. Here the absence of a vacancy-posting cost simpli…es the nature of the equilibrium. With such a cost, unmatched …rms would scrap capital at an earlier age than would matched …rms. 11 of the total surplus S (a) and a fraction (1 J (a) + W (a) U of a type a match goes to the worker V (a) ) goes to the …rm: W (a) = U + S (a) and J(a) = V (a) + (1 (13) )S (a) : Using the surplus-based de…nition (13) of the value of an employed worker W (a) in equation (8) and rearranging terms, we obtain the wage rate as w (a) = (r g)U + [(r S 0 (a)] : g + ) S (a) (14) Allocation of upgrading costs–The allocation of gains from upgrading is assumed to maximize the joint surplus.14 Firms of age a and workers thus split the gain from upgrading G (a) J (0) + W (0) Iu J (a) (15) W (a) ; according to the surplus sharing rule with parameter : Thus, they jointly solve max J (0) IuW (a);IuJ (a) s:t: J (a) IuJ (a) W (0) W (a) IuW (a) 1 (16) IuJ (a) + IuW (a) = Iu : The upgrading cost is then distributed according to IuW (a) = W (0) IuJ (a) = J (0) 3.2 W (a) J (a) G (a) ; (1 (17) ) G (a) : Characterizing the stationary equilibrium of the frictional economy We characterize the equilibrium of the matching model in terms of two variables: the age at which a …rm exits the market or upgrades its machine and the rate at which vacant …rms meet workers: (a; f ). The two variables are jointly determined by two key conditions. The …rst condition is labeled the job destruction or job upgrading condition. In the economy with creative destruction (upgrading) this condition expresses the indi¤erence between carrying on and scrapping (upgrading) the machine for a match with capital of age a. The second condition, labeled the job creation condition, expresses the indi¤erence for outside …rms between creating a vacancy with the newest vintage and not entering. This characterization is conditional on the steady-state employment and vacancy distributions. In Section 3.2.3 we show how these distributions can be characterized in terms of the two unknowns (a; 14 f ). In an earlier version of the paper, we also considered another assumption, namely that the …rm bears all of the upgrading costs and maximizes its own gains from upgrading. The quantitative di¤erence in results was small. 12 3.2.1 The economy with creative destruction The surplus function– It is useful to start by stating the (‡ow version of the) surplus equation. Using the de…nition of the surplus S (a) we arrive at (r g)S(a) = maxfe a S(a) f (1 )S(a) (r g)U + S 0 (a); 0g: (18) This asset-pricing-like equation is obtained by combining equations (7), (8), (9), and (13): the growth-adjusted return on surplus on the left-hand side equals the ‡ow gain on the right-hand side, where the ‡ow gain is the maximum of zero and the di¤erence between total inside minus total outside ‡ow values. The inside value includes: a production ‡ow e a ,a ‡ow loss due to the probability of a separation of the match S(a), and changes in the value for the matched parties, J 0 (a) + W 0 (a). The outside option ‡ows are: the ‡ow gain from the chance that a vacant …rm matches )S(a), the change in the value for the vacant f (1 …rm V 0 (a), and the ‡ow value of unemployment (r g)U . Note a key di¤erence with the traditional model: the value of a vacancy is positive, and it contributes towards a reduction of the rents created by the match.15 The solution of the …rst-order linear di¤erential equation (18) is the function Z a e (r g+ +(1 ) f )(~a a) e a~ (r g)U d~ a: S(a) = (19) a We have used the boundary condition associated with the fact that the surplus-maximizing decision is to keep the match alive until an age a when there is no longer any surplus to the match, S (a) = 0, and there is no gain from a marginal delay of the separation, S 0 (a) = 0. For lower a’s the match will have strictly positive surplus, and for values of a above a the surplus will be equal to zero.16 Intuitively, the surplus is decreasing in age a for two reasons: …rst, the time-horizon over which the ‡ow surplus accrues to the pair shortens with a; and second, the value of a job’s output declines with age relative to that of the new vacant jobs. Equation (19) contains a non-standard term associated with the non-degenerate distribution of vacancies: the non-zero …rm’s outside option of remaining vacant with its machine reduces the surplus by increasing the “e¤ective”discount rater through the term (1 ) f. Everything else being equal, the quasi-rents in the match are decreasing as the bargaining power of the idle …rm or its meeting rate is increasing. 15 In equation (18) we have S 0 (a) = J 0 (a) + W 0 (a) V 0 (a) : Straightforward integration of the right-hand side in (19) and further di¤erentiation shows that, over the range [0; a), the function S(a) is strictly decreasing and convex; moreover, S(a) will approach 0 for a ! a. 16 13 The job-destruction condition–The optimal separation rule S 0 (a) = 0 together with equation (19) implies that the exit age a satis…es a e = (r (20) g)U; for a given value of unemployment U . The left-hand side of (20) is the output of the oldest match in operation, whereas the right-hand side is the ‡ow-value of an idle worker. The idea is simple: …rms with old enough capital shut down because workers have become too expensive, since the average productivity of vacancies and, therefore, the workers’ outside option of searching, is growing at a constant rate. Note that this equation resembles the pro…t-maximization condition in the frictionless economy, with the worker’s ‡ow outside option, (r g)U , playing the role of the competitive wage rate. In fact, from the wage equation (14) it follows that the lowest wage paid in the economy (on machines of age a) exactly equals the ‡ow value of unemployment. We can now rewrite the surplus function (19) in terms of the two endogenous variables (a; f) only, by substituting for (r g)U from (20): Z a S(a; a; f ) = e (r g+ +(1 ) f )(~a a) e a ~ e a d~ a: (21) a In this equation, and occasionally below, we use a notation of values (the surplus in this case) that shows an explicit dependence of a and S(a; a; f) f. From (21) it is immediately clear that is strictly increasing in a and decreasing in f. A longer life-span of capital a increases the surplus at each age because it lowers the ‡ow value of the worker’s outside option, as evident from (20). A higher rate at which …rms, when idle, meet workers reduces the surplus because it increases the outside option for a …rm and shrinks the rents accruing to the matched pair. Using (10), (13), and (20) we obtain the optimal separation (or job destruction) condition Z a a (JD) e =b+ w (a; a; f )S(a; a; f )da; 0 which is an equation in the two unknowns (a; f ). The rates w (a) at which unemployed workers are matched with …rms also depend on the two endogenous variables; the explicit dependence on a and f is described below. The job-creation condition–We de…ne the value of a vacancy of age a using the new expression (21) for the surplus of a match S(a; a; 14 f) together with (13). The di¤erential equation for a vacant …rm (9) then implies that the net-present-value of a vacant …rm equals Z a V (a; a; f ) = f (1 ) e (r g)(~a a) S (~ a; a; f ) d~ a; (22) a where a equals the age at which the vacant …rm exits. Since vacant …rms do not incur any direct search cost, they will exit the market at an age such that this expression equals zero. This immediately implies that vacant …rms will exit at the same age a at which matched …rms exit and scrap their capital. Since in equilibrium there are no pro…ts from entry, we must have that V (0; a; = I0 , and we thus have the free-entry (or job creation) condition Z a I0 = f (1 ) e (r g)a S(a; a; f )da: (JCD) f) 0 This condition requires that the cost of creating a new job I0 equals the value of a vacant …rm at age zero, which is the expected present value of the pro…ts it will generate: a share 1 of the discounted future surpluses produced by a match occurring at the instantaneous rate f. The job creation condition is the second equation in the two unknowns (a; f ). In Hornstein, Krusell, and Violante (2003) we demonstrate that a solution to the two equations (JCD) and (JD) in the pair (a; f) exists and is unique under general conditions: in particular, the (JCD) condition traces a strictly decreasing curve in the (a; f) space, whereas the (JD) condition traces a strictly increasing relationship. 3.2.2 The economy with upgrading We now consider an economy where the …rm and worker jointly decide on when the machine should be upgraded and both parties share in the cost of the project. Optimal upgrading– A worker-…rm pair will not upgrade its machine as long as the gains from upgrading are negative. The match values, being the discounted expected present values of future returns, are continuous functions of the age. Since upgrading is instantaneous, at the time a machine is upgraded the gain from upgrading is then zero: G (a) = J (0) + W (0) Iu J (a) W (a) = 0: (23) At the optimal upgrading age not only is the gain from upgrading zero, but so is the marginal gain from a delay of the upgrading decision. This means that at the upgrading age the derivative of the gain function is zero, that is, J 0 (a) + W 0 (a) = 0. We can use this condition when we add the value function de…nitions of matched workers, (8), and …rms, (7): (r g) [J (a) + W (a)] = e a [J (a) + W (a) 15 V (a) U ] + J 0 (a) + W 0 (a) : (24) Together the two no-gains conditions then imply (r g) [J (0) + W (0) a Iu ] = e [J (0) + W (0) Iu V (a) U] . (25) This condition states that at the optimal upgrading age a the …rm-worker pair is indi¤erent between an upgrade and a marginal delay of the upgrade. The left-hand side of this expression is the matched pair’s return on an upgraded machine at a, and the right-hand side is the ‡ow return from a marginal delay of the upgrading decision: the production ‡ow minus the surplus capital loss from delay due to separation.17 Optimal upgrading depends on the value of a vacancy at the upgrading age, which in turn depends on the expected present value of a vacant …rm’s gain from upgrading upon meeting a worker (11). From the rule (17), which determines how the gains from upgrading (15) are shared, we obtain IuJ (a) J(0) V (a) = (1 ) [S (0) + I0 Iu V (a)] ; (26) where we have used the free-entry condition V (0) = I0 .18 Substituting this expression in the de…nition of the value of a vacancy (11) for a a ^, collecting terms, and solving the a di¤erential equation subject to the terminal condition V (^ a) = 0, we obtain an expression for the vacancy value for a a V (a) = with V (a; f) = 1 e V (^ a a) (1 V (a; ) f ) [S f= V (0) + I0 and V (27) Iu ] ; = (1 ) f +r g. In an equilibrium the value of a vacancy is non-negative at the upgrading age. The surplus and the vacancy value at the upgrading age and the surplus and vacancy value for a …rm with a new machine di¤er only through the cost of upgrading. To see this, use the surplus sharing rule (13) and the free-entry condition in the no-gains condition (23) and we arrive at S (a) + V (a) = S (0) + I0 Iu . (28) This means that the surplus at the upgrading age is given by S (a) = [1 V (a; f )] [S 17 (0) + I0 Iu ] : (29) In Appendix B we provide a heuristic derivation of this indi¤erence condition based on the limit of discrete time approximations. 18 Because we have imposed a maximum age for capital, there will always be machine exit, so entry must occur in any steady state. 16 In an equilibrium, both the surplus and vacancy value at the upgrading age are non-negative. Note that in the economy with upgrading, the surplus and vacancy value at the upgrading age can be strictly positive, unlike in the creative destruction economy where machines are scrapped at the exit age because the surplus of the match is zero. Since V the surplus and vacancy value are both zero or both strictly positive. 2 (0; 1) either In the de…nition of the job-upgrading and job-creation conditions below we use the surplus function S (a; a; 0 a f ), de…ned for machines which have not yet reached the upgrading age: a. In Appendix C we show that we can write the surplus as a function of the upgrading age and the worker meeting rate only. As a …rst step towards that result we derive an expression for the surplus value of a new machine as a function of (a; S (0; a; f) only, f ). The job-creation condition–The condition that ensures zero pro…ts at entry is always I0 = V (0; a; f) but we now have a di¤erent expression for the value of a vacant job. From (11) and (13), it is easy to see that, for a a, Z a V (a; a; f ) = f (1 ) e (r g)(~a a) S (~ a; a; a f ) d~ +e (r g)(a a) V (a; a; f) : (30) a Since the cost of upgrading a machine is independent of the age as long as the machine is not too old, a a ^, vacant machines older than a will be upgraded. Vacant machines at the upgrading age a therefore tend to have positive value, unlike vacant machines at the exit age in a creative-destruction economy. We can substitute (27) for the vacancy value at a and obtain the job-creation equilibrium condition as a function of the pair of unknowns (a; f) only: I0 = f (1 ) Z a e (r g)a (a; f ) [S S (a; a; (JCU) f ) da 0 +e (r g)a V (0; a; f) + I0 Iu ] : Also this equation is easily comparable with the job-creation condition (JCD) of the creativedestruction model, since it features only one additional positive term.19 This term captures the additional value of using the upgrading option when opening a job. This value is decreasing in Iu and also decreasing in f; the reason is that the marginal gain of the upgrading option, compared to scrapping, is large if the …rm’s meeting process is slow. 19 Note however that the expression for the surplus function S(a; a; f ) and the distributions w (a; a; f ) are not the same in the two economies. We discuss the invariant distributions below, in section 3.2.3, and as noted above we derive the expression for the surplus in Appendix C. 17 The job-upgrading condition–From the condition for the optimal delay of upgrading (25), together with the expression for the vacancy value at the threshold age a (27) and the surplus de…nitions (13), it is easy to derive that a e with J (a; f) =r g+ J [1 (a; f ) [S (0; a; (a; f )]. V f) + I0 Iu ] = (r (31) g) U; Now consider the ‡ow value of unemployment (12). Using the surplus sharing rules (13) in the …rst integral term and the upgrading cost sharing rule (17) in the second integral term of the right hand side, and substituting the expression (31) above for the ‡ow return on unemployment on the left hand side, we obtain an expression for the job upgrading condition entirely as a function of (a; Z a a e = b+ w (a; a; f )S(a; a; f )da f) : (JU) 0 + [S (0; a; f ) + I0 Z a^ + w (a; a; Iu ] f f ) [1 J (a; V f) (a; f )] da : a Comparing this equation with the job destruction condition in the economy with creative destruction (JD), we note again an additional term, always positive, implying that the upgrading age a is lower than the destruction age; how much lower it is depends on the size of the extra term. 3.2.3 Invariant employment and vacancy distributions The two kinds of economies we are considering— one with creative destruction and one with upgrading— have many parts in common but the distributions of matched …rm/worker pairs are quite di¤erent. In particular, in the former economy the density of matched pairs is sharply increasing in the age of the capital, unlike in the economy with upgrading. This is because new capital must be created prior to the creation of the match. We now describe how to construct the employment and vacancy distributions in some more detail. This characterization is interesting for understanding wage inequality, but it is also important for …nding explicit expressions for the matching probabilities in terms of the endogenous variables (a; f ). The economy with creative destruction– Denote with (a) the measure of matches between an a …rm and a worker, and denote total employment with 20 .20 The in‡ow of In Appendix D we derive the di¤erential equations which characterize the stationary employment dynamics, equations (32) and (33) below. 18 new …rms is v(0): new …rms acquire the new capital and proceed to the vacancy pool. Thereafter, these …rms transit stochastically back and forth between vacancy and match: …rms are matched with workers at rate f and they become vacant at rate . Finally …rms exit at age a, whether vacant or matched. The evolution of employment and vacancies in a stationary distribution is then determined by the di¤erential equations v 0 (a) = 0 (a) (a) = fv fv (a) (a) , for 0 a a (32) (a) , for 0 a a. (33) The evolution of matched machines is the mirror image of the evolution of vacancies, i.e., 0 (a) = v 0 (a). This implies that the number of vacant and matched machines of age a less than a remains constant: v (a) + (a) = v (0) + (0) , for 0 a (34) a. For a 2 [0; a), the evolution of (a) therefore follows 0 (a) = f [v(0) + (0)] ( + f) (35) (a): Because all …rms proceed …rst to the search pool with their new machines, we can solve this di¤erential equation subject to the initial condition (0) = 0. As Figure 1 reveals, the employment (vacancy) density is increasing and concave (decreasing and convex) in age a. The reason for this is that for every age a 2 [0; a) there is a constant number of machines, and older machines have a larger cumulative probability of having been matched in the past. This feature distinguishes our model from standard-search vintage models where the distribution of vacant jobs is degenerate at zero and the employment density is decreasing in age a at a rate equal to the exogenous destruction rate . With the vacancy distribution in hand, we now have the explicit expression for the value of w (a), w (a; a; f) = w (a) + fe = m( ; 1) v a + +f f (1 which depends only on the pair of endogenous variables (a; relation between 21 and f, ( + f ), e f )a ( + f )a ) ; (36) given the strictly decreasing equation (5).21 The closed form expressions for employment and vacancy densities, (E.2) and (E.3), are in Appendix E. 19 Figure 1. Employment and Vacancy Densities The economy with upgrading–In the economy with upgrading, the key di¤erence is that after the machine of a …rm reaches age a the …rm does not exit, but it upgrades the machine if it is matched to a worker and thereby resets the age to 0. Only vacant …rms that do not meet a worker by age a ^ exit the economy. For …rms with relatively young machines, i.e., …rms which do not upgrade their machines, the evolution of employment and vacancies in a stationary distribution continues to be determined by the di¤erential equations (32) and (33), and equation (34). With the possibility of the instantaneous upgrading of machines at age a and ongoing upgrading of vacant machines with age a a, the employment density of new machines is now strictly positive: (0) = (a) + f Z a ^ v (a) da: (37) a This means that in the economy with upgrading, the employment density is maximal for the newest vintage–it contains the machines which have just upgraded in addition to all vintages which immediately upgrade upon meeting a worker. Figure 1 illustrates how the two economies di¤er. Finally, the evolution of vacancies that are older than the upgrading 20 age is given by v 0 (a) = fv (a) , for a < a (38) a ^: Since matched machines never reach age a > a there are no additions from exogenous separations. Once the machine of a vacant …rm reaches age a ^, upgrading becomes infeasible and the …rm exits. In Appendix E we show how to solve the system of equations (32), (33), (34), (37), and (38) for the employment and vacancy distribution, (a; f ). (a) and v (a), conditional on the pair With the vacancy distribution in hand, we use (6) to obtain w (a), i.e., the rate at which workers meet vacant …rms. 4 The quantitative analysis The calibration proceeds as follows. For both the creative-destruction economy and for the upgrading economy, we target a common set of steady-state aggregate variables not including any measure of wage inequality. We then look at the implications for the level of frictional wage inequality given the resulting parameterizations. Next, we look at the e¤ect of changing the rates of disembodied and embodied technology growth on unemployment and other equilibrium labor market outcomes. 4.1 Calibration For each of the two model economies we choose the parameter values to match the same set of steady state values based on the U.S. economy. We choose a Cobb-Douglas speci…cation for our matching function, m = v u1 , where is the scale parameter. Hence, overall, we have 13 parameters to calibrate: (r; ; ; ; ; ; I0 ; Iu ; a ^; ; !; ; b). We set r to match an annual interest rate of 4%. We set the matching elasticity with respect to vacancies, , to 0:5, an average of the values reported in the comprehensive survey of empirical estimates of matching functions by Petrongolo and Pissarides (2001, Table 3). We have two sources of growth in the model: disembodied productivity change, occurring at rate , and capital-embodied productivity change. Hornstein and Krusell (1996) measure annual disembodied growth in the United States for 1954–1993 to be 0:8% per year, whereas more recently Cummins and Violante (2002) compute it to be 0:3% per year from 1965– 2000. We set = 0:5%. At least since Greenwood et al. (1997), a number of authors have suggested to measure the speed of embodied technical change through the (inverse of the) 21 rate of decline of the quality-adjusted relative price of capital. We argued above that in our environment embodied technical change is directly re‡ected in the rate at which the relative price of new capital declines, g. Gordon’s (1990) in‡uential work on quality-adjusted prices for durable goods suggests a value for the annual rate of embodied technical change in the United States around 3%. Given the observed 2% average U.S. output growth rate g, the rate of price decline for capital implies that = 5%: From the relation g = (1 !) , we obtain a capital elasticity parameter ! = 0:3. For the creative-destruction economy we simultaneously calibrate the remaining parameters ( ; ; ; I0 ; ; b) so that the steady state has (1) an unemployment rate of 4% (the U.S. historical average); (2) an average unemployment duration of 8–9 weeks, as reported by Abraham and Shimer (2001); (3) a labor income share of 0:685 (Cooley and Prescott (1995)); (4) an average vacancy duration of 4 weeks as estimated by Hall (2003) based on data from the Job Openings and Labor Turnover Survey; (5) an average age of capital of about 11:5 years, as reported by the Bureau of Economic Analysis (2002); and (6) an average replacement rate of 10%.22 The parameter b is supposed to summarize a wide range of bene…t policies that vary with unemployment duration and family status (none of which we model). The OECD Employment Outlook (1996) provides replacement rates for unemployment bene…ts in OECD countries from 1961 to 1995 for two earnings levels, three family types, and three durations of unemployment. The reported average replacement rate (in terms of the average wage) for the United States in that period was 10% and we choose b to replicate this number. For the economy with upgrading one can think of the calibration as matching the same targets but using Iu instead of I0 . This is because with upgrading we set the maximal age a ^ beyond which machines cannot be upgraded at 30 years. Given the matched steady state rate at which vacant machines meet workers, this high value of a ^ means that almost all machines will eventually upgrade even if initially vacant, and entry of new machines will essentially be zero. Since e¤ectively there is no entry, a ^ and I0 only have a minor e¤ect on the steady state (so I0 can be solved for residually): we again match six variables with six parameters. 22 The unemployment rate together with the average unemployment duration imply an annual separation rate for workers from employment to unemployment of 22% for either economy. This value of the separation rate is in line with the data reported in Alogoskou…s et al. (1995, page 10). 22 Table 1. Calibration of benchmark economies Common parameters r = 0:04; b I0 Iu a ^ w90 =w10 waverage =wmin = 0:5; = 0:005; = 0:05; ! = 0:3 Economy with Economy without upgrading investment upgrading investment Model speci…c parameters 8.911 8.923 0.061 0.064 0.255 0.214 0.802 0.767 0.048 0.048 4.599 4.281 4.278 NA 30.00 NA Wage Inequality 1.078 1.058 1.053 1.040 The parameter values are summarized in Table 1. A clear conclusion emerges: if we take the distance between parameters as a measure of closeness of the two economies, then it appears that the economies are remarkably similar: very small parametric di¤erences are needed to match the same set of facts. In our calibration we have not tried to match the wage inequality generated by the vintage induced productivity di¤erences in the two economies. We use two measures of wage inequality: the 90-10 wage ratio used widely in the empirical labor literature, and the ratio of average wages to minimum wages.23 From Table 1 we can see that the 90-10 wage ratio is about 8 percent (6 percent) in the economy with(out) upgrading, and the second inequality measure is about 5 percent (6 percent) in the economy with(out) upgrading. The fact that the economy with upgrading generates somewhat more wage inequality for both measures is related to the qualitative di¤erence of employment densities discussed in section 3.2.3. Even though the maximal and minimal wages of the youngest and oldest vintages are very similar in the two economies, there are relatively more high-productivity young vintages in the economy with upgrading. This feature increases the upper cut-o¤ wage for the 90-10 wage ratio and it increases average wages in the economy with upgrading. 23 The latter measure is useful because we show in Hornstein, Krusell, and Violante (2005) that the standard DMP model without capital makes very tight predictions on what this measure must be given the usual steady-state statistics. 23 To put these numbers on wage inequality in perspective, for observationally equivalent workers Katz and Autor (1999) report 90-10 wage ratios around 1.5 and Hornstein, Krusell and Violante (2005) report ratios of mean wages to minimal wages (proxied by the …rst percentile) around 1.3. These numbers are computed on the distribution of residuals in a typical wage regression, after controlling for observable characteristics of the workers (gender, race, education, experience) and …xed individual e¤ects to capture “unobserved ability.” Hence, the combination of matching frictions (“luck”) and vintage capital di¤erentials can only explain at best 1/6th of the observed inequality among ex-ante equal workers. 4.2 Creative-destruction vs. upgrading: a comparison We now analyze, for the baseline parametrization, the response of the two economies to accelerations in the rate of embodied technical change, , and in the rate of disembodied technical change, , of an empirically plausible magnitude. In our analysis we focus on the behavior of the unemployment rate, the average unemployment duration, the critical age at which upgrading/exit occurs, the wage income share, the ratio of average wages to lowest wages, and the 90-10 wage ratio. Embodied technical change– Krusell et al. (2000) and, more recently, Cummins and Violante (2002) have argued that the annual rate of embodied technical change in the U.S. has increased substantially in the past two decades, up to 6:5% over the years 1995-2000.24 This estimate, together with the assumption that and ! remain constant, means that has increased to 10% so as to generate a decline in the relative price of capital of 6:5% per year. The results of this experiment are reported in Figure 2. A faster rate of embodied technical change increases the unemployment rate and wage inequality and lowers the wage income share. The accelerated technical change shortens the useful life-time of a machine, that is, machines are either upgraded at a faster rate or they exit the economy at a faster rate. Although wages fall, re‡ected in the declining wage income share, they do not fall enough to compensate completely for the shortened life-time of machines. As a consequence, the value of …rms declines, but in equilibrium the value of a new machine has to equal the constant (normalized) cost of a new machines. Therefore the rate at which …rms meet workers has 24 Other authors, using measurement techniques di¤erent from quality-adjusted relative prices, arrived at very similar conclusions on the pace of embodied technical change in the postwar era (see for example Hobijn, 2000) for the United States. 24 to increase, and correspondingly the rate at which workers meet …rms declines, that is the average duration of unemployment increases. This means that the measure of active …rms and employment declines. Figure 2. Comparative Statics for Embodied Technical Change Employment in the economy without upgrading declines somewhat more than in the economy with upgrading because of the creative destruction e¤ect: upon …rm exit workers enter the unemployment pool. Overall, the di¤erences across the two economies are minor along all dimensions examined. Disembodied technical change– Mortensen and Pissarides (1998) have pointed out that there is a qualitative di¤erence between embodied and disembodied technical change. Whereas a higher rate of embodied technical change tends to lower the value of existing machines, a higher rate of disembodied technical change increases the value of machines because it increases the output over the life time of a machine. Machines become more valuable, vacancy values increase, more machines seek to match with workers, and therefore unemployment declines. Wage inequality declines and the wage income share increases. 25 Figure 3. Comparative Statics for Disembodied Technical Change From Figure 3 we see (1) that the economies with and without upgrading essentially respond in the same way to a change in the rate of disembodied technical change, and (2) that doubling the rate of disembodied technical change has a negligible quantitative e¤ect on labor market variables. 4.3 Interpreting the …ndings: an intuitive argument The facts that search frictions generate only a limited amount of wage inequality in our vintage model and that the nature of the investment decision does not a¤ect the comparative statics in a quantitatively important way, result from two important quantitative restrictions that calibration to the U.S. labor market imposes on the equilibrium of our model. First, on average a job/machine remains vacant in the U.S. economy for about four weeks, to be compared to the expected lifetime of a new machine, which is over 11 years. When the labor market frictions are so small from the …rm’s perspective, the …rm becomes roughly indi¤erent between scrapping and upgrading: for a given cost of new machines, the upgrading option is substantially better than scrapping only if the matching process is long 26 and costly. Second, the average duration of unemployment in the U.S. economy is very short, around 8 weeks. In search models, unemployment duration is directly related to the option value of search for the risk-neutral worker and the latter depends positively on wage dispersion. A duration of unemployment as short as it is in the data can therefore only be consistent with a minor amount of frictional inequality. Interestingly, our results are consistent with the conclusions of Ridder and van den Berg (2003) who estimate the severity of search frictions in the United States, United Kingdom, France, Germany, and Netherlands and …nd that the labor market in the United States is, by far, the least frictional. The (inverse of their) measure of search frictions— the arrival rate of job o¤ers to the unemployed workers— for the United States is 5 times larger than for the United Kingdom, 8 times larger than for Netherlands, 15 times larger than for France, and 22 times larger than for Germany. Technically, we show in Proposition 1 below that, for I0 = Iu and as matching frictions vanish, the upgrading model collapses to the creative-destruction economy. In addition, both economies converge to the frictionless benchmark of Section 2 where the equilibrium displays a unique market clearing wage. Proposition 1: If I0 = Iu and the meeting rate in the labor market becomes in…nitely high ( f ! 1), the economy with upgrading converges to the economy with creative destruc- tion. Moreover, the two economies converge to the frictionless benchmark which displays no wage inequality. Proof: See Appendix F. Finally, note that the result that the frictionless vintage capital model is obtained as the limit when frictions vanish is a desirable and natural property of our setup, which is not shared by the traditional framework. It is easy to verify that the equilibrium of the traditional matching model with vintage capital, where vacant …rms do not purchase a machine upon entry but simply pay a ‡ow cost to post their vacancy (recall our discussion of Section 3), converges in the limit to an economy where only the newest type of capital is in operation. Put di¤erently, not only do wage inequality and unemployment disappear as the matching frictions fade, which is natural, but the vintage capital component of the model vanishes as well. Therefore, in the traditional model the vintage capital structure is not a result of the 27 nature of technical change and of the investment decision, but its existence depends entirely on the fact that it is costly for …rms to replace workers, and hence that …rms hold on to their machines longer. This latter e¤ect is still present in our model, but it is not the only force generating heterogeneity in the age of capital in operation. 5 Extensions Still within the context of identical workers, we now consider two additional sources of inequality: productivity heterogeneity that is (i) machine-speci…c or is (ii) match-speci…c. In the former case, an entrepreneur who enters the economy buys a new machine but the new machine may not be a good match with the entrepreneur’s skills, and independently of the worker with whom the entrepreneur-machine pair is matched, the idiosyncratic productivity level stays intact (controlling for the aging of capital). Thus, the relevant idiosyncratic speci…city is between the entrepreneur and the machine; once the machine is scrapped (or upgraded), the speci…city disappears. In the latter case, in contrast, the relevant idiosyncratic speci…city is between the entrepreneur-machine and the worker; once the worker is gone, the speci…city disappears. In the economy with match-speci…c productivity, upgrading (as opposed to creative destruction) helps …rm-worker pairs keep the good matches for a longer time. In the economy with machine-speci…c productivity, this e¤ect is not present. 5.1 Model speci…cs We now brie‡y outline some speci…cs of each of the extensions. The details of each model can be found in the Appendix. 5.1.1 Machine-speci…c productivity In the economy with machine-speci…c productivity, once a new machine is created it has initial productivity ez , which is drawn from a probability density function f (z) with bounded support [0; z^]. As before, machines age and depreciate such that the productivity of an age a machine at time t relative to a new machine with initial productivity ez+gt is ez a . This productivity does not depend on the worker the machine/…rm is matched with. If the machine is ever upgraded it will receive a new productivity draw from f (z). 28 Equilibrium is straightforward to de…ne both in the cases of creative destruction and upgrading; we list the equations in Appendix G. Computation of an equilibrium is not more di¢ cult than before since machines can be ranked along a productivity dimension that captures both age and machine-speci…c productivity. In this sense, the model here is isomorphic to the one without machine-speci…c productivity. The di¤erence between models, of course, is quantitative: inequality in productivity across machines is now not limited to that implied by the relative price data for equipment of di¤erent vintages. 5.1.2 Match-speci…c productivity Once a new match is created it has productivity ez , with z drawn from a probability distribution with density f (z) and bounded support [0; z^]. As before, machines age and depreciate such that the productivity at age a relative to a machine with initial productivity ez+gt is ez a . If the machine is ever upgraded in an existing match it will maintain the same productivity draw z. Characterization of equilibria is less straightforward under match-speci…c productivity, because the …rm-worker pair now has two non-trivial state variables: age and productivity. Thus, whether there is scrapping/upgrading depends on the (a; z) pair. We now brie‡y discuss the characteristics of the stationary equilibrium in the case with creative destruction as well as in the upgrading case. The details of the analysis are to be found in Appendix H. In the economy without upgrading, the question is when to scrap a machine. For every z, there is a scrapping age a(z). Because scrapping means the dissolution of the match, a(z) is an increasing function: productive matches are dissolved later. In the economy with upgrading, matters are less transparent. There are three possible outcomes in a hypothetical (a; z) match: stay and do not upgrade, stay and upgrade, and separate. The option to separate now appears because of bad matches: if z is low enough, the machine is better employed elsewhere. For high enough values of z, however, separation never occurs, and the upgrading age aU (z) is now decreasing in z: the better the match, the more frequent is upgrading, because vintage and productivity are complementary. De…ne the cuto¤ for this region by z1 . When z is low enough, i.e., below a value z0 < z1 , upgrading never occurs but separation occurs for an age above aS (z). Thus, if the machine is very new, for these values of z the match remains intact and produces, but at a high enough age it is better for the machine to become vacant so as to …nd a better match, upon which upgrading could occur and the a would be reset to zero. The function aS (z), moreover, is increasing: 29 for a given a, an increase in z makes staying together more attractive. Figure 4. Optimal Separation and Upgrading Decisions Finally, when z is in an intermediate range, i.e., z 2 [z0 ; z1 ], then there are three regions: if a is low enough, the match remains intact; if it is in an intermediate range, the match separates and the machine looks for a better match; and if a is high enough, upgrading takes place. The three regions are thus de…ned by two cuto¤ functions: aS (z), for indi¤erence between staying and separating, and aU (z), for indi¤erence between separating and upgrading. Figure 4 summarizes the three regions. 5.2 Results The one-line summary of our experiments is that neither the introduction of machine-speci…c nor of match-speci…c productivity heterogeneity changes our main results: search frictions still generate only a limited amount of wage inequality in our vintage model, and the nature of the investment decision does not a¤ect the comparative statics in a quantitatively important way. We consider the four economies implied by the combination of the two elements: (1) productivity heterogeneity (machine-speci…c or match-speci…c) and (2) means of machine replacement (creative destruction or upgrading). For each economy we assume a two-point 30 support for productivity heterogeneity, z 2 fz1 ; z2 g. We normalize z1 = 0 and let z2 > 0 be the high-productivity machine/match. In particular, we consider economies with z2 2 Z2 = [0; z2 ] and f2 2 F2 = 0; f2 . For values of (f2 ; z2 ) 2 F2 Z2 , we calibrate each economy to the same steady-state values of the unemployment rate, unemployment duration, replacement rate, wage income share, and average machine age as in section 4.1.25 5.2.1 The comparative statics of technological change Recall that in the benchmark models we have shown that (1) embodied technical change increases the unemployment rate, the unemployment duration, and wage inequality, and that it lowers the wage income share; and that (2) disembodied technical change lowers the unemployment rate, the unemployment duration, and wage inequality, and that it increases the wage income share. The models with additional heterogeneity all display the same comparative statics as the benchmark models, both qualitatively and quantitatively.26 Figure 5 plots the comparative statics with respect to and in the economy with match-speci…c heterogeneity and upgrading. The graphs for the other three economies look remarkably similar, so we omit them. 5.2.2 Productivity heterogeneity and wage dispersion For the case of machine-speci…c productivity heterogeneity, we …nd that both the 90-10 wage ratio and the ratio of average wages to minimum wages are very insensitive to all the values of (f2 ; z2 ) that we consider; recall that our one-type model is reproduced by setting either of these two variables to zero. In the model with machine-speci…c heterogeneity, the wages paid by high and low productivity machines are very close since di¤erences in machine-speci…c 25 For the economies with machine-speci…c heterogeneity we match all steady-state targets for the range de…ned by z2 = 0:2 and f2 = 0:5. The calibration of economies with match-speci…c heterogeneity is somewhat more di¢ cult. For the economy without upgrading we match all steady-state targets except the average age of machines on the same range of (f2 ; z2 ). For low values of f2 and high values of z2 , the average age of machines increases to 13 years, which is larger than the 11.5 year target. For the economy with matchspeci…c heterogeneity and upgrading we match most of the steady-state targets on a smaller range de…ned by z2 = 0:2 and f2 = 0:2. Similarly to the case without upgrading, we have some problems here in matching the average age of machines for the combination of low f2 and high z2 values. In addition, for high z2 values our calibration procedure yields unemployment rates of 5 percent rather than the 4 percent target. It should also be pointed out that for this calibration exercise we are matching …ve variables with six parameters: we have dropped the average vacancy duration as a steady-state target. Nevertheless the calibration problem for the economy with match-speci…c heterogeneity is su¢ ciently non-linear that we are unable to mach all steady-state variables for all (f2 ; z2 ) combinations. 26 The comparative statics analysis of our four economies are all taken for the same point in the interior of our set F2 Z2 ; i.e. f2 = 0:1 and z2 = 0:1. 31 quality are re‡ected in the …rm’s productivity and in the …rm’s outside value option. These two e¤ects enter in the wage determination with opposite signs: high machine-productivity increases wages, but it also raises the outside option of the …rm, reducing wages. Therefore, wages (and wage dispersion) are largely una¤ected by this type of productivity dispersion. Figure 5. Comparative Statics for Embodied Technical Change Technical Change and Disembodied in the Model with Match-Speci…c Heterogeneity and Upgrading. Match-speci…c productivity-heterogeneity, on the other hand, does a¤ect our measures of wage inequality quite substantially. This is the only new feature of our economies with heterogeneity. In the economy with creative destruction the ratio of average wages to minimum wages is essentially independent of the (f2 ; z2 ) values, but the 90-10 wage ratio can increase from 5 percent to 10 percent for low values of f2 and high values of z2 . In the economy with upgrading both measures of wage inequality are a¤ected by productivity heterogeneity. The ratio of average wages to minimum wages increases only marginally from 5 percent to 7 percent, but the 90-10 wage ratio can increase up to 20 percent for low values of f2 and high values of z2 . See the last panel in Figure 5. In this economy the surplus in a high-productivity match is substantially higher than in the low-productivity match because the outside option of machines is independent of the current match-speci…c productivity, so we only have a positive productivity e¤ect on wages. The same vintage machine pays a substantially higher wage if it is in a high-productivity 32 match. Furthermore, due to upgrading, there is indeed a substantial positive mass of new machines with high z, since upgrading occurs most frequently for good matches (see Figure 4). All this together means that the top 10 percent of all wage earners are essentially all in good matches with young machines. Thus an increase of the high productivity level translates almost directly into an increase in the relative wage of the top percentile group, and in the 90-10 wage ratio, whereas the average-to-minimum wage ratio is less a¤ected. 6 Conclusions The existing literature has pointed out that it matters qualitatively for equilibrium unemployment whether technological progress bene…ts only new matches or also ongoing relationships (Aghion and Howitt, 1994; Mortensen and Pissarides, 1998). In this paper we have shown that, if one takes the view— common in modern macroeconomics— that economic models should be calibrated and tightly parameterized to replicate certain key features of aggregate data, then the qualitative ambiguity of the growth-unemployment relationship resolves into a stark quantitative answer: the various approaches to the capital replacement problem in frictional economies all yield near-equivalent quantitative results. This conclusion is reinforced once one includes into the picture the equilibrium income shares and wage inequality, beyond the unemployment rate. The driving force behind this result is that, quantitatively, the labor market frictions are very small: the average duration of vacancies and unemployment in the U.S. is just 8 weeks, whereas the average life of capital is over 11 years. As a result, upgrading gives only a very small advantage compared to innovating through creative-destruction. Moreover, we explained that the fact that the calibrated model is so close to the frictionless model, quantitatively, is also responsible for the …nding that equilibrium wage dispersion is tiny in the model. Only the model combining match-speci…c heterogeneity and upgrading generates sizeable frictional inequality. Three important caveats apply to our conclusions. First, in economies where frictions are more severe, like continental-European labor markets where average unemployment duration can reach 6-8 months, our equivalence result could be weaker. Second, the two replacement models may not be equivalent for the evaluation of the e¤ects of certain labor market policies. For instance, the two approaches have di¤erent implications for employment protection policies: in a world where the introduction of more 33 productive capital requires a re-organization of production and a …rm-worker separation, employment-protection policies can have a large impact on average productivity, whereas in a world where capital can be upgraded without shedding labor, the e¤ects of these policies will be minor.27 Third, throughout the analysis we maintained the hypothesis that workers are homogeneous and every source of heterogeneity that we analyzed originates from the …rm’s side. Random matching models with simultaneous …rm and worker heterogeneity are particularly di¢ cult to analyze because, generally, they have multiple steady-states (Sattinger 1995). Thus, the right modeling strategy seems to be to introduce directed search with segmented markets. One particularly interesting dimension of workers’ heterogeneity, in the context of our investigation, is age. The incentives to upgrade an existing machine are higher, the younger is the worker. Hence, creative destruction should occur mostly for older workers, whereas upgrading should be concentrated among younger workers. In line with this prediction, Bartel and Sicherman (1993) document that technical progress reduces employment for old cohorts and increases employment for younger workers. Future work should be directed toward evaluating the robustness of our …nding with respect to various dimensions of workers heterogeneity. Finally, our result has useful implications from the perspective of a recent literature that tries to identify the relative importance of disembodied technical change vis-a-vis capitalembodied productivity advances in the U.S. by exploiting the di¤erent implications these shocks have on job creation, job destruction, and the unemployment rate (see Pissarides and Vallanti 2003, and Lopez-Salido and Michelacci, 2003). In our analysis, all conclusions are based on steady-state comparisons. In other words, our equivalence result holds in the long run, but we have not yet studied the short-run predictions of the di¤erent models. In this sense, we provide a cautionary remark and a suggestion: our …ndings here suggest that it is likely very di¢ cult to disentangle the di¤erent sources of technical change from a lowfrequency analysis of the data. However, a high-frequency analysis of the response of labor market ‡ows to technology shocks might prove to be more informative. 27 See Ljungqvist (2002) for a discussion of the impact of …ring costs in a large class of environments, among others a standard matching model. 34 Appendix A Derivations of typical value functions The value functions of our continuous-time model can be derived as limits of a discrete time formulation. A typical derivation of the di¤erential equations for value functions (7)-(10) goes as follows. Consider the value of a vacant …rm with capital of age a a at time t, V~ (t; a). For a Poisson matching process, the probability that the vacant …rm meets a worker over a small …nite time interval [t; t + ] is as V~ (t; a) = f h ~ + J(t ;a + f. V~ (t + ) We can de…ne the vacancy value recursively i ) +e ;a + V~ (t + r ;a + ); where the …rst term is the expected capital gain from becoming a matched …rm with value J~ and the second term is the present value of remaining vacant at the end of the time interval. On a balanced growth path all value functions increase at the rate g over time, i.e., V~ (t; a) = egt V (a) and J~ (t; a) = egt J (a). Subtracting V~ (t + ; a) from both sides, ~ and dividing by eg(t+ ) , substituting the balanced growth path expressions for V~ and J, we can rearrange the value equation into e g V (a) eg 1 = [J(a + ) V (a + V (a + ) V (a) f + )] + e r di¤erential equation (9): B f [J(a) V (a)] V (a + ) : As we shorten the length of the time interval and take the limit for gV (a) = 1 ! 0, we obtain the rV (a) + V 0 (a): Derivation of optimal upgrading condition Consider the following discretization of the investment decision when a worker-…rm pair maximizes the joint value of the match.28 The length of a time period is . At a the …rm and worker prefer to upgrade at a rather than delaying it by one period: W (0) + J (0) Iu e a +e + (1 28 (r g) f( ) [V (a + ) [W (0) + J (0) ) + U] Iu ]g We consider the formulation of the problem after variables have been made stationary, that is normalized. 35 The left hand side is the joint capital value after upgrading at a, and the right hand side is the ‡ow return from production without upgrading plus the expected present value from upgrading in the next period. Note that the match separates with probability the upgrading opportunity. Rearranging terms and dividing by [W (0) + J (0) Taking the limit as Iu ] 1 (1 we get (r g) )e a e + e (r g) [V (a + ) + U] : ! 0 yields [W (0) + J (0) At age (a and loses Iu ] (r g+ ) e a + [V (a) + U ] : ) the …rm and worker prefer not to upgrade, but to delay until a: W (0) + J (0) Iu e (a ) + (1 Rearranging terms and taking the limit as [W (0) + J (0) +e (r g) f( ) [W (0) + J (0) ) [V (a) + U ] Iu ]g : ! 0 we get Iu ] (r g+ ) e a + [V (a) + U ] : Iu ] (r g+ )=e a + [V (a) + U ] ; Therefore we must have that [W (0) + J (0) which is equation (25) in the main text. C Derivation of the surplus function in the economy with upgrading In section 3.2.1 we have derived the di¤erential equation for the surplus value of a matched …rm-worker pair in a creative-destruction economy. This equation determines the surplus as a function of the age of the …rm’s machine and it is de…ned from the time of entry to the time of exit, 0 a a. The surplus value in an economy with upgrading satis…es the same di¤erential equation but the terminal condition for the surplus value di¤ers. In the creative-destruction economy the …rm/machine exits at age a and the surplus at the time of exit is zero, S (a) = 0. In the economies with upgrading the machine is upgraded at age a and the surplus at that age S (a) is de…ned in equation (29). 36 Substitute (31) for (r g) U into the di¤erential equation for the surplus value (18), and solve that equation subject to the terminal condition (29) for S (a) Z a S (a; a; f ) = e s (s a) e s e a ds (C.1) a + [S (0) + I0 Iu ] [r g + (1 V (a; f ))] Z a a e ss ds 0 + (1 with s =r g+ two unknowns (a; + (1 f) ) f. V (a; s (a f )) e a) ; This is an expression for the surplus as a function of the and S (0) : Now evaluate this expression at a = 0 and solve for the surplus value of a new machine S (0). We obtain S (0; a; f) 1J = + (I0 1 Iu ) 2J (C.2) ; 2J with 1J (a; f) Z a sa e e a e a da; 0 2J (a; D f) fr g + [1 V (a; f )]g Z a sa e da + [1 V (a; f )] e sa : 0 Derivation of the steady-state employment dynamics The equations describing employment dynamics are derived as follows. Consider the measure of matched vintage a …rms at time t. Over a short time interval of length , the approximate change in the measure is (t + Subtracting (t + ; a) = (t; a )(1 )+ (t; a) from both sides and dividing by ; a) Taking the limit for (t; a) = (t; a) (t; a f (t; a we obtain ) (t; a ! 0 we obtain t (t; a) = a (t; a) (t; a) + 37 ): f (t; a): )+ f (t; a ): At steady state, these measures do not change with t, and we obtain the result stated in (33). In the economy with upgrading the initial measure of matched …rms with new machines evolves according to (t + ; 0) = (t; a) + ( f) a ^= X v (t; 0) + ( f) v (t; ai ) : i=a= Taking the limit for E ! 0 we get (37) for the steady state. The invariant employment and vacancy distributions We solve the di¤erential equation (35) for matched pairs backwards and get Z a (a) = f [ (0) + v (0)] e ( + f )(a a~) d~ a + (0) e ( + f )a ; (E.1) 0 and v (a) = v (0) + (0) (a) for 0 a In the economy without upgrading a. (0) = 0 and we get closed form expressions for the employment and vacancy densities (a)= = v(a)=v = 1 a a + 1 + e (1 f ( + e + fe f (1 f + f )a ( + ( + e f )a ) (E.2) ; f )a ( + f )a ) (E.3) : In the economy with upgrading we have to solve for the employment density of new machines (0). We solve the di¤erential equation (38) for vacancies on the interval [a; a ^] backwards and get v (a) = e f (a a) (E.4) v (a) : The total measure of vacancies on [a; a ^] is then Z a ^ v (a) da = v (a) A2 with A2 a Z a ^ a e fa da: (E.5) 0 We evaluate expression (E.1) at a to get the measure of existing matches that upgrade: Z a + f )a ( (a) = [ (0) + v (0)] f A1 + (0) e with A1 e ( + f )a da: (E.6) 0 38 We substitute (E.6) and (E.5) into (37) and get (0) = (a) + f A2 v = (a) + f A2 h (a) [ (0) + v (0) (a)] f A2 ] f (0) + v (0)g = [1 + f A2 ( f A1 + (0) e + f )a [ (0) + v (0)] : i We can solve this expression for the density of new employed machines as a function of new vacant machines (0) = Bv (0) with (1 f A2 ) f A1 + f A2 B = ( + f )a 1 (1 f A2 ) f A1 + e (E.7) : f A2 Note that B can be simpli…ed to 1 a f (^ e a) B= e a f (^ n 1 h a) 1 h 1 e ( e ( + + )a f f )a i i f= ( + =( + f) o ) f : For the calibration of our economy B is very large since the denominator is close to zero. This will be important when we obtain numerical solutions of the steady state. Substituting (E.7) and (37) into the expression for the density of employed machines at the upgrade age a, (E.6), yields (a) = v (0) (1 + B) A + Bv (0) e ( + f )a or f 1 (a) = C1 v (0) with C1 = (1 + B) f A1 (E.8) + Be ( + Evaluating (37) at a and solving for v (a) we have v (a) = substitute (E.8) for (a) and (E.7) for f )a : (0) + v (0) (0) we have v (a) = C2 v (0) with C2 (a). After we (1 + B) (1 (E.9) f A1 ) Be ( + f )a : Integrating the employment density (E.1) over the interval [0; a] yields total employment Z a Z a Z a Z a + f )a ~ ( e ( + f )a~ d~ a: e d~ a da + (0) (a) da = f [ (0) + v (0)] 0 0 0 0 39 Substituting (E.7) for (0) in (37) yields Z a (a) da = C3 v (0) with (E.10) 0 C3 = (1 + B) f (a A3 ) = ( + Z a A3 = e ( + f )a da: f) + BA3 0 We can now calculate the total measure of vacancies on the interval [0; a]. Using (37) we get Z a Z a Z a v (a) da = [ (0) + v (0) (a)] da = [ (0) + v (0)] a (a) da; 0 0 0 and using equations (E.7) and (E.10) we get Z a v (a) da = C4 v (0) with (E.11) 0 C4 = (1 + B) [a (a A3 ) f= ( + f )] BA3 : Combining equations (E.5), (E.9), and (E.11) yields total vacancies as Z a ^ v (a) da = C5 v (0) with (E.12) 0 C5 (1 + B) [a (a A3 ) f = ( + h i + f )a ( B A3 + A 2 e : f) + A2 (1 f A1 )] To get the density of new …rms coming into the economy with new machines we use the de…nition of labor market tightness = and solve for v (0) R a^ v (a) da C5 v (0) = Ra 1 C3 v (0) (a) da 0 0 1 v (0) = C3 + C5 : Note that both C3 and C5 are linear in B, and since B is large for the calibration of the economy, entry is essentially zero. A good approximation of the employment and vacancy 40 densities is then obtained by multiplying v (0) with B and dividing all densities with B, or h i ~ ~ v~ (0) = Bv (0) = = C3 + C5 with C~5 = (1 + 1=B) [a (a A3 ) i h A3 + A2 e ( + f )a [a (a A3 ) C~4 = (1 + 1=B) f (a (a (a f (a C~1 = (1 + 1=B) +e ( (0) = v~ (0) , F f A1 + f) f + f + f) + A2 (1 + A2 (1 f) + A3 f) + A3 f A1 )] f A1 )] h A 3 + A2 e ( + f )a i + A3 + A3 e ( f A1 ) e ( f A1 ) f) A3 ) = ( + C~2 = (1 + 1=B) (1 f A1 f) A3 ) = ( + (1 + A3 ) = ( + A3 ) = ( + C~3 = (1 + 1=B) f f f= ( f= ( + f )a )a +e ( + f )a )a (a) = C~1 v~ (0) , v (a) = C~2 v~ (0) . Proof of Proposition 1 We model the disappearance of the matching friction by letting the shift parameter of the matching function endogenous variable limits as f ! 1. Given that all the relevant equations are written in terms of the f, which is increasing in , our line of proof will be based on taking ! 1: We use the key equilibrium conditions of the two replacement models (creative-destruction and upgrading) to show that (i) the economy with upgrading converges to the economy with creative destruction and that (ii) the latter converges to the frictionless economy when the instantaneous meeting rate for …rms becomes large enough and I0 = Iu . Precisely, we …rst show that as f ! 1 the “extra”terms that appear in the conditions (JCU) and (JU), but do not appear in the conditions (JCD) and (JD), vanish. Second, we show that the expressions for the surplus function converge as well. Third, we show that the (JCD) condition converges to the frictionless free-entry condition (4) and that the wage function w (a) implicitly de…ned in (14) collapses to the marginal product of labor e a ; i.e. the unique competitive wage. Finally, we show that the distribution of employed machines in the two economies converge to the competitive equilibrium distribution. 41 Proof: Consider the extra term in (JCU) and let converges to 1 and S (0; a; f) f ! 1. The expression V (a; f) converges to zero. The latter limit is clear from simple inspec- tion of (C.2) in Appendix C, since both 1J (a; f) and 2J (a; converge to zero as f) f gets large. Hence, the extra term in (JCU) converges to zero. Consider now the extra term in (JU) and let f ! 1. Since V (a; f) converges to 1 for all a’s, S (0; a; f) converges to zero, and I0 = Iu , then this term goes to zero as well. It is easy to see, from (C.1) in Appendix C, that the extra term (the second and third lines) in the surplus function of the economy with upgrading goes to zero as thus the expressions for the surplus in the two economies converge. Now consider how equation (JCD) changes as f ! 1, and ! 1. Using the surplus expression f (21) in (JCD) and integrating the right-hand side, yields I0 = (1 ) e (r g+ )a r g+ 1 ega + (1 ) f 1 2 +e 1 f 2a 2 =r Taking the limit of expression (F.1) as f ! 1, we get I0 = r (r g+ )a g+ e amin e (r r g 1 (r )a (F.1) ; 0 e e r 1 where we have introduced the notation 1 a1 e g+ , g)a = Z 1 = 0 + (1 f, ) and 2 = 1+ . a e (r g+ )a 1 e (a a) da; 0 which is the key equilibrium condition (4) of the frictionless model in section 2. Now, consider equation (18) that implicitly de…nes the surplus function. As see that the term f (1 ) S(a; a; equilibrium condition (r that (r w (a) = e g + )S(a) a f) g) U = e converges to e a a e a f ! 1, it is easy to . Using this result and the , we notice immediately that equation (18) implies S 0 (a) = 0: Using this result in the wage equation (14), we obtain that for every a, which is the competitive wage. It only remains to show that the vacancy and employment distributions converge, but this is trivial once it is recognized that as the meeting rate for …rms goes to in…nity, the measure of vacancies tends to zero and the employment density is simply in the frictionless economy. QED 42 (a) = = 1=a like G G.1 Equilibrium with machine-speci…c productivity differences The economy without upgrading Optimal entry and exit Capital values are functions of a machine’s age a and quality z. Machines age, but the quality of a machine does not change over time. (r g)J(a; z) = max ez (r (r a [J(a; z) V (a; z)] + Ja (a; z); (G.1) g)V (a; z)g ; g)W (a; z) = max fw(a; z) [W (a; z) U ] + Wa (a; z); g)U g ; (G.2) g)V (a; z) = max f f [J(a; z) V (a; z)] + Va (a; z); 0g ; Z z^ Z a^ (r g)U = b + U ] dadz: w (a; z) [W (a; z) (G.3) (r (r w(a; z) 0 (G.4) 0 The surplus value of a match is S (a; z) = [J (a; z) and the worker receives a share W (a; z) V (a; z)] + [W (a; z) U] ; of the surplus: U = S (a; z) and J (a; z) V (a; z) = (1 ) S (a; z) : The implied di¤erential equation for the surplus value is then [r g+ + f (1 )] S(a; z) = ez a (r g)U + Sa (a; z): (G.5) This di¤erential equation can be solved conditionally on the terminal condition that de…nes the optimal time of exit a (z) (the job-destruction condition) (G.6) S [a (z) ; z] = Sa [a (z) ; z] = 0: This implies that ez a = (r g)U or z a (z) = and the surplus capital value is Z a(z) S(a; z) = e [r g+ +(1 ) f ](~a a Z [z a ]= = e [r g+ +(1 0 = S (z a) : 43 a) ) ez f = log [(r a ~ ]a~ e(z g) U ] ; (r a) g)U d~ a a ~ (r g)U d~ a (G.7) The vacancy value is V (a; z) = f (1 ) Z a(z) e (r g)(~ a a) S (z a ~) d~ a a = f (1 ) Z a [z ]= e (r g)~ a S (z a a ~) d~ a 0 = V (z a) : The free-entry condition reads I0 = Z z^ f (z) V (z)dz: 0 Employment and vacancy measures va (a; z) = a (a; z) = We have (a; z) fv fv (a; z) (a; z) , for 0 a a (z) ; (G.8) (a; z) , for 0 a a (z) : (G.9) The evolution of matched machines is the mirror image of the evolution of vacancies, i.e., a (a; z) = va (a; z). This implies that the number of type z vacant and matched machines of age a < a (z) remains constant: v (a; z) + (a; z) = v (0; z) + (0; z) , for 0 a a (z) . Because all …rms proceed …rst to the search pool with their new machines, (G.10) (0; z) = 0. For a 2 [0; a (z)], the evolution of (a; z) therefore follows a (a; z) = ff (z) ef ( + f) (a; z): The solution of the di¤erential equations yields Z a (a; z) = f (z) ef f e ( + f )a~ d~ a; 0 Z a e ( + f )a~ d~ a v (a; z) = f (z) ef 1 f 0 for 0 a (z). Given the vacancy distribution v (a; z) the rate at which workers meet R machines is w (a; z) = w v (a; z) = v (a; z~) dz for 0 a a (z) and zero otherwise. a 44 G.2 The economy with upgrading Optimal entry and upgrading The de…nitions of the capital values for a matched machine and worker are the same as without upgrading, equations (G.1) and (G.2). The capital values of a vacancy and an unemployed worker are (r g)V (a; z) = max (r g)U = b + max J(a; z); Ez~ [J (0; z~)] f Z 0 IuJ (a; z) (G.11) V (a; z)] + Va (a; z); 0g ; z^ Z a ^ 0 w (a; z) [max fW (a; z); Ez~ [W (0; z~)] IuW (a; z) (G.12) U dadz; where Ez~ denotes the expectation with respect to the density f (~ z ). The gains from upgrading in an existing match are G (a; z) = Ez~ [J (0; z~) + W (0; z~)] = Ez~ [S (0; z~)] + I0 Iu Iu J (a; z) S (a; z) W (a; z) V (a; z) ; using the surplus sharing rule and the free entry condition. An existing match will upgrade the machine as soon as the gains from upgrading are non-negative and there are no gains from a marginal delay of the upgrading decision: G [a (z) ; z] = Ga [a (z) ; z] = 0: These two conditions imply Ez~ [S (0; z~)] + I0 (r (G.13) Iu = S [a (z) ; z] + V [a (z) ; z] ; g + ) S [a (z) ; z] = ez a(z) (r g) fV [a (z) ; z] + U g : (G.14) The di¤erential equation for the surplus of existing matches, S (a; z) for a de…ned as in (G.5). There are no existing matches with machines of age a a (z), is a (z). When a previously vacant type z machine of age a > a (z) meets an unemployed worker the gains from upgrading are Gm (a; z) = Ez~ [J (0; z~) + W (0; z~)] = Ez~ [S (0; z~)] + I0 45 Iu Iu V (a; z) V (a; z) : U Incorporating the optimal upgrading decision, the di¤erential equation for the vacancy value function becomes (r g) V (a; z) Va (a; z) = ) S (a; z) for a < a ^ (z) ; ) Gm (a; z) for a a ^ (z) : (1 f (1 f Using the surplus sharing rule the ‡ow value of unemployment (G.12) now becomes # Z z^ "Z a(z) Z a^ m (r g)U = b + w (a; z)S(a; z)da + w (a; z)G (a; z) da f (z) dz: 0 a(z) 0 Employment and vacancy measures De…ne the total measure of machines that are upgrad- ing at a point in time as (0) = Z z^ [a (z) ; z] + f Z a ^ v (a; z) da dz: a(z) 0 This is also the measure of all new machines. The distribution over new machines according to type z is then (0; z) = f (z) (0) and v (0; z) = f (z) ef : For type z machines the employment distribution for a a (z) is described in the same way as in the case of no upgrading, equations (G.8), (G.9), and (G.10). Vacant machines that are older than the critical upgrading age but younger than the critical age when they can no longer be upgraded, a (z) a a ^, stay in the vacancy pool until they …nd a worker and upgrade va (a; z) = fv (a; z) : (G.15) To solve for the employment and vacancy distributions we proceed the same way as in the upgrading case without machine-speci…c productivity heterogeneity. H H.1 Equilibrium with match-speci…c productivity di¤erences The economy without upgrading Optimal entry and exit The de…nitions of the capital values of employed and unemployed workers are the same as in the economy with machine-speci…c heterogeneity and no upgrading, equations (G.2) and (G.4). The capital value equations for a matched and a vacant 46 machine are g)J(a; z) = max ez (r (r (r a w(a; z) [J(a; z) V (a)] + Ja (a; z); (H.1) g)V (a)g ; g)V (a) = max f ~) f Ez~ [J(a; z (H.2) V (a)] + Va (a); 0g : The di¤erential equation for the surplus value is fr g + g S(a; z) = ez a (r g)U f (1 )Ez~ [S(a; z~)] + Sa (a; z): (H.3) Since productivity z is match-speci…c and random, the ‡ow return on the outside option of a machine vacancy is now with respect to the expected surplus, cf. equation (G.5). Optimal separation of a match depends on the match-speci…c productivity, and at the separation age a (z) the surplus is zero and there are no gains from a marginal delay of separation, equation (G.6). It is apparent that the ‡ow return on surplus is increasing in the match-speci…c productivity. Therefore, more productive matches will separate later, i.e., a(z) is increasing. The job-destruction condition is implied by the zero gain from a marginal delay of separation: ez a(z) = )Ez~ fS [a (z) ; z~]g + (r f (1 (H.4) g)U: Di¤erent from the economy with machine-speci…c productivity-heterogeneity there may now be a positive ‡ow return on the outside option of a vacancy since the machine might draw a higher match productivity. This chance of …nding a match with a higher productivity increases the cost of staying in a match with given productivity. Machines in a match of quality z separate at age a (z). A vacant …rm stays in the vacancy pool until it reaches the maximal age for active matches a = maxz a (z). Given the zero terminal value of the surplus at the separation age we can solve the surplus di¤erential equation forward for the surplus capital value S(a; z) = Z a(z) e (r g+ )(~ a a) ez a ~ f (1 )Ez~ [S(~ a; z~)] (r g)U d~ a: a Since all machines exit at age a, the terminal vacancy value is zero, V (a) = 0. Using the surplus sharing rule we can solve the di¤erential equation (H.2) for the vacancy value forward and conditionally on the terminal value obtain the capital value of a vacancy, Z a V (a) = f (1 ) e (r g)(~a a) Ez~ [S (~ a; z~)] d~ a: a 47 The free entry condition for new machines is I0 = V (0): Surplus and vacancy functions with a …nite set of z types Using the fact that ai < ai+1 we can de…ne a sequence of di¤erential equation systems for the surplus function fSi (a) = S (a; zi )g. On the interval ai ai the system de…nes the surplus functions a 1 Sj (a), for j = i; : : : ; Z, through Sj0 (a) = [r g+ + f (1 ) fj ] Sj (a) + f (1 ) X fk Sk (a) ezj a k>j +(r g)U: These systems can be solved sequentially working backwards using the terminal conditions Si (ai ) = 0. Conditional on the piecewise de…ned surplus functions we can then solve the di¤erential equation for the vacancy value recursively (r g)V (a) = f (1 ) X fj Sj (a) + Va (a) on [ai 1 ; ai ] : j i Employment and vacancy measures with …nite z types Without loss of generality assume that all matches operate at least for some time, that is a1 = a (z1 ) > 0. The total measure of young machines with age a a1 evolves according to (0) = 0 and v (0) = ef ; 0 (a) = v 0 (a) = Thus for a fv (a) (a) (a) ; fv (a) : a1 v (a) + (a) = v (0) + (0) : The measure of match type i is then given by i (a) = fi (a) : 48 In general, the distribution evolves according to " # " # X X (ai +) = (ai ) fj = fj ; j>i j>i 1 v (ai +) = v (ai ) + fi (ai ) ; X v 0 (a) = (a) fj for ai f v (a) a ai+1 ; j>i 0 (a) = (a) + f v (a) X fj for ai a j>i j fji (a) for j > i with fji = fj = 0 for j i; (a) = ai+1 ; P s>i fs ; where g (a+) denotes the right-hand side limit of the function g at a, g (a+) lim">0;"!0 g (a + "). The employment distributions are continuous functions of age and the total employment and vacancy functions are piecewise continuous with discontinuities at the critical exit ages. Again we normalize the employment and vacancy distribution by dividing through with the entry rate of new machines. H.2 The economy with upgrading Optimal entry, separation, and upgrading The de…nitions of the capital values for vacancy and an unemployed worker are (r g)V (a) = max (r g)U = b + f Z IuJ (a; z~) ; J(a; z~) Ez~ max J (0; z~) (H.5) V (a)g + Va (a); 0g a ^ w (a) Ez~ max W (0; z~) IuW (a; z~) ; W (a; z~) 0 U g da (H.6) and the capital value de…nitions of a matched machine and worker are the same as without upgrading, equations (H.1) and (G.2). We get the di¤erential equation for the surplus value from the surplus de…nition and the expression for the capital value of a matched worker and machine (r g + ) S (a; z) + (r g) [V (a) + U ] = ez a + Sa (a; z) + Va (a; z) : (H.7) Note that we can no longer eliminate the vacancy value from the surplus expression, rather we have to solve the di¤erential equation system for the surplus and the vacancy 49 value jointly. To this end, de…ne the gains from upgrading in a match with an age a machine and type z productivity G (a; z) = J (0; z) + W (0; z) Iu J (a; z) W (a; z) : We distinguish between the separation age of a match aS (z) and the upgrading age of a match aU (z), aS (z) aU (z). If aS (z) = aU (z) then existing matches do not separate. Using the surplus sharing rules, the expression for the gains from upgrading simpli…es to S (0; z) + I0 S (0; z) + I0 G (a; z) = Iu Iu S (a; z) for a aS (z) for a > aS (z) : V (a) V (a) Furthermore, the gains from a marginal delay of upgrading in an existing match, a aS (z), are Ga (a; z) = ez a (r g) [U + V (a)] (r g + ) S (a; z) . We can use the surplus sharing rule for existing matches and the gains from upgrading in the expression for the vacancy value and obtain (r g) V (a) = (1 ) f Ez~ [max fS (0; z~) + I0 Iu V (a) ; S (a; z~)g] + Va (a) : (H.8) Equations (H.7) and (H.8) de…ne a system of di¤erential equations in S (a; z) and V (a) which has to be solved jointly. For a …nite number of productivity types z the optimal separation/upgrading decisions are characterized by the critical age values aSi ; aUi : i = 1; : : : ; Z such that aSi < aUi : Si aSi = Si0 aSi = 0; Gi aSi < 0; Gi aUi = G0i aUi = 0 aSi = aUi : Si aSi 0; Gi aUi = G0i aUi = 0: Employment and vacancy measures De…ne the total measure of type zi matches that are upgrading at a point in time ( i (0) = R a^ aUi + fi f aU v (a) da for aSi = aUi ; i R a^ for aSi < aUi : fi f aU v (a) da i i The employment distributions evolve according to 0 i (a) = fi i (a) = 0 for ai < a fv (a) i (a) 50 a ^: for 0 a aSi ; The vacancy distribution evolves according to v 0 (a) = X i (a) fv (a) i X a aSi or a i and v aSi + = v aSi + subject to the initial condition v (0) = ef : 51 i aSi if aSi < aUi aUi fi References [1] Abraham, Katharine G. and Robert Shimer (2001); “Changes in Unemployment Duration and Labor Force Attachment,”in Alan B. Krueger and Robert M. Solow (eds), The Roaring Nineties, New York: Russell Sage Foundation, 367-420. [2] Aghion, Philippe and Peter Howitt (1994); “Growth and Unemployment,”Review of Economic Studies 61 (3), 477-494. [3] Akerlof, George A. (1969); “Structural Unemployment in a Neoclassical Framework,” Journal of Political Economy 77 (3), 399-407. [4] Bartel, Ann P. and Nachum Sicherman (1993); “Technological Change and Retirement Decisions of Older Workers,”Journal of Labor Economics 11 (1), 162-83. [5] Bureau of Economic Analysis (2002), “Fixed Assets and Consumer Durable Goods for 1925-2000,”Washington D.C.: U.S. Department of Commerce, www.bea.doc.gov. [6] Caballero, Ricardo J. and Mohamad L. Hammour (1998); “Jobless Growth: Appropriability, Factor Substitution, and Unemployment,”Carnegie-Rochester Conference Series On Public Policy 48 (1), 51-94. [7] George Alogoskou…s, Charles Bean, Giuseppe Bertola, Daniel Cohen, Juan Dolado, and Gilles Saint-Paul (eds.) (1995); “Unemployment: Choices for Europe,” Monitoring European Integration Series, vol.5. London: Centre for Economic Policy Research. [8] Cohen, Daniel and Gilles Saint-Paul (1994); “Uneven Technical Progress and Job Destruction,”mimeo CEPREMAP. [9] Cooley, Thomas F. and Edward C. Prescott (1995); “Economic Growth and Business Cycles,”in Tomas F. Cooley (ed), Frontiers of Business Cycle Research, Princeton NJ: Princeton University Press, 1-38. [10] Cummins, Jason and Giovanni L. Violante (2002); “Investment-Speci…c Technological Change in the U.S. (1947-2000): Measurement and Macroeconomic Consequences,”Review of Economic Dynamics 5 (2), 243-284. 52 [11] Gordon, Robert. J. (1990); The Measurement of Durable Good Prices, NBER Monograph Series, Chicago and London: University of Chicago Press. [12] Greenwood, Jeremy, Zvi Hercowitz and Per Krusell (1997);“Long-Run Implications of Investment-Speci…c Technological Change,” American Economic Review 87 (3), 342-62. [13] Hall, Robert (2003); “Wage Determination and Employment Fluctuations,” mimeo Stanford University. [14] Hobijn, Bart (2000); “Identifying Sources of Growth,” mimeo, Federal Reserve Bank of New York. [15] Hornstein, Andreas and Per Krusell, (1996); “Can Technology Improvements Cause Productivity Slowdowns?” NBER Macroeconomics Annual 1996, Cambridge MA and London: MIT Press, 209-259. [16] Hornstein, Andreas, Per Krusell and Giovanni L. Violante (2003); “Vintage Capital in Frictional Labor Markets,”mimeo. [17] Hornstein, Andreas, Per Krusell and Giovanni L. Violante (2005); “A Frictional Inequality Puzzle,”mimeo. [18] Jorgenson, Dale W. (2001); “Information Technology and the U.S. Economy,”American Economic Review 91 (1), 1-32. [19] Jovanovic, Boyan (1998); “Vintage Capital and Inequality,” Review of Economic Dynamics 1 (2), 497-530. [20] Katz, Lawrence and David Autor (1999); “Changes in the Wage Structure and Earnings Inequality,” in Orley Ashenfelter and David Card (eds.), Handbook of Labor Economics Vol. 3, Amsterdam: North-Holland, 1463-1555. [21] King, Ian and Linda Welling (1995); “Search, Unemployment and Growth,”Journal of Monetary Economics 35 (3), 499-507. [22] Krusell, Per, Lee Ohanian, José-Víctor Ríos-Rull and Giovanni L. Violante (2000); “Capital Skill Complementarity and Inequality: A Macroeconomic Analysis,” Econometrica 68 (5), 1029-1053. 53 [23] Ljungqvist, Lars (2002); “How do Lay-O¤ Costs A¤ect Employment?” The Economic Journal 112 (482), 829-853. [24] Lopez-Salido, David and Claudio Michelacci (2003); “Technology Shocks and Job Flows,”mimeo CEMFI. [25] Mortensen, Dale T. and Christopher A. Pissarides (1994); “Job Creation and Job Destruction in the Theory of Unemployment,”Review of Economic Studies 61 (3), 397-415. [26] Mortensen, Dale and Christopher Pissarides (1998); “Technological Progress, Job Creation, and Job Destruction,”Review of Economic Dynamics 1 (4), 733-53. [27] OECD (1996), “Employment Outlook,”Paris: OECD. [28] Petrongolo, Barbara and Christopher Pissarides (2001); “Looking into the Black Box: A Survey of the Matching Function,” Journal of Economic Literature, 39 (2), 390-431. [29] Pissarides, Christopher (2000); Equilibrium Unemployment Theory, Cambridge MA: MIT Press. [30] Pissarides, Christopher and Giovanna Vallanti (2003); “Productivity Growth and Employment: Theory and Panel Estimates,”mimeo LSE. [31] Ridder Geert and Gerard J. van den Berg (2003); “Measuring Labor Market Frictions: A Cross-Country Comparison,” Journal of the European Economic Association 1 (1), 224-244. [32] Sattinger, Michael (1995); “Search and the E¢ cient Assignment of Workers to Jobs,” International Economic Review 36 (2), 283-302. [33] Solow, Robert (1960); “Investment and technological progress,” in Kenneth Arrow, Samuel Karlin and Patrick Suppes (eds.), Mathematical Methods in the Social Sciences, Stanford CA: Stanford University Press, 89-104. 54
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