Capital Misallocation and Secular Stagnation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Capital Misallocation and Secular Stagnation Caggese and Perez-Orive (2017) 2017-009 Please cite this paper as: Caggese,AndreaandAnderPerez-Orive(2017). “CapitalMisallocationandSecularStagnation,”FinanceandEconomicsDiscussionSeries2017-009. Washington: BoardofGovernors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.009. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Capital Misallocation and Secular Stagnation* Andrea Caggese Ander PØrez-Orive Universitat Pompeu Fabra, Federal Reserve Board CREI, & Barcelona GSE (This Version: January 17, 2017) Abstract The widespread emergence of intangible technologies in recent decades may have signi(cid:133)cantly hurt output growth(cid:150)even when these technologies replaced considerably less productive tangible technologies(cid:150)because of structurally low interest rates caused by demographic forces. This insight is obtained in a model in which intangible capital cannot attract external (cid:133)nance, (cid:133)rms are credit constrained, and there is substantial dispersion in productivity. In a tangiblesintense economy with highly leveraged (cid:133)rms, low rates enable more borrowing and faster debt repayment, reduce misallocation, and increase aggregate output. An increase in the share of intangible capital in production reduces the borrowing capacity and increases the cash holdings of the corporate sector, which switches from being a net borrower to a net saver. In this intangibles-intense economy, the ability of (cid:133)rms to purchase intangible capital using retained earnings is impaired by low interest rates, because low rates increase the price of capital and slow down the accumulation of corporate savings. Keywords: Intangible Capital, Borrowing Constraints, Capital Reallocation, Secular Stagnation JEL Classi(cid:133)cation: E22, E43, E44 * A previous version of this paper was entitled "Reallocation of Intangible Capital and Secular Stagnation". We thank Andrew Abel, Fiorella de Fiore (discussant), Wouter Den Haan (discussant), Andrea Eisfeldt, Antonio Falato, Maryam Farboodi (discussant), Simon Gilchrist, Adam Guren, Matteo Iacoviello, Arvind Krishnamurthy, Tim Landvoigt (discussant), Claudio Michelacci (discussant), Guillermo Ordonez, Enrico Perotti, Vincenzo Quadrini, and Stephen Terry, and seminar participants at Boston University, Boston College, the Federal Reserve Board, the Bank of Spain, the CREI macro lunch, the 7th Meeting of the Macro Finance Society (UCLA 2016), the 2016 Barcelona Summer Forum Workshop on Financial Markets and Asset Prices, the 2016 NBER SI Workshop on Macro, Money and Financial Frictions, the Midwest Macro Meetings, the Cleveland Fed Day-AheadMeetingonProductivity,theBoEWorkshoponFinance,InvestmentandProductivity,andtheBank of Italy Workshop on Macroeconomic Dynamics for very helpful comments. We also thank Christoph Albert for excellentresearchassistance. AndreaCaggeseacknowledges(cid:133)nancialsupportfrom theMinistryofEconomicsof Spain and from Resercaixa. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System or of anyone else associated with the Federal Reserve System. All errors are, of course, our own responsibility.

1 Introduction Real interest rates have decreased in past decades, while economic growth has fallen short of previous trends, developments that have been linked to a process of (cid:146)secular stagnation(cid:146) (Summers(2015),Eichengreen(2015)). Atthesametime,thedevelopedworldhasexperienceda technological change toward a strongerimportance of information technology and of knowledge, human and organizational capital, which has gradually reduced the reliance on physical capital (Corrado and Hulten (2010a)) and has been linked to a signi(cid:133)cant decrease in corporate net borrowing (Falato, Kadyrzhanova, and Sim (2014), D(cid:246)ttling and Perotti (2015)).1 This paper argues that the increased reliance on intangible capital and the low real interest ratesinteracttohurtcapitalreallocationandreduceproductivityandoutputgrowth.Aggregate productivity depends on an e¢ cient reallocation of resources from declining or exiting (cid:133)rms to new entrants or expanding (cid:133)rms. The rise of intangible capital implies a growing importance of thereallocationofintangibleassetssuchaspatents,brandequity,andhumanandorganizational capital. These assets cannot be collateralized, and their acquisition has to be (cid:133)nanced mostly using retained earnings. As a result, the corporate sector borrows less, holds an increasing amount of cash, and switches from being a net borrower to a net saver. A decrease in interest rates increases the price of these intangible assets and reduces the ability of credit-constrained expanding (cid:133)rms to purchase them. Lower interest rates also decrease the rate at which noninvesting (cid:133)rms can accumulate savings to (cid:133)nance future expansions. We show that the rise in intangibles,viathesee⁄ects,altersthedynamicrelationshipbetweeninterestratesande¢ ciency in the allocation of capital. We formalize this intuition by developing a model of an economy in which (cid:133)rms use tangible capital,intangiblecapital,andlaborascomplementaryfactorsintheproductionofconsumption goods. A subset of (cid:133)rms have high productivity and su⁄er from (cid:133)nancing constraints that prevent them from issuing equity or from borrowing any amount in excess of the collateral value of their holdings of tangible and intangible capital. We follow Kiyotaki and Moore (2012) in assuming that these high-productivity (cid:133)rms can invest only occasionally. In equilibrium, they save as much as possible in non-investing periods, and invest all of their accumulated net savings plus their maximum available borrowing in investing periods. Any residual capital not absorbed by the high-productivity (cid:133)rms is used by low-productivity (cid:133)rms, which are (cid:133)nancially unconstrained. 1The decrease in corporate net borrowing has translated into a shift in the net (cid:133)nancial position of the non(cid:133)nancialcorporatesectorfromanetborrowingpositionroughlybeforetheyear2000toanetsavingposition from2000onward(ArmenterandHnatkovska(2016),Quadrini(2016),Chen,Karabarbounis,andNeiman(2016), Shourideh and Zetlin-Jones (2016)). 2

In our economy, the consumer sector is modeled as overlapping generations of households displaying a realistic life cycle, in a way that enables us to obtain an equilibrium interest rate in the steady state that is not necessarily equal to the household rate of time preference. This speci(cid:133)cation of the consumer sector allows us to consider some of the main structural forces that multiple studies have identi(cid:133)ed to have pushed real interest rates lower in recent decades. In particular, we focus on a higher propensity to save in the household sector due to an increase in household longevity and a decrease in the rate of time preference.2 The increased corporate net savings due to a higher intangibles usage also contributes to the downward pressure on real rates. We (cid:133)rst inspect the analytical solution of a simpli(cid:133)ed version of the model to describe four channels through which lower interest rates interact with the intensity of intangible capital in (cid:133)rms(cid:146)production function to a⁄ect the steady state equilibrium of our economy. First, a debt overhang channel allows net borrowing high-productivity (cid:133)rms to pay down their debt more easily when interest rates are low and enables them to absorb more capital. Second, and conversely, a savings channel operates when the (cid:133)rm sector is a net saver: reductions in the interest rate decrease the speed of accumulation of savings and hurt capital reallocation. Third, lower interest rates that increase the price of tangible and intangible assets reduce the amountofcapitalthathigh-productivity(cid:133)rmscanpurchaseforagivenamountofnetworthand borrowing capacity(cid:150)a capital purchase price channel. Fourth, a lower interest rate increases the present value of the collateral pledged next period, and reduces the size of the downpayment necessary to purchase capital, improving capital reallocation through a borrowing/collateral value channel. The analytical solution of the simpli(cid:133)ed model provides a clear illustration of the main theoretical (cid:133)nding of the paper: in an economy with a relatively low collateral value of capital, the savings and the capital purchase price channels dominate and a drop in the interest rate worsens the allocation of resources and reduces aggregate investment, productivity, and output. Intheremainingsectionsofthepaper, wecalibrateandsimulateourfullgeneralequilibrium model to study how the parallel developments in the household and corporate sectors have interacted to generate aggregate patterns consistent with the secular stagnation hypothesis. In thehouseholdsector, asdiscussedearlier, wemodelaprogressivedecreaseinindividuals(cid:146)rateof time preference and a progressive increase in their life expectancy, both of which put downward pressure on the equilibrium interest rate. In the corporate sector, we introduce a gradual shift 2We interpret our exercise as a shortcut for a collection of di⁄erent factors, such as population aging, wealth and income inequality, (cid:133)nancial deepening, and foreign-sector developments, which have contributed to increase households(cid:146)demand for savings in the past 40 years. 3

in the reliance on intangible capital of (cid:133)rms.3 We (cid:133)nd that while the household sector developments in isolation and the corporate sector developments in isolation are both expansionary, the combination of both developments is contractionary. The drop in the interest rate increases high-productivity (cid:133)rms(cid:146)ability to borrow and pay down their debt while (cid:133)rms still rely strongly on tangible capital. As (cid:133)rms use increasingly more intangible capital and become net savers, low rates reduce e¢ cient capital allocation by increasing capital prices and by slowing the accumulation of corporate savings. The share of output produced by the high-productivity (cid:133)rms drops signi(cid:133)cantly. The lower corporate borrowing itself also puts downward pressure on interest rates, which ampli(cid:133)es the misallocation of capital. Despite the fact that the economy is shifting toward a higher reliance on a more productive type of capital, aggregate productivity falls by 6.5%, and even though low rates encourage capital creation, output is 2% lower than in the case in which only household sector or only corporate sector developments occur. We interpret this comparative static exercise as capturing the developments in the U.S. economy following the rise in the share of intangible capital and the rise in net household and foreign-sector savings in the past 40 years. In this respect, this model is remarkably consistent with a series of well-documented trends during this period: (i) net corporate savings increased as a fraction of gross domestic product (GDP), (ii) household leverage increased as a fraction of GDP, (iii) the real interest rate fell, (iv) intra-industry dispersion in productivity has increased, and (v) output and productivity progressively declined relative to their previous trends. An important question is whether the trends identi(cid:133)ed in this paper are likely to persist, as in the secular stagnation hypothesis, or reverse. While the technological shift identi(cid:133)ed in the paper is likely to be permanent and possibly intensify, the developments that are keeping interest rates low may fade in the future. Some developments pushing down rates, such as the lower retirement age or the increase in the net demand for safe assets, may prove temporary, while others, such as population ageing, the decline in the growth rate of population, or the drop in the relative price of capital, might be more persistent.4 Overall, our results suggest that the interaction between low interest rates, intangible technologies, and corporate (cid:133)nancing patterns might be an important factor behind secular stagnation. 3We set the reliance on intangible capital to match its observed evolution from a pre-1980 value of 20% of aggregate capital to a post-2010 value of 60% of aggregate capital (Corrado and Hulten (2010a), Falato, Kadyrzhanova, and Sim (2014), D(cid:246)ttling and Perotti (2015)). Since we assume that intangible capital is more productive than tangible capital, this gradual shift is consistent with the notion of the transition to intangible capital as a privately optimal choice of (cid:133)rms adopting technologies that are more productive. 4For a detailed discussion of the causes of low real interest rates and the likelihood that these causes remain in the future, see Baldwin and Teulings (2014), Summers (2014), and Blanchard, Furceri and Pescatori (2014). 4

Related Literature The secular stagnation hypothesis as an explanation of recent economic trends has been proposed by, among others, Summers (2015) and Eichengreen (2015). One prominent example of a formalization of these ideas is Eggertsson and Mehrotra (2014), who show how a persistent tightening of the debt limit facing households can reduce the equilibrium real interest rate and, in the presence of sticky prices and a zero lower bound in nominal interest rates, generate permanent reductions in output.5 Our paper contributes to this literature by identifying and formalizing a novel misallocation e⁄ect of endogenously low real interest rates. Our alternative explanation of the secular stagnation hypothesis can account for a large drop in aggregate output, does not rely on the zero lower bound or sticky prices, and is consistent with a broad set of well-documented trends. The rising use of intangible capital has been documented by Corrado and Hulten (2010a), and its relation to the decrease in corporate borrowing and the rise in corporate cash holdings has been shown empirically by Bates, Kahle, and Stulz (2009). Giglio and Severo (2012), Falato, Kadyrzhanova, and Sim (2014) and D(cid:246)ttling and Perotti (2015) introduce models that describe how the rise in intangibles can lower the equilibrium interest rate by decreasing (cid:133)rms(cid:146) net borrowing. We add to this literature by describing a mechanism through which the rise in intangibles can have a negative e⁄ect on aggregate capital reallocation and growth. Our paper is also related to the literature on (cid:133)nancial frictions, (cid:133)rm dynamics, and misallocation (Buera, Kaboski, and Shin (2011), Caggese and Cuæat (2013), Moll (2014), Midrigan and Xu (2014), and Buera and Moll (2015)). With respect to these papers, our contribution is toprovidenoveltheoreticalinsightsontherelationbetweeninterestrates, thecollateralizability of capital, and misallocation.6 The rest of the paper is organized as follows. Section 2 introduces the empirical evidence that motivates this paper. We describe a very simple model in Section 3 that conveys the basic intuition of the mechanisms we introduce, and we develop a full-(cid:135)edged general equilibrium extension in Section 4. The steady state and calibration of the general equilibrium model are describedinSection5andthesimulationresultsarediscussedinSection7. Section8concludes. 5Other recent theoretical papers with alternative explanations of secular stagnation are Bacchetta, Benhima, and Kalantzis (2016) and Benigno and Fornaro (2015). 6Gopinathetal. (2016)alsoconsideramodelwith(cid:133)nancialfrictionsandheterogenous(cid:133)rmsinwhichdeclining interest rates cause an increase in the dispersion in the productivity of capital. However, their mechanism is fundamentalltydi⁄erentfrom ours. Intheirmodel,whentheinterestratefalls,all(cid:133)rmsinvestmoreandexpand aggregatecapitalandouput. Productivitydispersionincreasesbecauselarger(cid:133)rmsareabletogrowmorerapidly thansmallerandmore(cid:133)nanciallyconstrainedones. Inourmodel,instead,lowratestighten(cid:133)nancialconstraints of high-productivity (cid:133)rms that utilize intangible capital, and reduce their investment. 5

2 Empirical Motivation In this section, we summarize the key stylized facts that motivate our model. 1 - Developed economies are signi(cid:133)cantly more reliant on intangible capital now than in the 1980s, and this technological shift has been linked to the simultaneous transition of the corporate sector from net debtor to net saver. The developed world has experienced a technological change toward a stronger importance of information technology and of knowledge, human, and organizational capital, which has graduallyreducedtherelianceonphysicalcapital(Brown,FazzariandPetersen(2009),Corrado and Hulten (2010a), Falato, Kadyrzhanova, and Sim (2014)). In the United States, intangible capital as a share of total capital went from around 0.2 in the 1970s to 0.5 in the 2000s (Falato, Kadyrzhanova, and Sim (2014)). In parallel, there has been a shift in the net (cid:133)nancial position of the non(cid:133)nancial corporate sector from a net borrowing position roughly before the year 2000 to a net saving position from 2000 onward (Armenter and Hnatkovska (2016), Quadrini (2016), Chen, Karabarbounis, and Neiman (2016), Zetlin-Jones and Shourideh (2016)). The empirical evidence suggests that these two trends are related. The process of technological change has been linked to a lower availability of collateral for the corporate sector, which has lowered its debt capacity. Brown, Fazzari, and Petersen (2009) document that U.S. (cid:133)rms (cid:133)nance most of their research and development (R&D) expenditures out of retained earnings and equity issues, an observation in line with the conclusion in Hall (2002) that R&D-intensive (cid:133)rms feature much lower leverage, on average, than less R&D-intensive (cid:133)rms. Gatchev, Spindt, and Tarhan (2009) document that, in addition to R&D, marketing expenses and product developmentarealsomostly(cid:133)nancedoutofretainedearningsandequity. Incontrast, tangibleassets are mostly (cid:133)nanced with debt.7 The process of technological change has also been linked to an increase in the precautionary motives for cash accumulation to avoid future (cid:133)nancial shortages (Bates, Kahle, andStulz(2009), Falato, Kadyrzhanova, andSim(2014), FalatoandSim(2014), D(cid:246)ttling and Perotti (2015), Begenau and Palazzo (2016)).8 Furthermore,(cid:133)rm-levelempiricalevidencesuggeststhattheobservedlinkbetweenintangible intensity and high cash holdings is driven by (cid:133)nancial frictions. Begenau and Palazzo (2016) 7EisfeldtandRampini(2009)reportthatabigshareofmachinery,equipment,buildingsandotherstructures is(cid:133)nancedwithdebt. Inventoryinvestmentandothertangibleshort-termassetsattractsubstantialdebt(cid:133)nance intheformoftradecreditandbankcreditlines(PetersenandRajan(1997),Su(cid:133)(2009)). Finally,investmentin commercialrealestateisprimarily(cid:133)nancedwithmortgageloans(Benmelech,Garmaise,andMoskowitz(2005)). 8Lackofaccesstodebt(cid:133)nancingof(cid:133)rmsthatrelyon intangiblecapitalcould becompensated byeasyaccess toequity(cid:133)nancing. Whileeasyaccesstoequity(cid:133)nancingwouldbeconsistentwiththeobservedlowerleverageof these (cid:133)rms, it would be harder to reconcile with the remarkable accumulation of cash holdings. A large body of evidenceshowsthatexternalequity(cid:133)nancingissigni(cid:133)cantlycostly(AltinkilicandHansen(2000),Gomes(2001), Belo, Lin and Yang (2016)). 6

introduce evidence showing that an important determinant of the increase in cash holdings of public (cid:133)rms is the increase in frequency of new (cid:133)rms that are very R&D intensive, and they suggest that these trends are consistent with a model in which cash holdings are driven by (cid:133)nancial frictions of the R&D-intensive (cid:133)rms and costly equity (cid:133)nancing. Similarly, Falato, Kadyrzhanova,andSim(2014)showempiricallythattherelationbetweenrelianceonintangible capital and cash holdings is stronger among (cid:133)rms for which (cid:133)nancing frictions are more severe. 2 - Productivity dispersion has increased in intangibles sectors during recent decades, while it has remained roughly constant in tangibles sectors. Kehrig (2015) analyzes establishment-level manufacturing data from the U.S. census and documents a signi(cid:133)cant increasing trend in the dispersion of productivity across (cid:133)rms within sectors over the past 40 years. Earlier, we provided evidence that the rising intangible capital share is related to an increase in (cid:133)rm-level cash holdings to overcome external (cid:133)nance constraints. If the misallocation of resources caused by (cid:133)nancial constraints is a factor contributing to the increase in productivity dispersion, we should expect the latter to be more pronounced in sectors with higher intensity of intangible capital.9 In order to investigate the relation between the rise in intangibles and productivity dispersion, we use accounting data of 34,900 U.S. corporations obtained from Compustat, covering the period from 1980 to 2015, and containing 379,318(cid:133)rm-yearobservations. Wede(cid:133)neintangiblecapitalasthesumofknowledgecapitaland organizational capital. We consider two alternative productivity measures: labor productivity (y) and total factor productivity (TFP) (A) (see Appendix A for details). Our measure of misallocation, the productivity dispersion, is computed as the standard deviation of the di⁄erence between the logs of the productivity of (cid:133)rm i and the aggregate productivity of the industry s in which (cid:133)rm i operates. [FIGURE 1 ABOUT HERE] [FIGURE 2 ABOUT HERE] Figures 1 and 2 plot the dispersion of labor productivity and TFP, respectively, in 2-digit SICindustriesovertime(normalizedbythevaluein1980). Inboth(cid:133)gures, theleftgraphshows averagedispersionforallsectors,anditreplicatestheupward-slopingtrendalreadydocumented 9It is important to note that this paper, like Kehrig (2015), analyzes the dynamics of the cross-sectional dispersion of productivity, not the dispersion of business growth rates. Davis et al. (2006) focus on the latter, and using both (cid:133)rm- and establishment-level data document a negative trend instead. These opposite trends are consistent with the (cid:133)ndings of our model, in which a decline in the growth rate of expanding (cid:133)rms reduces reallocation of capital and increases steady state productivity di⁄erences. 7

by Kehrig (2015) using establishment-level data. In the right graph, the red dashed line shows the mean of the dispersion measure across industries (weighted by sales) in the top 50%, and the blue line in the bottom 50%, of the distribution of the industry-wide ratio of intangible capital to total capital (averaged across years).10 Both (cid:133)gures show that the constant rise in the within-industry dispersion of productivity is driven by the sectors with higher average shares of intangible capital. This evidence is consistent with the hypothesis that intangible capital exacerbates misallocation problems caused by (cid:133)nancial frictions. Appendix A discusses two additional exercises that provide robustness to these results. 3 Simple and Intuitive Explanation of the Mechanisms We introduce in this section the simplest possible model that can describe our proposed mechanisms and deliver analytical results. Our main interest is studying how exogenous interest rate variations a⁄ect the allocation of capital and aggregate output depending on the degree of tangibility of capital. This framework is extended in Section 4 in a full-(cid:135)edged general equilibrium setup that can be used for realistic quantitative analysis. Consider an in(cid:133)nite-horizon, discrete-time model of the (cid:133)nal goods producers of an economy. Firms use capital, which is in constant aggregate supply K, to produce a homogeneous consumption good using a constant-returns-to-scale technology. There are two types of (cid:133)rms, high-productivity and low-productivity. E¢ ciency is determined by the share of K allocated to high-productivity (cid:133)rms. Here we present the aggregate steady state equilibrium conditions and introduce the details of the derivation of this simple model in Appendix B. Aggregate output in the steady state is Y = Yp+Yu+Ye = zK +zu K K ; (1) (cid:0) (cid:0) (cid:1) where z captures the productivity of high-productivity (cid:133)rms and zu < z captures the productivity of low-productivity (cid:133)rms. Aggregate capital holdings K of the high-productivity (cid:133)rms, which are assumed to be (cid:133)nan- 10The sectors with high shares of intangible capital are: Chemicals and Allied Products; Industrial and Commercial Machinery and Computer Equipment; Electronic & Other Electrical Equipment & Components; Transportation Equipment; Measuring, Photographic, Medical, & Optical Goods, & Clocks; Miscellaneous Manufacturing Industries; Wholesale Trade - Durable Goods; Home Furniture, Furnishings and Equipment Stores; Miscellaneous Retail Business Services; and Engineering, Accounting, Research, and Management Services. The sectors with low shares of intangible capital are: Oil and Gas Extraction; Food and Kindred Products; Paper and Allied Products; Rubber and Miscellaneous Plastic Products; Stone, Clay, Glass, and Concrete Products; Primary Metal Industries; Fabricated Metal Products; Wholesale Trade - Nondurable Goods; General Merchandise Stores; Food Stores; Apparel and Accessory Stores; and Eating and Drinking Places. 8

cially constrained, are Ae(1+r)+Ye K = ; (2) q 1 (cid:18) (cid:0) 1+r (cid:16) (cid:17) where zu q = (3) r+(cid:24) is the price of capital. Low-productivity (cid:133)rms, which are (cid:133)nancially unconstrained, have aggregatecapitalholdingsofK K;arethemarginalbuyersofcapital, andpriceitaccordingtotheir (cid:0) marginal productivity. The parameter (cid:24) captures a pricing wedge (such as a risk premium).11 The numerator of (2) captures the total funds available to high-productivity (cid:133)rms to invest andisassumedtobepositiveinequilibrium. Itisequaltotheaggregatenetsavingsorliabilities of the high-productivity (cid:133)rms Ae(1 + r), including their return r this period, plus output generated this period, Ye.12 The denominator of (2) captures the downpayment necessary to purchase one unit of capital. High-productivity (cid:133)rms can borrow using one-period debt up to a fraction (cid:18) (0 (cid:18) 1) of the value of capital next period and have to pay q per unit. (cid:20) (cid:20) We capture reliance on intangible capital by two features: positive Ae and low (cid:18). Intangible capital is poor collateral (low (cid:18)), so (cid:133)rms that rely on intangible capital do not have a large borrowing capacity and instead accumulate retained earnings and are more likely to be net savers (Ae > 0). Tangible capital has a high collateral value (high (cid:18)), so (cid:133)rms that rely on tangible capital are able to borrow more and are more likely to be net borrowers (Ae < 0). Importantly, this negative relationship between the tangibility of capital (cid:18) and the (cid:133)nancial wealth of high-productivity (cid:133)rms is consistent with the empirical evidence, which we discussed in the previous section, and it arises endogenously in the full model derived in Section 4. We now describe the four mechanisms through which interest rates a⁄ect the allocation of capital and aggregate output. Inspecting dK=dr, which can be expressed as dK Ae Ae(1+r)+Ye 1 (cid:18) = + ; (4) dr q 1 (cid:18) q 1 (cid:18) r+(cid:24) (cid:0) (1+r (cid:18))(1+r) (cid:0) 1+r (cid:0) 1+r (cid:20) (cid:0) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) we can identify these four channels. If Ae > 0, an exogenous increase in r bene(cid:133)ts capital allocationbyincreasingavailablesavingstohigh-productivity(cid:133)rmstoinvest. Thatisthesavings channel. If Ae < 0, an increase in r hurts capital allocation by increasing the debt burden of high-productivity(cid:133)rms anddecreasingtheiravailablefunds. Thatisthedebt overhang channel. The (cid:133)rst term in (4) is positive if Ae > 0, capturing the savings channel, and is negative 11In the full general equilibrium model of Section 4, a positive wedge (cid:24) arises endogenously because of capital depreciation and because of decreasing returns to scale in the low-productivity (cid:133)rms(cid:146)production function. 12Equation (2) is derived in Appendix B from the equilibrium of a model in which overlapping generations of (cid:133)rms live for two periods, and receive an endowment of Ae(1+r)+Ye when they are born. 9

if Ae < 0, capturing the debt overhang channel. The capital purchase price channel is the mechanism through which increases in r bene(cid:133)t capital reallocation by decreasing q and making capital cheaper. The (cid:133)rst term inside the square brackets in (4) captures this channel and is always positive. Finally, the collateral value channel is the channel through which increases in r hurt capital reallocation by decreasing the value of (cid:133)rms(cid:146)collateral (the term (cid:18)=(1+r)) and tightening the borrowing constraint. The second term inside the brackets represents this channel and is always negative. How do these four channels depend on the intensity of intangible capital? In other words, how does the tangibility of capital matter for the e⁄ect of variations in r on the e¢ ciency of this economy? For clarity of exposition, assume that a tangibles-intensive economy is one in which Ae < 0 and (cid:18) > 0, and that an intangibles-intensive economy is one in which Ae > 0 and (cid:18) = 0. Then dK Ae Ae(1+r)+Ye 1 (cid:18) sign ( ) = + < 0 if (5) is met, tangible dr q 1 (cid:18) q 1 (cid:18) 2r+(cid:24)(cid:0)(1+r (cid:18))(1+r)3 (cid:20) (cid:21) (cid:0) 1+r (cid:0) 1+r >0 (cid:0) <0 (cid:16) <0 (cid:17) (cid:16) (cid:17) 4 5 and dK Ae Ae(1+r)+Ye 1 sign ( ) = + > 0 always. intangible dr q q 2r+(cid:24)3 (cid:20) (cid:21) >0 >0 4 5 The derivative dK is positive in an intangibles economy, meaning that a reduction in r is dr unambiguously contractionary. It is instead most likely expansionary in a tangibles economy, particularly if the responsiveness of q to r is limited ((cid:24) is high) and the borrowing capacity is large ((cid:18) is high). More speci(cid:133)cally, a decrease in r is unambiguously expansionary in a tangibles economy if the following condition is satis(cid:133)ed: 1+2r+r2 (cid:18) > : (5) 1+2r+(cid:24) Taken together, these results suggest that the degree of tangibility of capital in an economy matters importantly for how exogenous variations in the interest rate a⁄ect capital allocation and output and describe four important channels through which these e⁄ects occur. Crucially, they show that falling interest rates can become contractionary in an economy that relies on intangible capital. Section 4 provides a full-(cid:135)edged model in which we endogenize (cid:133)rms(cid:146)(cid:133)nancing constraints, saving and borrowing decisions, and investment, households(cid:146)consumption and savings, and the 10

interest rate, wages, and the prices of tangible and intangible capital. The Investment Demand Curve To provide a deeper understanding of how the features of the equilibrium of this economy change as a result of a transition from an economy reliant on tangible capital to one in which intangiblecapitalacquiresalargerimportance,werepresenttheequilibriuminthecreditmarket in Figure 3. The main objective is to provide an empirically relevant assessment of the slope of the investment demand curve for di⁄erent values of (cid:18). To do so, we calibrate the parameters at the annual frequency to be broadly consistent with observed moments of U.S. data. We postpone a more thorough calibration to the full model developed in Section 4. We study a range of the real interest rate between r = 6% and r = 0%, consistent with the observed evolution of real rates between the early 1980s and the present. We normalize the productivity of low-productivity (cid:133)rms to zu = 1 and the output endowment to Ye = 1. We consider a tangibles economy to feature a pledgeability parameter of capital (cid:18) equal to 0:9 and a net borrowing position equivalent to 20% of output (Ae = 0:2). We consider an intangibles (cid:0) economy to feature a pledgeability parameter of capital (cid:18) equal to 0:4 and a net saving position equivalent to 20% of output (Ae = 0:2). The interest rate wedge (cid:24) is set at 20% and is meant to capture a combination of factors such as risk premia, default premia, and capital depreciation. [FIGURE 3 ABOUT HERE] In Figure 3, the upward-sloping savings curve captures the combination of the (unmodeled) net savings of the household sector. Higher interest rates induce households to save more, under the empirically realistic assumption that the substitution e⁄ect dominates the income e⁄ect for them. The demand for capital by investing (cid:133)rms is equal to the amount borrowed by them plus (minus) the savings (debt) they carry over from the previous period. This curve can be upward or downward sloping depending on the relevance of intangible capital in the production function. In an economy where capital is interpreted to be of a tangible nature ((cid:18) = 0:9 and Ae = 0:2), an increase in aggregate savings has the e⁄ect of lowering interest (cid:0) rates and increasing capital purchases from expanding (cid:133)rms. When there is a shift outward in the savings curve, the economy moves from point A to point B. The collateral value channel and the debt overhang channel dominate. As a result, a larger share of the capital stock is in the hands of high-productivity (cid:133)rms, which improves the allocation of resources and increases aggregate productivity and output. Instead, in an economy where capital is interpreted to be of anintangiblenature((cid:18) = 0:4andAe = 0:2), thedemandforcapitalcurveisupwardslopingdue to the strength of the capital price and savings channels. As interest rates rise, (cid:133)rms demand 11

more capital because they have larger savings and the price of capital is lower. In this case, an outward shift in the savings schedule generates a decrease in the equilibrium capital purchases of high-productivity (cid:133)rms, because the decrease in interest rates the shift in savings generates hurtsthereallocationofcapitaltowardhigh-productivity(cid:133)rms. Theeconomymovesfrompoint A to point C, worsening the allocation of resources and reducing aggregate productivity and output. 4 General Equilibrium Model We introduce an in(cid:133)nite-horizon, discrete-time economy populated by an intermediate sector that produces capital; by a (cid:133)nal good sector in which (cid:133)rms use labor and capital to produce consumption goods; and by households, which provide labor and own both sectors. There are severalimportantextensionstothesimplemodelanalyzedinSection3,andwedescribethemain ones here. We introduce an intermediate capital producing sector that allows us to endogenize in equilibrium the aggregate stock of capital. In the (cid:133)nal good sector, we model explicitly tangible and intangible capital, and we derive endogenously the accumulation of (cid:133)nancial and physical assets of (cid:133)rms that live multiple periods. The household sector is modeled as a lifecycle framework, which allows us to endogenize the interest rate and study how it is a⁄ected by demographic changes and other demand-side factors. 4.1 The Capital-Producing Sector A representative (cid:133)rm in this sector chooses investment in tangible and intangible capital, respectively IT and II; in order to maximize pro(cid:133)ts: t t IJ ’ maxq IJ bJ t ; (6) IJ J;t t (cid:0) t ’ (cid:18) (cid:19) where ’ > 1;bJ > 0; and q is the price of the type of capital J T;I . We allow for bT and t J;t t 2 f g bI to be time varying in order to capture trends in the evolution of the relative price of capital. t ’ The (cid:133)rst order condition yields I t J = ’ (cid:16) q b J J t ;t (cid:17) ’ (cid:0) 1 1 ; and pro(cid:133)ts are (cid:25)J t = b q t J J ’ ’ ;t(cid:0) (cid:0) 1 1 1 (’ (cid:0) 1): At the beginning of period t, total capital available is K T and K I : New capital IT and II is t t t t producedandsoldinperiodtsothattheaggregatedividendsgeneratedbythecapital-producing sectors are Dk = (cid:25)T +(cid:25)I: (7) t t t During period t, tangible capital and intangible capital depreciate at the rates 0 (cid:14) < 1. (cid:20) 12

And the law of motion of aggregate capital is K J = IJ +(1 (cid:14))K J ; t+1 t t (cid:0) with J T;I : 2 f g 4.2 Final Good Sector There are two types of (cid:133)nal-good-producing (cid:133)rms: high-productivity and low-productivity. 4.2.1 The High-Productivity Firms There is a continuum of mass 1 of high-productivity (cid:133)rms. Technology and (cid:133)nancing opportunities High-productivity (cid:133)rms produce a (cid:133)nal good using a constant-returns-to-scale production function that is Cobb-Douglas in labor and capital. The (cid:133)rms use two di⁄erent types of complementary capital, tangible and intangible. For simplicity, we assume that they are perfect complements. The production function takes the following form: k k (cid:11) p (1 (cid:11)) T;t I;t y t = z t ((cid:22))n t (cid:0) min 1 (cid:22) ; (cid:22) ; (8) (cid:20) (cid:18) (cid:0) (cid:19)(cid:21) where 0 < (cid:11) 1 and 0 < (cid:22) < 1. The terms k and k represent tangible and intangible T;t I;t (cid:20) capital installed in period t 1 that produce output in period t, and n is labor. The Leontief t (cid:0) production structure implies that, in equilibrium, intangible capital as a share of total capital in high-productivity (cid:133)rms is equal to (cid:22): The productivity term z ((cid:22)) is increasing in the share t of intangible capital and captures the higher productivity of more intangibles-intensive technologies. We drop from now on reference to the dependence of z on (cid:22) for ease of notation and t defer discussion of their relationship to the calibration section. Thebudgetconstraintforhigh-productivity(cid:133)rmsisgivenbythefollowingdividendequation: p d = y +(1+r )a a q (k (1 (cid:14))k ) q (k (1 (cid:14))k ) w n ; (9) t t t f;t f;t+1 T;t T;t+1 T;t I;t I;t+1 I;t t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) where r is the interest rate paid or received in date t; q and q are the prices of tangible t T;t; I;t and intangible capital, respectively; and w is the wage. The term a > 0 indicates that the t f;t (cid:133)rm is a net saver, and a < 0 indicates that the (cid:133)rm is a net borrower. f;t High-productivity (cid:133)rms are subject to frictions in their access to external (cid:133)nance. They are 13

unable to issue equity, which means that dividends are subject to a non-negativity constraint: d 0: (10) t (cid:21) They can issue one-period riskless debt, subject to the constraint that they can pledge, as collateral, the fractions (cid:18)T and (cid:18)I of tangible capital and intangible capital, respectively. This constraint translates into the following inequality: (cid:18)Tq k +(cid:18)Iq k T;t+1 T;t+1 I;t+1 I;t+1 a ; (11) f;t+1 (cid:21) (cid:0) 1+r t+1 where 0 < (cid:18)T 1 and 0 < (cid:18)I < (cid:18)T: In reality, (cid:133)rms (cid:133)nance part of their investment with equity (cid:20) issues, which could be captured in the model by assuming that dividends can be negative up to a fraction of the (cid:133)rm(cid:146)s value. However, rather than complicating the model further, in the calibration section we consider equity (cid:133)nancing by assuming larger values of (cid:18)T and (cid:18)I than are normally assumed in the literature. This assumption is without loss of generality, because assuming instead negative dividends proportional to the (cid:133)rm(cid:146)s value and lower collateral values of capital would not change our qualitative and quantitative results. From the Leontief structure of the production function, it follows that k = 1 (cid:22)k . T;t (cid:0)(cid:22) I;t Therefore, from now on, we use this result to express all equations as a function of intangible capital only. At the beginning of each period, both types of capital are predetermined and in their optimal ratio k = 1 (cid:22)k ; therefore, the production function can be written as T;t (cid:0)(cid:22) I;t k (cid:11) p (1 (cid:11)) I;t y t = z t n t (cid:0) (cid:22) : (12) (cid:18) (cid:19) After producing, the (cid:133)rm(cid:146)s technology becomes obsolete with probability . In this case, the (cid:133)rm liquidates all of its capital, pays out as dividends all of its savings, including the liquidation value of capital, and exits. We follow Kiyotaki and Moore (2012) and assume that high-productivity (cid:133)rms can only invest each period with probability (cid:17). This assumption, in addition to capturing the realistic feature that (cid:133)rms(cid:146)investment is lumpy (Caballero (1999)), is meant to allow (cid:133)rms to have the opportunity to accumulate signi(cid:133)cant amounts of liquid savings, in line with the empirical evidence.13 Optimization Firms choose their investment and savings in order to maximize the net present value of their dividends. Let (cid:21) and # be the Lagrange multipliers of constraints (10) and (11), respect t 13In Section 6weinterpret and (cid:17) asshocksthatgeneratecreativedestruction: (cid:17) isthearrivalprobability of aninvestmentopportunitytoproduceanewproduct,and istheprobabilitythatthe(cid:133)rm(cid:146)stechnologybecomes obsolete because a competing (cid:133)rm enters the market and produces an improved version of its product. 14

tively. We de(cid:133)ne the value function conditional on having an investment opportunity, denoted V+(k ;a ), as follows: I;t f;t (cid:18)Tq k +(cid:18)Iq k V+(k ;a ) = max (1+(cid:21) )d +# a + T;t+1 T;t+1 I;t+1 I;t+1 t I;t f;t nt;dt;af;t+1;kI;t+1 t t t f;t+1 1+r t+1 ! 1 + (1 )V (k ;a )+ dexit ; (13) 1+r (cid:0) t+1 I;t+1 f;t+1 t+1 t+1 (cid:2) (cid:3) where 1 (cid:22) dexit = y p +(1+r )a +(1 (cid:14))q (cid:0) k +(1 (cid:14))q k w ; (14) t t t f;t (cid:0) T;t (cid:22) I;t (cid:0) I;t I;t (cid:0) t and V (k ;a ) is the value function conditional on continuation but before the investt+1 I;t+1 f;t+1 ment shock is realized: V (k ;a ) = (cid:17)V+(k ;a )+(1 (cid:17))V (k ;a ): (15) t+1 I;t+1 f;t+1 I;t+1 f;t+1 (cid:0) I;t+1 f;t+1 (cid:0) Thevaluefunctionofanon-investing(cid:133)rm,denotedV (k ;a ),isidenticaltoV+(k ;a ) (cid:0) I;t f;t I;t f;t but does not o⁄er the opportunity to choose k . I;t+1 The (cid:133)rm solves (13) (or its non-investing counterpart) subject to (9), (10), and (11). We next provide a characterization of high-productivity (cid:133)rms(cid:146)optimal choice under the assumption thattheyarepermanently(cid:133)nanciallyconstrained. Weclaim andchecklaterinourcalibrated (cid:0) simulations that, in equilibrium, the marginal return on capital for high-productivity (cid:133)rms (cid:0) is always higher than their user cost: @y t p +1 = (cid:11)z t+1 n ( t+ 1 (cid:0)1 (cid:11)) k I;t+1 (cid:11) (cid:0) 1 > q 1 (cid:0) (cid:22) +q (1 (cid:0) (cid:14)) q T;t+1 1 (cid:0)(cid:22) (cid:22) +q I;t+1 : @k (cid:22) (cid:22) T;t (cid:22) I;t (cid:0) (cid:16) 1+r (cid:17) I;t+1 t+1 (cid:18) (cid:19) (cid:18) (cid:19) (16) The implication of assumption (16) for investing (cid:133)rms is that the borrowing constraint (11) is binding, and that (cid:133)rms choose not to pay dividends, so the equity constraint (10) is also binding. Making d = 0 in budget constraint (9), using (9) to substitute for a in (11), t f;t+1 assuming (11) is binding, and solving for k , we obtain their level of investment: I;t+1 y p w n +(1+r )a +(1 (cid:14)) q 1 (cid:22) +q k t (cid:0) t t t f;t (cid:0) T;t (cid:0)(cid:22) I;t I;t (k invest) = : (17) I;t+1 j q 1 (cid:22) +q (cid:18)T qT;t+1 1 (cid:16)(cid:22) +(cid:18)I qI;t+1 (cid:17) T;t (cid:0)(cid:22) I;t (cid:0) 1+rt+1 (cid:0)(cid:22) 1+rt+1 (cid:16) (cid:17) Theright-handsideofequation(17)isthemaximumfeasibleinvestmentinintangiblecapital for a (cid:133)rm. The numerator is the total wealth available to invest. The denominator captures the downpayment necessary to purchase one unit of k and 1 (cid:22) units of k . The term I;t+1 (cid:0)(cid:22) T;t+1 15

q 1 (cid:22) +q represents the total cost necessary to purchase these amounts of both types of T;t (cid:0)(cid:22) I;t capital, and the term (cid:18)T qT;t+1 1 (cid:22)+(cid:18)I qI;t+1 is the amount that can be (cid:133)nanced by borrowing. 1+rt+1 (cid:0)(cid:22) 1+rt+1 Investing (cid:133)rms in equilibrium borrow as much as possible, and q 1 (cid:22) q (a invest) = (cid:18)T T;t+1 (cid:0) +(cid:18)I I;t+1 k < 0: (18) f;t+1 I;t+1 j (cid:0) 1+r (cid:22) 1+r t+1 t+1 (cid:18) (cid:19) The implication of assumption (16) for non-investing (cid:133)rms is that they will not sell any of their capital, and, for these (cid:133)rms, the law of motion of capital is (k not invest) = (1 (cid:14))k : (19) I;t+1 I;t j (cid:0) Non-investing (cid:133)rms always retain all earnings and select d = 0 because they face a positive t probability of being (cid:133)nancially constrained in the future, and hence the value of cash inside the (cid:133)rmisalwayshigherthanitsopportunitycost(seeAppendixCforaformalproof). Substituting d = 0 and (19) in (9): t p (a not invest) = y +(1+r )a w n : (20) f;t+1 t t f;t t t j (cid:0) Equations(18)and(20)determinethewealthdynamicsof(cid:133)rms. A(cid:133)rmthatinvestedinperiodt 1butisnotinvestinginperiodthasdebtequalto a = (cid:18)T qT;t+1 1 (cid:22) +(cid:18)I qI;t+1 k : (cid:0) (cid:0) f;t 1+rt+1 (cid:0)(cid:22) 1+rt+1 I;t+1 It uses current pro(cid:133)ts y p w n to pay the interest rate on debt(cid:16) r a and to reduce the(cid:17)debt t t t t f;t (cid:0) (cid:0) itself. As long as the (cid:133)rm is not investing, the debt a decreases until the (cid:133)rm becomes a net f;t (cid:0) p saver and has a > 0: At this point, wealth accumulation is driven both by pro(cid:133)ts y w n f;t t t t (cid:0) and by interest on savings r a ; until the (cid:133)rm has an investment opportunity and its accumut f;t lated wealth (1+r )a is used to purchase capital (see equation (17)). This discussion clari(cid:133)es t f;t that a lower interest rate r helps the non-investing (cid:133)rm repay existing debt (the debt hangover t channel), but it slows down the accumulation of savings after the (cid:133)rm has repaid the debt (the savings channel). Finally, the (cid:133)rst order condition for n ; for both investing and non-investing (cid:133)rms, imt plies that given the wage w and its predetermined capital k ; a (cid:133)rm will choose the pro(cid:133)tt I;t maximizing level of labor, which determines the optimal capital-labor ratio: 1 k I;t w t (cid:11) = (cid:22) : (21) n (1 (cid:11))z t t (cid:20) (cid:0) (cid:21) 16

4.2.2 The Low-Productivity Firms There is a mass 1 of identical low-productivity (cid:133)rms that have access to two production functions. Each production function combines capital k with specialized labor n using a uJ;t uJ;t constant-returns-to-scale technology, where J = I;T captures the tangibility of the capital f g used. The total amount yu of the homogeneous (cid:133)nal good produced is then t yu = z u;I n1 (cid:11)k(cid:11) +z u;T n1 (cid:11)k(cid:11) ; t t u(cid:0)I;t uI;t t u(cid:0)T;t uT;t where(cid:11)determinethecapitalshare. Wedonotintroducetheassumptionofperfectcomplementarity between tangible and intangible capital (which we do introduce for the high-productivity (cid:133)rms)togaintractabilityinthepricingofcapital, aswillbecomeclearinthenextsection. This is without loss of generality. This sector is assumed to be able to (cid:133)nance capital with equity from the household sector and to pay out all pro(cid:133)ts as dividends du to households every period: t du = yu wuIn wuTn q ku (1 (cid:14))ku q ku (1 (cid:14))ku : (22) t t t uI;t t uT;t I;t I;t+1 I;t T;t T;t+1 T;t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) In addition, the low-productivity (cid:133)rms sector is able to remunerate households for their labor services (wuIn +wuTn ). t uI;t t uT;t The(cid:133)rstorderconditionsforthetwotypesoflaborimplythatgivenwageswuI andwuTand t t a(cid:133)rm(cid:146)spredeterminedcapitalstocksku andku ,alow-productivity(cid:133)rmwillchoosethepro(cid:133)t- I;t T;t maximizing level of each type of labor, which determines the optimal capital-labor ratio: 1 k wuJ (cid:11) uJ;t = t : (23) n uJ;t "(1 (cid:11))z u;J # t (cid:0) Given that low-productivity (cid:133)rms are (cid:133)nancially unconstrained, and provided that their marginal return on each of the two types of capital is lower than for high-productivity (cid:133)rms, low-productivity(cid:133)rmsarewillingtoabsorballofthecapitalnotdemandedbyhigh-productivity (cid:133)rms, at a price equal to their marginal return on capital. 4.2.3 Aggregation of the Firm Sector and Pricing of Assets We assume (see Section 4.3) that the aggregate supply of all types of labor is normalized to N = N = N = 1: Since all high-productivity (cid:133)rms produce at the optimal capital-labor uI uT ratio determined by equation (21), and the production function is constant returns to scale , we can aggregate production across (cid:133)rms to obtain 17

K (cid:11) p I;t Y = z : (24) t t (cid:22) (cid:18) (cid:19) The wage is determined in competitive markets by the marginal return of labor: K (cid:11) I;t w = (1 (cid:11))z : (25) t t (cid:0) (cid:22) (cid:18) (cid:19) And aggregate wealth W of the high-productivity (cid:133)rms at the beginning of period t is t 1 (cid:22) p W Y w +(1+r )A +(1 (cid:14)) q (cid:0) +q K : (26) t (cid:17) t (cid:0) t t f;t (cid:0) T;t (cid:22) I;t I;t (cid:18) (cid:19) Aggregate capital is determined as follows. A fraction (1 ) of high-productivity (cid:133)rms (cid:0) continue activity, and a fraction (cid:17) of those have an investment opportunity. They have a fraction (1 )(cid:17) of total wealth W , which they use to buy the amount of capital given by t (cid:0) equation (17). A fraction of high-productivity (cid:133)rms exit, and are replaced by an equal number of (cid:133)rms with an initial endowment of W and no capital. A fraction (cid:17) of new entrants 0 invest. Therefore, we de(cid:133)ne total intangible capital in the hands of investing agents at the end of period t, expressed in aggregate terms, as (cid:17)KINV ; where KINV is I;t+1 I;t+1 (1 )W + W KINV = (cid:0) t 0 : (27) I;t+1 q (cid:18)T qT;t+1 1 (cid:22) +q (cid:18)I q I I ;t+1 T;t (cid:0) 1+rt+1 (cid:0)(cid:22) I;t (cid:0) 1+rt+1 (cid:16) (cid:17) The (1 (cid:17)) fraction of surviving (cid:133)rms that do not have an investment opportunity continue (cid:0) to hold their depreciated capital. Therefore, aggregate capital for the next period is equal to K = (cid:17)KINV +(1 (cid:14))(1 )(1 (cid:17))K (28) I;t+1 I;t+1 I;t (cid:0) (cid:0) (cid:0) and 1 (cid:22) K = (cid:0) K : (29) T;t+1 I;t+1 (cid:22) Furthermore, we can aggregate the output of low-productivity (cid:133)rms, substituting labor supply N = N = 1; and obtain uI uT Yu = z u;I K I K (cid:11) +z u;T K T K (cid:11) ; (30) t t I;t t T;t (cid:0) (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) wuJ = (1 (cid:11))z u;J K J K (cid:11) ; (31) t t J;t (cid:0) (cid:0) (cid:16) (cid:17) with J = I;T . f g The marginal return of capital in the high-productivity (cid:133)rms is as follows. In order obtain a marginal increase @Y t p = (cid:11)z KI;t (cid:11) (cid:0) 1 ; these (cid:133)rms purchase one unit of intangible capital @KI;t (cid:22) t (cid:22) (cid:16) (cid:17) 18

and 1 (cid:22) units of tangible capital. The equilibrium described earlier requires that the high- (cid:0)(cid:22) productivity (cid:133)rms have the highest return on capital, or (cid:11) z K I;t+1 (cid:11) (cid:0) 1 > z u;I (cid:11) K I K (cid:11) (cid:0) 1 + 1 (cid:0) (cid:22) z u;T (cid:11) K T K (cid:11) (cid:0) 1 ; (32) (cid:22) t (cid:22) t (cid:0) I;t (cid:22) t (cid:0) T;t (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) where the right-hand side of this inequality captures the marginal return of one unit of tangible capital and 1 (cid:22) units of intangible capital in the low-productivity (cid:133)rms. (cid:0)(cid:22) If condition (32) is satis(cid:133)ed, then it follows immediately that the prices of capital are q = z u;I (cid:11) K I K (cid:11) (cid:0) 1 + 1 (cid:0) (cid:14) q ; (33) I;t I (cid:0) I;t 1+r I;t+1 t+1 (cid:16) (cid:17) and q = z u;T (cid:11) K T K (cid:11) (cid:0) 1 + 1 (cid:0) (cid:14) q : (34) T;t t (cid:0) T;t 1+r T;t+1 t+1 (cid:16) (cid:17) If we substitute (33) and (34) into (32), it follows that (cid:11) K I;t+1 (cid:11) (cid:0) 1 1 (cid:14) 1 (cid:22) 1 (cid:14) z > q (cid:0) q + (cid:0) q (cid:0) q ; (35) t I;t I;t+1 T;t T;t+1 (cid:22) (cid:22) (cid:0) 1+r (cid:22) (cid:0) 1+r t+1 t+1 (cid:18) (cid:19) (cid:18) (cid:19) which implies that the claim (16) is correct. Aggregate (cid:133)nancial assets of the high-productivity (cid:133)rms (A ) are equal to the assets saved from the previous period by continuing (cid:133)rms, f;t+1 p (1 )((1+r )A ), plus their current retained earnings, (1 )(Y w ), plus the endowt f;t t t (cid:0) (cid:0) (cid:0) mentsofnew(cid:133)rms( W )minustotalinvestment, q 1 (cid:22) +q (K (1 (cid:14))(1 )K ): 0 T;t (cid:0)(cid:22) I;t I;t+1 (cid:0) (cid:0) (cid:0) I;t (cid:16) (cid:17) p A = (1 )(Y w +(1+r )A )+ W f;t+1 t t t f;t 0 (cid:0) (cid:0) 1 (cid:22) q (cid:0) +q (K (1 (cid:14))(1 )K ): (36) T;t I;t I;t+1 I;t (cid:0) (cid:22) (cid:0) (cid:0) (cid:0) (cid:18) (cid:19) Finally, total dividends paid out by exiting high-productivity (cid:133)rms to households are equal to 1 (cid:22) p p D = Y w +(1+r )A + q (cid:0) +q K W ; (37) t t (cid:0) t t f;t T;t (cid:22) I;t I;t (cid:0) 0 (cid:18) (cid:18) (cid:19) (cid:19) and the dividends paid by the low-productivity (cid:133)rms are: Du = Yu wuI wuT q K I K K I K q K T K K T K : t t t t I;t I;t+1 I;t T;t T;t+1 T;t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) h(cid:16) (cid:17) (cid:16) (cid:17)i h(cid:16) (cid:17) (cid:16) (cid:17)i 4.3 Households We consider a life-cycle model with two types of households, young and old(cid:150)with measures Hy and Ho, respectively(cid:150)whose sum is normalized to 1. Young households supply three types of 19

di⁄erentiated labor: high-productivity (cid:133)rm labor (in exchange for wage w ), low-productivity t intangible technology labor (in exchange for wage wuI), and low-productivity tangible technolt ogy labor (in exchange for wage wuT). There is an inelastic aggregate supply of one unit of each t type of labor. Young households receive a fraction (cid:13) of the aggregate dividends. Households remainyoungforN periodsandbecomeoldafterN+1periods, sothatthereisaconstantfraction (cid:30) = 1 of young households for every age between 1 and N, and, every period, a measure N (cid:30)Hy of households becomes old. Old households cannot work, receive a fraction (1 (cid:13)) of ag- (cid:0) gregate dividends, and die with probability %. The measure of old households Ho is determined as follows: Ho = (1 %)Ho+(cid:30)Hy: (38) (cid:0) At the same time, the measure of young households is Hy = (1 (cid:30))Hy +Ny; (39) (cid:0) whereNy istheconstantmeasureofnewbornhouseholds. FromtheassumptionthatHo+H y = t t 1, it follows that Ny = (cid:30)% ; Ho = (cid:30) , and H y = % . (cid:30)+% t (cid:30)+% t (cid:30)+% We follow Blanchard (1985) and Yaari (1965) in assuming that households participate in a life insurance scheme when old. For the detailed solution of the households(cid:146)maximization problem, see Appendix D. 5 Steady State 5.1 Equilibrium We consider a steady state equilibrium and drop reference to the time subscript t: Total output of the high-productivity and low-productivity (cid:133)rms is, respectively, K (cid:11) Yp = z I (40) (cid:22) (cid:18) (cid:19) and Yu = zu;I K I K (cid:11) +zu;T K T K (cid:11) : (41) I T (cid:0) (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) Dividends d are given by d = Dp+Du+Dk; (42) where Du = Yu wuI wuT q (cid:14) K I K q (cid:14) K T K ; t I I T T (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) 20

K (cid:11) 1 (cid:22) Dp = (cid:11)z I +(1+r)A + q (cid:0) +q K W ; f T I I 0 (cid:22) (cid:22) (cid:0) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19) and ’ ’ q’ 1 q’ 1 Dk = T;(cid:0)t (’ 1)+ I;(cid:0)t (’ 1): T 1 (cid:0) I 1 (cid:0) b ’ 1 b ’ 1 (cid:0) (cid:0) Aggregate cash holdings of the high-productivity (cid:133)rms in the steady state can be obtained by combining (36), (24), and (25) to obtain (cid:11) (1 (cid:0) )(cid:11)z t K (cid:22) I + W 0 (cid:0) q T 1 (cid:0)(cid:22) (cid:22) +q I [ +(cid:14)(1 (cid:0) )]K I A = : (43) f (cid:16) (cid:17) [1 (1 (cid:16) )(1+r)] (cid:17) (cid:0) (cid:0) Aggregate borrowing is equal to aggregate savings, or A = B; (44) f whereBisaggregatehouseholdborrowing, whichwederiveindetailinAppendixD.ByWalras(cid:146) Law, the aggregate resource constraint is satis(cid:133)ed. In order to determine the aggregate capital of the high-productivity (cid:133)rms, equation (28) in the steady state is equal to (1 )W + W 0 K = (cid:17) (cid:0) ; (45) I q 1 (cid:18)T 1 (cid:22) +q 1 (cid:18)I [1 (1 (cid:14))(1 )(1 (cid:17))] T (cid:0) 1+r (cid:0)(cid:22) I (cid:0) 1+r (cid:0) (cid:0) (cid:0) (cid:0) h (cid:16) (cid:17) (cid:16) (cid:17)i where W is de(cid:133)ned using equation (26) in steady state: K (cid:11) 1 (cid:22) I W (cid:11)z +(1+r)A +(1 (cid:14)) q (cid:0) +q K : (46) t f T I I (cid:17) (cid:22) (cid:0) (cid:22) (cid:18) (cid:19) (cid:18) (cid:19) We can also express (45) as (cid:11) (cid:17)(1 ) (cid:11)z KI +(1+r)A +(cid:17) W (cid:0) t (cid:22) f 0 K = ; I q 1 (cid:18)T 1 (cid:22) +q 1 (cid:18)I(cid:16) [(cid:14) (cid:16) + (cid:17) (1 (cid:14))] q (cid:17)(cid:18)T 1 (cid:22) +q (cid:18)I (cid:17)(1 (cid:14))(1 ) T (cid:0) 1+r (cid:0)(cid:22) I (cid:0) 1+r (cid:0) (cid:0) T1+r (cid:0)(cid:22) I1+r (cid:0) (cid:0) h (cid:16) (cid:17) (cid:16) (cid:17)i (cid:16) (cid:17) (47) which has an intuitive explanation. The numerator is the aggregate amount of liquid resources of investing (cid:133)rms. The denominator is the downpayment necessary to support one unit of capital in the steady state. It requires the replacement of the depreciated capital and the lost capital of exiting (cid:133)rms (a fraction (cid:14)+ (1 (cid:14))) and can bene(cid:133)t from using existing capital held (cid:0) by the investing (cid:133)rms as collateral (fraction (cid:17)(1 (cid:14))(1 )). (cid:0) (cid:0) Finally, the prices of capital are determined by recursively iterating forward equations (33) 21

and (34): q = 1 zu;I(cid:11) K I K (cid:11) (cid:0) 1 (48) I I r+(cid:14) (cid:0) (cid:16) (cid:17) and q = 1 zu;T(cid:11) K T K (cid:11) (cid:0) 1 ; (49) T T r+(cid:14) (cid:0) (cid:16) (cid:17) where aggregate capital and investment are given by IJ J K = (50) (cid:14) and q 1 IJ = ’ J ’ (cid:0) 1 ; (51) bJ (cid:16) (cid:17) respectively, for J I;T : 2 f g The steady state values of W, A , B, K , q , q , and r are jointly determined by equations f I I T (43), (44), (45), (46), (48), (49), and (93). 5.2 Discussion If we assume for simplicity that q = q = q; the collateral value of one unit of capital is T I q 1 (1 (cid:22))(cid:18)T +(cid:22)(cid:18)I . Since (cid:18)T > (cid:18)I; a technology that relies more on tangible capital 1+r(cid:22) (cid:0) (lower(cid:2)(cid:22)) places a high(cid:3)er weight on the collateral value of tangible capital (cid:18)T, thus increasing the overall collateral value of the (cid:133)rms(cid:146)capital. Such an economy has a lower downpayment in the denominator of (47) and more capital K for a given total wealth in the numerator. I Equation (43) determines (cid:133)nancial wealth A ; which is equal to the net earnings of the f high-productivity (cid:133)rms, in the numerator, multiplied by a multiplicative factor 1 ; 1 (1 )(1+r) (cid:0) (cid:0) which measures the future value of one unit of wealth saved today by these (cid:133)rms. The net earnings are the endowment of the new (cid:133)rms W plus the net earnings of continuing (cid:133)rms. 0 (cid:11) The term (1 )(cid:11)z KI is retained earnings, net of wage payments, and is concave in (cid:0) t (cid:22) K : The term q 1 (cid:22)(cid:16)+q(cid:17) [ +(cid:14)(1 )]K is total expenditures to replace the depreciated I T (cid:0)(cid:22) I (cid:0) I capital of conti(cid:16)nuing (cid:133)rms (cid:14)(cid:17)(1 )K ; and the capital liquidated by exiting (cid:133)rms K ; and is I I (cid:0) linear in K : A high average collateral value of capital in a tangible economy increases K and I I makesitlikelythatthesumofthetwolasttermsisnegative,andsince W isverysmall,italso 0 makes A negative: the high-productivity (cid:133)rms are, on aggregate, net borrowers. Conversely, f in an intangible (high (cid:22)) economy, A is likely to be positive. f The previous discussion clari(cid:133)es that the exogenous assumptions made in the simple model in Section 3 are endogenously derived in the full general equilibrium model. Moreover, even though a change in the interest rate a⁄ects aggregate capital K in (47) through the same I 22

four channels identi(cid:133)ed in the simple model in Section 3, it is important to emphasize that the endogeneity of (cid:133)nancial assets ampli(cid:133)es the strength of the savings channel. When A is f positive, a reduction in the interest rate reduces investment both through a reduction in the return on savings rA and through the multiplicative factor 1 : f 1 (1 )(1+r) (cid:0) (cid:0) 6 Calibration [TABLE 1 ABOUT HERE] For the purpose of evaluating the qualitative and quantitative importance of the channels explained earlier for the real economy, we calibrate the model to be broadly in line with recent U.S. data. We simulate the evolution of the economy from 1980 to the present as a sequence of steady states, and use this simulated time series to calculate the model-based moments. Our benchmark calibration is illustrated in Table 1. Our calibration strategy is twofold. We set most of our parameters to match key empirical moments of aggregate variables from 1980 to the present. A subset of parameters (cid:150)those which are key to the mechanisms introduced in our model (cid:150)are the basis of our comparative statics exercises and are set to change according to their observed variation or the observed variation of some direct moment they in(cid:135)uence during the 1980-present period. In this latter group we include the share of intangibles ((cid:22)), the cost of producing capital (driven by parameters bT and bI), the rate of time preference of households ((cid:12)), and the longevity of households (driven by %). We start discussing the calibration of parameters that remain constant across the di⁄erent steady states. In the (cid:133)rm sector, the elasticity of output with respect to capital (cid:11) is set equal to 0.4 for both types of (cid:133)rms, a common value used in most of the literature.14 The measures of high-productivity and low-productivity (cid:133)rms are assumed to be equal. This assumed share of high-productivity (cid:133)rms, which are (cid:133)nancially constrained in our model, matches the observed shareofcredit-constrained(cid:133)rmsintheUnitedStates, estimatedbyFarre-MensaandLjungqvist (2014) to be roughly 50%.15 The pledgeability parameters of tangible capital (cid:18)T and intangible capital (cid:18)I are equal to 1.00 and 0.35, respectively. Thus, we assume tangible capital to be fully collateralizable, in line with Falato, Kadyrzhanova, and Sim (2014). Moreover, we calibrate (cid:18)I to generate net leverage in the high-productivity (cid:133)rms on average equal to 6.4%, in line with the average net 14See King and Rebelo (1999) or Corrado, Hulten and Sichel (2009). 15They (cid:133)nd that roughly three quarters of privately held (cid:133)rms are (cid:133)nancially constrained. Within the sample of publicly listed (cid:133)rms, they report di⁄erent estimates of the share of (cid:133)nancially constrained (cid:133)rms that range between 10% and 45%. Given these estimates, we set the share of high productivity (cid:133)rms to be 50% in our simulations. 23

leverage ratio for Compustat publicly-listed (cid:133)rms.16 We set (cid:18)I relatively high compared with the literature to accommodate for the fact that we only allow (cid:133)rms to issue collateralized debt. As discussed in Section 2, in reality, (cid:133)rms (cid:133)nance their acquisitions in part with equity issues and other forms of external (cid:133)nancing beyond collateralized debt.17 In order to calibrate the exit probability and the investment probability (cid:17); we interpret them as shocks that generate creative destruction. Therefore, even though we do not model explicitly heterogeneous products, we interpret as the probability that the (cid:133)rm(cid:146)s technology becomesobsoletebecauseacompeting(cid:133)rmentersthemarketandproducesanimprovedversion of its product. Moreover, we interpret (cid:17) as the arrival probability of an investment opportunity to produce a new product. According to this interpretation, we set = (cid:17) = 13%, which generates yearly capital reallocation of 6% of total capital (tangible plus intangible). This is consistent with David (2014), which measures reallocation of capital generated by mergers and acquisitions to be around 5% of total capital in the past few decades, and with the reallocation datafromEisfeldtandRampini(2006).18 Theintuitionisthatwhena(cid:133)rm(cid:146)stechnologybecomes obsolete, it sells its capital to the new and more productive (cid:133)rms. The TFP of low-productivity (cid:133)rms, zu;T and zu;I; is normalized to 10. The TFP of highproductivity (cid:133)rms z is modeled as: t z = [1+((cid:22) 0:2)(cid:20)]z; (52) t (cid:0) so that for the early 1980s value of (cid:22) = 0:2, z = z for simplicity. We set z = 25 to match the t average interquartile productivity di⁄erential of the (cid:133)rms, which in our simulations is 2.54 over the 1980-present period, a number consistent with the cross-sectional dispersion in productivity for U.S. (cid:133)rms identi(cid:133)ed in Syverson (2004) for a similar time period.19 The parameter (cid:20) measures the increase in TFP associated with a stronger intensity of intangible capital in the production function. We choose (cid:20) = 0:1; which implies that an increase in (cid:22) is privately 16Bates et al. (2009) using data from 1980 to 2006; compute a value of 7.9%. They calculate net leverage as the ratio of total debt minus cash holdings to the book value of total assets, which maps in the model to A =(q K +q K ): f T T I I (cid:0)17An alternative approach would have been to assume a value of (cid:18) much closer to zero, in line with Falato, I Kadyrzhanova,andSim(2014),andintroduceequityissuesbyallowingdividendsd tobenegative,withanassot ciated equity issuance cost proportional to the amount (cid:133)nanced. This approach would have slightly complicated the model and yielded very similar quantitative results. 18Using capital reallocation data available at Andrea Eisfeldt(cid:146)s website ( https://sites.google.com/site/andrealeisfeldt/reallocation_data_eisfeldt.xlsx), we compute an average capital reallocation of 5.8% of total capital over the 1980-2013 period. 19Syverson (2004) examines plant-level data from 1977 and (cid:133)nds an average interquartile di⁄erence in labor productivity around 2 for 4-digit U.S. manufacturing sectors. Since the dispersion of productivity is larger for less narrowly de(cid:133)ned sectors, a value of 2.54 is probably a very conservative estimate of the dispersion of productivity across all (cid:133)rms. 24

optimal at the steady state equilibria obtained for most values of (cid:22).20 A positive value of (cid:20) is not necessary for our results. However, it is consistent with the notion of the rise of intangible capital as a privately optimal choice of (cid:133)rms, and allows us to be able to make conservative and robust statements about the potential for negative e⁄ects of the shift to intangibles. The depreciation factor (cid:14) is set equal to 15%. This value is consistent with the depreciation ratesusedfortheperpetualinventorymethodinSection2.21 Theinitialendowmentofnewborn (cid:133)rms W is equal to 5, and is the only one not to be calibrated to match a speci(cid:133)c moment due 0 to a lack of a clear empirical counterpart. It corresponds to 2% of average (cid:133)rm annual output. Our results show very little sensitivity to variations in our choice of W in the range 0.1%-20%. 0 The parameters associated to capital production are ’; bT and bI: The parameter ’ determines the elasticity of the capital stock to the price of capital (see equation 51), and we calibrate it so that the elasticity of the stock of capital to the user cost of capital is in line with the empirical evidence. Caballero, Engel and Haltiwanger (1995) estimate the short run elasticity of the capital stock to the user cost of capital to be between 0 and -0.1, and the long run elasticity to be between -0.3 and -1 for most 2-digit sectors. Since we do not model taxes, and the price of the consumption good is normalized to 1, the user cost of tangible capital in our model is (r+(cid:14))qT. We consider changes in the user cost of capital driven by exogenous changes in the interest rate: Our production sector implies that a decrease in r increases qT(see equation (49)). However the user cost of capital falls in equilibrium, because the increase in qT does not fully compensate the reduction in r: We choose a value of ’ = 9; which generates an elasticity equal to -0.23. Given the value of ’, the initial values of bT and bI determine the aggregate supply of tangible and intangible capital and their equilibrium prices. We calibrate them so that the relative price of tangible to intangible capital is normalized to 1 in our early 1980s steady state simulation, and so that output of the high-productivity (cid:133)rms is roughly 50% of total output. Therearetwohouseholdsectorparametersthatwekeepconstantacrosscomparativestatics. The share of dividends that are paid to the working-age population, (cid:13), is set to 40% in order to target a real interest rate of r = 6% in our simulation of the early 1980s, consistent with the 20Our results are robust to setting (cid:20) high enough so that increases in (cid:22) are always privately optimal. Our benchmark calibration, however, re(cid:135)ects the possibility that some of the technological changes that have driven an increase in the intensity of intangible capital are not always endogenous (cid:133)rm choices but the consequence of structural economic changes, such as secular changes in the sectoral specialization of di⁄erent countries. 21For tangible capital, this value is appropriate since we interpret it as a combination of more durable assets, such as equipment and structures, and less durable ones, such as inventories. For intangible capital, this value is consistent with existing literature regarding intangible and tangible capital, while possibly too low for other intangibleassetssuchascomputerisedinformationandbrandequity(Corrado,HultenandSichel2006). Assuming higherdepreciation rateforintangiblecapitaldoesnotsigni(cid:133)cantly changetheresultspresented in thefollowing sections. 25

real rate in that period. We set the number of years households remain young to N = 40, which corresponds to a working-age period between the ages of 25 and 65 years. Finally, we discuss the parameters that we vary in our comparative statics exercises. We follow Falato, Kadyrzhanova, and Sim (2014) in setting (cid:22), the reliance on intangible capital of (cid:133)rms, at 0:2 in our exercise for the early 1980s, so that the share of intangible capital over total capital is 20%. We introduce a gradual linear shift in (cid:22) so that our simulation matches the observed intangible to total capital ratio of 60% ((cid:22) = 0:6) in the 2010s (Corrado and Hulten (2010a), Falato, Kadyrzhanova, and Sim (2014), D(cid:246)ttling and Perotti (2015)). We vary the parameters bI and bT in the capital production function (6) to capture the observed evolution of capital prices. This is important because capital prices matter for the mechanism we describe, and because it has been well documented that tangible capital has experiencedasigni(cid:133)cantdecreaseinitsrelativeprice. KarabarbounisandNeiman(2014)estimate that the price of capital has fallen approximately by 30% between the late 1970s and the 2000s, and we match this trend by decreasing bT accordingly. Reliable measures of the change in the relative price of intangible capital are not available, however.22 Some authors have used instead the GDP de(cid:135)ator, which implies by construction no change in the relative price of intangible capital (Corrado, Hulten and Sichel (2009)). Other authors use an input cost approach. An important factor in the production of intangible capital is skilled labor (Dougherty, Inklaar, McGuckin, and van Ark (2007), Robbins, Belay, Donahoe, and Lee (2012)), which has experienced an important increase in its relative cost since the 1980s (Lemieux (2008)). An increase in input costs however might translate into lower intangible capital prices if the productivity of capital production increases substantially. This is the case for R&D, one of the types of intangible capital: Robbins, Belay, Donahoe, and Lee (2012) estimate an annual fall in the relative price of R&D of around 1.2% between 1998 and 2007 despite an increase in input costs. Computerised information, on the other hand, is estimated by Byrne and Corrado (2016) to have experienced an average annual real price change of -1% in the 1963-87 period, and of around -4% in the 1987-2015 period. Putting this evidence together, we change bI over time so that the relative price of intangible capital remains roughly constant over time. It is important to note that what matters for our purpose is how much interest rates a⁄ect the path of relative prices of capital, and not what the precise level of capital prices would be absent the observed signi(cid:133)cant decrease in real rates. The household sector parameters that we vary across our simulations are the discount factor (cid:12) and the probability of death after the age of 65, %. First, we vary % so that we match changes 22SeeCorrado,Haskel,Iommi,andJonaLasinio(2012)foradetaileddescriptionofthechallengesinobtaining a general price de(cid:135)ator for intangible capital. 26

in the life expectancy in the U.S. between the 1980s and the present.23 We vary the rate of time preference (cid:12) to match the evolution of real interest rate from around r = 6% in the 1980s to around 0% in the present, and so that the value of (cid:12) on average over our comparative statics exercises is in line with values used in the literature.24 7 Simulation Results In this section, we introduce two comparative static exercises that capture how parallel developments in the household and the corporate sector have interacted to generate aggregate patterns consistent with the secular stagnation hypothesis. First, we explore how an expansion of households(cid:146)savings a⁄ects economic outcomes in a tangibles-intensive economy compared to an intangibles-intensive one. Second, we introduce a simulation that replicates key trends in the United States between 1980 and 2015, a period characterized by an increase in households(cid:146) incentive to save and a rise in the reliance on intangible capital. 7.1 The E⁄ect of a Rise in Households(cid:146)Propensity to Save In order to clarify the di⁄erent e⁄ects at play, we (cid:133)rst conduct a counterfactual exercise in which households(cid:146)propensity to save and life expectancy both gradually increase, reducing the equilibrium interest rate. We run two simulations: one in which the share of intangible capital is kept constant at (cid:22) = 0:05 (a tangibles economy), and another in which it is kept constant at (cid:22) = 0:65 (an intangibles economy). The expansion in household savings is achieved by decreasingtherateofhouseholdtimepreference(increasing(cid:12))andbyincreasinglifeexpectancy (lowering %) to generate a decline in the interest rate from 6% to around 1%.25 The sequence of steady states associated to the set of di⁄erent values of (cid:12), % and (cid:22) is displayed in Figure 4. [FIGURE 4 ABOUT HERE] The left panel in the middle row of Figure4 shows thatthe net leverage ofhigh-productivity (cid:133)rms is positive in the tangibles economy and (cid:133)rms are on average net borrowers. Corporate net leverage is instead negative in the intangibles economy and (cid:133)rms are on average net savers. 23The Centers for Disease Control and Prevention (https://www.cdc.gov.htm) reports that life expentacy was around 70 years in 1970 and 78 years in 2016. 24Common values used in the literature range from 0.93 used in Jermann and Quadrini (2012) to 0.97 in Christiano,Eichembaum andEvans(2005). Weset(cid:12) torangefrom 0.9425in theearly1980sto0.9805inrecent years. 25All parameters are identical in the two cases except for the discount factor (cid:12); which is set so that in both casesthecomparativestaticsexercisestartswithavalueofr=6%:Therefore,whileinthetangibleseconomy(cid:12) changes from 0.9425 to 0.9805, in the intangibles economy it changes from 0.9375 to 0.9755. To avoid confusion we do not report these di⁄erent values of (cid:12) on the x-axis. 27

Correspondingly, households are net savers (borrowers) in a tangibles (intangibles) economy, as shown in the top-right panel. Household sector developments encourage households to save more in a tangibles economy and borrow less in an intangibles economy, pushing down the interest rate in both cases. The drop in the interest rate increases the price of capital and encourages capital creation, so that aggregate tangible and intangible capital stocks increase. The left and middle panels in the last row of Figure 4 analyze the changes in the allocation of capital and in e¢ ciency. In the tangibles economy, capital allocation improves and there is an expansion of capital and output of high-productivity (cid:133)rms. High-productivity (cid:133)rms have a high leverage and the decline in the interest rate bene(cid:133)ts them, both because it is easier to pay backdebt(thedebthangoverchannel)andbecausetheycanborrowmorewhentheyinvest(the collateral value channel).26 These two channels prevail over the capital price channel, which operatesintheoppositedirection, andimplythatthedropinr bene(cid:133)tshigh-productivity(cid:133)rms; they can absorb a higher share of existing capital, thus improving the allocation of resources.27 Conversely, in the intangibles economy, (cid:133)rms are net savers. As explained in Section 3, in this case the decline in the interest rate hurts their accumulation of wealth (the savings channel), and the collateral value channel is very weak because (cid:133)rms(cid:146)borrowing capacity is limited, so that a lower rate is strongly contractionary. The last panel shows that, overall, output increases by around 1.5% in the tangibles economy, bothbecauseofthepositivereallocatione⁄ectandbecauseoftheincreaseintheaggregate capital stock, while it declines by around 1.5% in the intangibles economy, because the contraction in the allocation of capital to the high productivity (cid:133)rms o⁄sets the positive e⁄ect of the increase in aggregate capital. [FIGURE 5 ABOUT HERE] Figure 5 shows that the dispersion in the marginal productivity of capital increases with lower rates in the intangibles economy, while it falls moderately in the tangibles economy. The dispersion in TFP shows similar diverging trends as well. The values of (cid:22) chosen for the tangibles and intangibles economies correspond to the 5% and 95% percentiles, respectively, of thecrosssectionaldistributionoftheaverageshareofintangiblecapitalin2-digitU.S.industrial sectors over the 1980-2015 period. Since interest rate movements are almost identical in both 26The drop in the interest rate increases corporate leverage in the tangibles economy via the collateral value channel and via a second, less intuitive, mechanism. As q and q go up and the productivity of capital in the T I high-productivity (cid:133)rms goes down because of the decreasing marginal product of capital and a (cid:133)xed aggregate labor supply, the share of capital (cid:133)nanced by debt as opposed to by the accumulation of past output goes up. 27Another indirect bene(cid:133)t of this positive reallocation is that since the low-productivity (cid:133)rms absorb less capital, their marginal return is higher, relative to the intangibles economy, driving up capital prices (middle row, middle graph), and stimulating capital production (middle row, left graph). 28

simulations, these can be interpreted as two sectors in an economy where capital and labor are sector speci(cid:133)c. In this respect, the simulated trends shown in Figure 5 are fully consistent with the empirical trends shown in Figure 2. 7.2 The Simultaneous Rise in Households(cid:146)Propensity to Save and in Intangible Capital (1980-2015) In Section 7.1 we explored an expansion in household savings but kept the intensity of intangible capital constant. In this section, in contrast, we reproduce the simultaneous rise in the propensity of households to save and in the reliance on intangible capital observed during the period from 1980 to 2015. To increase our understanding of the interaction between both developments, we also describe a sequence of steady states in which we only increase the reliance on intangiblecapital. OurresultsaredisplayedinFigure6. Sinceweabstractfromlong-rungrowth considerations, the graphs that show relative changes in total output should be interpreted as deviations from long-run trends. We (cid:133)rst focus on the exercise that explores the rise in intangibles in isolation. The gradual increase in (cid:22) pushes high-productivity (cid:133)rms to demand progressively more intangible capital and less tangible capital. Intangible capital attracts less external (cid:133)nance, which tightens (cid:133)rms(cid:146) borrowing constraints signi(cid:133)cantly and decreases corporate leverage. High-productivity (cid:133)rms switch from being net borrowers to being net lenders, consistent with evidence in the United Statesforcorporations(ArmenterandHnatkovska(2016), Quadrini(2016), Chen, Karabarbounis and Neiman (2016)). The increased reliance on a type of capital that (cid:133)rms cannot (cid:133)nance externally causes a contraction in the allocation of capital to the high productivity (cid:133)rms. Furthermore, the increase in net corporate savings reduces interest rates moderately, by about 1%, to ensure that households borrow more and absorb the excess savings. A lower interest rate also a⁄ects capital accumulation, but most of the misallocation is caused by high-productivity (cid:133)rms(cid:146)lower ability to borrow. Aggregate output rises initially driven by higher productivity of intangible capital (see equation (52)), but eventually levels o⁄and falls slightly as the negative e⁄ects of a decrease in corporate borrowing and lower interest rates dominate. Overall, a shift to intangibles is expansionary. [FIGURE 6 ABOUT HERE] When we consider corporate and household developments simultaneously, we observe instead a fall in aggregate output. The interest rate falls from 6% to around 0% and capital prices are generally higher than in the simulation of the rise in intangibles in isolation. Aggregate cap- 29

ital in the high-productivity (cid:133)rms (third row, middle panel) is roughly constant in the initial 1980-1990 period, while leverage is still positive and the increase in productivity driven by the rise of intangibles compensates the negative e⁄ects of the lower borrowing capacity. In this period, total output (bottom panel) expands by 2% until around the mid 1990s, thanks to the increase in productivity and aggregate capital.28 During the 1990s and 2000s, however, capital and output of high-productivity (cid:133)rms both fall substantially because their borrowing capacity declines further and the economy becomes more similar to the intangibles economy described in Section 7.1, an economy in which a decline in the interest rate causes a large contraction of the output of high productivity (cid:133)rms. By 2015, their output has fallen by 18%, compared to a fall of 10% in the economy in which only the rise in intangibles occurs (third row, right panel). Lower rates damage the high productivity (cid:133)rms both because of the savings channel, which becomes stronger the larger are their net savings, and because low rates imply relatively higher capital prices, which hurt (cid:133)rms through the capital price channel. Thus, the reduction in interest rates, which is expansionary for highly leveraged high-productivity (cid:133)rms, hurts capital reallocation and growth once the economy relies more on intangible and less collateralizable capital. It is important to note that while a decline in interest rates caused by household developmentsexpandsaggregateoutputinatangibleseconomy(seeFigure4),andashifttointangibles also expands it, the combination of the two developments is overall contractionary, with output in 2015 around 1% lower than in 1980 (bottom panel in Figure 6). [FIGURE 7 ABOUT HERE] The contraction in output happens despite our assumption that intangible capital is more productive, because of a strong misallocation e⁄ect in which too many resources are absorbed bylow-productivity(cid:133)rms. Figure7quanti(cid:133)estheconsequencesofmisallocationonproductivity more precisely. It shows that, in a counterfactual scenario in which the allocation of resources does not worsen, the rise in intangibles increases aggregate TFP by almost 3%. The rise in intangibles in isolation, but allowing for an endogenous allocation of capital between high- and low-productivity (cid:133)rms, generates a TFP drop of -3.5%, which almost doubles to -6.5% when the rise of intangibles coincides with increased incentives to save in the household sector. [FIGURE 8 ABOUT HERE] 28The values of bT and bI; which are calibrated to an empirically realistic evolution of capital prices in the simulation with both household and corporate developments, are also an important factor driving the increase in capital stock. 30

Finally, Figure 8 replicates the evolution of aggregate output in the benchmark case with both developments (bottom panel in Figure 6), and compares it to a counterfactual simulation in which the overhang/savings channel is eliminated. This counterfactual is constructed by assuming that interest rate changes can a⁄ect capital prices and the collateral constraint, but that (cid:133)rms(cid:146)interest rate on debt or return on savings is kept constant at the initial value of 6%. Inthesimulatedperiodinwhich(cid:133)rmsarenetborrowers(1980-1995),thedebtoverhangchannel implies that lower rates bene(cid:133)t high productivity (cid:133)rms, and shutting it down (the dashed line in Figure 8) lowers output relative to the benchmark. However, once (cid:133)rms become net savers, the savings channel implies that lower rates worsen reallocation, and signi(cid:133)cantly contributes to the decline in aggregate output. 8 Conclusion This paper highlights a novel misallocation e⁄ect of endogenously low interest rates that has potentially important policy implications. From a quantitative standpoint, our results are consistentwithseveraldevelopmentsthathavetakenplaceinthepast40years:(i)netcorporate savings increased as a fraction of GDP, (ii) household leverage increased as a fraction of GDP, (iii)therealinterestratefell,(iv)intra-industrydispersioninproductivityhasincreased,and(v) output and productivity progressively declined relative to their previous trends. Interestingly, the model shows that even though the shift to intangible technologies was already taking place in the 1970s, its net negative e⁄ects on output growth only started to gather pace from the mid-1980s onward. This (cid:133)nding is consistent with studies that show a decline in dynamism of U.S. businesses starting in the mid 1980s and gathering speed especially from 2000 onward (Haltiwanger (2015)). Morebroadly,ourresultssuggestthatthechangesin(cid:133)rms(cid:146)(cid:133)nancingbehaviorbroughtabout by technological evolution might help explain the subpar growth experienced in recent years, becausetheyhaveoccurredduringaperiodoflowinterestrates. Ourinsightscouldbeextended to develop interesting policy implications. On the one hand, the mechanisms described in this paper, operating mostly through the endogenous reaction of interest rates, suggest that the rise in intangibles might have important implications for monetary policy. On the other hand, the negative externality in households(cid:146)and (cid:133)rms(cid:146)excessive saving decisions might introduce a role for a (cid:133)scal policy that discourages such saving. 31

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1.8 1.8 1.6 1.6 ytiv ytiv itc itc u1.4 u1.4 d d o o rp rp ro b1.2 ro b1.2 a a l fo l fo .D .D .S 1 .S 1 0.8 0.8 1980 1985 1990 1995 2000 2005 2010 2015 1980 1985 1990 1995 2000 2005 2010 2015 Figure 1: Within-Industry Dispersion in Firm-Level Labor Productivity (Source: Compustat data, own calculations) 1.6 1.6 1.4 1.4 P F1.2 P F1.2 T T fo fo .D .S 1 .D .S 1 0.8 0.8 0.6 0.6 1980 1985 1990 1995 2000 2005 2010 2015 1980 1985 1990 1995 2000 2005 2010 2015 Figure 2: Within-Industry Dispersion in Firm-Level TFP (Source: Compustat data, own calculations) 1

7 6 5 A 4 r(%) 3 2 B 1 C 0 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 K Figure 3: Credit Market Equilibrium in the Simple Model of Section 3 2

Parameters that remain constant across comparative statics Parameter Symbol Value Capital share, high-productivity (cid:133)rms (cid:11) 0:4 Capital share, low-productivity (cid:133)rms, tangible capital (cid:31) 0:4 I Capital share, low-productivity (cid:133)rms, intangible capital (cid:31) 0:4 T u;T Low-productivity (cid:133)rms, TFP tangible technology z 10 t u;I Low-productivity (cid:133)rms, TFP tangible technology z 10 t Years households remain young N 40 High-productivity (cid:133)rms, TFP z 25 Collateral value of tangible capital (cid:18)T 1 Collateral value of intangible capital (cid:18)I 0:35 Probability of an investment opportunity (cid:17) 0:13 Additional productivity of intangible capital (cid:20) 0:1 Adjustment cost convexity ’ 9 Exit probability of high-productivity (cid:133)rms 0:13 Endowment of new (cid:133)rms W 5 0 Depreciation of capital (cid:14) 0:15 Share of dividends to young households (cid:13) 40% Parameters that change across comparative statics Value Parameter Symbol in 1980 in 2015 (cid:0) Discount factor (cid:12) 0:9425 0:9805 (cid:0) Intangible share of total capital (cid:22) 0:2 0:6 (cid:0) Probability of death of old households % 0:170 0:075 (cid:0) Adjustment cost parameter (intangible) b 3:2 10 6 15:3 10 6 I (cid:0) (cid:0) (cid:3) (cid:0) (cid:3) Adjustment cost parameter (tangible) b 2:1 10 10 0:6 10 10 T (cid:0) (cid:0) (cid:3) (cid:0) (cid:3) Table 1: Benchmark Calibration - Parameter Choices 3

80 8 200 60 6 100 % % % 40 4 0 20 2 0 0 100 100 30 4 3 50 20 % e g e g n n2 a a h h 0 c %10 c % 1 50 0 0 10 5 2 1 0 0 e e e g g g n n n 0 a a a h h h c % 10 c % 5 c % 1 20 10 2 =0.9425 > =0.9805 =0.9425 > =0.9805 =0.9425 > =0.9805 =0.170 > =0.075 =0.170 > =0.075 =0.170 > =0.075 Figure 4: Simulation Exercise: households(cid:146)propensity to save gradually increases because of (i) a decrease in the rate of time preference ((cid:12) increases) and (ii) a decrease in the likelihood of death of old households (% decreases) - comparison of the e⁄ects of the expansion in households savings in a tangibles economy ((cid:22) = 0:05) and an intangibles economy ((cid:22) = 0:65). 4

25 3 20 2.5 15 2 e e g g n n a10 a1.5 h h c c % % 5 1 0 0.5 5 0 =0.9425 > =0.9805 =0.9425 > =0.9805 =0.170 > =0.075 =0.170 > =0.075 Figure 5: Simulation Exercise: households(cid:146)propensity to save gradually increases because of (i) a decrease in the rate of time preference ((cid:12) increases) and (ii) a decrease in the likelihood of death of old households (% decreases) - comparison of capital misallocation and TFP dispersion in a tangibles economy ((cid:22) = 0:05) and an intangibles economy ((cid:22) = 0:65). 5

60 6 50 50 4 % % % 40 0 2 30 20 0 50 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 50 10 10 0 0 e 10 e 10 % g g 0 n a h 20 n a h 20 c c % % 30 30 40 40 50 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 30 20 10 0 20 0 e e e g g g n a n a 20 n a h h h c c c %10 % % 10 40 0 60 20 1980 1990 2000 2010 1980 1990 2000 2010 1980 1990 2000 2010 3 2 e g n a 1 h c % 0 1 1980 1990 2000 2010 Figure 6: Simulation Exercise: households(cid:146)propensity to save and the share of intangible capital both gradually increase - comparison of e⁄ects when both trends occur and when only the increase in the share of intangible capital occurs 6

3 2 1 0 1 e g n a 2 h c % 3 4 5 6 7 1980 1985 1990 1995 2000 2005 2010 2015 Figure7: SimulationExercise: households(cid:146)propensitytosaveandtheshareofintangiblecapital both gradually increase - comparison of e⁄ects on TFP when both trends occur, when only the increase in the share of intangible capital occurs, and in a counterfactual partial equilibrium scenario 2 1.5 1 % 0.5 0 0.5 1 1980 1985 1990 1995 2000 2005 2010 2015 Figure8: SimulationExercise: households(cid:146)propensitytosaveandtheshareofintangiblecapital both gradually increase - comparison of e⁄ects when we shut down the debt overhang/savings channel 7

APPENDICES For Online Publication A Robustness of Dispersion Evidence A.1 Construction of the Intangible Capital Measure We de(cid:133)ne intangible capital as the sum of knowledge capital and organizational capital.29 Following Falato et al. (2014), we measure the former by capitalizing R&D expenses and the latter by capitalizing selling, general and administrative (SG&A) expenses weighted by 0.2.30 The expenditures are capitalized by applying the perpetual inventory method with a depreciation rate of 15% for R&D and 20% for SG&A. In order to get a measure for tangible capital, we also use the perpetual inventory method to capitalize tangible capital expenses with a depreciation rate of 15%. We drop (cid:133)rms that are observed only once and (cid:133)rms that are not observed in a continuous time period, and we exclude regulated, (cid:133)nancial, and public service (cid:133)rms. We consider sectors at the 2-digit Standard Industrial Classi(cid:133)cation (SIC) level and drop those with less than 500 (cid:133)rm-year observations over the sample period. We measure output by sales, labor input by the number of employees, and total capital by the sum of capitalized tangible and intangible capital. TFP is de(cid:133)ned as the residual of a Cobb-Douglas production function with a capital share of income equal to 0.40. To control for outliers, we drop (cid:133)rms in the 1st and 99th percentiles of the distribution of labor productivity. A.2 Robustness Checks One alternative explanation of the results shown in Figures 1 and 2 in Section 2 could be that the sectors with a high intangibles share do not have a worse allocation of resources, but rather are more dynamic and fast growing, and that the increase in dispersion of productivity re(cid:135)ects this higher dynamism. However, in Figure A, we show that sectors with high average sales growth have lower productivity dispersion in the whole sample period. [FIGURE A ABOUT HERE] [TABLE A ABOUT HERE] Furthermore, Table A shows regression results where the dependent variable is a measure of productivity dispersion for each 2-digit sector-year observation. The regressors we consider are as follows: the dummy High share, which is equal to 1 if the sector belongs to the 50% 2-digit industries with the highest average intangible share and which is equal to 0 otherwise; a time trend; and year and sector (cid:133)xed e⁄ects. In columns 1 and 2, the dependent variable is the dispersion in TFP. Column 1 includes year (cid:133)xed e⁄ects and shows that the dispersion is signi(cid:133)cantly larger for sectors with higher intangible share. Column 2 includes a time trend, interactedwiththeHigh share variable,andbothsectorandtime(cid:133)xede⁄ects. Itshowsthatthe 29Falatoetal. (2014)alsoconsiderinformationalcapital. However,theystatethattheirresultsdonotdepend on its inclusion. As informational capital can be measured only at the industry level but not at the (cid:133)rm level using Compustat data, we choose not to include this type of capital. 30AportionofSG&Aexpensescapturesexpendituresthatincreasethevalueofintangiblecapitalitemssuchas brandnamesandknowledgecapital. PartofSG&Aexpenditures,however,doesnota⁄ectthevalueofintangible capital, so Falato et al. (2014) follow Corrado, Hulten, and Sichel (2009) and assume that the portion relevant to intangible capital is around 0.2. 37

trend in dispersion over time is signi(cid:133)cantly more positive in the 50% most intangible sectors than in the other sectors, con(cid:133)rming the signi(cid:133)cance of the result shown in Figures 1 and 2. Similar results, across the two groups of high and low intangibles sectors, are obtained using labor productivity, as shown in columns 3 and 4. B Derivation of the Simple Model This appendix solves a simple partial equilibrium model that delivers the equations introduced in Section 3. Consider an in(cid:133)nite-horizon, discrete-time model of the (cid:133)nal good-producing sector of an economy. Firms use capital, which is in constant aggregate supply K, to produce a homogeneous consumption good. There are two types of (cid:133)rms, high-productivity and lowproductivity,eachcomposedofacontinuumofmass1. High-productivity(cid:133)rmslivefor2periods, and there are overlapping generations of these types of (cid:133)rms. E¢ ciency in this economy is determinedbytheshareofK allocatedtohigh-productivity(cid:133)rms. Ourmaininterestisstudying how exogenous interest rate variations a⁄ect the allocation of capital and aggregate output depending on the degree of tangibility of capital. This framework is extended in Section 4 in a full-(cid:135)edged general equilibrium setup that can be used for realistic quantitative analysis. B.1 High-Productivity Firms Technology and Financing High-productivity(cid:133)rmslivefortwoperiods,whichwedenotewithy (young)ando(old). An old (cid:133)rm that dies in period t 1 leaves a a (cid:133)nancial endowment or liability ae ( < ae < ); (cid:0) (cid:0)1 1 which translates into net worth ae(1+r ) for the newborn young (cid:133)rm in period t. The young t (cid:133)rm is able to produce ye units of the (cid:133)nal good in period t; and has access to a technology to produce the (cid:133)nal good in period t+1 using the following linear production function: p y = z k ; (53) t+1 t+1 t+1 where k represents capital purchased in t that produces output in t + 1, and z is a t+1 t+1 productivity parameter. The (cid:133)rm can borrow b to purchase capital, subject to a constraint: t+1 (1+r )b (cid:18)q k ; (54) t+1 t+1 t+1 t+1 (cid:20) where 0 < (cid:18) 1. The collateral value of capital (cid:18) is the parameter in this stylized model that (cid:20) captures capital tangibility. A shift toward a stronger reliance on intangible capital will be captured as a decrease in (cid:18). 31 Firms cannot issue equity. The budget constraint for a high-productivity young (cid:133)rm is: q k = ae(1+r )+b +ye: (55) t t+1 t t+1 31Note that a standard collateral constraint of the form (cid:18)q k b t+1 t+1; t+1 (cid:20) 1+r t+1 would not work when W <0. This is because in that case the (cid:133)rm would have to be assumed to borrow more 0 than q per unit of capital: t (cid:18)q q < t+1 ; t 1+r t+1 tobeabletopurchasecapitalandpaydownthedebt,andbymakingthecollateralconstraintafunctionofk , t+1 it would make the (cid:133)rm (cid:133)nancially unconstrained, and the problem would not have a solution. 38

Amature(cid:133)rmrealizesoutput, paysbackanydebts, sellsitsholdingsofcapital, andpaysthe residual, netoftheendowmentforthenextgenerationae;asadividendd toitsshareholders: t+1 d +ae = y p +q k b (1+r ): t+1 t+1 t+1 t+1 (cid:0) t+1 t+1 Optimal Solution Productive young (cid:133)rms in t = 0 maximize the present value of the dividend d . We claim, t+1 and later verify, that their marginal product of capital is greater than its user cost: q t+1 z > q (56) t t (cid:0) 1+r t+1 (cid:133)rms are credit constrained ((54) is binding), so that ae(1+r )+ye t k = : (57) t+1 q (cid:18) qt+1 t (cid:0) 1+rt+1 in which investment is equal to the total wealth available to invest divided by the downpayment necessary to purchase one unit of capital. B.2 Low-Productivity Firms There is a mass 1 of identical (cid:133)rms in the unproductive sector that produce the same homogeneous (cid:133)nal good as the high-productivity (cid:133)rms using a linear production function: yu = zuk ; (58) t t u;t where k represents capital installed in period t 1 that produces output in period t, and zu u;t t (cid:0) is a productivity parameter. This sector is assumed to be (cid:133)nancially unconstrained, and to pay out all pro(cid:133)ts as dividends: du = yu q ku ku (59) t t t t+1 t (cid:0) (cid:0) to their shareholders every period. (cid:0) (cid:1) B.3 Aggregation From (57) it follows that the aggregate stock of capital held by the high-productivity (cid:133)rms is Ae(1+r )+Ye t K = : t+1 q (cid:18) qt+1 t (cid:0) 1+rt+1 and aggregate output is Y = Y p +Yu+Ye = z K +zu K K : t t t t t t t (cid:0) (cid:0) (cid:1) Under the assumption that the high-productivity (cid:133)rms have the highest return on capital (z > zu), but their resources are insu¢ cient to absorb all the capital, K < K , it follows t t t+1 that the low-productivity (cid:133)rms are willing to absorb all the capital not demanded by the highproductivity (cid:133)rms at a price equal to their marginal return on capital. the price of capital is: 1 q = zu+ q ; (60) t t+1 1+r +(cid:24) t+1 which, together with the assumption that z > zu; proves the claim (56). (cid:24) 0 is a wedge t t (cid:21) 39

that reduces the sensitivity of the price of capital to the interest rate, and summarizes the e⁄ect of factors included in the full model developed later, such as decreasing marginal return to capital in the unproductive sector. B.4 Steady State We consider a steady state equilibrium and drop reference to the time subscript t. Total output is Y = Yp+Yu+Ye = zK +zu K K : (61) (cid:0) Aggregate capital holdings of the high-productivity (cid:133)r(cid:0)ms in th(cid:1)e steady state are: Ae(1+r)+Ye K = ; (62) q 1 (cid:18) (cid:0) 1+r (cid:16) (cid:17) where zu q = : (63) r+(cid:24) Equations (1), (2), and (3) in the steady state equilibrium of Section 3 correspond to equations (61), (62) and (63) in this section. C Optimal Dividend and Cash Accumulation Policy Given equation (13), the (cid:133)rst order condition for cash holdings a for non investing (cid:133)rms is: f;t+1 (1+(cid:21) ) = (1 ) (cid:17)(1+(cid:21)+ +# )+(1 (cid:17))(1+(cid:21) +# ) + : (64) t (cid:0) t+1 t (cid:0) (cid:0)t+1 t (cid:2) (cid:3) If we substitute (64) recursively forward, it is clear that if the (cid:133)rm expects # to be positive t now or in the future, then (cid:21) > 0; and a non-investing (cid:133)rm will always retain all earnings and t d = 0. It is important to note that this is so because there is no cost of holding cash. t D Households Wederivebelowthesolutionofhouseholds(cid:146)optimizationproblemunderthesteadystate,sothat wages, dividends and interest rate are constant. Households have log utility. A representative old household still living at time t maximizes the following objective function: 1 Vo(bo) = max (1 %)j(cid:12)jlog(c ) (65) t t t+j co;bo (cid:0) t t+1j=0 X subject to (1+r) co = bo +(1 (cid:13))d bo. t t+1 (cid:0) (cid:0) (1 %) t (cid:0) Working backward, we next consider the optimization problem of a young agent of age N in period t; who will become old in period t+132: V y b y = max u c y +(cid:12)(1 %)Vo bo (66) t;N t;N cy ;bo t;N (cid:0) t+1 t+1 (cid:16) (cid:17) t;N t+1 (cid:16) (cid:17) (cid:0) (cid:1) 32We assume that an agent can also die with probability % in the transition between young and old. 40

subject to c y = (cid:13)d+wTOT (1+r)b y +bo : (67) t;N (cid:0) t;N t+1 where wTOT is de(cid:133)ned as: wTOT w+wuI +wuT (cid:17) Then we consider the optimization problem for a young household of age j < N : y y y y y V b = max u c +(cid:12)V b (68) t;j t;j cy ;by t;j t+1;j+1 t+1;j+1 t;j t+1;j+1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) subject to c y = (cid:13)d+wTOT (1+r)b y +b y (69) t;j (cid:0) t;j t+1;j+1 D.1 Individual Problem of Old Households We follow Blanchard (1985) and Yaari (1965) in assuming that households participate in a life insurance scheme when old. The insurance scheme works within a cohort so that the survivors within a cohort pay the debt of the dying (if they are in debt) or, alternatively, receive the savings of the dying. An old household begins a period with net debt (1+r )bo. The insurance t t contract speci(cid:133)es that the % fraction of old households that die transfer their assets (or debt) (1+r )bo to the life insurer. Among the fraction (1 %) of households that survive, if they are t t (cid:0) net savers (bo < 0), then they receive a return 1 (1+r )bo on their assets, while, if they are t 1 % t t net debtors (bo > 0), they make a payment of 1(cid:0) (1+r )bo to the life insurer. t 1 % t t The (cid:133)rst order condition with respect to bo (cid:0) is t+1 co = (cid:12)(1+r)co: (70) t+1 t We guess a consumption policy rule: co = (cid:1)d+(cid:2)bo; t t and plug it into the FOC (1+r) (cid:1)d+(cid:2) co (1 (cid:13))d+ bo = (cid:12)(1+r)((cid:1)d+(cid:2)bo) t (cid:0) (cid:0) (1 %) t t (cid:20) (cid:0) (cid:21) (cid:12)(1+r)(cid:1) (cid:1) 1 co = +(1 (cid:13)) d+(1+r) (cid:12) bo; t (cid:2) (cid:0) (cid:0) (cid:2) (cid:0) (1 %) t (cid:20) (cid:21) (cid:20) (cid:0) (cid:21) and then solve for the unknown coe¢ cients (cid:12)(1+r)(cid:1) (cid:1) (cid:1) = +(1 (cid:13)) (cid:2) (cid:0) (cid:0) (cid:2) 1 (cid:2) = (1+r) (cid:12) (cid:0) (1 %) (cid:20) (cid:0) (cid:21) (cid:12)(cid:1) (cid:1) (cid:1) = +(1 (cid:13)) (cid:12) 1 (cid:0) (cid:0) (1+r) (cid:12) 1 (cid:0) (1 %) (cid:0) (1 %) (cid:0) (cid:0) (h1 (cid:13))(1+i r)[1 (cid:12)(1 %)] h i (cid:1) = (cid:0) (cid:0) (cid:0) %+r 41

The policy rule is: (1 (cid:13))(1+r) (1+r) co = (1 (1 %)(cid:12)) (cid:0) d bo : (71) t (cid:0) (cid:0) (%+r) (cid:0) (1 %) t (cid:20) (cid:0) (cid:21) and the evolution of the wealth of old households is given by (1 %)[1 (cid:12)(1+r)] bo = (cid:0) (cid:0) (1 (cid:13))d+(1+r)(cid:12)bo; (72) t+1 (%+r) (cid:0) t which says that old households slowly consume their savings if (cid:12)(1+r) < 1, and do so at a faster rate the higher the dividends. In our simulations typically (cid:12)(1+r) < 1.33 D.2 Individual Problem of Young Households We (cid:133)rst consider the optimization problem of an agent of age N in period t; who will become old in period t+1: V y b y = max u c y +(cid:12)(1 %)Vo bo (73) t;N t;N cy ;bo t;N (cid:0) t+1 t+1 (cid:16) (cid:17) t;N t+1 (cid:16) (cid:17) (cid:0) (cid:1) such that: c y = (cid:13)d+wTOT (1+r)b y +bo : (74) t;N (cid:0) t;N t+1 The (cid:133)rst order condition implies that 1 @Vo bo @Vo bo @bo +(cid:12) t+1 t+1 + t+1 t+1 t+2 = 0: c y @bo @bo @bo t;N t(cid:0)+1 (cid:1) t(cid:0)+2 (cid:1) t+1! And applying the envelope theorem we obtain: co = (cid:12)(1+r)c y : t+1 t;N We substitute co using (71) and we obtain: t+1 1 (1 (cid:13))d bo c y = (1 %) (cid:0) t+1 : (75) t;N (cid:12) (cid:0) (cid:0) (%+r) (cid:0) (1 %) (cid:18) (cid:19)(cid:18) (cid:0) (cid:19) Then we consider the optimization problem for a young household of age j<N : y y y y y V b = max u c +(cid:12)V b ; (76) t;j t;j cy ;by t;j t+1;j+1 t+1;j+1 t;j t+1;j+1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) such that: c y = (cid:13)d+wTOT (1+r)b y +b y ; (77) t;j (cid:0) t;j t+1;j+1 which yields the standard Euler equation: c y = [(cid:12)(1+r)] (N j)c y : (78) t;j (cid:0) (cid:0) t+N j;N (cid:0) Equations (75) and (78) fully characterize the life-cycle path of consumption of a household 33To see this more clearly, denote a= b as savings, and write (cid:0) (1 %)[1 (cid:12)(1+r)] ao t+1 =(1+r)(cid:12)ao t (cid:0) (cid:0) (%+ (cid:0) r) (1 (cid:0) (cid:13))d: 42

as a function of its assets when entering old age in period t+1; bo : t+1 D.3 Value of Savings of Oldest Young: bo t+1 We use the above equations, the budget constraint (69), and the assumption that newborn y households have no endowment (b = 0) to determine the value of savings for retirement t;1 bo b y : t+1 (cid:17) t+1;N+1 We use the budget constraint for j = 1 (a young household of age = 1), in which the debt y brought over, b , is zero: t;1 (cid:13)d+wTOT c y +b y b y = (cid:0) t;1 t+1;2 = 0; t;1 (1+r) (cid:0) (cid:1) and we solve forward: (cid:13)d+wTOT c y (cid:13)d+wTOT c y +b y b y = (cid:0) t;1 + (cid:0) t+1;2 t+2;3 t;1 (1+r) (1+r)2 (cid:0) (cid:1) (cid:0) (cid:1) N 1 N c y b y = (cid:13)d+wTOT t+j (cid:0) 1;j + t+N;N+1 (1+r)j (cid:0) (1+r)j (1+r)N j=1 j=1 (cid:0) (cid:1)X X Making use of the FOC: c y = [(cid:12)(1+r)] (N j)c y (79) t;j (cid:0) (cid:0) t+N j;N (cid:0) we get c y = [(cid:12)(1+r)] (N j)c y t;j (cid:0) (cid:0) t+N j;N (cid:0) c y = [(cid:12)(1+r)] (N 1)c y t;1 (cid:0) (cid:0) t+N 1;N (cid:0) c y = [(cid:12)(1+r)] (N 2)c y = [(cid:12)(1+r)] (N 2)c y t+1;2 (cid:0) (cid:0) t+1+N 2;N (cid:0) (cid:0) t+N 1;N (cid:0) (cid:0) c y = [(cid:12)(1+r)] (N 3)c y = [(cid:12)(1+r)] (N 3)c y t+2;3 (cid:0) (cid:0) t+2+N 3;N (cid:0) (cid:0) t+N 1;N (cid:0) (cid:0) c y = [(cid:12)(1+r)] (N N)c y = c y ; t+N 1;N (cid:0) (cid:0) t+N 1+N N;N t+N 1;N (cid:0) (cid:0) (cid:0) (cid:0) and plug in and simplify N c y c y c y c y c y t+j 1;j t;1 t+1;2 t+2;3 t+N 1;N (cid:0) = + + +:::+ (cid:0) (1+r)j (1+r) (1+r)2 (1+r)3 (1+r)N j=1 X = c ( y t 1 + + N (cid:0) r 1 ) ; N N 2 N (cid:0) 1 (cid:12) (cid:0) (N (cid:0) 1 (cid:0) j) 3 = (cid:0) c ( y t 1 + + N (cid:0) r 1 ) ; N N (cid:12)N (cid:12)N 1( (cid:0) (cid:12) 1 1) : X j=0 (cid:0) (cid:0) 4 5 We substitute back in and keep simplifying 1 1 c y (cid:12)N 1 b y b y t;1 = (cid:13)d+wTOT r " 1 (cid:0) (1+r)N # (cid:0) ( t 1 + + N (cid:0) r 1 ) ; N N (cid:12)N (cid:0) 1( (cid:0) (cid:12) 1) + ( t 1 + + N;N r) + N 1 (cid:0) (cid:0) (cid:1) 43

1 1 (cid:12)N 1 (1 (cid:13))d 0 = (cid:13)d+wTOT (1+r)N 1 (1 %) (cid:0) (cid:0) r (cid:0) (cid:0) (cid:12) (cid:0) (cid:0) (cid:12)N 1((cid:12) 1) (%+r) h i (cid:18) (cid:19) (cid:0) (cid:0) (cid:0) 1 (cid:1) 1 (cid:12)N 1 y + (1 %) (cid:0) +1 b (cid:12) (cid:0) (cid:0) (1 %)(cid:12)N 1((cid:12) 1) t+N;N+1 (cid:20)(cid:18) (cid:19) (cid:0) (cid:0) (cid:0) (cid:21) 1 (cid:12)N 1 (1 (cid:13))d 1 (1 %) (cid:0) (cid:0) (cid:13)d+wTOT (1+r)N 1 (cid:12) (cid:0) (cid:0) (cid:12)N 1((cid:12) 1) (%+r) (cid:0) r (cid:0) (cid:18) (cid:19) (cid:0) (cid:0) h i 1 1 (cid:12)N(cid:0) 1 (cid:1) y = (1 %) (cid:0) +1 b (cid:12) (cid:0) (cid:0) (1 %)(cid:12)N 1((cid:12) 1) t+N;N+1 (cid:20)(cid:18) (cid:19) (cid:0) (cid:0) (cid:0) (cid:21) Finally, we solve to get: (cid:13)d+wTOT 1 (1+r)N 1 (cid:9) (1 (cid:0) (cid:13))d y r (cid:0) (cid:0) (%+r) b = (80) (cid:0) t+N;N+1 (cid:0) (cid:1) h (cid:9) +1 i 1 % (cid:0) 1 (cid:12)N 1 (cid:9) (1 %) (cid:0) (cid:17) (cid:12) (cid:0) (cid:0) (cid:12)N 1((cid:12) 1) (cid:18) (cid:19) (cid:0) (cid:0) Equation (80) is very intuitive. Savings for retirement bo increase in the di⁄erence (cid:0) t+1 between income before and after retirement. Moreover, an increase in life expectancy (a drop in %) reduces the value of the term (cid:9) and therefore increases bo : (cid:0) t+1 D.4 Aggregate Savings of the Young y y The previous section determines a sequence of optimal consumption at every age, c ;:::;c ; t;1 t;N and applying the budget constraint (69) we can determine a sequence of assets for every age y y b ;:::;b ; which is constant for every period t. In equilibrium there is a measure 1 of houset;2 t;N holds, a fraction (cid:30) old, and a fraction % young. Moreover there is a measure % 1 of (cid:30)+% (cid:30)+% (cid:30)+%N young households for each age. Therefore, after dropping the subscript t; we can de(cid:133)ne aggregate savings of the young households as: N % 1 By = b y : (81) (cid:30)+%N j+1 j=1 X Savings of a young household are: b y = c y (cid:13)d wTOT +(1+r)b y (82) j+1 j (cid:0) (cid:0) j y We solve for b (from now on for simplicity omit the superscript y) : N 1 1 1 b = (cid:13)d+wTOT c + b ; (83) N N N+1 1+r (cid:0) 1+r 1+r (cid:0) (cid:1) where both b and c are determined by (75) and (80) above. At age N 1 (we use N+1 N c = [(cid:12)(1+r)] 1c ): (cid:0) t (cid:0) t+1 1 1 1 1 1 1 b = (cid:13)d+wTOT c + (cid:13)d+wTOT c + b N 1 N 1 N N+1 (cid:0) 1+r (cid:0) 1+r (cid:0) 1+r 1+r (cid:0) 1+r 1+r (cid:20) (cid:21) (cid:0) (cid:1) (cid:0) (cid:1) 44

1 1 b = (cid:13)d+wTOT [(cid:12)(1+r)] 1c N 1 (cid:0) N (cid:0) 1+r (cid:0) 1+r 1 (cid:0) 1 (cid:1) 1 1 + (cid:13)d+wTOT c + b N N+1 1+r 1+r (cid:0) 1+r 1+r (cid:20) (cid:21) 1 (cid:0) 1(cid:1) = (cid:13)d+wTOT + (cid:13)d+wTOT 1+r (1+r)2 1 (cid:0) (cid:1) 1(cid:0) (cid:1)1 [(cid:12)(1+r)] 1c c + b (cid:0)1+r (cid:0) N (cid:0) (1+r)2 N (1+r)2 N+1 1 1 1 c 1 = + (cid:13)d+wTOT +1 N + b ; 1+r (1+r)2 (cid:0) (cid:12) (1+r)2 (1+r)2 N+1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) and therefore at age N 2 : (cid:0) 1 1 1 b = (cid:13)d+wTOT c + b N 2 N 2 N 1 (cid:0) 1+r (cid:0) 1+r (cid:0) 1+r (cid:0) = 1 (cid:0) (cid:13)d+wTOT (cid:1) 1 [(cid:12)(1+r)] 1[(cid:12)(1+r)] 1c (cid:0) (cid:0) N 1+r (cid:0) 1+r 1 (cid:0) 1 (cid:1) 1 1 1 1 + + (cid:13)d+wTOT + c + b 1+r 1+r (1+r)2 (cid:0) (cid:12)(1+r)2 (1+r)2 N (1+r)2 N+1 (cid:18)(cid:18) (cid:19) (cid:18) (cid:19) (cid:19) 1 1 1 (cid:0) (cid:1) 1 1 c 1 = + + (cid:13)d+wTOT + +1 N + b ; 1+r (1+r)2 (1+r)3 (cid:0) (cid:12)2 (cid:12) (1+r)3 (1+r)3 N+1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) and at a generic age N t : (cid:0) t (cid:13)d+wTOT c t 1 b N N+1 b = + (84) N (cid:0) t (1+r)j+1 (cid:0) (1+r)t+1 (cid:12)j (1+r)t+1 j=0 j=0 X X t t 1 We use general formulas: xj = 1 and xj = 1 xt+1 1 , or 1 = 1 x (cid:0) 1 x (1+r)j (cid:0) (cid:0) j=0 j=0 j=0 X X (cid:0) (cid:1) X t 1 1 1+r and 1 = 1 1 1 , so that (cid:0) (1+r)t+1 r (1+r)j+1 (cid:0) (1+r)t+1 r (cid:16) (cid:17) X j=0 (cid:16) (cid:17) t (cid:13)d+wTOT 1 (cid:13)d+wTOT = 1 (1+r)j+1 (cid:0) (1+r)t+1 r j=0 (cid:18) (cid:19) X t c 1 c 1 (cid:12) N N = 1 (1+r)t+1 (cid:12)j (1+r)t+1 (cid:0) (cid:12)t+1 (cid:12) 1 j=0 (cid:18) (cid:19) (cid:0) X hence: 1 (cid:13)d+wTOT c 1 (cid:12) b N N+1 b = 1 1 + (85) N (cid:0) t (cid:0) (1+r)t+1 r (cid:0) (1+r)t+1 (cid:0) (cid:12)t+1 (cid:12) 1 (1+r)t+1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:0) 45

Now we add up the savings/borrowing over all ages from b to b to get: 2 N N (cid:0) 2 (cid:13)d+wTOT N (cid:0) 2 1 (cid:12) N (cid:0) 2 1 1 b = 1 c 1 N (cid:0) t r (cid:0) (1+r)t+1 (cid:0) (cid:12) 1 N (1+r)t+1 (cid:0) (cid:12)t+1 t=0 t=0 (cid:18) (cid:19) (cid:0) t=0 (cid:18) (cid:18) (cid:19)(cid:19) X X X N 2 (cid:0) 1 +b N+1 (1+r)t+1 t=0 (cid:18) (cid:19) X The value of each summation term is: N 2 (cid:0) 1 1 = 1 r (r+1) N+1 Nr+1 (cid:0) (1+r)t+1 (cid:0)r (cid:0) (cid:0) (cid:0) X t=0 (cid:18) (cid:19) (cid:16) (cid:17) 1 1 = 1+r(1 N) (cid:0)r (cid:0) (cid:0) (1+r)N (cid:0) 1 ! 1 1 = +r(N 1) 1 r (1+r)N (cid:0) 1 (cid:0) (cid:0) ! N 2 N 2 N 2 (cid:0) 1 1 (cid:0) 1 (cid:0) 1 1 = (1+r)t+1 (cid:0) (cid:12)t+1 (1+r)t+1 (cid:0) [(1+r)(cid:12)]t+1 t=0 (cid:18) (cid:18) (cid:19)(cid:19) t=0 t=0 (cid:18) (cid:19) X X X 1 (1+r) N+1 1 [(1+r)(cid:12)] N+1 (cid:0) (cid:0) = (cid:0) (cid:0) (1+r) 1 (cid:0) [(1+r)(cid:12)] 1 (cid:0) (cid:0) N (cid:0) 2 1 1 (1+r) (cid:0) N+1 = (cid:0) (1+r)t+1 (1+r) 1 t=0 (cid:18) (cid:19) (cid:0) X Substituting them back: N (cid:0) 2 (cid:13)d+wTOT 1 1 b = +r(N 1) 1 t=0 N (cid:0) t r r (1+r)N (cid:0) 1 (cid:0) (cid:0) ! X (cid:12) 1 (1+r) N+1 1 [(1+r)(cid:12)] N+1 1 (1+r) N+1 (cid:0) (cid:0) (cid:0) c (cid:0) (cid:0) +b (cid:0) N N+1 (cid:0)(cid:12) 1 (1+r) 1 (cid:0) (1+r)(cid:12) 1 (1+r) 1 " # (cid:0) (cid:0) (cid:0) (cid:0) We rename terms: N (cid:0) 2 (cid:13)d+wTOT b = A A c +A b N t 1 2 N 3 N+1 (cid:0) r (cid:0) t=0 X 1 1 A +r(N 1) 1 1 (cid:17) r (1+r)N (cid:0) 1 (cid:0) (cid:0) ! (cid:12) 1 (1+r) N+1 1 [(1+r)(cid:12)] N+1 (cid:0) (cid:0) A (cid:0) (cid:0) 2 (cid:17) (cid:12) 1 (1+r) 1 (cid:0) (1+r)(cid:12) 1 " # (cid:0) (cid:0) (cid:0) 1 (1+r) N+1 (cid:0) A (cid:0) 3 (cid:17) (1+r) 1 (cid:0) 46

D.5 Aggregate Savings of the Old In equilibrium there are % 1 households that become old every period, and % 1(1 %)j (cid:30)+%N (cid:30)+%N (cid:0) households that survived for j periods . Therefore aggregate savings of the old households are: % 1 1 Bo = (1 %)jb0 (86a) (cid:30)+%N (cid:0) j j=1 X also note that b0 is the initial savings from young age as de(cid:133)ned in (80) 1 Recall that (from (72)): bo = A+(cid:12)(1+r)bo; (87) j+1 j and (1 %)[1 (cid:12)(1+r)] A (cid:0) (cid:0) (1 (cid:13))d; (cid:17) (%+r) (cid:0) so that: bo = A+(cid:12)(1+r)bo; (88) 2 1 bo = A+(cid:12)(1+r)bo = A+(cid:12)(1+r)A+(cid:12)2(1+r)2bo; (89) 3 2 1 bo = A+(cid:12)(1+r)A+:::+(cid:12)t 2(1+r)t 2A+(cid:12)t 1(1+r)t 1bo; (90) t (cid:0) (cid:0) (cid:0) (cid:0) 1 t 2 (cid:0) bo = A (cid:12)j(1+r)j +(cid:12)t 1(1+r)t 1bo; (91) t (cid:0) (cid:0) 1 2 3 j=0 X 4 5 ((cid:12)(1+r))t 1 1 bo t = (cid:12)(1+r) (cid:0) 1 (cid:0) A+[(cid:12)(1+r)]t (cid:0) 1bo 1 ; (92) (cid:0) 1 (1 (cid:0) %)tb0 t = 1 (1 (cid:0) %)t ((cid:12) (cid:12) (r (1 + + 1) r ) ) t (cid:0) 1 1 (cid:0) 1 A+[(cid:12)(1+r)]t (cid:0) 1bo 1 " # t=1 t=1 (cid:0) X X = A 1 (1 %)t ((cid:12)(r+1))t 1 1 +bo 1 (1 %)t[(cid:12)(1+r)]t 1 (cid:12)(1+r) 1 (cid:0) (cid:0) (cid:0) 1 (cid:0) (cid:0) (cid:0) X t=1 h i X t=1 = A +bo 1 (1 %)t((cid:12)(r+1))t 1 A 1 (1 %)t (cid:12)+r(cid:12) 1 1 (cid:0) (cid:0) (cid:0) (cid:12)+r(cid:12) 1 (cid:0) (cid:20) (cid:0) (cid:21)t=1 (cid:0) t=1 X X = A +bo 1 1 [(1 %)(cid:12)(1+r)]t A 1 (1 %)t; (cid:12)+r(cid:12) 1 1 (cid:12)(1+r) (cid:0) (cid:0) (cid:12) +r(cid:12) 1 (cid:0) (cid:20) (cid:0) (cid:21) t=1 (cid:0) t=1 X X 1 [(1 %)(cid:12)(1+r)]t = 1 1 = (1 (cid:0) %)(cid:12)(1+r) ; (cid:0) 1 (1 %)(cid:12)(1+r) (cid:0) 1 (1 %)(cid:12)(1+r) t=1 (cid:0) (cid:0) (cid:0) (cid:0) X and 1 1 1 % (1 %)t = 1 = (cid:0) : (cid:0) 1 1+% (cid:0) % t=1 (cid:0) X Finally: 1 A (1 %) A 1 % (1 %)tb0 = +bo (cid:0) (cid:0) (cid:0) t (cid:12) +r(cid:12) 1 1 1 (1 %)(cid:12)(1+r) (cid:0) (cid:12)+r(cid:12) 1 % t=1 (cid:20) (cid:0) (cid:21) (cid:0) (cid:0) (cid:0) X 47

D.6 Summing Up Aggregate Household Borrowing Aggregate household borrowing is: B = Bo+By (93) where savings of the old is: % 1 A (1 %) A 1 % Bo = +bretirement (cid:0) (cid:0) (cid:30)+%N (cid:12)+r(cid:12) 1 1 (1 %)(cid:12)(1+r) (cid:0) (cid:12)+r(cid:12) 1 % (cid:20)(cid:18) (cid:0) (cid:19) (cid:0) (cid:0) (cid:0) (cid:21) and (1 %)(1 (1+r)(cid:12)) A (cid:0) (cid:0) (1 (cid:13))d; (cid:17) (%+r) (cid:0) (cid:18) (cid:19) (cid:9) (1 (cid:0) (cid:13))d (cid:13)d+wTOT 1 (1+r)N 1 bretirement (%+r) (cid:0) r (cid:0) ; (cid:17) (cid:0) (cid:9) +(cid:1)1 h i 1 % (cid:0) and 1 (cid:12)N 1 (cid:9) (1 %) (cid:0) : (cid:17) (cid:12) (cid:0) (cid:0) (cid:12)N 1((cid:12) 1) (cid:18) (cid:19) (cid:0) (cid:0) And savings of the young is: % 1 N (cid:0) 2 % 1 (cid:13)d+wTOT By = b = A A c +A bretirement ; N t 1 2 N 3 (cid:30)+%N (cid:0) (cid:30)+%N r (cid:0) t=0 (cid:20) (cid:21) X where 1 1 A +r(N 1) 1 ; 1 (cid:17) r (1+r)N (cid:0) 1 (cid:0) (cid:0) ! (cid:12) 1 (1+r) N+1 1 [(1+r)(cid:12)] N+1 (cid:0) (cid:0) A (cid:0) (cid:0) ; 2 (cid:17) (cid:12) 1 (1+r) 1 (cid:0) (1+r)(cid:12) 1 " # (cid:0) (cid:0) (cid:0) 1 (1+r) N+1 (cid:0) A (cid:0) ; 3 (cid:17) (1+r) 1 (cid:0) and 1 (1 (cid:13))d bretirement c = (1 %) (cid:0) : N (cid:12) (cid:0) (cid:0) (%+r) (cid:0) (1 %) (cid:18) (cid:19)(cid:18) (cid:0) (cid:19) 48

For Online Publication (Appendix Tables and Figures) 1

1.6 1.4 1.5 1.3 ytiv1.4 itc u d o1.3 P F 1.2 rp ro b1.2 T fo .D a l fo .S 1.1 .D1.1 .S 1 1 0.9 0.9 1980 1985 1990 1995 2000 2005 2010 2015 1980 1985 1990 1995 2000 2005 2010 2015 Figure A: Within-Industry Dispersion in Firm-Level Labor Productivity and TFP: Robustness Exercise that Groups Sectors According to Average Sales Growth Rates (Source: Compustat data, own calculations) (1) (2) (3) (4) Variables TFP TFP y y Time trend 0:000892 0:00316 (cid:0) (cid:3)(cid:3)(cid:3) (0:000837) (0:000737) Time trend*High share 0:00558 0:00370 (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (0:000622) (0:000548) High share 0:0897 0:110 (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (0:00990) (0:0144) Observations 828 828 828 828 R-squared 0:112 0:632 0:134 0:869 Industry FE no yes no yes Year FE yes yes yes yes Table A: Relationship Between the Intangible Share and the Dispersion in Productivity - Regression Analysis 2