Reallocation of Intangible Capital and Secular Stagnation

Reallocation of Intangible Capital and Secular Stagnation Ander Perez-Orive Federal Reserve Board (joint with Andrea Caggese - Pompeu Fabra & CREI) Workshop on Finance, Investment and Productivity BoE, BHC, CEPR and CFM London, 15-16 December 2016 Disclaimer: The views expressed here are of the authors, and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System, or of anyone else associated with the Federal Reserve System. 1 / 47

MOTIVATION AND QUESTIONS Several trends: 1 Secular stagnation (Summers (2015), Eichengreen (2015)) Decrease in real interest rates Economic growth short of previous trends 2 / 47

DECLINING REAL INTEREST RATE FIGURE: Long-term Nominal Interest Rates and 2-year ahead Inflation Expectations (Source: Federal Reserve Bank of St. Louis) 3 / 47

LOW GROWTH 22 21 Actual and Potential GDP Year Estimated: 2007 2008 Trillions of 2013 Dollars 20 19 18 17 Potential GDP Estimates Actual 2009 2010 2011 2012 2013 2014 16 15 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 FIGURE: Actual and Potential GDP in the United States (Source: Congressional Budget Offi ce, Bureau of Economic Analysis, and Summers (2014)) 4 / 47

RISE IN INTANGIBLES (a) Intangible to Net Total Asset 1 0.24 (b) Cash to Total Asset 0.22 (c) Net Leverage 0.9 0.22 0.2 0.8 0.2 0.18 0.7 0.18 0.16 0.6 0.16 0.14 0.5 0.14 0.12 0.4 0.12 0.1 0.3 0.1 0.08 0.2 1970 1980 1990 2000 2010 0.08 1970 1980 1990 2000 2010 0.06 1970 1980 1990 2000 2010 FIGURE: Rise in intangible intensity reduction in net leverage in U.S. non-financial listed firms (Source: Falato, Kadyrzhanova and Sim (2014)) 5 / 47

MOTIVATION AND QUESTIONS Several trends: 1 Secular stagnation (Summers (2015), Eichengreen (2015)) Decrease in real interest rates Economic growth short of previous trends 2 Rise in intangibles (Corrado and Hulten (2010)) Stronger importance of knowledge, human and organizational capital, and reduced reliance on physical capital 3 Decrease in corporate net borrowing (Armenter and Hnatkovska (2016), Quadrini (2016), Chen, Karabarbounis and Neiman (2016), Zetlin-Jones and Shourideh (2016)) 2 / 47

NET FINANCIAL POSITION OF THE US CORPORATE SECTOR 25% 15% Corporate Net financial assets (In percent of nonfinancial assets) 5% 5% 1980 1985 1990 1995 2000 2005 2010 15% 25% 35% FIGURE: Net financial asset position of US nonfinancial corporations (Source: Quadrini (2016)) 6 / 47

MISALLOCATION AND INTANGIBLES INTENSITY FIGURE: Mean labor productivity dispersion in low intangible vs high intangible industries (U.S. Compustat firms). 7 / 47

MISALLOCATION AND INTANGIBLES INTENSITY FIGURE: Mean TFP dispersion in low intangible vs high intangible industries (U.S. Compustat firms). 8 / 47

MOTIVATION AND QUESTIONS Several trends: 1 Secular stagnation: low real interest rates, low growth 2 Decrease in corporate net borrowing 3 Rise in intangibles 4 Increase in productivity dispersion in intangibles industries relative to tangibles industries Existing literature Rise in intangibles corporate net borrowing (Falato, et al. (2014)) Rise in intangibles low interest rates (Döttling and Perotti (2016)) Low rates low growth in presence of ZLB (Eggertsson and Mehrotra (2014)) Decrease in relative price of capital low rates (Thwaites (2015)) 9 / 47

MAIN INSIGHTS Low interest rates can hurt capital reallocation, and as a result aggregate productivity and output, in economies that rely strongly on intangible technologies 10 / 47

MAIN INSIGHTS Low interest rates can hurt capital reallocation, and as a result aggregate productivity and output, in economies that rely strongly on intangible technologies Mechanism: Intangible capital significantly less collateralizable than tangible capital: financed mostly with retained earnings Low interest rates cause (i) high price of intangible assets, and (ii) slow accumulation of savings Reduced ability of credit constrained expanding productive firms to purchase capital from exiting or unproductive firms: increased misallocation 10 / 47

OUTLINE OF TALK 1 Simple Analytical Intuition 2 Model 3 Simulation of Rise in Intangibles and Increase in Household Net Savings (U.S. 1970s-present) 11 / 47

SIMPLE ANALYTICAL INTUITION Consider infinite-horizon, discrete-time economy in which: final good firms use capital K (constant aggregate supply K ) 2 types of firms: high-productivity and low-productivity effi ciency: share of K allocated to high-productivity firms Aggregate K holdings of high-productivity (credit constrained) firms in SS: K = A e (1 + r) + Y ( ) e where q = z u q 1 1+r θ r + ξ. Question 1: how does r affect the allocation of K? 1) savings/debt overhang channel, 2) capital purchase price channel, 3) collateral value channel. Question 2: how does θ (and A e ) affect the response of K to r? 12 / 47

GRAPHICAL INTUITION 7 6 5 Credit Market Equilibrium Investment Curve ( =0.4, A e =0.2) Investment Curve for ( =0.9, A e -0.2) Savings Curve Savings Curve after Savings Increase 4 r(%) A 3 2 C 1 B 0 0.9 0.95 1 1.05 1.1 K Demand for capital becomes upward sloping in interest rate with high intangibles reliance 13 / 47

OUTLINE OF TALK 1 Simple Analytical Intuition 2 Model 3 Simulation of Rise in Intangibles and Increase in Household Net Savings (U.S. 1970s-present) 14 / 47

MODEL Infinite-horizon, discrete-time economy Agents Final good producers use labor and tangible and intangible capital to produce consumption goods 2 types: high-productivity and low-productivity Capital producers Households provide labor and own firms No aggregate uncertainty: comparison of SS under different calibrations 15 / 47

HIGH-PRODUCTIVITY FIRMS Produce consumption goods according to [ ( yt p = z t n (1 α) kt t min,t 1 µ, k )] α I,t, µ where µ = k I,t k I,t +k T,t captures optimal intangible capital ratio 16 / 47

HIGH-PRODUCTIVITY FIRMS Produce consumption goods according to [ ( yt p = z t n (1 α) kt t min,t 1 µ, k )] α I,t, µ where µ = k I,t k I,t +k T,t captures optimal intangible capital ratio Maximize PV dividends paid out to shareholders: ( ) d t = y p t w tn t +(1 + r t )a f,t a f,t+1 q j,t k j,t+1 (1 δ)k j,t j=t,i Financial constraints Unable to issue equity: dt 0. Can issue one-period riskless debt, subject to: a f,t+1 θt q T,t+1 (1 δ) k T,t+1 + θ I q I,t+1 (1 δ) k I,t+1 1 + r t+1 θ T > θ I 16 / 47

HIGH-PRODUCTIVITY FIRMS Given Leontief structure, optimal capital ratio is k T,t = 1 µ µ k I,t 17 / 47

HIGH-PRODUCTIVITY FIRMS Given Leontief structure, optimal capital ratio is k T,t = 1 µ µ k I,t Firm dynamics and timing: A firm enters a period with predetermined capital, and produces 17 / 47

HIGH-PRODUCTIVITY FIRMS Given Leontief structure, optimal capital ratio is k T,t = 1 µ µ k I,t Firm dynamics and timing: A firm enters a period with predetermined capital, and produces Exit shock: technology becomes useless with probability ψ each period Firm liquidates all its capital, and pays out as dividends all of its savings, and exits Replaced with new firm with no capital and small amount of wealth W0 17 / 47

HIGH-PRODUCTIVITY FIRMS: VALUE FUNCTION Investing firm value function V + t (k I,t, a f,t ) = max a f,t+1,k I,t+1 d t + 1 ψ 1 + r t+1 ηv + t+1 (k I,t+1, a f,t+1) Non-investing firm value function + 1 ψ (1 η) V 1 + r t+1 (k I,t+1, a f,t+1)+ ψdexit t+1 t+1 1 + r t+1 V t (k I,t, a f,t ) = max a f,t+1 d t + 1 ψ 1 + r t+1 ηv + t+1 (k I,t+1, a f,t+1 ) + 1 ψ (1 η) V 1 + r t+1 (k I,t+1, a f,t+1 )+ ψdexit t+1 t+1 1 + r t+1 18 / 47

HIGH-PRODUCTIVITY FIRMS: CONSTRAINED INVESTMENT CHOICE Claim (check later) - in equilibrium marginal return of capital always higher than marginal cost: ( ) y p ( ) 1 µ t+1 1 µ (1 δ) q T,t+1 µ > q k T,t + q I,t+1 µ I,t + q I,t+1 1 + r t+1 Therefore, firms invest as much as possible, subject to a binding borrowing constraint: ( ) k I,t+1 = y t p w 1 µ tn t + (1 + r t )a f,t + (1 δ) q T,t µ + q I,t k I,t ( ) q T,t θt q T,t+1 1 µ 1+r t+1 µ + q I,t θ I q I,t+1 1+r t+1 = Available wealth Downpayment 19 / 47

HIGH-PRODUCTIVITY FIRMS: BORROWING/SAVINGS Firms always retain all earnings (d t = 0) Investing firms borrow as much as possible: ( a f +,t+1 = θ T q T,t+1 1 µ + θ I q I,t+1 1 + r t+1 µ 1 + r t+1 ) k I,t+1 < 0 And non-investing firms save as much as possible: a f,t+1 = y p t + (1 + r t )a f,t w t n t 20 / 47

REST OF THE ECONOMY Unproductive sector of final good producers financially unconstrained, absorb all capital not demanded by productive marginal buyers, capital priced by them Capital-producers representative financially unconstrained firm produce tangible and intangible capital Household sector Life-cycle with two types of households, young and old (measures H y and H o, H y + H o = 1) Young households: work and receive dividends Old households: cannot work, receive dividends, die with probability ϱ (Blanchard (1985) and Yaari (1965) framework)) 21 / 47

STEADY STATE Total amount of steady state intangible capital K I held by the productive firms: ( ( ) α ) η(1 ψ) αz KI t µ + (1 + r)af + ηψw 0 K I = (Q Q θ ) [δ + ψ (1 δ)] Q θ η(1 δ)(1 ψ), where price of capital : Q = q T 1 µ µ +q I collateral value of capital : Q θ = q T θ T 1 + r 1 µ µ +q θ I I 1 + r 22 / 47

OUTLINE OF TALK 1 Simple Analytical Intuition 2 Model 3 Simulation of Rise in Intangibles and Increase in Household Net Savings (U.S. 1970s-present) 23 / 47

CALIBRATION Parameter Symbol Value Discount factor β 0.95 Capital share, productive firms α 0.4 Capital share, unproductive firms, tangible capital χ I 0.4 Capital share, unproductive firms, intangible capital χ T 0.4 Intangible share of total capital µ 0.20 Unproductive firms, TFP tangible technology zt u,t 10 Unproductive firms, TFP tangible technology zt u,i 10 Years households remain young N 40 Probability of death of old households ϱ 0.25 Productivity parameter z 25 Collateral value of tangible capital θ T 1 Collateral value of intangible capital θ I 0.6 Probability of an investment opportunity η 0.07 Additional productivity of intangible capital κ 0.25 Adjustment cost convexity ϕ 4 Adjustment cost parameter (intangible) b I 0.00018 Adjustment cost parameter (tangible) b T 0.00004 Exit probability of high-productivity firms ψ 0.19 Endowment of new firms W 0 5 Depreciation of capital δ 0.15 Share of dividends to young households γ 50.2% 24 / 47

RISE IN INTANGIBLES AND INCREASE IN HOUSEHOLD NET SAVINGS (U.S. 1970S-PRESENT) 1. Increase in firms reliance on intangible capital Follow Corrado and Hulten (2010a), Falato et al (2013), Döttling and Perotti (2015): from µ = 0.2, 1970s ratio of intangible to tangible of 20% to µ = 0.6 2010 s ratio of intangible to tangible 60% Shortcut for endogenous process of adoption of more productive technologies 2. Household sector increase in net savings Captures demand side factors such as demographic forces, higher inequality, and higher saving by emerging market governments, over last three decades (Rachel and Smith, 2015) Increase in longevity and decrease in rate of time preference Achieve transition from 6% to 0% real interest rate 25 / 47

% INCREASE IN HOUSEHOLD SAVINGS INCREASE IN HOUSEHOLD SAVINGS 21 Share of Intangibles (%) 70 House. Saving (% Income) 55 Corporate Leverage 20.5 60 50 % 20 % 50 % 45 19.5 40 40 19 1970 1980 1990 2000 2010 30 1970 1980 1990 2000 2010 35 1970 1980 1990 2000 2010 6 Interest Rate 20 Capital Prices (% change) 6 Aggregate Capital (% change) 4 % 15 % 10 4 % 2 5 2 0 1970 1980 1990 2000 2010 0 1970 1980 1990 2000 2010 0 1970 1980 1990 2000 2010 Capital in Prod Sector (% change) 3 1.5 High Prod Firms Y (% change) 2 Total Y (% change) 2 % 1 1.5 % 1 1 0.5 0.5 0 1970 1980 1990 2000 2010 0 1970 1980 1990 2000 2010 0 1970 1980 1990 2000 2010 Tangible Capital Intangible Capital 26 / 47

% RISE IN INTANGIBLES INCREASE IN INTANGIBLE INTENSITY 80 Share of Intangibles (%) 50 House. Saving (% Income) 50 Corporate Leverage 60 % 40 % 0 % 0-50 20 1970 1980 1990 2000 2010-50 1970 1980 1990 2000 2010-100 1970 1980 1990 2000 2010 5.5 Interest Rate 20 Capital Prices (% change) 40 Aggregate Capital (% change) % 5 % 0 20 % 4.5-20 0 4 1970 1980 1990 2000 2010-40 1970 1980 1990 2000 2010-20 1970 1980 1990 2000 2010 Capital in Prod Sector (% change) 0 5 High Prod Firms Y (% change) 0.8 Total Y (% change) -10-20 -30 0 % -5 0.6 % 0.4 0.2-40 1970 1980 1990 2000 2010-10 1970 1980 1990 2000 2010 0 1970 1980 1990 2000 2010 Tangible Capital Intangible Capital 27 / 47

% INCREASE IN HOUSEHOLD SAVINGS AND RISE IN INTANGIBLES INCREASE IN INTANGIBLE INTENSITY AND HOUSEHOLD SAVINGS 80 Share of Intangibles (%) 40 House. Saving (% Income) 50 Corporate Leverage 60 % 20 % 0 % 0 40-20 20 1970 1980 1990 2000 2010-40 1970 1980 1990 2000 2010-50 1970 1980 1990 2000 2010 6 Interest Rate 40 Capital Prices (% change) 30 Aggregate Capital (% change) 4 % 20 % 20 % 10 2 0 0 0 1970 1980 1990 2000 2010-20 1970 1980 1990 2000 2010-10 1970 1980 1990 2000 2010 Capital in Prod Sector (% change) 0 5 High Prod Firms Y (% change) 1 Total Y (% change) -20 0 % -5 0 % -40-10 -1-60 1970 1980 1990 2000 2010-15 1970 1980 1990 2000 2010-2 1970 1980 1990 2000 2010 Tangible Capital Intangible Capital 28 / 47

OUTPUT IN THE 3 SIMULATIONS 2 0-2 -4 % -6-8 -10-12 OUTPUT IN THE THREE SIMULATIONS High Prod Firms Y (% change) 2 1.5 1 0.5 % 0-0.5-1 Total Y (% change) -14 1970 1975 1980 1985 1990 1995 2000 2005 2010-1.5 1970 1975 1980 1985 1990 1995 2000 2005 2010 HH Savings Increase Only Intangibles Intensity Increase HH Savings and Intangibles Intensity Increase 29 / 47

MISALLOCATION 120 Capital Misallocation (% change) MISALLOCATION 30 TFP Dispersion (% change) 100 25 80 % 60 20 % 15 40 10 20 5 0 1970 1980 1990 2000 2010 0 1970 1980 1990 2000 2010 Household Savings Increase Only Household Savings and Intangibles Intensity Increase 30 / 47

CONCLUSION Changes in firms financing behavior brought about by technological evolution might help explain the subpar growth associated with secular stagnation These changes interact with low interest rates behind secular stagnation to amplify negative effects Insights could be extended to develop interesting policy implications: negative externality in households and firms saving decisions might introduce a role for a fiscal policy that discourages such saving 31 / 47

APPENDIX 32 / 47

FINAL GOOD PRODUCERS: PRODUCTIVE FIRMS Produce consumption goods according to [ ( yt p = z(u t )n (1 α) kt t min,t 1 µ, k )] α I,t, µ where µ = k I,t k I,t +k T,t captures optimal intangible capital ratio Maximize PV dividends paid out to shareholders: ( ) d t = y p t w tn t +(1 + r t )a f,t a f,t+1 q j,t k j,t+1 (1 δ)k j,t j=t,i Financial constraints Unable to issue equity: dt 0. Can issue one-period riskless debt, subject to: a f,t+1 θt q T,t+1 (1 δ) k T,t+1 + θ I q I,t+1 (1 δ) k I,t+1 1 + r t+1 θ T > θ I 33 / 47

FINAL GOOD PRODUCERS: PRODUCTIVE FIRMS Given Leontief structure, optimal capital ratio is k T,t = 1 µ µ k I,t 34 / 47

FINAL GOOD PRODUCERS: PRODUCTIVE FIRMS Given Leontief structure, optimal capital ratio is k T,t = 1 µ µ k I,t Firm dynamics and timing: A firm enters a period with predetermined capital, and produces Exit shock: technology becomes useless with probability ψ each period Firm liquidates all its capital, and pays out as dividends all of its savings, and exits Replaced with new firm with no capital and small amount of wealth W0 34 / 47

PRODUCTIVE FIRMS: VALUE FUNCTION Investing firm value function V + t (k I,t, a f,t ) = max a f,t+1,k I,t+1 d t + 1 ψ 1 + r t+1 ηv + t+1 (k I,t+1, a f,t+1) Non-investing firm value function + 1 ψ (1 η) V 1 + r t+1 (k I,t+1, a f,t+1)+ ψdexit t+1 t+1 1 + r t+1 V t (k I,t, a f,t ) = max a f,t+1 d t + 1 ψ 1 + r t+1 ηv + t+1 (k I,t+1, a f,t+1 ) + 1 ψ (1 η) V 1 + r t+1 (k I,t+1, a f,t+1 )+ ψdexit t+1 t+1 1 + r t+1 35 / 47

INVESTING FIRMS: CONSTRAINED INVESTMENT CHOICE Claim (check later) - in equilibrium marginal return of capital always higher than marginal cost: ( ) y p ( ) 1 µ t+1 1 µ (1 δ) q T,t+1 µ > q k T,t + q I,t+1 µ I,t + q I,t+1 1 + r t+1 Therefore, firms invest as much as possible, subject to a binding borrowing constraint: ( ) k I,t+1 = y t p w 1 µ tn t + (1 + r t )a f,t + (1 δ) q T,t µ + q I,t k I,t ( ) q T,t θt q T,t+1 1 µ 1+r t+1 µ + q I,t θ I q I,t+1 1+r t+1 = Available wealth Downpayment 36 / 47

BORROWING AND SAVINGS CHOICE Firms always retain all earnings (d t = 0) Investing firms borrow as much as possible: ( a f +,t+1 = θ T q T,t+1 1 µ + θ I q I,t+1 1 + r t+1 µ 1 + r t+1 ) k I,t+1 < 0 And non-investing firms save as much as possible: a f,t+1 = y p t + (1 + r t )a f,t w t n t 37 / 47

FINAL GOOD PRODUCERS: UNPRODUCTIVE SECTOR Mass one of identical, financially unconstrained firms in unproductive sector Production function: y u t = zt u,i ki u,t + zu,t t kt u,t 38 / 47

FINAL GOOD PRODUCERS: UNPRODUCTIVE SECTOR Mass one of identical, financially unconstrained firms in unproductive sector Production function: y u t = zt u,i ki u,t + zu,t t kt u,t Budget constraint: dt u = yt u ( q I,t k u I,t+1 (1 δ)ki u ) (,t qt,t k u T,t+1 (1 δ)kt u ),t Unproductive sector willing to absorb all capital not demanded by productive sector at price q I,t = zt u,i q + I,t+1 1 + r t+1 + ξ t+1 q T,t = zt u,t q + T,t+1 1 + r t+1 + ξ t+1 ξ t+1 : positive wedge (shortcut to a risk premium factor) 38 / 47

AGGREGATION OF FINAL GOOD SECTOR (1) Aggregate across productive firms and substitute aggregate labor supply N = 1 to obtain: ( ) α Yt p KI = z,t t µ 39 / 47

AGGREGATION OF FINAL GOOD SECTOR (1) Aggregate across productive firms and substitute aggregate labor supply N = 1 to obtain: ( ) α Yt p KI = z,t t µ Wage determined in competitive markets by marginal return of labor: ( ) α KI w t = (1 α) z,t t µ Equilibrium aggregate capital in productive sector: where: KI INV,t+1 = ηki,t+1 + (1 δ) (1 ψ) (1 η) K I,t, KI INV,t+1 = (1 ψ)w t + ψw 0 ( W t =available wealth of continuing firms W 0 =initial endowment of new firms q T,t θ T q T,t+1 1+r t+1 ) 1 µ µ + q I,t θ I q I I,t+1 1+r t+1 39 / 47

AGGREGATION OF FINAL GOOD SECTOR (2) Aggregate firms in unproductive sector: Y u t = z u,i t (K I K I,t ) + z u,t t (K T K T,t ). 40 / 47

AGGREGATION OF FINAL GOOD SECTOR (2) Aggregate firms in unproductive sector: Y u t = z u,i t (K I K I,t ) + z u,t t (K T K T,t ). Assumption: z t is suffi ciently large & KI,t+1 suffi ciently small so that: MPK in the productive sector {}}{ ( α K ) α 1 µ z I,t+1 t > µ MPK in the unproductive sector {}}{ zt u,i + 1 µ µ zu,t t 40 / 47

HOUSEHOLD SECTOR (1) Life-cycle with two types of households, young and old (measures H y and H o, H y + H o = 1) Young households supply one unit of labor, receive fraction γ of dividends remain young for N periods Old households Cannot work, receive fraction (1 γ) of dividends die with probability ϱ Follow Blanchard (1985) and Yaari (1965) and assume life insurance scheme: survivors within a cohort pay the debt (or receive the savings) of the dying 41 / 47

HOUSEHOLD SECTOR (2) Representative old household maximizes: subject to Vt o (bt o ) = max ct o,bo t+1 c o t = b o t+1 + (1 γ)d t Young household of age j < N maximizes: ( ) ( V y t,j b y t,j c y t,j = max u c y t,j,by t+1,j+1 (1 ϱ) j β j log ( ) c t+j j=0 (1 + r) (1 ϱ) bo t ) ( ) + βv y t+1,j+1 b y t+1,j+1 subject to c y t,j = γd + w (1 + r)by t,j + by t+1,j+1 42 / 47

STEADY STATE EQUILIBRIUM Aggregate household borrowing/savings equal to aggregate corporate savings/borrowing: B = B o + B y = A f, where aggregate savings (debt) of the productive sector is A f = (1 ψ) αz t ( ) α ( ) KI µ + 1 µ ψw0 q T µ + q I [ψ + δ(1 ψ)] K I, [1 (1 ψ) (1 + r)] and the total amount of capital in the productive sector is K I = η where W is: ( ) [q T 1 1+r θt 1 µ µ + q I W αz t ( KI µ (1 ψ)w + ψw ( 0 1 1+r θi )] [1 (1 δ)(1 ψ) (1 η)] ) α ( ) 1 µ + (1 + r)a f + (1 δ) q T + q µ I K I., 43 / 47

STEADY STATE EQUILIBRIUM Dividends d are given by: d = D p + D u + D k. Prices of capital: q I = q T = Investment in capital creation is equal to: ( qt I t = ϕ b and aggregate capital is equal to: 1 r + δ zu,i 1 r + δ zu,t ) ϕ 1 1 (, and It I qi ) = ϕ ϕ 1 1, b K T = I T δ and K I t = I I δ. 44 / 47

RISE IN INTANGIBLES: INTUITION. A closed form solution for K I can be obtained if α = 1, ψ = 1 and η = 1 (firms only live for one period, and all firms can invest): K I = ( ) z u,i r + ξ } {{ } q I + 1 µ µ r lower downpayment for tangible capital: W ( 0 z u,t ) r + ξ } {{ } q T ( 1 θ 1+r ( ). 1 1+r θ ). This positive effect dominates if µ is small (µ = 0.2 1 µ µ = 4) So if µ suffi ciently large, capital purchase price channel prevails: r q I and q T : 45 / 47

SIMPLIFIED STEADY STATE Simplified steady state: α = 1, and fixed aggregate capital K (δ = 0) Total amount of steady state intangible capital K I held by the productive firms: K I = [ ( 1 µ ψ q T µ 1 1+r θt internal funds of all productive firms {[ ( ) }} ] { KI η(1 ψ) z + (1 + r)a µ f + ηψw 0 ) ( )] ( + q I 1 1+r θi θ q T 1 µ T 1+r µ + q I ) θ I 1+r η (1 ψ) where prices of capital: q I = zu,i r + δ, and q T = zu,t r + δ 46 / 47

SIMPLIFIED STEADY STATE Simplified steady state: α = 1, and fixed aggregate capital K (δ = 0) Total amount of steady state intangible capital K I held by the productive firms: K I = [ ( 1 µ ψ q T µ 1 1+r θt share of productive firms alive and that can invest {}}{ η(1 ψ) ) + q I ( 1 θi 1+r [ ( ) ] z KI µ + (1 + r)a f )] ( θ q T 1 µ T 1+r µ + q I ) θ I 1+r η (1 ψ) where prices of capital: q I = zu,i r + δ, and q T = zu,t r + δ 47 / 47