Financing Durable Assets

Transcription

1 Duke University Hebrew University of Jerusalem Finance Seminar May 30, 2018

2 Effect of Durability on Financing Durability essential feature of capital Fixed assets comprise as much as 72% of aggregate capital stock Includes structures (both residential and non-residential), infrastructure, equipment, and consumer durable goods Focus on tangible durable assets but intangible assets similar Variation in durability Depreciation rates vary from as low as 1% (new residential structures) to as high as 31% (computing equipment) How does durability affect financing? Can durable assets serve as collateral and facilitate financing? Intuitively, as the assets become more durable, they provide the creditor with the security to wait longer before being repaid.... And hence the debtor need not set aside as much of his initial borrowing to finance early debt repayments, leaving more to finance the initial investment Hart/Moore (1994)

3 Effect of Durability on Financing (Cont d) To the contrary: durable assets are harder to finance Durability increases resale/collateral value allowing more borrowing But durability also raises price of asset and hence financing need Net effect: raises down payments Implications Durability choice: more vs. less durable types of capital Choice between vintages: new, durable vs. used, non-durable assets Legal enforcement affects... technology adoption: dominated technologies may be adopted trade in used capital goods Rent vs. buy decision Critical distinction between durability and pledgeability Durability impedes financing Pledgeability facilitates financing Hart/Moore (1994) is about pledgeability not durability

4 Model of Durable Asset Financing Discrete time, infinite horizon, deterministic productivity Entrepreneurs (or firms ) Measure ρ (0, 1) enter each period; survive with probability 1 ρ; measure 1 alive Preferences: t=0 βt d t with β (0, 1), d t 0, and β ˆβ(1 ρ) Initial net worth w 0 (cash-in-hand) In stationary equilibrium, unconstrained entrepreneurs are financiers Equilibrium (gross) interest rate R = β 1 (net: r R 1) Limited enforcement Entrepreneurs can default and abscond with all cash flows and fraction 1 θ of assets without exclusion where θ [0, 1) Rampini/Viswanathan (2010, 2013, 2018) Optimal dynamic contract can be implemented with one-period ahead debt subject to collateral constraints

5 Modeling Durability: Two Approaches Capital with different geometric depreciation rate Lower depreciation rate means more durable Capital with finite useful life (one-horse shay depreciation) Capital provides same service flow for finite number of periods New capital assets are more durable than used ones

6 Financing and Durability: Geometric Depreciation Durability choice Two types of capital, more and less durable, j J {d, nd} Geometric rate δ j per period, δ d < δ nd Cost q j (in output goods) linear capital production technology Law of motion for capital where i j,t is investment k j,t = k j,t 1 (1 δ j ) + i j,t Perfect substitutes: k t j J k j,t Output Af(k t ) next period where A is TFP and f is (strictly) increasing, concave, and lim kt 0 f k (k t ) = +

7 Financing and Durability: Firm s Problem Firm s (recursive) problem: Given w, q d, q nd, and R, solve v(w) max d + βv(w ) (1) {d,k d,k nd,b,w } R 3 + R2 subject to budget constraints and collateral constraint Af(k) + j J θ j J w + b d + q j k j j J (2) q j k j (1 δ j ) Rb + w (3) q j k j (1 δ j ) Rb (4) and k j J k j Endogenous state variable: net worth w Properties, FOCs, and envelope condition

8 User Cost and Down Payment Frictionless user cost Jorgenson (1963) s frictionless one-period rental rate Paid at beginning of period u j R 1 q j(r + δ j) Down payment Minimal amount of internal funds needed to deploy unit of asset Present value of collateral value: R 1 θq j(1 δ j) Down payment per unit of capital Relation to frictionless user cost j = j q j R 1 θq j(1 δ j) u j }{{} + R 1 (1 θ)q j(1 δ j) }{{} > u j Jorgensonian user cost non-pledgeable part of resale value

9 Investment with Collateral Constraints Investment Euler equation Combining first-order conditions With equality if k j > 0 (ν j = 0) Firm s discount factor β µ µ 1 β µ Af k (k) + (1 θ)q j(1 δ j) µ j Determinants of investment First-order condition for investment with collateral constraints u j + λ µ j βaf k(k) Key determinants: (frictionless) user cost and down payment FOCs and envelope condition

10 Trade-off between User Cost and Down Payment Suppose both two types of capital used in equilibrium If u j u j and j j, j j, type j capital weakly dominated Therefore: u j > u j and j < j, j j, Suppose u d > u nd Then R 1 q d (r + δ d ) > R 1 q nd (r + δ nd ) implies q d > q nd d = u d +R 1 (1 θ)q d (1 δ d ) > u nd +R 1 (1 θ)q nd (1 δ nd ) = nd Contradiction Durable capital has lower user cost but higher down payment u d < u nd and d > nd More durable capital must be more expensive (q d > q nd ) q d (1 R 1 θ(1 δ d )) > q nd (1 R 1 θ(1 δ nd )) so effect of durability on price must exceed effect on collateral value Harder to finance

11 Constrained Firms Adopt Dominated Technologies Unconstrained never use less durable low quality capital User cost of potentially constrained firm u j(w) β µ µ Af k(k): u j(w) u j + β λ (1 θ)qj(1 δj) µ If λ = 0, then u j(w) = u j. Unconstrained firms simply compare user costs Severely constrained adopt dominated technology Alternatively, define user cost of potentially constrained firm as u j(w) j β µ qj(1 θ)(1 δj) µ Using the investment Euler equation, 1 β µ Af k (k) + (1 θ)q j(1 δ j) µ j as w 0, µ j kj 0 and hence β 0, so uj(w) j µ Severely constrained firms simply compare down payments Continuum of technologies Model with vintage capital and geometric depreciation

12 Dynamics of Firm Financing, Payout, and Investment Assume initial net worth w 0 low Young (severely constrained) firms Pay no dividends Compare down payments and invest only in less durable capital Unconstrained firms Pay dividends Compare user costs and invest only in more durable capital

13 Adopting Dominated Technologies and Legal Enforcement Two countries with strong and weak legal enforcement: θ H > θ L Technology choice: two types of capital with δ d < δ nd Both used in equilibrium: u d < u nd ; d (θ L) > nd (θ L) (q d > q nd ) Less durable capital dominated with strong legal enforcement Note: j(θ L) > j(θ H) and d (θ i) nd (θ i) decreasing in θ i As θ H 1, d (θ H) nd (θ H) u d u nd < 0 More dominated technology use with weak legal enforcement Firm indifferent between two types of capital Af k (k) + (1 θ i)q d (1 δ d ) d (θ i) = Af k(k) + (1 θ i)q nd (1 δ nd ), nd (θ i) Lower θ i implies indifferent at higher total investment k(θ i) and hence higher net worth

14 Durability and Composition of Investment Suppose two types of capital are imperfect substitutes Aggregator for capital with constant elasticity of substitution (CES) ( k σ jk γ j j J ) 1/γ Type-j capital k j, j J; factor shares σ j > 0, j J; j J σj = 1 Substitution coeff. γ; < γ < 1; elasticity of subst. 1/(1 γ) First order conditions yield Firm s problem ( ) 1 γ u nd (w) u d (w) = kd σ nd k nd σ d Composition of investment determined by... (frictionless) user costs u j(w) = u j for unconstrained firms... down payments u j(w) = j for severely constrained firms Constrained firms substitute away from durable assets Composition

15 Composition of Investment and Legal Enforcement Legal enforcement affects investment composition Ratio of down payments nd (θ i)/ d (θ i) and hence k d /k nd increasing in θ i No effect on unconstrained firms

16 Hart/Moore (1994): Pledgeability not Durability We say that the assets become longer lived, or more durable, if [the liquidation value] L(t) rises for all 0 t T. Durability liquidation value (price and use value fixed) Interpretation of their liquidation value L in our model L j k j θq j k j (1 δ j ) Effect of L j is effect of pledgeability θ in our model not durability

17 Durability and Financing: Two-Period Assets Technology New assets last for two periods (one-horse shay depreciation) New, durable assets k d last two periods; price q d = 1 (exogenous) Used, non-durable assets k nd one period of useful life left; price q nd Perfect substitutes in production: k k d + k nd ; output Af(k) where A is TFP; f is (strictly) increasing, concave, lim k 0 f k (k) = +

18 One-Horse Shay Depreciation The Deacons Masterpiece: or the Wonderful One-Hoss-Shay Oliver Wendell Holmes (1858)

19 Durability and Financing: Two-Period Assets Technology New assets last for two periods (one-horse shay depreciation) New, durable assets k d last two periods; price q d = 1 (exogenous) Used, non-durable assets k nd one period of useful life left; price q nd Perfect substitutes in production: k k d + k nd ; output Af(k) where A is TFP; f is (strictly) increasing, concave, lim k 0 f k (k) = + Price of used capital q nd determined in (stationary) equilibrium In equilibrium, New assets have lower frictionless user cost: u d u nd New assets have larger down payments: d > nd (= q nd ) Properties (Un)constrained firms buy used (new) assets Weak legal enforcement economies are net importers of used assets

20 Entrepreneur s Problem with Two-Period Assets Given net worth w, entrepreneur solves v(w) max d + βv(w ) (5) {d,k d,k nd,b,w } R 3 + R2 subject to budget constraints for current and next period and the collateral constraint w + b d + k d + q nd k nd (6) Af(k) + q nd k d Rb + w (7) θq nd k d Rb (8) and k k d + k nd Endogenous state variable: net worth w Properties, FOCs, and envelope condition

21 Equilibrium Market clearing condition in used capital market (1 ρ) p(w)k d (w) = p(w)k nd (w) }{{}}{{} supply of used assets demand for used assets where p(w) is stationary net worth distribution

22 User Cost and Down Payment User cost for unconstrained firm New, durable assets Used, non-durable assets u d 1 R 1 q nd = R 1 (r + (1 q nd )) u nd q nd = R 1 q nd (r + 1) Down payment New, durable assets Used, non-durable assets Relation to frictionless user cost d 1 R 1 θq nd nd q nd d = u d + R 1 (1 θ)q nd > u d

23 Investment with Collateral Constraints Investment Euler equation New, durable assets Used, non-durable assets 1 β µ Af k (k) + (1 θ)q nd + ν d µ d µ d 1 β µ Af k (k) + ν nd µ nd µ nd Determinants of investment (unchanged) First-order condition for investment with collateral constraints u j + λ µ j βaf k(k) Key determinants: (frictionless) user cost and down payment FOCs and envelope condition

24 Equilibrium in Used Asset Market In equilibrium u d u nd Otherwise d > u d > u nd = nd, so no new investment In equilibrium d > nd Otherwise u nd = nd d > u d, so used assets dominated Intuition: if d nd, new assets require less funds up front and give extra payoff next period ( arbitrage ) New assets have lower user cost but higher down payment u d u nd and d > nd Clearly new assets more expensive: 1 > q nd Recall q nd endogenous

25 Durability and Financing Need Financing need of new, durable and used, non-durable assets Time t t + 1 t + 2 Used, non-durable assets Value q nd 0 Collateral value 0 Borrowing 0 Internal funds required ( nd ) q nd New, durable assets Value 1 q nd 0 Collateral value θq nd Borrowing R 1 θq nd Internal funds required ( d ) 1 R 1 θq nd New, durable assets require more internal funds d > nd... despite ability to borrow R 1 θq nd > 0... because they cost more 1 > q nd larger financing need Example

26 Equilibrium Price of Used Capital In frictionless economy frictionless user cost u would be 1 = u + R 1 u q nd = u = R 1 + R Marginal investor in used capital Indifferent between investing in new and used capital: ν d = ν nd = 0 Investment Euler equations imply Down payment d {}}{ 1 R 1 θq nd q nd } {{ } Incr. cost of new capital = β µ µ (1 θ)q nd }{{} Valuation of resale value If constrained firm prices used assets, then q nd > q nd = If β µ µ = R 1, q nd = q nd and u d = u nd = u If β µ µ < R 1, q nd > q nd and u d < u nd R 1+R

27 Dividend Paying Firm Once firm starts to pay dividends, it always pays dividends and is unconstrained Marginal value of net worth v w(w) = µ = 1 Firm is unconstrained going forward v w(w ) = µ = 1 and λ = 0 If q nd > R 1+R, unconstrained firm purchases only new assets u d = βaf k (k) + ν d u nd = βaf k (k) + ν nd Unconstrained firms simply compare user costs Unconstrained firms sell assets once they are old/used Capital stock of unconstrained firm k d solves 1 = β[af k (k d ) + q nd ]

28 Investment Euler Equation and Constrained Investment Severely constrained firm purchases only used capital As w 0 so does k d + k nd and hence f k + Using Investment Euler equations 1 = β µ Af k (k) + (1 θ)q nd + ν d µ d µ d 1 = β µ Af k (k) + ν nd µ q nd µq nd so as w 0, β µ µ 0 and combining Euler equations ν d > 0 d nd }{{} Incr. cost of new capital = β µ µ (1 θ)q nd + ν d µ ν nd µ }{{} Valuation of resale value Severely constrained firms simply compare down payments User cost with financial constraints Imperfect substitutes

29 Trade World market for used capital (at price q nd > q nd ) Strong vs. weak legal enforcement economies: θ H > θ L Used capital may be dominated with strong legal enforcement In equilibrium, u d u nd and d (θ L) > nd But: if θ H 1, d (θ H) u d < nd (all used capital shipped) Weak legal enforcement economies net used capital importers Unconstrained firms investment identical: 1 = β[af( k d ) + q nd ] Constrained firms indifferent between new and used assets when Af k (k(θ i)) + (1 θ)q nd d (θ i) = Af k(k(θ i)) q nd Lower θ i implies indifferent at higher capital of capital k(θ i) and higher level of net worth More firms use used assets in weak legal enforcement economies

30 Renting Durable Assets Renting/leasing Benefit: higher leverage due to repossession advantage Cost: monitoring cost Two types of capital k j (different δ j ) required (fixed proportions) Severely constrained firms rent both types of assets Moderately constrained rent only more durable assets Renting/leasing model

31 Literature Hart/Moore (1994) See above Eisfeldt/Rampini (2007) Used capital is cheaper but requires ex-post maintenance cost Durability plays no role Empirical evidence on used capital International trade: Sen (1962), Navaretti/Soloaga/Takacs (2000) Legal enforcement: Benmelech/Bergman (2011) Literature on durable goods theory

32 Conclusion Durable assets are harder to finance Durability raises financing need and required down payments Financial constraints especially salient for durable assets Structures (residential and non-residential) and infrastructure Equipment, especially aircraft and ships Consumer durables, especially motor vehicles and household durables Similarly, durable intangible capital (θ = 0), e.g., organization capital Constrained firms (and households) buy used assets, adopt dominated technologies, and rent durable assets Weak legal enforcement economies invest in dominated technologies and import used assets Distinguish durability from pledgeability

33 Characterization of Entrepreneur s Problem Well-behaved dynamic program Return function concave; constraint set convex Operator defined by (1) to (4) satisfies Blackwell s sufficient conditions Solution:! value function v; strictly increasing; concave First order conditions Denote multipliers on (2) and (3) by µ and βµ and on (4) by βλ µ = 1 + ν µq j = βµ [Af k (k) + q j(1 δ j)] + βλ θq j(1 δ j) + ν j, j J µ = µ + λ µ = v w(w ) Envelope condition: v w (w) = µ Value function continuously differentiable Back

34 Characterization of Entrepreneur s Problem First order conditions Denote multipliers on (2) and (3) by µ and βµ and on (4) by βλ µ = 1 + ν µq j = βµ [Af k (k) + q j(1 δ j)] + βλ θq j(1 δ j) + ν j, j J µ = µ + λ µ = v w(w ) Envelope condition: v w (w) = µ Value function continuously differentiable Back

35 Effect of Durability on Down Payment and User Cost Effect of durability on down payment When collateral constraint binds, b = R 1 θqk(1 δ) Minimal down payment per unit of capital δ = q R 1 θq(1 δ) = R 1 θq > 0; more durable capital has lower down payment? But keeping q fixed, lower δ reduces frictionless user cost of capital! u R 1 q(r + δ)

36 Effect of Durability on Down Payment and User Cost More durable capital requires larger down payment Fixing frictionless user cost u instead, using u = R 1 q (δ)(r + δ) price is q (δ) = Ru r + δ Down payment keeping user cost fixed (δ) δ = q (δ) 1 θ < 0 r+δ Down payment decomposition (δ) = q (δ) R 1 θq (δ)(1 δ) (δ) = }{{} u + R 1 q(δ)(1 θ)(1 δ) }{{} Jorgensonian user cost non-pledgeable part of resale value In general, arguably more durable capital is more expensive q(δ) with q(δ) δ < 0 (δ) = q(δ) R 1 θq(δ)(1 δ) (δ) δ = q(δ) (1 δ R 1 θ(1 δ)) + qr 1 θ 0 (ambiguous) User cost and sensitivity to financial constraints

37 Effect of Durability on Down Payment and User Cost User cost depends on firm s discount factor u(w)(δ) = (δ) β µ q(δ)(1 θ)(1 δ) µ = u + β λ Ru (1 θ)(1 δ) µ r + δ Durable capital user cost sensitive to financial constraints u(w)(δ) = q(δ) λ 1 θ < 0 δ µ r+δ u(w)(1) = u independent of w (also if θ = 1 but we assume θ < 1) Back

38 Continuum of Capital Types with Varying Durability Continuum of technologies with durability 1 δ Production cost (and price) for type δ: φ(δ) with φ δ (δ) < 0 < φ δδ (δ) Unconstrained firms minimize frictionless user cost u(δ) δ arg max δ [0,1] R 1 φ(δ)(r + δ) In frictionless economy (or if θ = 1), only durability 1 δ produced Constrained firms purchase less durable types of capital Possibly continuum of less durable types of capital produced Back

39 Model with Vintage Capital and Geometric Depreciation Vintage capital with geometric depreciation New capital: cost q d ; depreciation δ d ; fraction η becomes used Used capital: price q nd ; depreciation δ nd > δ d Previous vintage model: δ d = 0, δ nc = 1, η = 1, q d = 1, q nd = q Endogenous price of used capital q nd If marginal investor is unconstrained, u d = u nd ; frictionless price q nd = r + δ d + η(1 δ d ) r + δ nd + η(1 δ d ) q d < q d If marginal investor is constrained, q nd > q nd Down payments d > nd d q d R 1 θq d (1 δ d ) + R 1 θη(q d q nd )(1 δ d ) nd q nd R 1 θq nd (1 δ nd ) Back

40 Composition Determined by User Cost or Down Payments Firm s problem with two types of capital: Given w, solve v(w) max {d,k d,k nd,b,w } R 3 + R2 d + βv(w ) subject to budget constraints and collateral constraint w + b d + j J (( ) 1/γ ) Af σ j k γ j + q j k j (1 δ j ) Rb + w j J j J θ j J q j k j (1 δ j ) Rb q j k j First order condition for type j capital u j (w) = β µ µ Af k(k) k k j Back

41 Distortion of Financially Constrained Investment Investment distorted away from durable assets if u nd u d > nd d d u d > nd u nd Down payment to user cost ratio higher for durable assets j = R 1 q j (r + δ j ) + R 1 q j (1 θ)(1 δ j ) u j R 1 = 1 + (1 θ) 1 δ j q j (r + δ j ) r + δ j ( j ) Decreasing in δ j: δ j u j < 0 Why? Residual value is larger fraction of value j u j j = ( (1 θ) 1 δ j r+δ j ) 1 Back

42 Characterization of Entrepreneur s Problem Well-behaved dynamic program Return function concave; constraint set convex Operator defined by (5) to (8) satisfies Blackwell s sufficient conditions Solution:! value function v; strictly increasing; concave First order conditions Denote multipliers on (6) and (7) by µ and βµ and on (8) by βλ µ = 1 + ν µ = βµ [Af k (k) + q nd ] + βλ θq nd + ν d µq nd = βµ Af k (k) + ν nd µ = µ + λ µ = v w(w ) Envelope condition (marginal value of net worth): v w (w) = µ Value function continuously differentiable Back

43 Characterization of Entrepreneur s Problem First order conditions Denote multipliers on (6) and (7) by µ and βµ and on (8) by βλ µ = 1 + ν µ = βµ [Af k (k) + q nd ] + βλ θq nd + ν d µq nd = βµ Af k (k) + ν nd µ = µ + λ µ = v w(w ) Envelope condition (marginal value of net worth): v w (w) = µ Value function continuously differentiable Back

44 Durability and Financing Need Financing need of new, durable and used, non-durable assets Time t t + 1 t + 2 Used, non-durable assets Value q nd 0 Collateral value 0 Borrowing 0 Internal funds required ( nd ) q nd New, durable assets Value 1 q nd 0 Collateral value θq nd Borrowing R 1 θq nd Internal funds required ( d ) 1 R 1 θq nd New, durable assets require more internal funds d > nd... despite ability to borrow R 1 θq nd > 0... because they cost more 1 > q nd larger financing need Example: If R 1, θ = 0, and q nd = 0.5, then d = 1 Back

45 Durability and Depreciation Rate In first period, new assets depreciate at depreciation rate δ d 1 q nd 1 1 R 1 + R = R < 50%... and in second period (when used) at rate δ nd q nd 0 q nd = 100% Back

46 User Cost of New Capital for Financially Constrained Firm User cost of durable assets depends on firm s discount factor u d (w) = d β µ µ (1 θ)q nd = u d + β λ µ (1 θ)q nd User cost of new, durable assets for severely constrained firm As w 0, β µ µ 0 so u d (w) d Severely constrained firms simply compare down payments Back

47 Composition of Investment with Two Period Assets Suppose two period assets are imperfect substitutes First order conditions yield (as in case with geometric depreciation) ( ) 1 γ u nd u d (w) = kd σ nd k nd σ d Composition of investment nd / d (u nd /u d ) determines composition for (un)constrained firms For used, non-durable assets: nd u nd = q nd q nd = 1 For new, durable assets: d u d = 1 R 1 θq nd > 1 1 R 1 q nd Constrained firms substitute away from new capital Intuition: d u d d = R 1 (1 θ)q nd 1 R 1 θq nd Especially with weak legal enforcement > 0 = nd u nd nd Back

48 Renting Durable Assets Capital goods with different durability Capital k j, j J, with depreciation rate δ d < δ nd and price q j Leontief aggregator k = min{ k d+kd l σ d, k nd+knd l σ nd } Implies k j + kj l = σ jk where k is aggregate capital Rental price u l j R 1 q j(r + δ j + m) where m is monitoring cost Firm cannot abscond with rented capital (repossession advantage) Entrepreneur s problem with rentals: Given w, solve v(w) max d + βv(w ) (9) {d,{k j,kj l } j J,b,w } R 5 + R2 subject to budget, collateral, and technological constraints w + b d + q j k j + u l jkj l (10) j J j J Af(k) + q j k j (1 δ j ) Rb + w (11) j J θ q j k j (1 δ j ) Rb (12) j J k j + kj l σ j k (13)

49 Renting Durable Assets (Cont d) First order conditions with multipliers as before and βµ η j on (13) σ j η j = Af k (k) j J µq j = βµ η j + βµ q j (1 δ j ) + βλ θq j (1 δ j ) + ν j µu l j = βµ η j + νj l Investment Euler equation (IEE) 1 = β µ η j + (1 θ)q j (1 δ j ) + ν j µ j µ j More constrained firms rent less durable assets Suppose rent and own some type j capital, so ν j = ν l j = 0 and IEE λ (1 θ)(1 δj) = m µ If δ j higher, so multiplier on collateral constraint λ µ higher Back

50 Literature: Durable Goods Theory Optimal durability and planned obsolescence Conventional wisdom: monopolists produce less durable goods Fallacy: Swan (1970, 1972), Sieper/Swan (1973): cost minimization Resuscitated: Barro (1972): impatience; Rust (1986): scrappage Bulow (1982, 1986): escape Coase (1972) time-inconsistency Role of rental markets Coase (1972), Bulow (1982), Stokey (1981): Coasian dynamics Hendel/Lizzeri (2002): leasing ameliorates adverse selection Key driving force: market power Our theory: optimal competitive response to financial constraints More literature Back

51 Literature: Durable Goods Theory Optimal durability and planned obsolescence Conventional wisdom: monopolists produce less durable goods Fallacy: Swan (1970, 1972), Sieper/Swan (1973): cost minimization Resuscitated: Barro (1972): impatience; Rust (1986): scrappage Bulow (1982, 1986): escape Coase (1972) time-inconsistency Role of rental markets Coase (1972), Bulow (1982), Stokey (1981): Coasian dynamics Hendel/Lizzeri (2002): leasing ameliorates adverse selection Key driving force: market power Our theory: optimal competitive response to financial constraints Other features Adverse selection: Akerlof (1970) Illiquidity Back