## 1. Estimated Physical Depreciation Rate

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### Introduction and key empirical observations
- Sample and data:
  - Sample period: 1979 to 2009.
  - Data sources: Standard and Poor’s Compustat industrial file and CRSP monthly stock file.
  - Physical assets imputed using perpetual inventory method; intangible assets proxied by accumulated SG&A (Compustat item XSGA) via a perpetual inventory equation.
  - Benchmark depreciation rate chosen: δ = 0.08.
- Core empirical regularities:
  - Firms with higher share of intangible assets in total assets start smaller, grow faster, and have higher market value per unit of assets.
  - Asset tangibility explains:
    - 34% of the variance in the size of physical assets of entrant firms,
    - 13% of their variance in employment,
    - 17% of their variance in assets growth,
    - 7% of their variance in the market value per unit of total assets.
  - Predictive power of asset tangibility remains strong for firms up to 30 years old but eventually ceases to matter with age.
  - Market value per unit of fixed assets (Tobin’s q) falls with age among young firms; Tobin’s q levels off eventually but does not converge across firms with different asset tangibility. In contrast, market value per unit of total assets (intangible adjusted q) converges among older firms.

### Theoretical model: limited contract enforcement with heterogeneous assets
- Framework overview:
  - Builds on Albuquerque and Hopenhayn (2004), extended to two asset types (physical k and intangible a) and heterogeneous technologies.
  - Three key assumptions:
    1. Young firms have insufficient internal funds and are partly financed by external debt.
    2. Financial frictions arise because debt contracts have limited enforceability; a repudiation-free contract sets an endogenous borrowing limit.
    3. Physical and intangible assets affect the borrowing limit asymmetrically because intangible assets have higher residual value to the firm upon repudiation.
- Contracting setup and enforcement:
  - Repudiation/diversion function: D(k,a) = η_k k + η_a a with 0 ≤ η_k < η_a ≤ 1.
  - Enforcement constraint: β(d_0 + EV'_c) ≥ D(k_0,a_0).
  - Debtholders and firm share discount rate r; survival probability to next period is 1−φ.
- Dynamic programming and efficient frontier:
  - Define total surplus W(V,z) = B(V,z) + V and equivalent Bellman problem subject to V ≥ D(k′,a′) and V ≥ β E[V(z′)].
  - Existence of efficient frontier Ṽ(z): for V ≥ Ṽ(z) enforcement constraint no longer binds and W(V,z) = W̃(z).
- Properties of optimal contract (selected results):
  - Lemma 1: W(V,z) weakly increasing in V; for V ≥ Ṽ(z) W(V,z) = W̃(z); for V < Ṽ(z) W(V,z) < W̃(z).
  - Lemma 2: W(V,z) strictly concave in V when V < Ṽ(z).
  - Lemma 3: W(V,z) strictly increasing in V when V < Ṽ(z).
  - Proposition 1: If V < β E[Ṽ(z′)], the optimal policy requires no dividends be distributed until reaching efficient frontier.

### Model implications for firm dynamics, Tobin’s q, and asset composition
- Asset accumulation and q definitions:
  - i_k,t = k_{t+1} − (1−δ_k) k_t
  - i_a,t = a_{t+1} − (1−δ_a) a_t
  - q_Tobin ≡ W / (I_0 + k_0)
  - q_adj ≡ W / (I_0 + k_0 + a_0)
- Age effects and comparative statics:
  - Proposition 2: Conditional on revenue state z, firm size increases with age before the firm matures (V < Ṽ(z)).
  - Proposition 3: Conditional on revenue state, Tobin’s q decreases with age before the firm reaches its optimal size.
  - Proposition 4: Conditional on revenue state, share of intangible assets increases with age before firm reaches optimal size (constrained firms distort investment toward physical assets; as constraints relax share of intangible rises).
  - Cobb–Douglas example: F(k,a,γ) = e^z (k^γ a^{1−γ})^θ with firm-specific γ drawn at birth; efficient frontier Ṽ(γ) depends on γ.
  - Proposition 5: Entrants with higher share of intangible assets have lower equity value V_0 and smaller asset size k_0 and a_0.
  - Proposition 6 and Corollary 1: Entrant firms with higher share of intangible assets have higher Tobin’s q; conditional on revenue state and age, higher intangible share ⇒ higher Tobin’s q.
  - Proposition 7: Among young age categories, firms with higher intangible share on average have higher growth rate in firm value V and intangible assets a; correlation diminishes with age.

### Numerical simulations and model calibration
- Calibration and key parameter choices:
  - Period = one year.
  - Interest rate r 0:04.
  - Exit probability φ 0:10, which implies discount rate β 0:865.
  - Return to scale θ 0:25.
  - Depreciation rates: δ_k 0:08 and δ_a 0:08.
  - Productivity process: z′ = ρ z + ε′ with ρ 0:40 and σ 0:05 (truncated normal); discretization uses Tauchen (1991) with two points.
  - State grids: 90-point uniform grids for V, V′, k′, a′.
  - γ drawn from CDF H(γ) with support [0:1; 0:9] discretized into 5 points.
  - Estimated repudiation values: η_k 0:52 and η_a 0:87.
- Solution and simulation:
  - Numerical solution by value function iteration on a multigrid scheme.
  - Simulate a model economy with 10,000 firms over 200 periods.
  - Set I0 = 0 and jointly calibrate {η_k, η_a} to match simulated moments, particularly the size and growth rate premium associated with intangible assets.
- Numerical insights:
  - Simulated model reproduces stylized facts: high average Tobin’s q among young firms that decreases with age and levels off; intangible-intensive entrants start smaller and grow faster; intangible-adjusted q converges among older firms.
  - Comparative experiments: lowering financial frictions or reducing spread between repudiation values changes speed to efficient frontier and compresses cross-firm differences.

### Empirical implementation and main tests
- Measurement and construction:
  - Physical assets imputed from investment (Compustat item CAPX minus SPPE) via perpetual inventory; implied firm-level depreciation δ_k,i computed from equation (22); firms with δ_k,i < 0 or > 1 eliminated.
  - Intangible assets a_it constructed by perpetual inventory a_it = (1−δ_a) a_{it−1} + SGA_{it−1}; initial intangible set to first reported INTAN; benchmark δ_a = δ_k = δ = 0.08.
  - Tangibility τ_it ≡ k_it / (k_it + a_it); share of intangible = 1 − τ_it.
  - Age approximated by earliest of CRSP appearance, Compustat inclusion, or valid CRSP-Compustat link (LINKDT).
  - Tobin’s q = market value / physical assets (market value defined in Appendix II).
  - Windsorization: growth and financial measures (including Tobin’s q and market value to total assets) windsorized at 2% and 98%.
  - Timing convention: use year t−1 balance sheet for stock variables and year t income/cash flow for flow variables.
- Stylized empirical facts (figures summarized):
  - Distribution of τ among entrants spans below 5% to above 95%.
  - Entrant firms with low τ (high intangible share) are on average 6 times smaller than firms with high τ; size premium diminishes with age.
  - Entrant firms with higher intangible share grow faster on average; growth premium diminishes with age.
  - Among young firms, higher intangible share ⇨ higher market value per unit of assets; Tobin’s q remains permanently higher for low-τ firms, while intangible-adjusted q converges among older firms.
- Regression evidence (selected numerics from entrant regressions, Table 2; No. of obs. often 2,593):
  - Tangibility coefficient estimates (Table 2):
    - Physical assets: 5.6203*** (SE 0.1376)
    - Total assets: 3.2812*** (SE 0.1368)
    - Employment: 2.9554*** (SE 0.1449)
    - Sales: 2.5936*** (SE 0.1614)
    - Growth (physical assets): −0.2579*** (SE 0.0261)
    - Growth (total assets): −0.5308*** (SE 0.0229)
    - Tobin’s q: −29.9738*** (SE 0.8465)
    - Intangible adjusted q: −2.5915*** (SE 0.1732)
  - R^2 examples (Table 2):
    - Physical assets model R^2 = 0.6237; R^2 for tangibility = 0.3368.
    - Tobin’s q model R^2 = 0.5213; R^2 for tangibility = 0.2569.
  - Age-interaction regressions (Table 3 and 4; example sample sizes):
    - No. of observations: e.g., 63,907 for pooled size and sales regressions; No. of firm fixed effects: e.g., 465.
    - Example coefficients (Table 3, column 1): Tangibility (τ) = 5.3048*** (SE 0.1112); τ×Age 2 = −0.3383*** (SE 0.1462); τ×Age 31 = −1.1126*** (SE 0.3545).
    - Example coefficients (Table 4, column 3): Tangibility (τ) = −30.2926*** (SE 0.6939); τ×Age 3 = 10.9938*** (SE 0.8853).
- Robustness and selection:
  - Alternative tangibility measure PPEGT / AT (τ̂) yields coefficients in same direction but smaller magnitudes and lower R^2.
  - Binomial selection model indicates firms with higher intangible share are less likely to exit; selection unlikely to explain size-growth patterns.

### Quantitative identification and additional measurement details
- Industry estimated mean physical depreciation rates (examples from Table 1):
  - Food: 0.072
  - Oil: 0.098
  - Automobile: 0.063
- Appendix II variable definitions (selected):
  - Total debt = DLTT + DLC.
  - Market value = total debt + CRSP December market capitalization + PSTKRV − CHE − INVT.
  - Cash flow = IB + DP.
  - Cash flow rate = cash flow / physical assets.
- Identification strategy:
  - Use entrant size differentials across γ classes to identify repudiation values η_k and η_a, then set I0 such that B_L = B_M = B_H = I0.
  - I0 affects initial entrant size and age when firms become unconstrained; η_k and η_a affect entrant size and time to mature in similar manners as I0.

### Conclusions and implications
- Modeling accumulation of intangible assets is crucial to explain level and age dynamics of Tobin’s q and firm heterogeneity.
- Financial frictions from limited contract enforcement with asset-specific repudiation values generate:
  - Discouraged entry of intangible-intensive technologies,
  - Prolonged inefficiently small sizes for intangible-intensive firms,
  - Distortion of investment away from intangible assets (misallocation between physical and intangible).
- Empirical evidence from U.S. public firms (1979–2009) supports model predictions: intangible asset share predicts entrant size, growth, and market valuation patterns; predictive power diminishes with firm age.
- Suggested model extensions discussed:
  - Allow aggregate fluctuations to analyze cyclicality of firm dynamics.
  - Distinguish firm from production technology to allow acquisitions or technology switching.

*Source: _wp1488 - 1. Estimated Physical Depreciation Rate*

### 1. Estimated Physical Depreciation Rate ................................................................................

### _wp1488 - 1. Estimated Physical Depreciation Rate ................................................................................

### Major sections
- 1. Estimated Physical Depreciation Rate .................................................................................26
- 2. Entrant Asset Tangibility, Firm Dynamics, and Q ..............................................................27
- 3. The Age Effects of Asset Tangibility on Firm Size .............................................................29
- 4. The Age Effects of Asset Tangibility on Firm Growth and Q .............................................30

### Figures listed
- 1. Intangible Assets, Firm Dynamics, and Q: Stylized Facts...................................................4
- 2. Timing of Events..................................................................................................................8
- 3. Tobin’s Q and Intangible Adjusted Q: Data ......................................................................14
- 4. Tobin’s Q and Intangible Adjusted Q: Simulated Model ..................................................15
- 5. Intangible Asset, Firm Dynamics and Q: Simulated Model ..............................................18
- 6. Asset Tangibility Distribution Among Entrants ................................................................21

*Source: _wp1488 - 1. Estimated Physical Depreciation Rate*

### References .............................................................................................................

### _wp1488 - References .............................................................................................................

### Introduction and key empirical observations
- External financing is critical for firm creation and expansion; enforceability of financial contracts is a major obstacle.
- Investment in intangible assets (research and development, employee training, marketing, strategy consultants) is more susceptible to financing frictions than investment in physical assets because intangible assets are inalienable and firm-specific.
- New firm-level measures of physical and intangible assets are constructed from Compustat (U.S. public firms).
- Empirical regularities documented:
  - Firms with higher share of intangible assets in total assets start smaller, grow faster, and have higher market value per unit of assets.
  - Asset tangibility explains:
    - 34% of the variance in the size of physical assets of entrant firms,
    - 13% of their variance in employment,
    - 17% of their variance in assets growth,
    - 7% of their variance in the market value per unit of total assets.
  - Predictive power of asset tangibility remains strong for firms up to 30 years old but eventually ceases to matter with age.
  - Market value per unit of fixed assets (Tobin’s q) falls with age among young firms; Tobin’s q levels off eventually but does not converge across firms with different asset tangibility. In contrast, market value per unit of total assets (intangible adjusted q) converges among older firms.
- Sample and data notes:
  - Sample period: 1979 to 2009.
  - Data sources: Standard and Poor’s Compustat industrial file and CRSP monthly stock file.
  - Physical assets imputed using perpetual inventory method; intangible assets proxied by accumulated SG&A (Compustat item XSGA) via a perpetual inventory equation.

### Theoretical model: limited contract enforcement with heterogeneous assets
- Framework overview:
  - Builds on Albuquerque and Hopenhayn (2004), extended to two asset types (physical k and intangible a) and heterogeneous technologies.
  - Three key assumptions:
    1. Young firms have insufficient internal funds and are partly financed by external debt.
    2. Financial frictions arise because debt contracts have limited enforceability; a repudiation-free contract sets an endogenous borrowing limit.
    3. Physical and intangible assets affect the borrowing limit asymmetrically because intangible assets have higher residual value to the firm upon repudiation.
- Contracting setup and enforcement:
  - Firm can repudiate contract and divert D(k,a) = η_k k + η_a a; assume 0 ≤ η_k < η_a ≤ 1, so repudiation value is higher for intangible assets.
  - Enforcement constraint: β(d_0 + EV'_c) ≥ D(k_0,a_0).
  - Debtholders and firm have same discount rate r; survival probability to next period is 1−φ.
- Dynamic programming characterization:
  - Define total surplus W(V,z) = B(V,z) + V; equivalent Bellman problem (equation (11)) subject to:
    - V ≥ D(k',a') = η_k k' + η_a a'
    - V ≥ β E[V(z')]
  - Existence of an efficient frontier Ṽ(z): once V ≥ Ṽ(z) the enforcement constraint no longer binds and total surplus W(V,z) is constant at W̃(z).
- Properties of optimal contract:
  - Lemma 1: W(V,z) weakly increasing in V; for V ≥ Ṽ(z) W(V,z) = W̃(z); for V < Ṽ(z) W(V,z) < W̃(z).
  - Lemma 2: W(V,z) strictly concave in V when V < Ṽ(z).
  - Lemma 3: W(V,z) strictly increasing in V when V < Ṽ(z).
  - Proposition 1: If V < β E[Ṽ(z')], the optimal policy requires no dividends be distributed (all earnings allocated to debt repayment) until reaching efficient frontier.

### Model implications for firm dynamics, Tobin’s q, and asset composition
- Asset accumulation and q definitions:
  - Asset accumulation:
    - i_k,t = k_{t+1} − (1−δ_k) k_t
    - i_a,t = a_{t+1} − (1−δ_a) a_t
  - Market value W_t relates to future profits and assets; Lemma 4: W = market surplus = W + k_0 + a_0 + I_0 (constitutive relation in model).
  - Tobin’s q and intangible-adjusted q:
    - q_Tobin ≡ W / (I_0 + k_0)
    - q_adj ≡ W / (I_0 + k_0 + a_0)
- Age effects:
  - Proposition 2: Conditional on revenue state z, firm size increases with age before the firm matures (V < Ṽ(z)).
  - Proposition 3: Conditional on revenue state, Tobin’s q decreases with age before the firm reaches its optimal size.
  - Intuition: While constrained (V < Ṽ(z)) assets are small but market value W is forward-looking; as V grows with age (via retained earnings and debt repayment), q falls and levels off once unconstrained.
- Asset composition dynamics:
  - Proposition 4: Conditional on revenue state, share of intangible assets increases with age before firm reaches optimal size (constrained firms distort investment toward physical assets, and as constraints relax share of intangible rises).
- Heterogeneity in technology (Cobb-Douglas example):
  - Profit F(k,a,γ) = e^z (k^γ a^{1−γ})^θ; γ is firm-specific parameter drawn at birth; lower γ means more intangible-intensive technology.
  - Efficient frontier Ṽ(γ) is decreasing in γ; firms more intensive in intangible assets have higher Ṽ(γ) (tighter constraints).
  - Proposition 5: Conditional on revenue state z_0, entrant firms with higher share of intangible assets have lower equity value V_0 and smaller asset size k_0 and a_0.
  - Proposition 6 and Corollary 1: Entrant firms with higher share of intangible assets have higher Tobin’s q; conditional on revenue state and age, higher intangible share ⇒ higher Tobin’s q.
  - Proposition 7: Among young age categories, firms with higher share of intangible assets on average have higher growth rate in firm value V and intangible assets a; correlation diminishes with age as firms become unconstrained.
- Numerical simulations:
  - Simulated model reproduces stylized facts:
    - High average Tobin’s q among young firms, decreasing with age and leveling off.
    - Firms with higher intangible share start smaller, grow faster, converge over time; they have higher Tobin’s q and intangible-adjusted q.

### Empirical implementation and main tests
- Data construction and measurement:
  - Physical assets imputed from investment (Compustat item CAPX minus SPPE) via perpetual inventory; implied firm-level depreciation δ_k,i computed from equation (22); firms with δ_k,i < 0 or > 1 eliminated.
  - Intangible assets a_it constructed by perpetual inventory a_it = (1−δ_a) a_{it−1} + SGA_{it−1} where SGA is Compustat item XSGA; initial intangible set to first reported INTAN; benchmark δ_a = δ_k = δ = 0.08.
  - Tangibility τ_it ≡ k_it / (k_it + a_it); share of intangible = 1 − τ_it.
  - Age defined as years since IPO approximated by earliest of CRSP appearance, Compustat inclusion, or valid CRSP-Compustat link (LINKDT).
- Stylized empirical facts (visuals summarized from Figures):
  - Distribution of τ among entrants spans below 5% to above 95%; substantial cross-sectional variation.
  - Entrant firms with low τ (high intangible share) are on average 6 times smaller than firms with high τ; size premium diminishes with age.
  - Entrant firms with higher intangible share grow faster on average; growth premium diminishes with age.
  - Among young firms, higher intangible share ⇨ higher market value per unit of assets; market value per physical assets (Tobin’s q) remains permanently higher for low-τ firms, while market value per total assets (intangible-adjusted q) converges among older firms.
- Hypotheses tested:
  - Hypothesis 1 (asset tangibility and firm dynamics): Among entrants, higher intangible share ⇒ higher growth, smaller size, higher Tobin’s q, higher intangible-adjusted q.
  - Hypothesis 2 (diminishing effect): Sensitivity of growth and size to intangible share diminishes with age.
- Regression evidence:
  - Entrant regressions (equation (25)) with industry and year fixed effects:
    - Tangibility τ has positive and significant coefficients for measures of firm size (physical assets, total assets, sales, employment).
    - τ explains 13% of variance in entrants’ employment and 8% in sales (R^2 for tangibility reported).
    - τ has positive and significant coefficients for growth in physical and total assets; τ alone explains 4% of variance in growth of physical assets and 17% in growth of total assets.
    - τ coefficients negative and significant for Tobin’s q and intangible-adjusted q regressions; τ alone explains 26% of variance in Tobin’s q and 7% in intangible-adjusted q (Table 2).
  - Age-interaction regressions:
    - Pooled regressions with Age dummies and τ × Age interactions show: marginal effect of tangibility on firm size is positive but τ × Age coefficients are negative and increasingly smaller for older age categories, indicating diminishing sensitivity with age (Table 3).
    - Similar diminishing patterns hold for total assets, sales, growth rates, Tobin’s q, and intangible-adjusted q (Table 4).
  - Robustness:
    - Using reported PPEGT / AT as alternative tangibility measure (τ̂) gives coefficients in same direction but generally smaller magnitudes and lower R^2.
    - Selection explanation tested: binomial model indicates firms with higher intangible share are less likely to exit (so selection is unlikely to explain observed size-growth patterns).

### Quantitative and estimation specifics (selected numeric values and sample notes)
- Sample period: 1979–2009.
- Benchmark depreciation rate chosen: δ = 0.08.
- Entrant sample sizes and R^2 examples (Table 2):
  - No. of obs. in entrant regressions: 2,593 for many specifications.
  - Tangibility coefficient estimates (Table 2):
    - Physical assets: 5.6203*** (SE 0.1376)
    - Total assets: 3.2812*** (SE 0.1368)
    - Employment: 2.9554*** (SE 0.1449)
    - Sales: 2.5936*** (SE 0.1614)
    - Growth (physical assets): −0.2579*** (SE 0.0261) [note sign convention in table headings]
    - Growth (total assets): −0.5308*** (SE 0.0229)
    - Tobin’s q: −29.9738*** (SE 0.8465)
    - Intangible adjusted q: −2.5915*** (SE 0.1732)
  - R^2 values and R^2 attributable to tangibility reported in Table 2 (examples):
    - Physical assets model R^2 = 0.6237; R^2 for tangibility = 0.3368.
    - Tobin’s q model R^2 = 0.5213; R^2 for tangibility = 0.2569.
- Industry estimated mean physical depreciation rates (Table 1): examples
  - Food: 0.072
  - Oil: 0.098
  - Automobile: 0.063
  - Note: industry table reports mean depreciation rates and number of observations by 17 Fama–French industries.
- Age-interaction regression sample sizes (Table 3 and 4):
  - No. of observations: e.g., 63,907 for pooled size and sales regressions; No. of firm fixed effects: e.g., 465.
  - Example coefficients (Table 3, column 1): Tangibility (τ) = 5.3048*** (SE 0.1112); τ×Age 2 = −0.3383*** (SE 0.1462); τ×Age 31 = −1.1126*** (SE 0.3545).
  - Example coefficients (Table 4, column 3): Tangibility (τ) = −30.2926*** (SE 0.6939); τ×Age 3 = 10.9938*** (SE 0.8853).

### Conclusions and implications
- Modeling accumulation of intangible assets is crucial for explaining the level and age dynamics of Tobin’s q and firm heterogeneity.
- Financial frictions arising from limited contract enforcement with asset-specific repudiation values generate:
  - Discouraged entry of intangible-intensive technologies,
  - Prolonged inefficiently small sizes for intangible-intensive firms,
  - Distortion of investment away from intangible assets (misallocation between physical and intangible).
- Empirical evidence from U.S. public firms (1979–2009) supports model predictions: intangible asset share predicts entrant size, growth, and market valuation patterns; predictive power diminishes with firm age.
- Suggested model extensions (discussed in text):
  - Allow aggregate fluctuations to analyze cyclicality of firm dynamics.
  - Distinguish firm from production technology to allow acquisitions or technology switching.

*Italic: Source — contents extracted from the provided PDF content unit.*

### Appendix I. Proofs

### _wp1488 - Appendix I. Proofs

### Main theoretical results and proof structure
- Lemma 1
  - (i) π(V; z) is weakly increasing in V following monotonicity of the constraint set.
  - (i) W(V; z) is weakly increasing in V following monotonicity of π(V; z) and standard dynamic programming arguments.
  - (ii) If for all Ṽ V(z), W(V; z′) = W̃(z′) and V(z′) = Ṽ(z′), then (11) implies W(V; z) = E_{z′|z}[π(z) + β W̃(z′)] = W̃(z).
  - (iii) If V < Ṽ(z) then V < V^n(z) for some n. Induction on n is used: for n = 1, V < V^1(z) implies (by the model constraints and notation) W(V; z) < W̃(z). The inductive step uses either V < V^1(z) or V < β E_{z′|z} V^{n-1}(z′) and follows similarly.

- Lemma 2
  - π(V; z) is strictly concave in V by strict concavity of F(:) and convexity of the constraint set.
  - W(V; z) is concave by Theorem 9.8 in Stokey, Lucas and Prescott (1989).
  - If V < Ṽ(z) then there exists n with V < V^n(z). Induction on n shows strict concavity of W(V; z) in a neighborhood of V. For n = 1 strict concavity follows from π(V; z); the inductive step uses that the optimal continuation value V′(z′) < V^{n-1}(z′) on a subset with positive measure.

- Lemma 3
  - If V2 < V1 < Ṽ(z) and W(V2; z) = W(V1; z), concavity of W implies W(V2; z) = W(Ṽ(z); z), contradicting Lemma 1. Hence strict monotonicity.

- Lemma 4
  - Substituting (18) into (19) and comparing with (11) establishes the claimed equality.

- Remarks
  - The proofs for (ii) and (iii) of Lemma 1 follow from Albuquerque and Hopenhayn (2004).

### Results under Cobb-Douglas production (η_k = 0, deterministic z)
- Production technology: F(k; a; γ) = e^{z} (k^{γ} a^{1−γ})^{θ}, with deterministic z and η_k = 0.
- Proposition 5 (comparative statics in γ)
  - Suppose γ2 < γ1 and V exists such that V < Ṽ(γ1) < Ṽ(γ2).
  - Indirect profit function:
    - π(V; γ) = max_{k′, a′} [−R_k (k′ − k) − R_a (a′ − a) + β F(k′; a′; γ)], subject to V ≥ η_a a′.
  - Equality of the enforcement constraint implies a′(V; γ1) = a′(V; γ2) = V / η_a.
  - First-order condition for k′: R_k = β ∂F(k′; a′; γ)/∂k′ implies k′(V; γ) is increasing in γ.
  - π(V; γ) = β(1 − θ γ) F(k′(V; γ); a′(V; γ); γ) − R_a a′(V; γ): π(V; γ) is increasing in γ given V.
  - W(:; γ) is increasing in γ following monotonicity of π(:; γ) in γ. B(V; γ) = W(V; γ) − V is increasing in γ.
  - By Lemma 2 and 3, W(V; γ) is strictly increasing and strictly concave in V for V < Ṽ(γ); thus B(V; γ) is strictly concave in V.
  - For entrants: if debtholders’ initial participation constraint satisfies B(V0; γ) ≥ I0 for some V0 ≥ 0, the initial contract is signed.
  - The optimal initial equity value V0(γ) lies in a region where B(V; γ) is decreasing in V. Strict concavity of B implies V maps one-to-one to B(V; γ) for feasible V.
  - Define transformation V = B^{-1}(B(V; γ); γ). Total differentiation yields ∂V(B; γ)/∂γ = ∂B^{-1}(B; γ)/∂γ > 0.
  - Conclusion: ∂V0(I0; γ)/∂γ > 0. Given V0 = η_a a0 and monotonicity of k′ and a′ in γ, entrant asset size is increasing in γ.

- Proposition 4 (asset composition and shadow value λ)
  - Let λ ≥ 0 be the Lagrange multiplier for the enforcement constraint. Optimality conditions imply:
    - k_a = (γ / (1 − γ)) (R_a + λ η_a) / (R_k + λ η_k).
  - Conditional on γ, the size of physical assets relative to intangible assets is increasing in λ if η_a > η_k and R_a = R_k.
  - As the firm ages, total surplus W is strictly concave and strictly increasing in V; the shadow value λ decreases until the firm reaches the efficient frontier and λ = 0.
  - Consequence: a firm’s share of physical assets is higher than its unconstrained level and falls monotonically until the firm becomes unconstrained.

### Appendix II — Variable measurement
- Total debt = long-term debt (item DLTT) + short-term debt (item DLC).
- Market value = total debt + market value of common equity (CRSP December market capitalization) + book value of preferred stock (item PSTKRV) − cash and short-term investments (item CHE) − inventory (item INVT).
- Tobin’s q = market value / physical assets.
- Cash flow = income before extraordinary items (item IB) + depreciation (item DP).
- Cash flow rate = cash flow / physical assets.
- To alleviate impacts of outliers, growth and financial measures (including Tobin’s q and market value to total assets) are windsorized at 2% and 98%.
- Timing convention: Compustat reports both stock and flow variables at the end of year t. The model requires stock variables at the beginning of year t and flow variables over year t. Any year t stock variable (e.g., k_t) is taken from the year t−1 balance sheet; any year t flow variable is taken from the year t income or cash flow statement.

### Appendix III — Numerical solution, calibration, solution strategy, and identification

- Calibration (parameters and model setup)
  - Period = one year.
  - Interest rate r = 0:04.
  - Exit probability φ = 0:10, which implies discount rate β = 0:865.
  - Production function: F(k; a; z) = Z e^{z} (k^{γ} a^{1−γ})^{θ}.
  - Parameter choices and rationale:
    - Return to scale θ = 0:25 (set so labor share = 0:7 given a mapping with labor; example ε = 0:95 used in illustration).
    - Depreciation rates: δ_k = 0:08 and δ_a = 0:08.
    - Productivity process: z′ = ρ z + ε′, with ε truncated normal mean zero, standard deviation σ, finite support [−2σ; 2σ].
    - Discretization: Tauchen (1991) with two points; pick ρ = 0:40 and σ = 0:05 to match mean and variance of investment rate among old firms.
    - Discretize state spaces V, V′, k′, a′ into 90-point uniform grids.
    - Parameter Z scales average level of assets.
    - Firm-specific production parameter γ is set to match the distribution of physical and intangible assets in the data; γ is drawn from a CDF H(γ) with support [0:1; 0:9] and discretized into 5 points.
  - Notes on calibration checks:
    - Simulation with θ = 0:25 and benchmark parameters yields variation of unconstrained total assets size with respect to γ: standard deviation 0:11 and average size of medium class is 7% lower than low class.

- Table 5: Parameter values (as reported)
  - Calibrated parameters
    - Interest rate r 0:04
    - Exit rate φ 0:10
    - Return to scale θ 0:25
    - Depreciation rate (physical) δ_k 0:08
    - Depreciation rate (intangible) δ_a 0:08
    - Shock persistence ρ 0:40
    - Stochastic shock variance σ 0:05
  - Estimated parameters
    - Repudiation value (physical) η_k 0:52
    - Repudiation value (intangible) η_a 0:87

- Solution strategy
  - Numerical solution by value function iteration on a multigrid scheme.
  - Simulate a model economy with 10,000 firms over 200 periods.
  - Set I0 = 0 and jointly calibrate {η_k, η_a} to match simulated moments, particularly the size and growth rate premium associated with intangible assets.

- Identification
  - The set up cost I0 and repudiation values η_k and η_a have similar implications in a representative-firm model but can be separately identified using heterogeneous-firm observations.
  - Identification argument using entrant firm sizes for three classes L, M, H (different γ):
    - B_L(η_k k_L + η_a a_L; η_k, η_a) = B_M(η_k k_M + η_a a_M; η_k, η_a) = B_H(η_k k_H + η_a a_H; η_k, η_a)
    - This identifies η_k and η_a using observed size differentials associated with γ.
    - Once η_k and η_a are identified, setup cost I0 is the level such that B_L(:) = B_M(:) = B_H(:) = I0.
  - Role of I0: affects initial entrant size and age when firms become unconstrained; η_k and η_a affect entrant size and time to mature in similar manners as I0.

- Numerical illustration and model intuition (Figure 7 description)
  - With parameter values in Table 5 except η_k = 0:3 and η_a = 0:95:
    - W as a function of V is strictly increasing and strictly concave until V reaches threshold Ṽ(γ) (efficient frontier). For V < Ṽ(γ) the marginal change in W with respect to V is larger for higher γ.
    - B as a function of V: as V increases, B decreases more than one-to-one with respect to V due to postponed dividend payment until V = Ṽ(γ). The maximum debt achievable under the optimal contract  B̄(γ) is increasing in γ; if B̄(γ) ≥ I0 then the optimal contract picks B0(γ) = I0, otherwise no contract satisfies debtholders’ initial incentive constraint.
    - V0(γ) is monotone in γ: firms with a higher share of physical assets have higher V0.
    - Continuation value V′(z_l) as a function of V: for low current z and three γ values, V′(γ) increases at a constant rate until V reaches Ṽ(γ); threshold Ṽ(γ) is higher for firms with higher γ, reflecting more restrictive borrowing constraints.
    - Regions where V ≥ Ṽ(γ): V is irrelevant for total surplus and many payment schedules possible; example illustrated where firm pays interests and keeps debtholder value constant.
  - Comparative parameter experiments:
    - Lower financial frictions (η_k = 0:05, η_a = 0:95): firms with low share of intangible assets are most affected, reach efficient frontier sooner, and have higher B̄(γ).
    - Decreased spread between repudiation values (η_k = 0:5, η_a = 0:75): reduces differences between firms; efficient frontiers Ṽ(γ) are closer across γ and variation in entrant value V0 is smaller.

*Source: _wp1488 - Appendix I. Proofs (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp1488.pdf_