## wp17176

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### Agents and Maximization Problem (Sections 3.1, 3.2)
- General framework
  - Firms in a small open economy produce one differentiated variety in monopolistic competition.
  - World output: Y^w_t = ( sum_{j=0}^{N^w} (y^j_t)^{(σ−1)/σ} d_j )^{σ/(σ−1)}.
  - σ is elasticity of substitution; firm-level variables lowercase, aggregates uppercase.
  - Real global income D_t is exogenous: deterministic growth rate g ≥ 0 times a stochastic stationary component.
  - Firm revenue relation: p^j_t y^j_t = (D_t)^{1/σ} (y^j_t)^{(σ−1)/σ}.
  - Spanish manufacturing: maximum number of domestic varieties N constant and N ≤ N^w.
- Production technology
  - Firms use labor l, tangible capital k, private intangible capital a, and public intangible capital A; fixed cost T each period.
  - Production function (Cobb-Douglas): y = A^{1−α−β} a^{β} k^{α} l^{1−α}, with α, β ∈ (0,1) and α + β < 1.
  - Assumption: (1 + β) ( (σ−1)/σ ) < 1 to ensure private optimal firm size is real.
  - Physical capital law of motion: k′ = ( k + i_k − φ_k(i_k / k) k ) (1 − δ_k).
  - Private intangible law of motion: a′ = ( a + i_a − φ_a(i_a / a) a ) (1 − δ_a z′).
  - Adjustment cost: φ_x(i_x / x) = η_x ( ( (i_x / x) − ( δ_x / (1 − δ_x) ) ) / ( 1 + ( δ_x / (1 − δ_x) ) − η_x ) )^2 for x ∈ {k, a}.
  - Idiosyncratic depreciation shock: log(z′) = min{ ρ log(z) + ν, log(1/δ_a) }, ν ∼ N(0, θ), ρ ∈ [0,1), process normalized to mean one.
  - Fraction κ ∈ (0,1] of private depreciation diffuses to other firms each period; public intangible accumulation: A′ = ( (F − A) / A ) κ δ_a a + A.
  - Normalize BGP: stocks grow at rate g; choose normalization so in BGP (F − A)/A = 1 ⇒ A = κ δ_a a / g < F for g > 0.
  - Assume κ = 1 henceforth for cyclical analysis (quantitative analysis also considers A exogenous at rate g).
- Exit and entry
  - Exit and entry endogenous; liquidation implies divesting all capital stocks and losing fraction η_x of capital x (i_x = −(1 − η_x) x).
  - Potential entrants up to N; entrants occupy varieties of exited firms.
  - Entrant endowment: initial equity e_0 = −b_0 > 0; entry cost γ e_0.
  - Entrant observes z_0; entrants with lower z_0 have priority if idle varieties limited.
  - Entrant constraint: k_0 (1 − δ_k) + a_0 (1 − z_0 δ_a) = −(1 − γ) b_0; entrants do not issue debt in period zero.
- Firm financing and default
  - Firms finance via internal profits and defaultable one-period bonds b priced q_b; investors observe firm and aggregate states.
  - Recoveries in liquidation up to fraction ψ_k of tangible assets and ψ_a of intangible assets.
  - Collateralizable maximum: b̄ ≡ (1 − η_k) (1 − ψ_k) k + (1 − η_a) (1 − ψ_a) a. Debtholders recover min{ b, b̄ } in liquidation.
  - Key empirical assumption: ψ_k > ψ_a (physical capital easier to seize than intangible).
  - Firms can save in risk-free bond (b < 0) with net risk-free rate R_f and no divestment cost.
  - No equity issuance in equilibrium (new equity issuance cost prohibitive).
- Aggregate shocks and asset pricing
  - Aggregate state M_t ∈ { M_n, M_c, M_f } evolves via transition matrix Π_M.
  - Aggregate state determines global income D(M) and equity premium R_e(M): D(M_c) < D(M_n), D(M_f) < D(M_n); R_e(M_n) = R_e(M_c) < R_e(M_f).
  - Real wages normalized to W = 1 for all states.
  - Perfect diversification against idiosyncratic risk ⇒ agents price only aggregate risk via state-price matrix S where S(M, M′) is price of next period’s state M′ conditional on current state M.
- Dynamic firm problem and equilibrium
  - Firm state: k, a, A, b, z, M, production status.
  - Incumbent value: V(k,a,A,b,z,M) = max{ C(k,a,A,b,z,M), L(k,a,b) }.
  - Continuation value C chooses { i_k, i_a, l, b′ } to maximize d + Σ_{M′} S(M,M′) E[V(k′,a′,A′,b′,z′,M′)] subject to d ≥ 0 and laws of motion.
  - Dividends: d = v + q_b b′ − b with net profits v ≡ (D(M))^{1/σ} y^{(σ−1)/σ} − W(M) l − i_k − i_a − T.
  - Liquidation value: L = (1 − η_k) k + (1 − η_a) a − min{ b, b̄ }.
  - Entrant problem: max_{k_0,a_0} V(k_0,a_0,A,0,z_0,M) subject to k_0 (1 − δ_k) + a_0 (1 − z_0 δ_a) = −(1 − γ) b_0.
  - Competitive equilibrium: debt price q_b, quantities and decisions satisfy firm optimization, investor pricing given S(M), and laws of motion for k, a, A.

### Numerical computation and characterization (Appendix A.1)
- Labor and FOCs
  - Optimal labor analytically solved: l = [ (σ−1) (1−α) / σ W D^(1/σ) ( A^(1−α−β) a^β k^α )^(σ−1)/σ ]^(σ / (σ−(σ−1)(1−α))). (Equation 19)
- Debt issuances and pricing
  - In equilibrium firms never pay positive dividends ⇒ d = 0 and b′ = (q_b)^−1 (b − v). (Equation 20)
  - Debt price: q_b = Σ_{M′} S(M,M′) E[R(k′,a′,A′,b′,z′,M′)]. (Equation 21)
  - Repayment rate R = 1 if C ≥ L; R = min(b̄/b, 1) if C < L. (Equation 22)
  - q_b determined by expected repayment rate and covariance of repayment with aggregate state via S(M,.).
- Numerical algorithm steps
  - Construct grid for k, a, k+a, z, A, and M; grid denser where continuation and liquidation values are close.
  - Assume parametric form for continuation C linear in {r, zr, k, a, b}, where r ≡ py − T − Wl; include zr term to capture z value conditional on continuation.
  - Use non-uniform discretization for z′ expectations; approximate A′ = f(M) + A and iterate on f with simulated data.
  - Maximize numerically over i_k and i_a per grid point subject to labor optimality, d = 0, and bond pricing.
  - Iterate until continuation coefficients converge (correlation > 99.99 percent).
  - Parametric investment policy for simulation linear in a rich set: {1, py, r, r_k, r_a, rz, rb_{k+a}, 1/r, k, k^2, a_{k+a}, (a_{k+a})^2, A, b_{k+a}, (b_{k+a})^2, 1_{k+a−b}, z, zb_{k+a}, zk, za_{k+a}}.
  - Simulate an economy with firm heterogeneity matching data and iterate on parameters to minimize SMM distance.

### Debt supply intuition and comparative statics (text description of Figure 2)
- Competitive debt interest rate (1/q_b − 1) increases in debt issuance b′.
- For b′ below b̄′ default probability is zero and inverse loan supply is flat; beyond that default probability and covariance with aggregate state increase rates.
- Intangible-intensive firms face steeper supply curve once default probability > 0 due to lower recovery and higher covariance with aggregate state prices.
- Financial shocks raise aggregate risk price and steepen both curves, more so for intangible firms.

### Firm life-cycle, investment, leverage, entry and exit (numerical patterns)
- Life-cycle
  - Firms start small with no debt; invest heavily when young issuing bonds; leverage rises then falls as internal finance dominates.
  - Mature firms: intangible investment rates > tangible investment rates because private depreciation of intangibles larger.
  - Young firms: tangible investment can exceed intangible due to stricter constraints and higher intangible adjustment costs.
  - High leverage reduces affordability of external finance and compresses intangible investment.
  - Investment rates not monotone in leverage; moderate leverage can raise investment due to owners’ default option.
- Entry/exit
  - Default implies exit; exit can also occur absent default due to idiosyncratic z shocks erasing private intangible stock.
  - Entrants have lower z (higher creativity) on average; threshold z_0 exists above which entrants do not enter.
  - Maximum number of varieties fixed; adverse aggregate shocks → negative net entry → fewer varieties, amplifying GDP fall in financial shocks.

### State-price construction (S) and aggregate shock parametrization (Section 4.1 and Table 1)
- Identification from data: infer S from R_f, R_e(M), and realized market returns x_m without specifying a utility function.
- Procedure:
  - Estimate reduced 2-by-2 state-price matrix ̄S for the two non-financial-shock states combined using risk-free price 1/(1+R_f) and market price 1/(1+R_f+R_e(M)), assuming market payoffs 1+m and 1−m with variance matching empirical variance of x_m. (Equation 23)
  - Approximate 3-by-3 S by allocating reduced-state prices across two non-financial states proportionally to unconditional probabilities. (Equation 24)
- Table 1 reported values:
  - Aggregate Shock Values (rows: normal, non-finan. shock, financial shock)
    - global income10.98 0.92
    - equity premium 4.0% 4.0% 13.9%
  - Transition matrix Π_M (rows = current state; columns = next period)
    - normal → normal 0.84, non-finan. shock′ 0.11, financial shock′ 0.05
    - non-finan. shock → 0.38, 0.38, 0.24
    - financial shock → 0.14, 0.53, 0.33
  - State Prices S (rows current state; columns next period)
    - normal → normal 0.33, non-finan. shock′ 0.04, financial shock′ 0.60
    - non-finan. shock → 0.18, 0.19, 0.60
    - financial shock → 0.02, 0.09, 0.85
- Remarks: for each current state row, state prices across columns sum to 1/(1+R_f); composition varies with equity premium and financial shock increases state price on next-period financial state.

### Data, estimation, and parameter estimates (Section 4.2, Tables 2 and 3)
- Data sources and construction
  - Main microdatasets: Survey on Business Strategies (ESEE): panel of 1,800 Spanish manufacturing firms 1990–2013; Public Registry of Firms (DIRCE) for representativity weights and aggregate entry/exit.
  - Intangible investment defined as sum of R&D, marketing and advertising, workers training and technology imports (Corrado et al. (2009) approach).
  - Sample used: after outliers, 23,380 firm-year observations 1992–2013.
- Aggregate shock classification and calibration
  - Global income variable: real GDP in the E.U. excluding Spain.
  - Normal years: GDP growth weakly higher than period average 2 percent; shocks otherwise.
  - Financial shock if any year in same shock sequence has expected equity premium > 10 percent.
  - Identified periods: non-financial shock: 1981-1983, 1992-1994, 2003-2004; financial shock: Great Recession 2009-2013.
  - Mapping: normalize D(M_n) = 1; global income falls by 2 percent in a non-financial shock and by 8 percent in a financial shock.
  - Real risk-free rate R_f = 3.3 percent constant.
  - R_e(M) increases from 4 to 14 percent between normal and financial shock states.
- Estimation approach
  - Structural parameters estimated by Simulated Method of Moments (SMM); compare model-simulated moments to data for 1992–2013 feeding empirical M history.
  - Estimation minimizes squared distance weighted by optimal GMM matrix; uses cross-sectional and within-firm moments.
- Key parameter estimates (Table 2 highlights)
  - α = 0.19 (s.e. 0.02); β = 0.19 (s.e. 0.01); exponent on public stock 1 − α − β = 0.62.
  - δ_k = 0.08 (s.e. 0.02); δ_a = 0.12 (s.e. 0.01).
  - T = 0.45 (s.e. 0.06); γ = -0.26 (s.e. 0.01) (entry cost 26 percent of initial debt b_0).
  - entrants net debt b_0 = -1.49 (s.e. 0.38).
  - ψ_k = 0.65 (s.e. 0.23); ψ_a = 0.26 (s.e. 0.09).
  - η_k = 0.08 (s.e. 0.02); η_a = 0.18 (s.e. 0.10).
  - ρ = 0.09 (s.e. 0.03); θ = 0.85 (s.e. 0.16).
  - σ = 4 (calibrated).
  - Adjustment costs imply upon liquidation firms lose 8 percent of tangible value and 18 percent of intangibles.
- Moments fit (selected comparisons, Table 3 highlights)
  - labor costs / value added: data 60.9, s.e. 1.76, model 61.1
  - tangible / fixed assets: data 55.6, s.e. 5.35, model 51.6
  - average ik/kmature firms: data 11.9, s.e. 0.61, model 11.7
  - average ia/amature firms: data 15.6, s.e. 4.11, model 6.2
  - average entry and exit rates: data 5.4, s.e. 0.6, model 5.3
  - within sd. ik/(a+k): data 20.0, s.e. 0.4, model 5.7
  - within sd. ia/(a+k): data 11.4, s.e. 0.5, model 3.1
  - autocorrelation ia/(a+k): data 56.2, s.e. 9.64, model 49.8
  - average leverage: data 18.3, s.e. 0.72, model 22.8
  - Goodness of fit: model fits majority of targeted moments but understates within-firm volatility; lowering adjustment costs or raising idiosyncratic shock variance would raise volatility but also increase average leverage.

### Macroeconomic analysis: role of endogenous intangibles and the Great Deviation (Section 6, Table 4)
- Model comparison: full model (endogenous intangible investment, higher financing costs for intangible investment, spillovers) vs standard/exogenous-intangible-investment model.
- Main macro findings
  - For financial shock (2009-2013), full model fits extent of GDP fall between 2008 and 2013 and predicts persistent deviation from trend even with immediate return to normal in 2014.
  - Standard model misses more than half of GDP fall in Great Recession and rebounds quickly.
  - For smaller non-financial shock, models evolve more similarly though full model reacts more due to additional capital subject to frictions.
- GDP change decomposition (percentual contributions to deviation in GDP w.r.t. trend) — Table 4 (columns summarized)
  - Columns: full model 2008-13, full model 2008-18, exog. intan. 2008-13, exog. intan. 2008-18, no spillov. 2008-13, same ψ 2008-13
  - global income: -1.90, -1.90, -1.9, -1.9
  - physical capital: -2.0, -0.1, 0.2, 0.3, -1.7, -0.2
  - private intan. capt.: -1.6, -0.3, -0.2, 0.0, -1.7, -0.4
  - public intan. capt.: -0.1, -0.1, 0.0, 0.0, 0.0, 0.0
  - labor: -11.4, -0.8, -3.8, 0.5, -11.0, -4.6
  - allocative efficiency: -2.0, -0.1, -0.5, 0.1, -2.0, -0.5
  - entrants: -6.8, 3.5, -0.8, 0.0, -5.8, 1.4
  - incumbents: -7.8, -0.6, -5.1, 1.0, -9.1, -6.8
  - exiters: -4.5, -4.1, -0.5, -0.2, -3.1, -2.1
  - total: -19.0, -1.2, -6.3, 0.8, -18.0, -7.5
- Key amplification and persistence channels (Section 3.1.2, Section 6.2)
  - Lower collateralizability of intangibles: ψ_k > ψ_a implies intangibles provide less recoverable collateral; intangible-intensive firms face higher borrowing costs and higher default sensitivity to aggregate risk.
  - Spillovers to public intangible capital: fraction κ of private depreciation diffuses; public intangible A evolves with mean reversion implying lower effective social depreciation and persistence.
  - Higher adjustment costs for intangibles: η_a > η_k slows recovery of private intangible stock and dampens reallocation.
  - Complementarity across inputs and convex adjustment costs transmit financial constraints on intangibles to physical investment and labor (intensive margin) and through net entry/exit reduce varieties (extensive margin).
  - Distinction from standard financial accelerator: persistence arises from increased discount on aggregate risk disproportionately affecting intangible-intensive firms rather than a uniform rise in interest rates.

### Decomposition by firm type and policies (Section 6.2–6.3)
- Decomposition by firm type
  - Compute difference between each group's contribution and BGP average to account entrants create VA and exiters destroy it by definition.
  - Bulk of changes during financial shock due to incumbents; exiters generate persistence because balance-sheet deterioration and liquidation take time.
  - Compared to exogenous-intangible model, exiters account for larger differences over whole decade.
- Policy objectives and mechanism
  - Welfare proportional to sum of existing firm value (agents hold diversified portfolios).
  - Decentralized inefficiencies from borrowing constraints (no-equity-issuance, limited commitment) and intangible spillovers.
  - Policy principle: increase investment rate of firms with higher marginal returns via budget-neutral transfers conditional on observable characteristics correlated with returns: age and size.
- Age-based transfers (one-time, unanticipated in 2009, budget-neutral)
  - Optimal transfers proportional to equilibrium interest rate (q_b)^{-1} by age.
  - Transfer pattern: positive for young firms and negative for older firms; most constrained firms are age between 4 and 8.
  - Aggregate effects: avoids 3.3 percentual points of the 2008-2013 GDP fall (17 percent of the total fall) and accelerates return to trend; benefits accrue as constrained firms deleverage and increase investment while a few old firms exit in 2009.
- Size-based (employment) transfers
  - Optimal one-time unanticipated budget-neutral transfers proportional to labor l in 2009.
  - Less effective than age targeting: avoids 1.7 percentual points of the 2008-2013 GDP drop.
  - Size less correlated with marginal return; targeting small firms reduces inefficient exit because small firms more affected by fixed costs.
- Comparison with credit subsidies (e.g., Juncker Plan)
  - Credit subsidies dominated by well-targeted outright transfers in heterogeneous-firm model with endogenous borrowing constraints.
  - Lowering interest rates moderately increases leverage capacity only marginally; subsidized credit crowds out private unsubsidized credit; excessive discounts may distort unconstrained firms into excessive leverage.
- First-best vs second-best
  - Targeted transfers are second-best; first-best would be legal reforms increasing pledgeability or facilitating external equity financing.

### Robustness, benchmark, and additional empirical patterns (Section 1, Benchmark model)
- Benchmark (exogenous intangible investment) model
  - Equivalent to Midrigan and Xu (2014) with endogenous default; i_a fixed, costless, no adjustment costs; A fixed to 1 to suppress spillovers.
  - Assumptions made to reduce gap with full model (e.g., dividend payments) are conservative.
- Empirical patterns and additional moments
  - Tangibility average k/(k+a) = 57 percent in ESEE sample.
  - Average leverage (value-added weighted) = 23 percent in Great Recession discussion; average leverage targeted in estimation is 18.3 (data) vs 22.8 (model).
  - Aggregate TFP (value added net of labor and physical capital) fell by 5 percentage points between 2008 and 2013 in the ESEE sample.
  - Cross-industry: industries with lower average asset tangibility reduced investment rates more after the Great Recession.
- Numerical solution practicalities
  - Use non-uniform grid, parametric continuation function, dense discretization where liquidation option close, iterate until continuation coefficients correlation > 99.99 percent.
  - Simulation and block-bootstrapped standard errors: block bootstrap with 500 samples for firm-level moments; aggregate entry/exit s.e. from assumed VAR(1); parameter var-cov matrix accounts for simulation error.

### Conclusions and suggested avenues for research (Section 7)
- Modeling intangible assets introduces amplification and persistence channels to financial shocks because:
  - Intangibles are less collateralizable.
  - Intangibles take longer to recover after firm exit.
  - Intangibles have lower social depreciation rates via spillovers.
- Empirical estimation with Spanish manufacturing data:
  - Endogenous intangible investment helps explain the magnitude and persistence of the Great Recession output fall; lower intangible investment and increased exit rates are key channels.
- Policy implications:
  - Targeted transfers to younger firms more efficient than targeting small firms or subsidizing credit.
  - First-best would involve reforms to increase asset pledgeability or external equity access.
- Future research directions:
  - Frictions in public and private equity issuances for intangible-investing firms.
  - Imperfect portfolio diversification by firm owners.
  - Longer lags in formation of intangible capital; differentiated parameters by intangible type.
  - Empirical linking of firm-level debt interest rates to intangible investment and tangibility; explicit modeling of cyclical changes in aggregate risk prices.

*Source: wp17176 - IMF Working Paper (Sections 1, 3.1–3.2, 4.2, 6.2, Appendix A.1).*

### 3.1  Agents and Maximization Problem

### 3.1  Agents and Maximization Problem

### 3.1.1 General Framework
- Model of firms in a small open economy; each firm produces one differentiated variety in monopolistic competition.
- World output Y^w_t defined as a CES composite of the N^w final good varieties:
  - Y^w_t = ( sum_{j=0}^{N^w} (y^j_t)^{(σ−1)/σ} d_j )^{σ/(σ−1)}.
  - σ is the elasticity of substitution across varieties; y^j_t is value added output of variety j.
- Variables at firm level are lowercase; aggregates are uppercase.
- Real global income D_t is exogenous: deterministic component growing at rate g ≥ 0 times a stochastic stationary component.
- Given D_t, world consumers maximize Y^w_t taking individual prices p^j_t as given; firm revenue function:
  - p^j_t y^j_t = (D_t)^{1/σ} (y^j_t)^{(σ−1)/σ}.
- Shocks to D_t affect firm revenues via (D_t)^{1/σ} and endogenous y^{(σ−1)/σ}.
- Spanish manufacturing sector: maximum number of domestic varieties N is constant and N ≤ N^w.

### 3.1.2 Production Technology
- Firms use labor l, tangible capital k, and private intangible capital a; public intangible capital is A. Fixed cost T each period.
- Production function (Cobb-Douglas):
  - y = A^{1−α−β} a^{β} k^{α} l^{1−α}, with α, β ∈ (0,1) and α + β < 1.
- Features:
  - Constant returns to scale in (k, l); increasing returns when including private intangible investment.
  - Assumption: (1 + β) ( (σ−1)/σ ) < 1 to ensure private optimal firm size is real.
  - Exponent on A chosen so exponents on (k, a, A) sum to one → simplifies to AK Growth Model from social planner perspective with fixed labor.
- Physical capital accumulation:
  - k′ = ( k + i_k − φ_k(i_k / k) k ) (1 − δ_k).
- Private intangible accumulation (includes depreciation shock z′):
  - a′ = ( a + i_a − φ_a(i_a / a) a ) (1 − δ_a z′).
- Adjustment cost function (quadratic form) for x ∈ {k, a}:
  - φ_x(i_x / x) = η_x ( ( (i_x / x) − ( δ_x / (1 − δ_x) ) ) / ( 1 + ( δ_x / (1 − δ_x) ) − η_x ) )^2.
  - Parameter η_x ensures zero cost when maintaining x constant.
- Idiosyncratic depreciation shock z′:
  - log(z′) = min{ ρ log(z) + ν, log(1/δ_a) }, ν ∼ N(0, θ), ρ ∈ [0,1).
  - Process normalized to mean one; z′ unknown when firm invests.
  - Interpretation: z is inverse of firm creativity; lower z → lower expected depreciation.
- Private intangible depreciation δ_a z′ includes diffusion (spillovers) and technological depreciation:
  - Fraction κ ∈ (0,1] of private depreciation diffuses to other firms each period.
- Public intangible accumulation:
  - A′ = ( (F − A) / A ) κ δ_a a + A.
  - F is the world technology frontier; a is cross-firm average of (a + i_a − φ_a(i_a / a) a) z′.
  - A not subject to technological depreciation; mean reversion term (F − A)/A keeps model stationary.
- Normalization and BGP:
  - All stock variables grow at rate g in long run.
  - Normalize F so that in BGP (F − A)/A = 1 ⇒ A = κ δ_a a / g < F for g > 0.
  - A never reaches F; effective social depreciation of A is g.
- Assumption on κ:
  - As long as κ > 0, κ does not affect cyclical behavior; model focuses on fluctuations relative to trend.
  - Henceforth assume κ = 1 (quantitative analysis also considers no spillovers with A exogenous at rate g).

### 3.1.3 Exit and Entry
- Exit and entry are endogenous. Liquidation implies divesting all capital stocks.
- When liquidating, fraction η_x of capital x is lost in adjustment (i_x = −(1 − η_x) x).
- Potential entrants number equals N (potential varieties), but entrants can only occupy varieties of exited firms.
- Entrant endowment: initial equity e_0 = −b_0 > 0; entry cost γ e_0.
- Entrant observes initial idiosyncratic shock z_0 drawn from stationary distribution; entrants with lower z_0 have priority if idle varieties limited.
- If enter, entrant uses initial equity net of entry cost to invest in k and a and start production; otherwise receives outside value e_0.
- Assumption: depreciation of initial investments equals δ_k and z_0 δ_a; initial investments subject to same proportional adjustment costs η_k and η_a.

### 3.1.4 Firm Financing
- Firms finance via internal profits and by issuing defaultable one-period bonds b at price q_b. Bonds traded competitively; investors observe firm and aggregate states.
- Bond contract: firm receives q_b b′ today and must repay b′ next period to avoid liquidation; q_b can depend on current aggregate and firm states and current firm actions.
- Default option: firms may default and liquidate.
- Recoveries in liquidation: debtholders can recover up to fraction ψ_k of tangible assets and ψ_a of intangible assets.
  - Maximum collateralizable amount b̄:
    - b̄ ≡ (1 − η_k) (1 − ψ_k) k + (1 − η_a) (1 − ψ_a) a.
  - In liquidation debtholders recover min{ b, b̄ }.
  - Owners capture remaining value: (1 − η_k) k + (1 − η_a) a − min{ b, b̄ }.
- Key empirical assumption: ψ_k > ψ_a (physical capital easier to seize than intangible).
- Firms can save in risk-free bond (b < 0) with net risk-free rate R_f and no divestment cost.
- No agency conflict between managers and owners; equity trading possible but new equity issuance cost is prohibitive.

### 3.1.5 Aggregate Shocks
- Aggregate state M_t ∈ { M_n, M_c, M_f } evolves as a Markov Chain with transition matrix Π_M.
- Aggregate state determines global income D(M) and equity premium R_e(M).
  - Both financial and non-financial shocks feature lower global income than normal: D(M_c) < D(M_n), D(M_f) < D(M_n).
  - Equity premium higher in financial shock: R_e(M_n) = R_e(M_c) < R_e(M_f).
- Real wages normalized to W = 1 for all states (wages roughly acyclical in Spanish manufacturing).
- Section 4.1 (estimation) uses E.U. data to estimate Π_M, D(M), R_e(M), and R_f.

### 3.1.6 Asset Pricing
- Perfect diversification against idiosyncratic risk (tradeable equity) ⇒ agents only price aggregate risk.
- Define unique matrix of state prices S where S(M, M′) is price of next period’s state M′ conditional on current state M.
- S(M) prices any asset conditional on M.
- Section 4.1.1 infers S from data on R_f, R_e(M), and realized market returns x_m without specifying a utility function.
- Changes in S over time reflect changes in beliefs about consumption distribution across aggregate states or changes in risk preferences.

### 3.1.7 Dynamic Firm Problem
- Firm state variables: k, a, A, b, z, M, and production status of variety.
- Incumbent firm value:
  - V(k, a, A, b, z, M) = max{ C(k, a, A, b, z, M), L(k, a, b) } where C is continuation value and L is liquidation value.
- Continuation problem: choose { i_k, i_a, l, b′ } to maximize dividends d plus discounted next-period value:
  - C = max_{i_k, i_a, l, b′} [ d + sum_{M′} S(M, M′) E[ V(k′, a′, A′, b′, z′, M′) ] ]
  - Subject to no-equity-issuance condition d ≥ 0 and laws of motion for k, a, A and z.
  - Expectation over pdf of z.
- Dividends:
  - d = v + q_b b′ − b.
- Net profits v:
  - v ≡ (D(M))^{1/σ} y^{(σ−1)/σ} − W(M) l − i_k − i_a − T.
  - Output y given by production function in 3.1.2.
- Liquidation value:
  - L = (1 − η_k) k + (1 − η_a) a − min{ b, b̄ }.
- If no default (b̄ ≥ b), owners recover post-liquidation asset value after repaying b; if b̄ < b, owners repay b and may divert fraction of post-liquidation value.
- Entrant problem (no debt in period zero):
  - max_{k_0, a_0} V(k_0, a_0, A, 0, z_0, M) subject to:
    - k_0 (1 − δ_k) + a_0 (1 − z_0 δ_a) = −(1 − γ) b_0.
  - Entrants do not issue debt in period zero; they start issuing debt after incorporation.

### 3.2 Equilibrium

#### 3.2.1 Definition
- Competitive equilibrium: debt price q_b, quantities { k, a, A, b, i_k, i_a, l }, decision { C, L } for each firm period and number of varieties produced each period such that:
  - i) Firms maximize value given q_b, z, D(M), S(M), W, A, and d ≥ 0.
  - ii) Investors price firm debt competitively given S(M).
  - iii) Laws of motion for k, a, A in equations (4), (5), (8) are satisfied.
- Only endogenous aggregate variable is A. Financial markets globally integrated; labor flows across sectors.
- Small open economy assumption: Spanish manufacturing does not affect global prices or wages → no market clearing conditions needed.

*Source: wp17176 - 3.1  Agents and Maximization Problem*

### Appendix A.1 lists the steps for the numerical computation of the equilibrium.

### wp17176 - Appendix A.1 lists the steps for the numerical computation of the equilibrium

### Characterization: labor and first-order conditions
- Optimal labor is purely static and solves analytically from the firm first order condition:
  - l = [ (σ−1) (1−α) / σ W D^(1/σ) ( A^(1−α−β) a^β k^α )^(σ−1)/σ ]^(σ / (σ−(σ−1)(1−α))). (Equation 19)

### Debt issuances and debt prices
- In equilibrium, firms never pay out positive dividends; therefore d = 0 and dividends definition yields:
  - b′ = (q_b)^−1 (b − v). (Equation 20)
- Debt issuances b′ are pinned down by investment in physical and intangible assets i_k and i_a, which determine q_b and net profits v given optimal labor.
- Investors’ pricing implies the equilibrium price of debt:
  - q_b = Σ_{M′} S(M,M′) E[R(k′,a′,A′,b′,z′,M′)]. (Equation 21)
- Repayment rate R is:
  - R = 1 if C ≥ L; R = min(b̄/b, 1) if C < L. (Equation 22)
  - Interpretation: R = 1 if firm does not exit (liquidate production); otherwise R equals the maximum recoverable amount from assets divided by outstanding debt.
- Two components determine q_b:
  - Expected repayment rate next period.
  - Covariance of repayment rate with the aggregate state (market beta of firm debt) via state prices S(M,.).
- Comparative static illustrated in Figure 2 (text description):
  - Competitive debt interest rate (1/q_b − 1) increases in debt issuances b′.
  - For b′ below b̄′, default probability is zero and the inverse loan supply curve is flat.
  - As b′ grows, default probability and covariance with aggregate state increase, interest rate rises.
  - The supply curve is steeper for the intangible-only investor: for any given b′ once default probability > 0, the intangible-intensive firm faces higher interest rates due to lower recovery in default and higher covariance with aggregate state prices.
  - When aggregate risk price rises (financial shock), both curves steepen, with the intangible firm's curve steepening more.

### Investment, firm life cycle, and leverage
- Equilibrium investments i_k and i_a are obtained by numerical optimization (Appendix A.1).
- Firm life-cycle patterns (Figure 3, described):
  - Firms start small and with no debt; invest at high rates when young, issuing bonds to finance investment—leverage rises until firms mature and deleverage as internal finance dominates.
  - Mature firms: intangible investment rates > tangible investment rates because private depreciation of intangibles is larger (see estimated parameters in Table 2).
  - Young firms: tangible investment rates can exceed intangible due to stricter financial constraints and higher adjustment costs on intangible investment.
  - High leverage reduces external financing affordability and particularly compresses intangible investment rates.
  - Note: investment rates are not monotonically decreasing in leverage; near moderate leverage, higher leverage can raise investment rates due to the real option from owners’ possibility to default and capture assets.

### Exit, entry, and varieties
- Default leads to exit (liquidation); exit can also occur absent default due to idiosyncratic shock z that can erase private intangible stock.
- Entrants differ only by initial intangible depreciation shock z0; threshold z0 exists above which potential entrants do not enter.
- Entrants tend to have lower z (higher creativity) because they face higher adjustment and external financing costs.
- Maximum number of varieties is fixed; fluctuation in actual number produced arises from net entry/exit. Adverse aggregate shocks → negative net entry → lower varieties → deepen GDP fall in financial shock states.

### Amplification and persistence of financial shocks
- Key modeled features producing amplification and persistence: lower collateralizability of intangibles, spillovers to public intangible capital (social depreciation), and higher adjustment costs for intangibles.
- Lower collateralizability and financial constraints:
  - ψ_k > ψ_a implies intangible assets less collateralizable; intangible-intensive firms have lower expected repayment rates in shock states.
  - Debt financing costs for intangible investment are higher, exacerbated in financial shock states where the discount on aggregate risk increases.
  - Convex adjustment costs and production complementarity prevent concentrating adjustment solely on intangibles → financial constraint transmits to physical investment and labor.
  - Higher aggregate risk price increases default and exit rates, especially for intangible-intensive firms; since entrants do not automatically replace exiters, the fall in number of varieties deepens GDP decline.
  - Amplification operates via intensive (investment) and extensive (entry/exit) margins.
- Persistence channels:
  - Gradual spillovers and lower social depreciation: public intangible stock increases persistence because intangibles’ social depreciation is lower; aggregate risk price rises disproportionately affect capital with lower social depreciation.
  - Adjustment costs and life-cycle growth: η_a > η_k implies higher adjustment costs for intangible investment. Higher exit rates and higher adjustment costs for intangibles slow recovery of aggregate intangibles on the extensive margin as entrants (starting small) take longer to rebuild intangible capital.
- Distinction from standard financial accelerator:
  - The model’s persistence after financial shocks arises not merely from higher average interest rates but because the discount on aggregate risk rises and intangible-intensive firms bear more aggregate risk. An increase in the risk-free rate would resemble a non-financial shock, affecting tangibles and intangibles similarly.

### Data: aggregate shocks and estimation
- Aggregate shocks estimated with E.U. data; other structural parameters estimated with Spanish manufacturing firm-level data (1990-2013).
- Classification of years into M ∈ {normal, non-financial shock, financial shock}:
  - Global income shock variable: real GDP in the E.U. excluding Spain.
  - Normal years: GDP growth weakly higher than period average: 2 percent; remaining years are “shock” years.
  - Distinguish financial vs non-financial shocks using expected equity premium in E.U. stock market: a year is a financial shock if any year in same shock sequence has expected equity premium > 10 percent.
  - According to algorithm: non-financial shock periods are 1981-1983, 1992-1994 and 2003-2004. The only financial shock identified is the Great Recession, 2009-2013.
- Expected equity premium computed a la Cochrane (1999) by regressing realized returns on one-year-lagged dividend-price ratio.
- Observed patterns: Great Recession accompanies a large, persistent increase in expected returns and higher real interest rates for Spanish non-financial firms’ debt (bonds and loans); dot-com years have very low expected returns.
- Mapping M to global income D(M) and equity premium R_e(M):
  - Normalize D(M_n) = 1.
  - Global income falls by 2 percent in a non-financial shock and by 8 percent in a financial shock.
  - Real risk-free rate R_f set constant at 3.3 percent.
  - R_e(M) increases on average from 4 to 14 percent between normal and financial shock states.
- Transition matrix estimation:
  - Two-stage estimation due to limited years. Conditional on normal state, estimate transition probabilities from sample. For shock states, probability of transitioning to the other shock type is set equal to unconditional frequency of that shock divided by unconditional probability of both shock states.
  - Transition matrix used to simulate M time series; agents’ behavior determined by state prices S.

### State prices (S)
- Under no-arbitrage, a vector of positive state prices exists and is inferred using observed R_f, R_e(M), and realized market returns; but data yields payoffs/prices for only two independent assets (risk-free bond and market security).
- Procedure:
  - Estimate reduced 2-by-2 state-price matrix ̄S for the two non-financial-shock states combined using:
    - Price of risk-free security = 1/(1+R_f).
    - Price of market security = 1/(1+R_f+R_e(M)).
    - Assume market payoff = 1 + m in states without a financial shock and 1 − m in financial shock, where m chosen so variance of {1+m,1−m} equals empirical variance of realized market returns x_m.
    - ̄S = ( [[1,1],[1+m,1−m]] )^−1 * [1/(1+R_f), 1/(1+R_f+R_e(M_n)); 1/(1+R_f), 1/(1+R_f+R_e(M_f)) ]^T. (Equation 23)
  - Approximate 3-by-3 S by allocating the reduced-state prices across the two non-financial states proportionally to their unconditional probabilities:
    - S = ( π1/(π1+π2) ̄S1 ; π2/(π1+π2) ̄S1 ; ̄S2 ). (Equation 24)
- Table 1 (reported estimates)
  - Aggregate Shock Values
    - normal   non-finan. shock   financial shock
    - global income10.980.92
    - equity premium    4.0%4.0%13.9%
  - Transition Matrix (Π_M) (rows = current state M; columns = next period’s state M′)
    - normal′  non-finan. shock′  financial shock′
    - normal0.840.110.05
    - non-finan. shock    0.380.380.24
    - financial shock0.140.530.33
  - State Prices (S) (rows = current state; columns = next period’s state)
    - normal′  non-finan. shock′  financial shock′
    - normal0.330.040.60
    - non-finan. shock    0.180.190.60
    - financial shock0.020.090.85
- Remarks on S:
  - For each current state (row), state prices across columns sum to the price of a risk-free bond 1/(1+R_f).
  - Composition varies with the equity premium: in a current financial shock, next-period financial shock state has higher state price, consistent with a low equity payoff and higher equity premium.

*Source: wp17176 - Appendix A.1 lists the steps for the numerical computation of the equilibrium.*

### 4.2  Firm-Level Data

### 4.2  Firm-Level Data

### Data sources and construction
- Main microdatasets:
  - Survey on Business Strategies (ESEE): panel of 1,800 Spanish manufacturing firms from 1990 to 2013; surveys whole population of firms with more than 200 employees and a representative sample of firms between 10 and 200 employees; sample covers around 35 percent of value added in Spanish manufacturing.
  - Public Registry of Firms (DIRCE): whole population of manufacturing firms, used to compute firm representativity weights and aggregate entry and exit rates.
- Intangible investment definition and measurement:
  - Intangible investment is the sum of four accounting entries: R&D expenditures, marketing and advertising, workers training and technology imports (following Corrado et al. (2009)).
  - The first two entries account for almost all intangible investment and both predict future productivity growth.
  - Spanish accounting law (Royal Decree 1514/2007) imputes book value of intangible assets at cost of purchase or internal development; depreciation is asset-specific.
- Sample used for estimation:
  - After eliminating outliers, remaining firm-year observations from 1992 to 2013: 23,380.
  - Firm representativity weights constructed from DIRCE by sector (20 categories in manufacturing) and employment size bin (1-19, 20-49, 50-99, 100-199 and 200+).
  - Employment share of firms with less than 10 employees is 12 percent of the sector.

### Key empirical patterns around the Great Recession
- Tangibility and investment responses:
  - Tangibility defined as physical capital divided by total fixed assets (k/(k+a)); average tangibility = 57 percent (implying 43 percent share of intangible assets in total fixed assets).
  - Intangible-intensive firms were more affected by the recession: they drastically reduced debt issuances and cut total investment disproportionately more.
  - Average leverage (debt over total assets) weighted by value added = 23 percent.
  - Fall in debt issuances coincided with an increase in interest rates for debt and the equity premium.
- Investment magnitudes and volatility:
  - Tangible and intangible investment have comparable magnitudes and both dropped after the Great Recession.
  - Aggregate total factor productivity (defined as value added net of labor and physical capital) fell by 5 percentual points between 2008 and 2013 in the ESEE sample.
  - Tangible investment is more volatile; intangibles have a higher private depreciation rate and a higher adjustment cost, which prevents firms from disproportionately cutting intangible investment after an aggregate shock.
- Entry and exit:
  - Net entry rates shifted from positive to negative since the onset of the Great Recession.
  - Majority of job losses in Spain were due to firm closures rather than downsizing (Bentolila et al. (2013)); similar qualitative pattern in U.S. (Decker et al. (2017)) though U.S. rise in exit rates is less persistent.

### Estimation method
- Estimation approach:
  - Structural parameters estimated by the Simulated Method of Moments (SMM).
  - Model-simulated and actual data compared for the same period, feeding in empirical history of aggregate shocks estimated in Section 4.1.
  - Estimation minimizes squared distance in the moments weighted by the estimated optimal GMM matrix.
  - Estimation uses cross-sectional and within-firm moments; overidentifying check compares model to Spanish aggregate time series in Section 6.
- Heterogeneity:
  - Heterogeneity across firms arises from different current and past realizations of the idiosyncratic intangible depreciation shock z.

### Parameter estimates and identifying moments (Table 2 summary)
- Production function and shares:
  - physical capital share α = 0.19 (s.e. 0.02); identified by labor costs / value added.
  - intangible share β = 0.19 (s.e. 0.01); identified by tangible / fixed assets.
  - Exponent on public stock of intangibles (1 − α − β) = 0.62.
- Depreciation and investment:
  - depreciation tangibles δk = 0.08 (s.e. 0.02); identified by average ik/kmature firms.
  - depreciation intangibles δa = 0.12 (s.e. 0.01); identified by average ia/amature firms.
- Entry, exit and fixed costs:
  - fixed cost T = 0.45 (s.e. 0.06); matches average entry and exit rates.
  - entry cost γ = -0.26 (s.e. 0.01); identified by average entry-exit gap; γ estimated to be 26 percent of initial debt of entrants b0.
  - entrants net debt b0 = -1.49 (s.e. 0.38); set to match life-cycle growth in employment by firm age (age ≤ 5 vs age > 5).
- Creditor recovery and adjustment costs:
  - recovery creditor tangible ψk = 0.65 (s.e. 0.23); identified by leverage and slope (ik/(a+k), ba+k).
  - recovery creditor intangible ψa = 0.26 (s.e. 0.09); identified by leverage and slope (ia/(a+k), ba+k).
  - adjustment cost tangibles ηk = 0.08 (s.e. 0.02); identified by within std. dev. ik/(a+k).
  - adjustment cost intangibles ηa = 0.18 (s.e. 0.10); identified by within std. dev. ia/(a+k).
  - Adjustment costs imply upon liquidation firms lose 8 percent of value of tangible assets and 18 percent of intangibles.
- Idiosyncratic shock process:
  - autocorrelation shock ρ = 0.09 (s.e. 0.03); identified by autocorrelation of ia/(a+k).
  - std. dev. shock θ = 0.85 (s.e. 0.16); identified by correlation (ia/a, Δaa).
- Other:
  - entrants net debt relative employment entrants: relative employment entrants = 64.0 (data moment targeted).
  - elasticity of substitution across varieties σ = 4 (calibrated).

### Moments fit and model diagnostics (Table 3 highlights)
- Select data vs model comparisons (data, s.e., model):
  - labor costs / value added: 60.9, 1.76, 61.1
  - tangible / fixed assets: 55.6, 5.35, 51.6
  - average ik/kmature firms: 11.9, 0.61, 11.7
  - average ia/amature firms: 15.6, 4.11, 6.2
  - average entry and exit rates: 5.4, 0.6, 5.3
  - average entry-exit gap: 1.7, 1.1, 1.2
  - within std. dev. ik/(a+k): 20.0, 0.4, 5.7
  - within std. dev. ia/(a+k): 11.4, 0.5, 3.1
  - correlation (ia/a, Δaa): 29.0, 4.4, 18.6
  - relative employment entrants: 64.0, 3.7, 65.5
  - slope coeff. ik/(a+k) on ba+k*: -0.05, 0.01, -0.02
  - slope coeff. ia/(a+k) on ba+k*: -0.10, 0.02, -0.02
  - autocorrelation ia/(a+k): 56.2, 9.64, 49.8
  - average leverage: 18.3, 0.72, 22.8
  - Notes: Asterisks (*) refer to coefficients from a regression of ix/(a+k) on log assets, tangibility, leverage, age, sector of activity and year fixed effects, where x={k,a}.
- Goodness of fit summary:
  - Model fits the majority of targeted moments accurately but has difficulty fitting high within-firm volatility in the data.
  - Lower adjustment costs or higher standard deviation of idiosyncratic shock would improve volatility fit but would generate excessive average leverage.
  - Average value-added-weighted leverage in the model is a bit too high due to greater firm size dispersion in the data and very large firms having lower leverage.

### Macroeconomic analysis: role of endogenous intangibles and the Great Deviation
- Model comparison and shocks:
  - Full model includes endogenous intangible investment, higher financing costs for intangible investment, and intangible spillovers.
  - Standard model (exogenous intangible investment) uses same idiosyncratic and aggregate shocks.
  - Aggregate shocks fed into models: financial shock (2009-2013) and non-financial shock (2003-2004), with return to normal state assumed post-2013.
- Main macro findings:
  - For the financial shock, the full model fits the extent of the GDP fall between 2008 and 2013 and predicts a persistent deviation from trend even under an immediate return to the normal state in 2014.
  - The standard model misses more than half of the GDP fall in the Great Recession and generates a quick rebound to trend.
  - For the smaller non-financial shock, the two models evolve more similarly, though the full model still produces a larger reaction due to additional capital subject to borrowing frictions and spillovers.
- GDP change decomposition (percentual contributions to deviation in GDP w.r.t. trend) — Table 4
  - Columns: full model 2008-13, full model 2008-18, exog. intan. 2008-13, exog. intan. 2008-18, no spillov. 2008-13, same ψ 2008-13
  - global income: -1.90, -1.90, -1.9, -1.9
  - physical capital: -2.0, -0.1, 0.2, 0.3, -1.7, -0.2
  - private intan. capt.: -1.6, -0.3, -0.2, 0.0, -1.7, -0.4
  - public intan. capt.: -0.1, -0.1, 0.0, 0.0, 0.0, 0.0
  - labor: -11.4, -0.8, -3.8, 0.5, -11.0, -4.6
  - allocative efficiency: -2.0, -0.1, -0.5, 0.1, -2.0, -0.5
  - entrants: -6.8, 3.5, -0.8, 0.0, -5.8, 1.4
  - incumbents: -7.8, -0.6, -5.1, 1.0, -9.1, -6.8
  - exiters: -4.5, -4.1, -0.5, -0.2, -3.1, -2.1
  - total: -19.0, -1.2, -6.3, 0.8, -18.0, -7.5

*Source: wp17176 - 4.2 Firm-Level Data (IMF working paper PDF).*

### 6.2  Components

### 6.2  Components

### Decomposition by production factors
- Table 4 decomposes the GDP change with respect to trend for the periods 2008-13 and 2008-18 to distinguish amplification (while the economy is hit by a financial shock) and persistence (after the aggregate state goes back to normal).
- GDP (aggregate value added by domestic firms) is defined and decomposed as:
  - N∑_{j=1} p_j y_j = D^{1/σ} E (A^{1−α−β} a^{β} k^{α} l^{1−α})^{(σ−1)/σ}
  - Variables without the superscript j denote aggregates; E denotes allocative efficiency (Olley-Pakes residual).
- First-row contribution: the exogenous global income shock D (change in GDP if domestic producers did not react) — identical across all models.
- Labor accounts for the largest GDP changes due to one-period labor contracts, the magnitude of the labor share, and wage rigidity.
- Intangible vs. physical capital dynamics:
  - The stock of intangible capital has more persistent dynamics (recovers more slowly) than physical capital.
  - Public intangible capital has a limited contribution but is the most persistent component: its fall is larger for the whole decade than for the crisis period and continues to reduce GDP when the aggregate state returns to normal.
- Allocative efficiency substantially worsens in the Great Recession due to firm heterogeneity and frictions to reallocate capital across firms after shocks.
- Measured TFP in revenues is equal to aggregate GDP divided by the contribution of physical capital and labor: (k^{α} l^{1−α})^{(σ−1)/σ}.
  - The measured TFP fall between 2008-13 is 5.6 percent in the model and 5.0 percent in the ESEE data.
- Comparing models:
  - Full model (endogenous intangible investment) vs. exogenous-intangible-investment model: endogenous model predicts an additional 12.7 percentual points of GDP fall between 2008-2013, and a persistent difference of 2 points by 2018.
  - Model without intangible capital spillovers (A = 1) and model where intangible assets are as collateralizable as tangible assets (ψ_h = ψ_k = 0.65) are used to identify causal contributions.
  - Spillovers: keeping public intangible capital constant relative to trend adds one percentage point to the 2008-13 GDP fall.
  - Making intangibles as collateralizable as tangibles eliminates a great chunk of the Great Depression and brings results close to the standard model, implying financial constraints are the primary cause of amplification.
- Cross-country interpretation: countries with deeper private and public equity markets (e.g., the U.S.) experienced shallower and less persistent falls in GDP and TFP after the Great Recession compared to most European countries.

### Decomposition by firm type (entrants, incumbents, exiters)
- Method: compute difference between each group's contribution and the average contribution on the BGP to account for the fact that entrants create value added and exiters destroy it by definition.
- Key observations:
  - All groups contribute less than normal in the Great Recession.
  - The bulk of changes when the financial shock hits is due to incumbents.
  - Exiters generate more persistence because balance sheet deterioration and liquidation take time.
  - Compared to the exogenous-intangible-investment model, exiters account for larger differences over the whole decade.

### Policy (section 6.3) — objectives and mechanism
- Policy debate context: need to speed up recovery after the Great Recession; European Investment Plan (Juncker Plan) implemented as subsidized credit to risky investments to relieve credit constraints.
- Model implication: expected welfare is proportional to the sum of existing firm value (all agents hold a perfectly diversified portfolio). First best equalizes marginal product of physical and intangible capital across firms.
- Decentralized equilibrium inefficiencies arise from:
  - Financial frictions: no-equity issuance constraint and limited commitment.
  - Intangible capital spillovers.
- Policy principle: any policy that increases investment rate of firms with higher marginal returns increases efficiency. Policies considered are budget-neutral transfers conditional on observable firm characteristics correlated with investment returns: age and size.

### Policy results: age-based and size-based transfers
- Age-based transfers:
  - Transferring equity to younger firms relaxes borrowing constraints of firms with higher marginal products of tangible and (especially) intangible capital.
  - Analysis restricted to a one-time unanticipated transfer in 2009 to avoid manipulation of firm age; transfers are budget neutral (sum to zero across firms).
  - Optimal transfers are sought within schemes proportional to the average firm net debt interest rate (q_b)^{-1} by age.
  - By firm optimality, the interest rate approximates expected marginal private return to tangible and intangible investment; reallocating resources toward firms with higher equilibrium interest rates maximizes sum of firm value and reduces inefficient exit.
  - Pattern of optimal transfers by age:
    - Positive for young firms and negative for older firms.
    - For very young firms, transfer is increasing in age: the most financially constrained firms are those between 4 and 8 years of age due to investment adjustment costs.
  - Aggregate effects:
    - The one-time policy generates a persistent increase in GDP relative to no-policy benchmark.
    - It avoids 3.3 percentual points of the 2008-2013 fall (17 percent of the total fall) and accelerates return to trend.
    - Benefits accrue gradually as constrained firms reduce leverage, increase investment, and avoid exit; a few old firms are forced to exit in 2009.
- Size-based (employment) transfers:
  - Same procedure: optimal one-time unanticipated budget-neutral scheme of transfers proportional to labor l in 2009.
  - Size is less correlated with marginal return than age: small firms may be small due to negative idiosyncratic shocks rather than high marginal returns.
  - Targeting small firms helps reduce inefficient exit further because small firms are proportionally more affected by fixed production costs.
  - Numerically, conditioning on firm size avoids 1.7 percentual points of the 2008-2013 GDP drop (less effective than age-based transfers).
- Implementation notes and caveats:
  - Basing policy on age or employment avoids the need for government collection of harder-to-observe variables (loan interest rates, leverage, assets, investments).
  - Operational implementation could use corporate income or social security tax credits conditional on the age of the firm’s oldest plant; challenges remain.
  - One-time unanticipated nature avoids manipulation (e.g., changing name or establishment location) but policies are not time consistent.
  - Anticipation could increase investment but also induce inefficient entry, spinoffs, or screen companies.
- Comparison with credit subsidies (e.g., Juncker Plan):
  - In heterogeneous-firm framework with endogenous borrowing constraints, credit subsidies are dominated by well-targeted outright transfers.
  - Lowering interest rates moderately increases leverage capacity only marginally via income effect on net worth; incentives to default barely change — subsidized credit crowds out private unsubsidized credit.
  - Excessive interest rate discounts can distort unconstrained firms into excessive leverage.
- First-best vs second-best:
  - Targeted transfers are second-best because correlation between marginal returns and observable characteristics is imperfect.
  - First-best would be microeconomic legal reforms to increase pledgeability of assets or facilitate external equity financing channels.

### Conclusions (summary points from section 7)
- Standard macro models with financial frictions struggle to generate large real effects and to explain long durations of financial recessions.
- Modeling intangible assets adds channels for amplification and persistence to a financial shock because:
  - Intangibles are less collateralizable.
  - Intangibles take longer to recover after firm exit.
  - Intangibles have lower social depreciation rates due to spillovers.
- These properties make financing costs for intangible investment disproportionately sensitive to financial conditions and make shocks to intangibles more persistent.
- Empirical/estimation evidence:
  - Model estimated structurally with Spanish manufacturing firm data.
  - Endogenous intangible investment can explain extent and components of output fall after the Great Recession; lower intangible investment and increased exit rates are important channels.
- Policy implication reiterated: transferring funds to younger firms could be more efficient than targeting small firms or subsidizing credit.
- Suggested avenues for future research:
  - Frictions related to public and private equity issuances for intangible-investing firms.
  - Allow imperfect portfolio diversification by firm owners.
  - Longer lags in formation of intangible capital.
  - Differentiated parameters by intangible type.
  - Empirical linking of firm-level debt interest rates to intangible investment and asset tangibility.
  - Explicit consideration of cyclical changes in the price of aggregate risk.

*Source: wp17176 - 6.2  Components*

### 1. Construct a grid of state values fork,

### wp17176 - 1. Construct a grid of state values fork,

### Numerical solution algorithm (grid, continuation function, iteration)
- Construct a grid of state values for k, a, k+a, z, A, and the three possible values for M. The grid is denser in regions of the state space where the continuation and the liquidation value are closer.
- Assume a parametric form for the continuation function C in each aggregate state M. Replicate the form of the analytical value function without frictions and idiosyncratic shocks. The parametric form is linear in {r, zr, k, a, b}, where r ≡ py − T − Wl are revenues net of fixed and labor costs.
  - Rationale: If the firm continues production forever, the analytical solution is linear on debt b and r, which is a function of the current state vector (k, a, A, b, z, M) given the optimal choice for labor. If the firm liquidates production, the analytical solution is linear on the stocks of physical k and intangible a capital and debt b (if b < b). To capture the value of the idiosyncratic shock to depreciation z, which correlates with next period’s z and thus to accumulation of private intangible capital, add the term zr to the linear specification. The term is interacted with r because z is more valuable if the firm continues operating.
  - Iterating on the continuation function rather than on the current-period value function allows analytical solution for the non-differentiable continuation-liquidation decision in the current period.
- Use non-uniform discretization to compute expectations over realizations of z′ given an aggregate state M′. The discretization is denser for regions where the liquidation option is closer (i.e. for high z′).
- To compute expectations for A′, approximate A′ = f(M) + A, and iterate on the values of f with the simulated data.
  - Footnote: This is only an approximation because growth in A′, which depends on the distribution of private intangible investment i_a across firms, is affected by the distribution of state variables other than M. However, fluctuations in A are typically two orders of magnitude smaller than A, so approximation error does not substantially affect the investment policy functions.
- Maximize numerically over investment in the two types of capital i_k and i_a for each grid point, taking into account:
  - (a) labor optimality in equation (19),
  - (b) dividends optimality, i.e., d = 0,
  - (c) bond pricing by investors given by equation (21).
- Iterate until the coefficients in the parametric guess for the continuation value function converge. The correlation between the grid values produced by the old and the new guess of the continuation value function must be higher than 99.99 percent.
- Compute investment policy functions for i_k_{k+a} and i_a_{k+a} with a rich parametric form, linear on:
  - {1, py, r, r_k, r_a, rz, rb_{k+a}, 1/r, k, k^2, a_{k+a}, (a_{k+a})^2, A, b_{k+a}, (b_{k+a})^2, 1_{k+a−b}, z, zb_{k+a}, zk, za_{k+a}}.
  - The parametrization of investment rates can be richer than the one for the continuation function because it is only used for simulation, not iteration.
- Simulate an economy with as many heterogeneous firms as in the data for many periods, using the parametric investment policy functions. Compute the moments of interest both in the simulated and in the real data for the period 1992-2013, using the aggregate shock process for M inferred from the E.U. data.
- Iterate on the model’s fundamental parameters until the GMM optimal distance measure between simulated and data moments is minimized.

### Benchmark Model: Exogenous Intangible Investment
- The benchmark model is equivalent to Midrigan and Xu (2014) with the addition of endogenous default. It has no endogenous intangible investment and no spillovers.
- Intangible investment i_a is fixed at the same level for all firms, costs no resources and has no adjustment costs. The level of i_a is chosen to replicate the level of private intangible capital in the full model.
- Entrants start with the same level of intangible capital as in the full model. Spillovers are suppressed by fixing the (post-normalization) public stock of intangibles to A = 1.
- The idiosyncratic intangible depreciation shock z is maintained as an exogenous productivity shock.
- A fraction of a firm’s profits is paid out in the form of dividends every period, such that the post-dividend frictionless profit level is the same as in the full model. This assumption is conservative: it reduces the gap with the full model by offsetting the absent intangible investment expenditures that would otherwise make firms more profitable and less leveraged.
- For comparability:
  - Calibrate the initial level of equity −b_0 to match the average size of entrants in the full model.
  - Reestimate the entry cost γ to allow for realistic changes in net entry over time.
  - The rest of parameters are unchanged.

### Firm-level data variables and summary statistics (Table 5)
- Summary statistics (panel (a)):
  - value added (EUR*) py: 2,023,511 mean; 200,465 10th pctile; 2,130,631 90th pctile; winsorizing: -
  - phys. capital (EUR*) k: 1,720,838 mean; 26,948 10th pctile; 2,100,430 90th pctile; winsorizing: > EUR100
  - tangibility (%) k/(k+a): 73.0 mean; 28.9 10th pctile; 100.0 90th pctile; winsorizing: INM > EUR100
  - leverage (%) (b/(k+a))+: 26.80 mean; 63.4 90th pctile; winsorizing: < 100%
  - tang. invest. rate (%) i_k/(k+a): 11.10 mean; 34.7 90th pctile; winsorizing: <±200%
  - intang. invest. rate (%) i_a/(k+a): 6.90 mean; 18.8 90th pctile; winsorizing: <±200%
  - age (years): -22.66 mean; 42 90th pctile; winsorizing: -
  - labor share (%) Wl/py: 62.3 mean
  - entry rate (%): -5.2 mean
  - exit rate (%): -2.7 mean
  - entrants rel. empl. (%): -57.1 mean
  - firm weights: -0.84 mean; 0.04 10th pctile; 2.95 90th pctile
- Variable definitions (panel (b)):
  - value added (EUR*) py: VA
  - phys. capital (EUR*) k: IN - INM
  - tangibility (%) k/(k+a): (IN - INM)/IN
  - leverage (%) (b/(k+a))+: FACC/PASIVO
  - tang. invest. rate (%) i_k/(k+a): (CIM - VIM)/IN
  - intang. invest. rate (%) i_a/(k+a): (GEFT + GTID + GPV*VENTAS/100 + IMPTEC)/IN
  - age (years): - year - AEMP
  - labor share (%) Wl/py: CP/VA
  - entry rate (%): DIRCE Tables 287-292, Altas/Total
  - exit rate (%): DIRCE Tables 287-292, Bajas/Total
  - entrants rel. empl. (%): PERTOT
  - firm weights: DIRCE Tables 296, 297
- Notes: (*) monetary units are 2008 Euros. Entrants (age =< 5) employment is relative to older firms (age > 5).

### Additional non-targeted moments and empirical patterns
- Firm life cycle:
  - Figure 12 replicates Figure 3 in the data: plots tangible and intangible investment rates and leverage rates as a function of firm age. Data averaged every two years to minimize sampling error for some firm ages.
  - Qualitative patterns are similar between data and model; intangible investment rate is much lower for very young firms, probably due to lack of reporting.
- Leverage distribution:
  - Figure 13: more than 35 percent of firms in the data, weighting by value-added contribution, have a leverage ratio below 5 percent; firms with leverage ratios above 50 percent contribute very modestly.
  - The model produces a bimodal leverage distribution with some bunching around the endogenous borrowing constraint of each firm. Presence of heterogeneity and endogenous default yields a flatter distribution than models without default.
  - Estimation targets average leverage only, so fit in the distribution is not perfect.
- Cross-industry evidence:
  - Figure 14: plots change in average total investment rate (i_k + i_a)/(k+a) from 1991-2008 to 2008-2013 against average tangibility for 1991-2008 across 20 manufacturing sectors.
  - Firms in industries with lower average asset tangibility reduced investment rates more after the Great Recession.
  - Notes: Investment rate change between 1991-2008 and 2008-2013. Average tangibility computed for 1991-2013. Sample: Spanish manufacturing firms. Source: ESEE.

### Computation of standard errors
- Variance-covariance matrix for firm-level moments in the data and in the model computed by block bootstrap with 500 samples. Bootstrap samples are generated with random draws at the firm-level.
- Standard errors for aggregate entry and exit rates: assume these variables follow a joint VAR(1) process. For tractability, assume cross-covariances between firm-level and aggregate moments are zero.
- Variance-covariance matrix for parameters given by:
  - (H′ Σ^{-1} H)^{-1} + (H′ ̃Σ^{-1} H)^{-1},
  - where H is the Jacobian matrix (computed as the average of the 1 and -1 percent numerical gradient), Σ is the estimated variance-covariance matrix of the moments in the data, and ̃Σ is the equivalent of Σ for the model-simulated data.
  - The second term accounts for error due to simulation of one finite sample of observations from the model.
- For parameters whose estimated standard error is smaller than the minimum grid step in the maximization algorithm, report the grid step as an upper bound for the standard error. This applies to ψ_k, η_k and θ.
- The weighting matrix used to measure the distance between data and model-simulated moments is Σ^{-1}.

*Source: wp17176 - 1. Construct a grid of state values fork, (PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17176.pdf_