## wpiea2019029

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### Research question and contribution
- Research question:
  - Do temporary tax reliefs to business affect the economy?
- Main novelties and motivations:
  - Extends Ricardian equivalence analysis to the firm side, focusing on corporate taxes.
  - Incorporates a constraint on external financing to assess interaction of capital market imperfections with tax policies.
  - Considers both lump-sum and proportional business taxes to isolate stimulative effects from inter-temporal substitution effects.
  - Policy experiment: government reduces business taxes for one period, issues bonds; one period later raises corporate taxes enough to cover current spending and full servicing of outstanding debt; then resets tax rate to initial steady-state with zero public debt.

### Model structure and agents
- Agents and technology:
  - Representative household, representative firm, government.
  - Firm production: Cobb-Douglas CRS, Y_t = K_t^α N_t^{1−α}.
  - Capital law of motion: K_{t+1} = I_t + (1−δ)K_t.
- Firm financing and valuation:
  - Equity issuance/repurchase Z_{t+1}; cum-dividend equity value V_{t+1} = P_{t+1} − Z_{t+1} + D_{t+1}.
  - Firm value as present discounted value of payouts d_t = D_t − Z_t.
  - Resource constraint: D_t = F(K_t,N_t) + Z_t − w_t N_t − I_t − τ_t, where τ is a lump-sum tax on corporate profits.
  - First-order conditions:
    - w_t = F_2(K_t,N_t)
    - r_{t+1} = F_1(K_{t+1},N_{t+1}) − δ
- Household:
  - Utility U(C) = log C, discount factor 0 < β < 1.
  - Budget constraint and no-Ponzi conditions govern bond holdings; Euler equations link marginal utility to returns.
- Government:
  - Constant spending G>0 each period financed by lump-sum corporate taxation τ_t or one-period debt B_{t+1}.
  - Budget constraint: (1+r_t)B_t + G = τ_t + B_{t+1} and no-Ponzi condition; present-value constraint with B_0 = 0.

### Ricardian equivalence with corporate taxation (Proposition 1)
- Statement (paraphrased structure preserved):
  - Given initial (B_0, K_0) and spending G^2, two policy sequences {τ_t, B_{t+1}} and {ˆτ_t, ˆB_{t+1}} that leave the present value of lump-sum corporate taxes unchanged:
    - +∞∑_{t=0} t∏_{i=0} (1+ˆr_i)^{−1} ˆτ_t = +∞∑_{t=0} t∏_{i=0} (1+r_i)^{−1} τ_t,
    imply identical real allocations and prices across equilibria:
    - {ˆC_t, ˆN_t, ˆK_{t+1}, ˆθ_t}_t = {C_t, N_t, K_{t+1}, θ_t}_t
    - {ˆw_t, ˆr_t, ˆP_t}_t = {w_t, r_t, P_t}_t
  - Additional equalities for payouts and government budget close characterization.
- Interpretation:
  - Temporal pattern of lump-sum taxes that leaves present value unchanged is irrelevant for real variables under perfect capital markets and infinitely-lived, foresighted agents.
  - Financing policy shifts between taxes and debt do not affect firm investment or household wealth in this baseline.

### Departures from Ricardian equivalence: financing frictions and proportional taxation
- Main deviations examined:
  - Dividend lower bound (ˆD) and costly external financing (cost parameter κ) create financing frictions.
  - Proportional corporate taxation (tax rate τ on profits) with depreciation tax-deductible.
- Financing frictions:
  - Dividend floor: firm expected to pay minimum dividend ˆD each period.
  - External equity issuance cost: each additional dollar of new equity implies additional cost κ (0≤κ≤...).
- Qualitative mechanisms:
  - Without frictions: Modigliani–Miller irrelevance holds for lump-sum taxes.
  - With frictions:
    - Direct stimulative channel: tax cut increases retained earnings available for investment, reducing costly equity issuance.
    - Expectations channel: anticipated higher future taxes reduce expected marginal return on investment (especially under proportional taxation).
  - Net investment response depends on:
    - Whether tax is lump-sum or proportional.
    - Pre-existing marginal cost of external financing κ.
    - Size of the tax stimulus.

### Firm financial and investment policies (key equations and implications)
- Financial first-order condition (identical under lump-sum and proportional):
  - 1 = (1−κ)(1+λ^d_t)+λ^z_t (equations (15d)/(18d))
  - Interpretation:
    - Issuing one unit of new equity delivers 1−κ to shareholders.
    - Corner cases:
      - If κ = 0 then λ^d_t = λ^z_t = 0 and financial policy irrelevant (Ricardian equivalence holds if tax is lump-sum).
      - If κ > 0:
        - λ^d_t = 0 ⇒ λ^z_t = κ > 0 → dividend constraint not binding (internal financing/retained-earnings regime).
        - λ^z_t = 0 ⇒ λ^d_t = κ/(1−κ) > 0 → dividend constraint binding (external financing/equity issuance regime).
    - Firm uses external financing iff dividend constraint binds.
- Investment marginal cost depends on marginal source of finance:
  - Dividend-constrained (external finance): q_t = 1/(1−κ) (equation (19))
  - Not dividend-constrained (internal finance): q_t = 1 (equation (20))
- Optimal investment dynamics under taxes:
  - Lump-sum: 1+λ^d_t = (1+λ^d_{t+1})/(1+r_{t+1})(αK_{t+1}^{α−1}N_{t+1}^{1−α}+1−δ) (equation (21))
  - Proportional: 1+λ^d_t = (1+λ^d_{t+1})/(1+r_{t+1})[(1−τ_{t+1})(αK_{t+1}^{α−1}N_{t+1}^{1−α}−δ)+1] (equation (22))
- Key comparative insight:
  - Lump-sum: tax policy influences investment only if it changes marginal source of finance between adjacent periods.
  - Proportional: tax policy affects investment even if marginal source of finance unchanged because expected after-tax return depends on future τ.

### Government budget under proportional taxation
- R_t B_t + G = τ_t[K_t^α N_t^{1−α} − w_t N_t − δK_t] + B_{t+1} (equation (23))
- Inter-temporal budget constraint:
  - B_0 + G + ∑_{t=0}^{+∞} ∏_{i=0}^t R_i^{−1} = ∑_{t=0}^{+∞} ∏_{i=0}^t R_i^{−1} τ_t[K_t^α N_t^{1−α} − w_t N_t − δK_t] (equation (24))

### Steady state properties, propositions, and baseline calibration
- Selected steady-state results:
  - Proposition 2: optimal steady state stock of capital does not depend on the lower bound on dividend.
  - Proposition 3:
    - Higher proportional tax rate implies lower capital stock at steady state.
    - Lump-sum tax does not affect long-run capital stock.
    - Lump-sum steady state: K^{lump}_{ss} = (αβ/(1−β(1−δ)))^{1/(1−α)}
    - Proportional steady state: K^{prop}_{ss} = [αβ(1−\bar{τ})/(1−β+δ β(1−\bar{τ}))]^{1/(1−α)} and ∂K^{prop}_{ss}/∂\bar{τ} < 0
  - Lemma 2 (Laffer curve): threshold ˆτ_p with maximum feasible tax revenue ˆG_p = α(1−α)(α^2 β/(1−β+δ β))^{α/(1−α)} and ˆτ_p ≡ (1−β+δ β)(1−α)/(1−β+δ β(1−α)).
  - Proposition 8: existence of critical tax rate ̄τ_p such that firm dividend-constrained under proportional taxation iff τ > ̄τ_p.
- Baseline parameterization (top panel of Table 1):
  - Capital income share α = 0.33
  - Discount factor β = 0.96
  - Depreciation rate δ = 0.101
  - Marginal cost of external financing κ = 0.027
  - Lower bound on dividend ˆD = 0.113
- Calibration notes and thresholds:
  - κ = 0.027 close to Gomes (2001) estimate 0.028.
  - The proportional tax threshold ̄τ_p above which firm dividend-constrained at steady state equals highest US federal statutory corporate rate of 39%.
  - Threshold ˆτ_p corresponding to maximum feasible government spending equals 87%.
  - Feasible tax-rate interval where dividend constraint binds under both tax types is non-empty (] ̄τ_p, ˆτ_p [ ≠ {φ}).

### Policy experiments: design and representative scenario
- Experimental design highlights:
  - Tax is lump-sum in experiments 1 and 2; proportional in experiments 3 to 7.
  - Lump-sum tax rates in experiments 1 and 2 set to equal government revenue under proportional experiments 3 and 4.
  - Table 2 entries preserved exactly (selected values):
    - Steady-state rates: 16.35, 52.50, 15.00, 41.00, 41.00, 41.00, 41.00
    - Tax rate at period of tax cut: 10.58, 5.14, 105.00, 5.00, 5.00, 5.00, 5.00
    - Equity issuance cost κ: 0.027, 0.027, 0.027, 0.027, 0.01, 0.04, 0.06
    - Constraint Status initially: N B B N B B B B B
  - Note: lump-sum tax rate in column 1 (respectively 2) is the tax revenue implied by the proportional tax rate in column 3 (respectively 4) divided by before-investment profit.
- Representative experiment (illustrated in Figure 2):
  - Economy initially steady-state with dividend constraint binding.
  - Period 2: unexpected tax cut large enough to relax dividend constraint → firm switches to retained-earnings financing regime; government issues one-period debt.
  - Period 3: government increases tax to repay debt; tax and debt revert after period 4 and economy converges to initial steady state.

### Transitional dynamics — selected quantitative findings
- Experiments 1 and 3 (initially unconstrained; tax cut does not change marginal source of finance):
  - Lump-sum (black lines): firm devotes all tax cut to dividends at period of tax cut; investment unchanged; households maintain consumption and invest extra dividend in government bonds.
  - Proportional (red lines): firm reduces investment and increases dividends by more than tax cut due to expectation of higher future tax reducing after-tax return; temporary cut in proportional tax boosts consumption at expense of investment.
- Experiments with initially binding financing constraint (initial steady state τ = 41% and tax cut to 5%):
  - Both tax instruments: tax cut temporarily releases dividend constraint; firm switches from equity-issuance regime to internal financing regime; equity issuance falls to zero; marginal cost of investment decreases.
  - Lump-sum: investment rises initially by about 8%, then declines when tax increases and gradually returns to steady state.
  - Proportional: investment increases by less than 1% in benchmark; expectation of future high tax reduces after-tax return, mitigating investment response; firm increases dividends relatively more than under lump-sum.
  - Across both taxes: higher demand for capital raises interest rate, inducing households to save more and consume less; once constraint rebinds at period 3 firm reduces investment and interest rate falls back.

### Sensitivity analysis: cost of equity issuance κ
- Simulated proportional tax cut from 41% to 5% for κ ∈ {0.01, 0.04, 0.06}.
- Key numeric responses at period 2:
  - Dividend payments increase by about 70% when κ = 0.01; 50% when κ = 0.04; 35% when κ = 0.06.
  - Investment responses:
    - decreases by more than 3% when κ = 0.01;
    - increases by around 5% when κ = 0.04;
    - increases by around 10% when κ = 0.06.
- Mechanism:
  - Two opposing forces determine investment response under proportional taxation:
    1. Tax cut reduces marginal cost of investment by allowing switch to internal financing; strength ∝ κ.
    2. Expected future higher tax reduces expected after-tax return on current investment.
  - For low κ (e.g., 0.01) the second force dominates → investment falls.
  - For higher κ (e.g., 0.04, 0.06) the first force dominates → investment rises.

### Conclusions and policy implications
- Key conclusions:
  - Firms behave Ricardian when capital markets are perfect or financing constraints not binding and taxes are lump-sum.
  - Lump-sum tax cuts affect firm decisions only if they change the marginal source of finance between adjacent periods.
  - Under proportional taxation, temporary tax cuts can reduce investment by lowering expected after-tax return even when they relax financing constraints.
  - When firms are dividend-constrained, temporary tax cuts can stimulate investment by reducing marginal cost of investment; the net effect under proportional taxes depends on relative strength of reduced marginal cost versus expected future tax-induced reduction in returns.
  - For a given tax cut size, the higher the marginal cost of external financing κ, the larger the positive effect on investment.
- Policy recommendation:
  - Temporary tax incentives should be carefully designed (size and instrument) and well-targeted to potentially constrained firms to maximize stimulative benefits.
- Suggested extension:
  - Incorporate firm heterogeneity so different firms in different financing regimes can be analyzed for aggregate and distributional effects.

### Appendix B — numerical algorithm (transition dynamics with lump-sum tax)
- Algorithm steps used to compute transition dynamics:
  1. Solve for the steady state and fix a length for the transition.
  2. Make a guess on the status of the constraint for each period of the transition.
  3. Given the initial stock of capital and the tax rate after tax cut, make a guess for initial consumption.
  4. Given the guess for consumption, use the household’s resource constraint to derive the dividend.
  5. Given conjecture on the status of the constraint, use the firm’s resource constraint to derive the investment.
  6. Given the capital choice of next period and the status of the constraint, use the firm’s Euler equation to determine the interest rate.
  7. Given the interest rate, use the household’s Euler equation to have a guess of the future consumption and tax rate.
  8. Given the interest rate, the tax cuts and the zero debt government condition for the next period, use the government budget constraint to update the conjecture of the future tax rate. This step is necessarily only for the period after the tax cut. For the other periods, the tax rate is the same as in steady state.
  9. Repeat steps 3 to 8 until convergence of the investment decision to its steady state value. If the investment decision does not converge before the end of the transition, go back to step 3 and update the guess on consumption.
  10. After convergence, use the path of dividend and investment to check the conjecture of the status of the constraint. If the conjecture is correct then stop. Otherwise, go back to step 3 and update the conjecture on the status of the constraint.

*Source: wpiea2019029 (PDF chapter/section).*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31

### wpiea2019029 - References . . . . . . . . . . . . . . .  31

### Appendices and supporting material
- Appendices list:
  - A. Proofs (pages 33–37) with subsections:
    - A.1. Ricardian equivalence result with Corporate Taxation
    - A.2. Proof of lemma 1
    - A.3. Proof of proposition 4
    - A.4. Proof of Proposition 5
    - A.5. Proof of Proposition 6
    - A.6. Proof of Lemma 2
    - A.7. Proof of Proposition 7
    - A.8. Proof of Proposition 8
    - A.9. Proof of lemma 3
  - B. Numerical Algorithm (page 37)
- Tables and Figures inventory (with page references):
  - Table 1. Baseline Parameterization (page 20)
  - Table 2. Description of Experiments (page 23)
  - Figure 1. Laffer Curve (page 21)
  - Figure 2. Illustration of Tax Policy Experiment (lump-sum tax) (page 22)
  - Figure 3. Transitional Dynamics without Financing Frictions (page 25)
  - Figure 4. Transitional Dynamics with Financing Frictions (page 26)
  - Figure 5. Sensitivity analysis (page 28)

### Key research question and contribution (Introduction)
- Research question:
  - Do temporary tax reliefs to business affect the economy?
- Main novelties and motivations:
  - Extends analysis of Ricardian equivalence to the firm side, focusing on corporate taxes.
  - Incorporates a constraint on external financing to assess interaction of capital market imperfections with tax policies.
  - Considers both lump-sum and proportional business taxes to isolate stimulative effects from inter-temporal substitution effects.
  - Policy experiment: government reduces business taxes for one period, issues bonds; one period later raises corporate taxes enough to cover current spending and full servicing of outstanding debt; then resets tax rate to initial steady-state with zero public debt.

### Model structure (Section II)
- Agents and technology:
  - Representative household, representative firm, government.
  - Firm production: Cobb-Douglas CRS, Y_t = K_t^α N_t^{1−α}.
  - Capital law of motion: K_{t+1} = I_t + (1−δ)K_t (equation (1)).
- Firm financing and valuation:
  - Equity issuance/repurchase variable Z_{t+1}; cum-dividend equity value V_{t+1} = P_{t+1} − Z_{t+1} + D_{t+1} (equation (3)).
  - Firm value as present discounted value of payouts d_t = D_t − Z_t (equation (5) and (7)).
  - Resource constraint: D_t = F(K_t,N_t) + Z_t − w_t N_t − I_t − τ_t (equation (6)), where τ is a lump-sum tax on corporate profits.
  - First-order conditions (investment and labor):
    - w_t = F_2(K_t,N_t) (equation (8a))
    - r_{t+1} = F_1(K_{t+1},N_{t+1}) − δ (equation (8b))
- Household:
  - Utility U(C) = log C, discount factor 0 < β < 1.
  - Budget constraint (equation (10a)); no-Ponzi conditions (10b), (10c).
  - First-order conditions linking intertemporal marginal rate of substitution and returns (equations (11a), (11b)).
- Government:
  - Constant spending G>0 each period financed by lump-sum corporate taxation τ_t or one-period debt B_{t+1}.
  - Budget constraint: (1+r_t)B_t + G = τ_t + B_{t+1} and no-Ponzi condition (12).
  - Present value budget constraint with B_0 = 0 (equation (13)).

### Ricardian equivalence result with corporate taxation (Section II.E)
- Proposition 1 (Ricardian Equivalence Statement):
  - Given initial conditions (B_0, K_0) and a government spending G^2, let {C_t, N_t, K_{t+1}, B_{t+1}, θ_t, d_t}_t and prices {w_t, r_t, P_t}_t be an initial equilibrium under policy {τ_t, B_{t+1}}_t.
  - If another equilibrium under policy {ˆτ_t, ˆB_{t+1}}_t satisfies:
    - +∞∑_{t=0} t∏_{i=0} (1+ˆr_i)^{−1} ˆτ_t = +∞∑_{t=0} t∏_{i=0} (1+r_i)^{−1} τ_t,
    then real allocations and prices are identical across equilibria:
    - {ˆC_t, ˆN_t, ˆK_{t+1}, ˆθ_t}_t = {C_t, N_t, K_{t+1}, θ_t}_t
    - {ˆw_t, ˆr_t, ˆP_t}_t = {w_t, r_t, P_t}_t
  - Additional equalities (as stated in proposition) for payouts and government budget close the characterization.
- Interpretation:
  - Temporal pattern of lump-sum taxes that leaves present value unchanged is irrelevant for real variables under perfect capital markets and infinitely-lived, foresighted agents.
  - Ricardian forces imply financing policy shifts between taxes and debt do not affect firm investment or household wealth in this baseline.

### Departures from Ricardian equivalence (Section III)
- Two main deviations examined:
  - Imposing a lower bound on dividend payments with costly external financing (financing frictions).
  - Replacing lump-sum corporate taxation with proportional taxation.
- Financing frictions (Section III.A):
  - Dividend floor: firm expected to pay a minimum dividend ˆD each period (motivations: established dividend practice, signaling, agency considerations; Allen and Michaely (2003); Bianchi (2016); Poterba, Hall, and Hubbard (1987)).
  - When dividend constraint binds, firm issues new equity to finance investment.
  - Cost of external equity issuance: each additional dollar of new equity implies an additional cost of κ, 0≤κ≤ (text truncated in source at this point).
- Qualitative mechanisms identified:
  - Without financing frictions: Modigliani and Miller irrelevance holds; lump-sum temporary corporate tax reductions have no real effects if present value of tax burden is unchanged and capital markets are perfect.
  - With financing frictions:
    - Direct stimulative channel: a tax cut increases retained earnings available for investment, reducing marginal cost of investment (avoids costly equity issuance); when tax is lump-sum, this direct benefit tends to increase investment.
    - Expectations channel: anticipation of higher future taxes reduces expected marginal return on investment—particularly relevant when tax is proportional to profits—counteracting direct stimulative effects.
    - Net investment response depends on:
      - Whether tax is lump-sum or proportional.
      - Pre-existing marginal cost of external financing.
      - Size of the tax stimulus.
- Modeling implications:
  - The paper computes transitional dynamics following the policy experiment and compares cases with and without financing frictions (Figures 3 and 4 referenced).
  - Sensitivity analysis conducted (Figure 5).

*Source: wpiea2019029 - References (excerpted content unit).*

### 1. This cost of equity issuance can be interpreted as transaction costs stemming from capital

### wpiea2019029 - 1. This cost of equity issuance can be interpreted as transaction costs stemming from capital

### Financial frictions and corporate taxation
- Firm problem specified under lump-sum taxation: maximize ∑_{t=0}^{+∞} Λ_{0,t}(D_t − Z_t) subject to constraints (14a)-(14d) with first-order conditions (15a)-(15d).
- Firm problem specified under proportional taxation: maximize ∑_{t=0}^{+∞} Λ_{0,t}(D_t − Z_t) subject to constraints (17a)-(17d) with first-order conditions (18a)-(18d).
- Under proportional taxation depreciation is tax deductible (equation (17a)).
- Key first-order conditions preserved:
  - (15a)/(18a) Labor: w_t = (1−α)K_t^α N_t^{−α}
  - (15b)/(18b) Investment: q_t = 1 + λ^d_t
  - (15c) Lump-sum K_{t+1}: q_t = (1+λ^d_{t+1})/(1+r_{t+1})(αK_{t+1}^{α−1}N_{t+1}^{1−α}+1−δ)
  - (18c) Proportional K_{t+1}: q_t = (1+λ^d_{t+1})/(1+r_{t+1})[(1−τ_{t+1})(αK_{t+1}^{α−1}N_{t+1}^{1−α}−δ)+1]
  - (15d)/(18d) Equity issuance Z_t: 1 = (1−κ)(1+λ^d_t)+λ^z_t

### Firm’s financial policy
- Equations (15d) and (18d) are identical: type of tax does not matter for the financial decision of the firm.
- Interpretation:
  - Raising one unit of new equity delivers 1−κ to shareholders (equity issuance involves marginal cost κ).
  - Left-hand side of (18d) is marginal cost to shareholder (one unit lowers equity value by one).
  - Equilibrium requires marginal benefit = marginal cost.
- Corner cases:
  - If κ = 0 then λ^d_t = λ^z_t = 0 and financial policy is irrelevant (Ricardian equivalence holds if tax is lump-sum).
  - If κ > 0:
    - If λ^d_t = 0 then λ^z_t = κ > 0 → dividend constraint not binding, share repurchase constraint binding (internal financing/retained-earnings regime).
    - If λ^z_t = 0 then λ^d_t = κ/(1−κ) > 0 → firm issuing new equity and dividend constraint binding (external financing/equity issuance regime).
- Summary statement: firm uses external financing if and only if the dividend constraint is binding.

### Firm’s investment policy
- Marginal cost of investment depends only on the marginal source of finance (λ^d_t, λ^z_t), not on tax instrument per se.
- Investment conditions:
  - Dividend-constrained (external finance): q_t = 1/(1−κ) (equation (19))
  - Not dividend-constrained (internal finance): q_t = 1 (equation (20))
- Implication: marginal cost of investment is higher in an equity-issuing regime than in an internal-financing regime.
- Optimal investment dynamics under taxes:
  - Lump-sum: 1+λ^d_t = (1+λ^d_{t+1})/(1+r_{t+1})(αK_{t+1}^{α−1}N_{t+1}^{1−α}+1−δ) (equation (21))
  - Proportional: 1+λ^d_t = (1+λ^d_{t+1})/(1+r_{t+1})[(1−τ_{t+1})(αK_{t+1}^{α−1}N_{t+1}^{1−α}−δ)+1] (equation (22))
- Key comparative insights:
  - Under lump-sum taxation, tax policy influences investment only if it changes the marginal source of finance between adjacent periods (factors 1+λ^d_t and 1+λ^d_{t+1} can cancel).
  - Under proportional taxation, tax policy affects investment even if marginal source of finance remains unchanged because expected after-tax return is affected by future τ.

### Other agents’ problems
- Household problems under both tax types are identical to the basic framework.
- Government budget under proportional taxation:
  - R_t B_t + G = τ_t[K_t^α N_t^{1−α} − w_t N_t − δK_t] + B_{t+1} (equation (23))
  - Government inter-temporal budget constraint:
    B_0 + G + ∑_{t=0}^{+∞} ∏_{i=0}^t R_i^{−1} = ∑_{t=0}^{+∞} ∏_{i=0}^t R_i^{−1} τ_t[K_t^α N_t^{1−α} − w_t N_t − δK_t] (equation (24))
- Government may run short- to medium-term deficits but cannot play a Ponzi game.

### Model solution — steady state properties and parameterization
- Use closed-form steady state for long-run equilibrium; transition solved by backward induction.
- Propositions and lemmas (selected):
  - Proposition 2: optimal steady state stock of capital does not depend on the lower bound on dividend.
  - Proposition 3: higher proportional tax rate implies lower capital stock at steady state while lump-sum tax does not affect long-run capital stock.
    - Lump-sum steady state: K^{lump}_{ss} = (αβ/(1−β(1−δ)))^{1/(1−α)}
    - Proportional steady state: K^{prop}_{ss} = [αβ(1−\bar{τ})/(1−β+δ β(1−\bar{τ}))]^{1/(1−α)} and ∂K^{prop}_{ss}/∂\bar{τ} < 0
  - Lemma 1: Under lump-sum tax, feasible G at zero debt steady state iff G < \hat{G}_l, with \hat{G}_l expression provided.
  - Proposition 5: firm issues equity iff G > \bar{G}_l ≡ \hat{D}^{hi}_l − \hat{D}.
  - Lemma 2 (Laffer curve): threshold \hat{τ}_p with maximum feasible tax revenue \hat{G}_p = α(1−α)(α^2 β/(1−β+δ β))^{α/(1−α)} and \hat{τ}_p ≡ (1−β+δ β)(1−α)/(1−β+δ β(1−α)).
  - Proposition 8: existence of critical tax rate \bar{τ}_p such that firm dividend-constrained under proportional taxation iff τ > \bar{τ}_p, with expression for \bar{τ}_p and corresponding \bar{G}_p provided.
- Baseline parameterization (top panel of Table 1):
  - Capital income share α = 0.33
  - Discount factor β = 0.96
  - Depreciation rate δ = 0.101
  - Marginal cost of external financing κ = 0.027
  - Lower bound on dividend \hat{D} = 0.113
- Calibration notes:
  - κ = 0.027 close to Gomes (2001) estimate 0.028.
  - The proportional tax threshold \bar{τ}_p above which firm dividend-constrained at steady state equals highest US federal statutory corporate rate of 39%.
  - Threshold \hat{τ}_p corresponding to maximum feasible government spending equals 87%.
  - Feasible tax-rate interval where dividend constraint binds under both tax types is non-empty (] \bar{τ}_p, \hat{τ}_p [ ≠ {φ}).
- Figure 1: Laffer Curve plotted for parameter set with \bar{τ}_p and \hat{τ}_p indicated.

### Effects of temporary tax cuts — policy experiments
- Experimental design:
  - Tax is lump-sum in experiments 1 and 2 and proportional in experiments 3 to 7.
  - Lump-sum tax rates in experiments 1 and 2 set to equal government revenue under proportional experiments 3 and 4.
  - Table 2 summary (selected entries preserved exactly):
    - Steady-state rates: columns show values including 16.35, 52.50, 15.00, 41.00, 41.00, 41.00, 41.00
    - Tax rate at period of tax cut: values include 10.58, 5.14, 105.00, 5.00, 5.00, 5.00, 5.00
    - Equity issuance cost κ: 0.027, 0.027, 0.027, 0.027, 0.01, 0.04, 0.06
    - Constraint Status initially: entries include N B B N B B B B B
    - Constraint Status after tax cut: alternating N/B pattern noted
  - Note: lump-sum tax rate in column 1 (respectively 2) is the tax revenue implied by the proportional tax rate in column 3 (respectively 4) divided by before-investment profit.
- Representative experiment illustrated (Figure 2):
  - Economy initially steady-state with dividend constraint binding.
  - Period 2: unexpected tax cut large enough to relax dividend constraint → firm switches to retained-earnings financing regime; government issues one-period debt.
  - Period 3: government increases tax to repay debt; tax and debt revert after period 4 and economy converges to initial steady state.

### Transitional dynamics after tax cuts (selected findings)
- Experiments 1 and 3 (initially unconstrained, tax cut does not change marginal source of finance):
  - Lump-sum (black lines): firm devotes all tax cut to dividends at period of tax cut; investment unchanged; households maintain consumption and invest extra dividend in government bonds. Adjustment occurs through dividend policy.
  - Proportional (red lines): firm reduces investment and increases dividends by more than tax cut due to expectation of higher future tax reducing after-tax return on current investment; temporary cut in proportional tax boosts consumption at expense of investment.
- Experiments where financing constraint initially binding (initial steady state τ = 41% and tax cut to 5%):
  - Both tax instruments: tax cut temporarily releases dividend constraint; firm switches from equity-issuance regime to internal financing regime; equity issuance falls to zero; marginal cost of investment decreases.
  - Lump-sum: investment rises initially by about 8%, then declines when tax increases and gradually returns to steady state.
  - Proportional: investment increases by less than 1% in benchmark; expectation of future high tax reduces after-tax return, mitigating investment response; firm increases dividends relatively more than under lump-sum.
  - Across both taxes: higher demand for capital raises interest rate, inducing households to save more and consume less; once constraint rebinds at period 3 firm reduces investment and interest rate falls back.

### Sensitivity analysis (cost of equity issuance κ)
- Simulated proportional tax cut from 41% to 5% for κ ∈ {0.01, 0.04, 0.06}.
- Key numeric responses at period 2:
  - Dividend payments increase by about 70% when κ = 0.01; 50% when κ = 0.04; 35% when κ = 0.06.
  - Investment responses: decreases by more than 3% when κ = 0.01; increases by around 5% when κ = 0.04; increases by around 10% when κ = 0.06.
- Mechanism:
  - Two opposing forces determine investment response under proportional taxation:
    1. Tax cut reduces marginal cost of investment by allowing switch to internal financing; strength ∝ κ.
    2. Expected future higher tax reduces expected after-tax return on current investment.
  - For low κ (e.g., 0.01) the second force dominates → investment falls.
  - For higher κ (e.g., 0.04, 0.06) the first force dominates → investment rises.

### Conclusion and policy implications
- Firms behave Ricardian when capital markets are perfect or financing constraints not binding and taxes are lump-sum.
- Lump-sum tax cuts affect firm decisions only if they change the marginal source of finance between adjacent periods.
- Under proportional taxation, temporary tax cuts can reduce investment by lowering expected after-tax return even when they relax financing constraints.
- When firms are dividend-constrained, temporary tax cuts can stimulate investment by reducing marginal cost of investment; the net effect under proportional taxes depends on relative strength of reduced marginal cost versus expected future tax-induced reduction in returns.
- For a given tax cut size, the higher the marginal cost of external financing κ, the larger the positive effect on investment.
- Policy recommendation: temporary tax incentives should be carefully designed (size and instrument) and well-targeted to potentially constrained firms to maximize stimulative benefits.
- Suggested extension: incorporate firm heterogeneity so different firms in different financing regimes can be analyzed for aggregate and distributional effects.

*Italic: Source — wpiea2019029 (PDF chapter/section) — content unit provided.*

### References

### References and Appendices (wpiea2019029 - References)

### Key bibliographic references
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- Altinkilic, Oya, and Robert S. Hansen, 2000, “Are there economies of scale in underwriting fees? Evidence of rising external financing costs,” Review of Financial Studies, Vol. 13, No. 1, pp. 191–218.
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- Bianchi, Javier, 2016, “Efficient Bailouts?” American Economic Review, Vol. 106, No. 12, pp. 3607–59.
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- Gourio, François, and Jianjun Miao, 2011, “Transitional Dynamics of Dividend and Capital Gains Tax Cuts,” Review of Economic Dynamics, Vol. 14, No. 2, pp. 368–383.
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### Appendix A — Proofs: structure and main results
- A.1. Ricardian equivalence result with Corporate Taxation
  - Defines Z = {τ_t, B_{t+1}}_{t=0}^{+∞} and ˆZ = {ˆτ_t, ˆB_{t+1}}_{t=0}^{+∞}.
  - Equilibrium allocations under policy Z: {Ξ_t, B^h_{t+1}, d_t}_{t=0}^{+∞} where Ξ ≡ {C_t, N_t, K_{t+1}, θ_{t+1}} and B^h_{t+1} is household investment in government bonds.
  - Shows V_0 = ˆV_0 and that {K_{t+1}, N_t}_{t=0}^{+∞} = {ˆK_{t+1}, ˆN_t}_{t=0}^{+∞}.
  - Demonstrates feasibility of {C_t, θ_{t+1}, ˆB^h_{t+1}}_{t=0}^{+∞} under ˆZ via household and firm budget/resource constraints.
  - Establishes ˆB^h_{t+1} = ˆB_{t+1}, ∀t, and verifies the transversality condition: lim_{T→∞} ∏_{t=0}^{T} R_t^{-1} ˆB_{t+1} = 0.

- A.2. Proof of lemma 1
  - Under lump sum tax, consumption at steady-state:
    C_{ss} = (αβ / (1−β(1−δ)))^{α/(1−α)} [1−α + α(1−β) / ((1−β(1−δ))(1−κ1_{D_{ss}<ˆD})) ] − ( ̄τ + ˆD) κ1_{D_{ss}<ˆD}
  - Condition C_{ss} = 0 ⇒ G = (αβ / (1−β(1−δ)))^{α/(1−α)} [ (1−α)(1−κ) + α(1−β)/ (1−β(1−δ)) ] − ˆD κ ≡ ˆG_l.

- A.3. Proof of proposition 4
  - Under lump-sum tax:
    D_{ss} − ˆD = 1 / (1−κ1_{D_{ss}<ˆD}) [ (1−β)/β (αβ / (1−β(1−δ)))^{1/(1−α)} − G_l − ˆD κ1_{D_{ss}<ˆD} ]
  - Concludes ˆD < ˆD_{l hi} ⇒ (D_{ss} − ˆD < −G_l < 0).

- A.4. Proof of Proposition 5
  - Under lump-sum tax:
    D_{ss} = 1 / (1−κ1_{D_{ss}<ˆD}) [ α(1−β)/(1−β(1−δ)) (αβ / (1−β(1−δ)))^{α/(1−α)} − G_l − ˆD κ1_{D_{ss}<ˆD} ]
  - D_{ss} < ˆD ⇔ G_l > 1−β / β (αβ / (1−β(1−δ)))^{1/(1−α)} − ˆD ≡ ̄G_l.

- A.5. Proof of Proposition 6
  - ˆG_l − ̄G_l expression:
    ˆG_l − ̄G_l = (αβ / (1−β(1−δ)))^{α/(1−α)} [ (1−α)(1−κ) + α(1−β)/(1−β(1−δ)) ] − ˆD κ − α(1−β)/(1−β(1−δ)) (αβ / (1−β(1−δ)))^{α/(1−α)} + ˆD
  - ˆG_l − ̄G_l > 0 ⇐⇒ ˆD > −(1−α) (αβ / (1−β(1−δ)))^{α/(1−α)} ≡ ˆD_{l lo}.

- A.6. Proof of Lemma 2 (proportional tax)
  - Steady state government debt under proportional tax:
    B_{ss} = β/(1−β) [ (αβ(1− ̄τ) / (1−β+δ β(1− ̄τ)))^{α/(1−α)} α(1−β) ̄τ / (1−β+δ β(1− ̄τ)) − G ]
  - B_{ss} = 0 ⇔ G(τ) = (αβ(1−τ) / (1−β+δ β(1−τ)))^{α/(1−α)} α(1−β) τ / (1−β+δ β(1−τ))
  - Shows ∂G(τ)/∂τ > 0 ⇔ τ < (1−β+δ β)(1−α) / (1−β+δ β(1−α)) ≡ ˆτ_p.
  - Gives G(ˆτ_p) = ˆG_p and verifies ˆτ_p and 1−ˆτ_p expressions:
    ˆτ_p = (1−β+δ β)(1−α) / (1−β+δ β(1−α)) and 1−ˆτ_p = (1−β)α / (1−β+δ β(1−α)).

- A.7. Proof of Proposition 7
  - Establishes ordering of threshold values:
    1. ˆD_{lo l} = (1−α) (αβ / (1−β(1−δ)))^{α/(1−α)} < ˆD_{lo p} = (1−β)/β [ α^2 β / (1−β+αδ β^2 + δ β(1−β)) ]^{1/(1−α)}
    2. ˆD_{lo p} < ˆD_{hi p} = (1−β)/β (α/δ)^{1/(1−α)}
    3. ˆD_{lo p} < ˆD_{hi l} = (1−β)/β (αβ / (1−β(1−δ)))^{1/(1−α)}
    4. ˆD_{hi l} < ˆD_{hi p} = (1−β)/β (α/δ)^{1/(1−α)}

- A.8. Proof of Proposition 8
  - Assumes ˆD < ˆD_{hi l} and deduces ˆD < ˆD_{hi p}.
  - Shows D_{ss} < ˆD ⇐⇒ τ > αβ α (1−β) / (1−α) − ˆD^{1−α} (1−β+δ β) αβ α (1−β)^{1/(1−α)} − δ β ˆD^{1−α} ≡ ̄τ_p.
  - Derives G(̄τ_p) = α (β ˆD^{1−β})^{α} + δ β ˆD^{1−β} − ˆD ≡ ̄G_p.
  - Shows ˆD < ˆD_{hi l} ⇒ ̄τ_p > 0.

- A.9. Proof of Lemma 3
  - Uses results ˆτ_p < 1 and ̄τ_p > 0.
  - Establishes interval:
    ˆD_{lo p} = (1−β)/β [ α^2 β / (1−β+αδ β^2 + δ β−δ β^2) ]^{1/(1−α)} < ˆD < ˆD_{hi p} = (1−β)/β (α/δ)^{1/(1−α)} ⇐⇒ ̄τ_p < ˆτ_p.
  - Introduces ̄ ̄τ_p = (1−α)(1+δ β) / (1+δ β(1−α)) and shows ̄τ_p < ̄ ̄τ_p < ˆτ_p consistent with bounds on ˆD.

### Appendix B — Numerical algorithm for transition dynamics (lump sum tax)
- Algorithm steps used to compute transition dynamics:
  1. Solve for the steady state and fix a length for the transition.
  2. Make a guess on the status of the constraint for each period of the transition.
  3. Given the initial stock of capital and the tax rate after tax cut, make a guess for initial consumption.
  4. Given the guess for consumption, use the household’s resource constraint to derive the dividend.
  5. Given conjecture on the status of the constraint, use the firm’s resource constraint to derive the investment.
  6. Given the capital choice of next period and the status of the constraint, use the firm’s Euler equation to determine the interest rate.
  7. Given the interest rate, use the household’s Euler equation to have a guess of the future consumption and tax rate.
  8. Given the interest rate, the tax cuts and the zero debt government condition for the next period, use the government budget constraint to update the conjecture of the future tax rate. This step is necessarily only for the period after the tax cut. For the other periods, the tax rate is the same as in steady state.
  9. Repeat steps 3 to 8 until convergence of the investment decision to its steady state value. If the investment decision does not converge before the end of the transition, go back to step 3 and update the guess on consumption.
  10. After convergence, use the path of dividend and investment to check the conjecture of the status of the constraint. If the conjecture is correct then stop. Otherwise, go back to step 3 and update the conjecture on the status of the constraint.

*Content derived from “References” (wpiea2019029 - References).*

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