A. Firm

A representative firm combines labor N_t and capital K_t to produce a final good using Cobb-Douglas production technology with a Constant Return to Scale (CRS), $Y_t=K_t^{\alpha },N_t^{1-\alpha }$. Physical capital depreciates geometrically at rate δ leading to the following law motion of capital:

where I_t is the investment at period t and K_t the capital stock at the beginning of period t.

The firm may issue new shares or repurchase old shares. Thus, the price of the firm at date t + 1 satisfies: $P_{t+1}=P_{t+1}^0+Z_{t+1}$, where Z_t+1 denotes the value of new shares issued (repurchased) at period t + 1 if Z_t+1 ≥ (<)0, and $P_{t+1}^0$ is the period t + 1 value of equity outstanding in period t. The managers of the firm act on behalf of the stockholders in order to maximize the value of the firm. The net after-tax return to the owners of the firm at time t comprises current dividends and capital gains. Under the assumption of no uncertainty and perfect capital markets, the return on firms’ shares must equal the return on government bonds, r_t+1, if the owners are to be content holding their shares in equilibrium. Let’s denote by D_t+1 the dividend payments of the firm at period t +1,

where the term in parentheses represents the capital gains component of the return. The equity value of the firm stems from the non-arbitrage condition governing the valuation of its shares and can be derived as:

I follow Gourio and Miao (2010) and define the cum- dividend equity value, V_t+1, at period t + 1 as:

Using (2):

In the absence of any bubbles, solving (4) forward and using the transversality condition on stock prices, $$\underset{T\to \infty }{\lim }{\displaystyle \prod _{t=0}^T{(1+r_t)}^{-1}{P}_{T+1}=0}$$, yields the firm’s market value at period 0 given by:

Here $${\Lambda }_{0,t}={\displaystyle \prod _{t=0}^{t-1}{\Lambda }_{i,i+1}}={\displaystyle \prod _{i=0}^{t-1}\frac{1}{1+r_{t+1}}}$$ is the firm’s discount factor between periods 0 and t. The value of the firm is simply the present discounted value of its future dividend stream. This specification implies that dividends are paid out at the end of the period. The firm takes employment, investment and financing decisions to maximize its market value (5) subject to the law motion of capital (1), and its resource constraint:

where τ is a lump-sum tax on corporate profits. Dividends are defined as the the firm’s residual profits after expenditures. Cash inflows include current output and undepreciated capital, while cash outflows consist of dividend and factor payments, tax liability and investment expenditures. Under the assumption of perfect capital markets, the capital structure does not matter for the firm’s value. In other words, it does not matter for the firm’s value and investment policy how much earnings to retain for use as internal financing, rather than distributing dividends and raising new equity in the external equity market. More precisely in the firm’s problem, the payout D_t-Z_t can be determined but D_t and Z_t will be undetermined. This is the Modigliani and Miller (1958) dividend policy irrelevance theorem. Thus, for the remainder of this section, I define for simplicity the payouts of the firm as d_t:

Substituting sequentially (7) into (6) and (5) for d_t, and using (1) to eliminate I_t from the problem, the first-order conditions of the firm’s maximization problem are:

Equation (8a) implies that wage equals the marginal product of labor. The left-hand side of equation (8b) shows the marginal cost of investing one unit of output in physical capital. Equation (8b) implies that the return on investment in physical capital equals the return on investment in government bonds.

B. Household

The representative household derives utility from consumption alone, according to a standard time-additive utility function U(C) = logC, with future utility discounted at rate 0 < β < 1. A time endowment of 1 is supplied inelastically every period.

The household trades equity shares in the firm, as well as a government bond. Let θ_t denote the shareholding at the start of period t valued at price $P_t^0$, P_t the equity price at the end of period t (including capital gain in t), and B_t+1 the government bondholding paying interest rate r_t.

The problem of the representative household can be derived as follows:

subject to

Equation (10a) is the household’s budget constraint. The household’s income consists of labor earnings, plus the income from government bondholding, plus shareholding income. The consumer spends his resources in consumption, government bonds, and equity purchases.

Equations (10b) and (10c) represent the no-Ponzi game constraints on government bonds and firm’s shares holdings, respectively.

C. Government

Government spends a constant amount of resources G > 0 at each time t, funded either by lump sum corporate taxation τ_t, or by issuing one period debt B_t+1 held by the representative consumer. The government’s budget constraint is

together with the no Ponzi game condition:

Requiring that the government wastes no resources and therefore satisfies the no Ponzi game condition with equality, I obtain the present value budget constraint:

where I assume B₀ = 0. The government’s policy will be a sequence ${\{{\tau }_t,B_{t+1}\}}_{t=0}^{\infty }$ of tax rates and debt issuance which satisfies (13).

D. Competitive Equilibrium

Given the government policy ${\{{\tau }_t,B_{t+1}\}}_{t=0}^{+\infty }$, a competitive equilibrium consists of a set of prices ${\{w_t,r_t,P_t\}}_{t=0}^{+\infty }$, and allocations $${\{C_t,N_t,K_{t+1},B_{t+1},d_t,{\theta }_{t+1}\}}_{t=0}^{+\infty }$$, such that: the household optimizes given prices, the firm optimizes given prices and the tax rate, the government satisfies its inter-temporal budget constraint and all markets (good, firm’s shares, bond and labor) clear.

E. Ricardian Equivalence with Corporate Taxation

Proposition 1. Ricardian Equivalence Statement

Given initial conditions (B₀,K₀), and a government spending G², let the set of allocations $${\{C_t,N_t,K_{t+1},B_{t+1},{\theta }_t,d_t\}}_{t=0}^{+\infty }andprice{\{w_t,r_t,P_t\}}_{t=0}^{+\infty }$$, be an initial equilibrium under a government policy $${\{{\tau }_t,B_{t+1}\}}_{t=0}^{+\infty }$$.

If $${\{{\hat{C}}_t,{\hat{N}}_t,{\hat{K}}_{t+1},{\hat{B}}_{t+1},{\hat{\theta }}_t,{\hat{d}}_t\}}_{t=0}^{+\infty }andprice{\{{\hat{w}}_t,{\hat{r}}_t,{\hat{P}}_t\}}_{t=0}^{+\infty }$$ constitute another equilibrium with a government policy $${\{{\hat{\tau }}_t,{\hat{B}}_{t+1}\}}_{t=0}^{+\infty }$$:

$$\begin{array}{c}\underset{t=0}{\overset{+\infty }{\Sigma }}\underset{i=0}{\overset{t}{{{\displaystyle \prod }}^{\text{}}}}{(1+{\hat{r}}_i)}^{-1}{\hat{\tau }}_t=\underset{t=0}{\overset{+\infty }{\Sigma }}\underset{i=0}{\overset{t}{{{\displaystyle \prod }}^{\text{}}}}{(1+r_i)}^{-1}{\tau }_t\end{array}$$. Then:

According to the theorem, the temporal pattern of lump-sum tax required to finance a particular public expenditure stream is irrelevant to the determination of real variables of the economy. That is, substituting taxes today for taxes plus interest tomorrow via debt financing affects neither the investment decision of the firm, nor the wealth of the individuals. The reason is that under perfect capital markets, infinitely lived and perfect foresighted private agents can exactly undo any financing policy undertaken by the government³. Given the Barro (1974)’s Ricardian equivalence result, the statement established here is not surprising but it has some implications which are absent in the standard Ricardian equivalence analysis. For instance, it helps characterizing the effects of the policy on the optimal decisions of the firm and their transmission channel to other variables in the economy. In addition, it is useful for understanding the dynamics of the economy when we depart from the Ricardian framework.

A. Financing Frictions

The analysis of the effects of temporary tax cuts on firm decisions requires a model that departs from Modigliani-Miller. To achieve this, we impose two standard constraints on firm financing decisions.

First, we impose a lower bound on dividend payments. A possible interpretation is that the firm has an established dividend practice that leads stockholders to expect a minimum dividend payment, $$\hat{D}$$, after each operating year. Such a restriction can be justified by appealing to the notion that minimum dividend payments may help reduce agency problems between shareholders and managers and work as a signaling device (see Allen and Michaely (2003) for a review). Bianchi (2016) also adopts a similar dividend payout policy. Poterba, Hall, and Hubbard (1987) presented evidence of the remarkable stability of dividend payouts throughout periods of extensive tax changes in the U.S.. Since the stock of capital K_t is given at the beginning of each period, the only variables under the control of the firm are the input of labor, N_t, the dividend payment, D_t, and the equity issuance Z_t. Thus, If the dividend constraint is binding, the firm will resort to new equity issuance to finance investment. We follow Gomes (2001) and assume that each additional dollar of new equity implies an additional cost of κ, 0 ≤ κ ≤ 1. This cost of equity issuance can be interpreted as transaction costs stemming from capital market frictions which increase the cost of outside capital relative to internally generated funds (Greenwald, Stiglitz, and Weiss, 1984; Myers and Majluf, 1984). Without this constraint, firms could raise enough equity to finance desired investment and the dividend constraint will be useless. Although share repurchases are allowed in the United States, we following the literature (Auerbach, 2002; Desai and Goolsbee, 2004; Gomes, 2001; Gourio and Miao, 2010; Hennessy and Whited, 2005) and impose no share-repurchases constraint for simplicity. In addition, we abstract from debt as in the literature⁴ and assume the firm is all equity financed. Incorporating debt financing would complicate the analysis since we would need to include debt as an additional state variable in the dynamic programming problem of the firm.

B. Financial Frictions and Corporate Taxation

We start by presenting the optimal decisions of the firm under both lump-sum and proportional taxation, and discuss the financial and the investment policies of the firm later on. When tax is lump-sum, the representative firm’s problem is as follows:

Firm’s Problem with Lump-sum Taxation

subject to:

Let q_t, ${\lambda }_t^{d}and{\lambda }_t^z$ be the Lagrange multipliers associated with the constraints (14b)-(14d), respectively. Using (14a) to eliminate D_t, the first-order conditions are:

• First Order Conditions

The usual transversality conditions and the complementary slackness conditions are omitted from now and onward for simplicity.

C. Firm’s Financial Policy

Holding the investment decision of the firm unchanged, the financial policy of the firm is determined by equations (15d) (respectively (18d)) under lump-sum (respectively proportional) taxation. The two equations are identical, suggesting that the type of the tax does not matter for the financial decision of the firm, and have the following interpretation: raising one unit of new equity to pay dividends relaxes the dividend constraint and the share repurchase constraint. In addition, when the firm raises one unit of new equity, the shareholder receives 1-κ because equity issuance involves an additional marginal cost of κ. Thus, the expression on the right-hand side of equation (18d) gives the marginal benefit to the shareholder. On the other hand, a one-unit increase in new shares lowers equity value by one unit, and the expression on the left-hand side of equation (18d) represents the marginal cost to the shareholder. Equation (18d) requires that the marginal benefit and the marginal cost must be equal at equilibrium.

If external financing is costless, meaning κ = 0, there is no difference between internal financing and external financing. Equation (18d) implies that $${\lambda }_t^d={\lambda }_t^z=0$$. In this case, the firm’s financial policy is irrelevant. Thus, it does not matter for the firm’s value and investment policy how much earnings to retain for use as internal financing, rather than distributing dividends and raising new equity in the external equity market. In this case, the Ricardian equivalence result holds if tax is lump-sum.

In contrast, if new equity issuance is costly, i.e. κ > 0, the firm’s financial policy matters. In this case, it follows from equation (18d) that it can not be true that $${\lambda }_t^d={\lambda }_t^z=0$$. For instance, equation (18d) shows first that if $${\lambda }_t^d=0,then{\lambda }_t^z=\kappa >0$$. This means that if the dividend constraint is not binding, then the share repurchase constraint is binding. In other words, it is not optimal for the firm to issue new equity if it is not dividend-constrained because equity issuance is costly. We refer to the case where the dividend constraint is not binding as the internal financing or retained-earnings financing regime. Second, equation (18d) shows that if $${\lambda }_t^z=0,then{\lambda }_t^d=\frac{\kappa }{1-\kappa }>0$$. That is, if the firm is issuing new equity, then the dividend constraint is binding. In this case, the firm is in the external financing or equity issuance regime.

In summary, equation (18d) implies that the firm will use external financing if and only if the dividend constraint is binding. The reason is that it is not optimal for the firm to simultaneously distribute more dividend than $$\hat{D}$$ and issue new equity. In fact, one unit of new equity reduces the equity value by one unit, whereas shareholders receive only 1 — κ. If the firm is not dividend-constrained, using retained earnings to finance investment is costless.

D. Firm’s Investment Policy

For a given financial policy, the firm’s investment policy is governed by equations (15c-15b) (respectively (18c-18b)) under lump-sum (respectively proportional) taxation. Equations (18b) and (15b) show that the nature of the tax does not affect the marginal cost of investment, which depends only on the dividend constraint, and the share-repurchase constraint. In other words, the marginal cost of investment depends solely on the marginal source of finance.

If the dividend constraint is binding, the marginal source of finance is new equity. In this case, Z_t > 0 and $${\lambda }_t^z=0$$. Using equation (18d), $${\lambda }_t^d=\frac{\kappa }{1-\kappa }$$. We can hence derive the investment equation:

Thus a dividend-constrained firm stops investing when the marginal product of investment, q_t, equals the cost of financing one additional unit of investment by new equity, $$\frac{\kappa }{1-\kappa }$$.

On the other hand, if the firm is not dividend-constrained, the marginal source of finance is retained-earnings. In this case, $$D_t>\hat{D}and{\lambda }_t^d=0$$. We can derive the investment equation:

Thus, the shareholder will stop investing when he or she has no preference between receiving one dollar of additional dividend and investing one dollar in firm’s shares. That is, he stops investing when the marginal benefit of investing q_t equals the cost of investment. Comparing equations (18b) and (15b) reveals that the marginal cost of investment is higher in an equity-issuing regime than in an internal-financing regime.

We now turn to the effect of corporate tax on investment. To this end, we use (15b-15c), respectively (18b-18c), to obtain the optimal investment condition (21), respectively (22), when tax is lump-sum, respectively proportional to corporate profit.

Under lump-sum taxation, equation (21) shows that the tax policy influences investment only if it changes the marginal source of finance between two adjacent periods. In fact, if the current and the next marginal sources of finance are new equity, i.e ${\lambda }_t^z={\lambda }_{t+1}^z=0$ (implying ${\lambda }_t^d>0and{\lambda }_{t+1}^d>0$), then $1+{\lambda }_t^d=1+{\lambda }_{t+1}^d=\frac{1}{1-\kappa }$. Similarly, if the current and the next marginal sources of finance are retained-earnings, i.e ${\lambda }_t^d={\lambda }_{t+1}^d=0$. In both cases, the factors $1+{\lambda }_t^{d}and{\lambda }_{t+1}^d$ cancel out in equation(21).

In contrast, tax policy affects investment policy even if the marginal source of finance remains unchanged when tax is proportional to business profits. More precisely, a tax cut financed by a tax increase one period later will reduce the expected marginal after-tax return on investment while the marginal cost of investment remains unchanged. In response, the firm will reduce investment.

Now, let us consider a situation where a tax cut changes the marginal source of finance between two adjacent periods. Assume that the firm is initially constrained and the tax cut is enough to temporarily relax the constraint, and the tax increase is such that the dividend constraint binds again. In this case, the firm uses internal cash-flow at the period of tax cuts but switches to external financing one period later. Under lump sum taxation, the tax cuts temporarily reduce the marginal cost of investment from $\frac{1}{1-\kappa }$ to 1. Since the tax policy does not affect the expected marginal return on investment, current investment should increase. When tax is proportional to corporate profits, two forces are present. The marginal cost of investment from $\frac{1}{1-\kappa }$ to 1 as before, but the higher future tax rate pins down the after-tax return on current investment. Thus, the net effect on investment will depend on the relative intensity of each force. If the expected reduction in the return on investment is more than proportional to the reduction in the marginal cost of investment, investment will decrease. Otherwise, investment will increase because the decrease in the marginal cost of investment more than fully offsets the reduction in after-tax return on investment. Equation (19) indicates that the higher the marginal cost of equity issuance, the higher the positive effect will be.

E. Other Agents’ Problems

Again, it may run a deficit in the short to medium term, but is not allowed to play a Ponzi game with other agents. Accordingly, its inter-temporal budget constraint is:

The household’s problems under the two types of tax are identical to the household problem in the basic framework. In contrast, under proportional taxation the government problem changes as follows:

A. Steady State Properties⁵

Proposition 2. The optimal steady state stock of capital does not depend on the lower bound on dividend.

Proposition 3. A higher proportional tax rate implies lower capital stock at steady state while lump-sum tax does not affect long-run capital stock.

Proof: With lump-sum tax: $K_{ss}^{lump}={\left(\frac{\alpha \beta }{1-\beta 1-\delta )}\right)}^{\frac{1}{1-\alpha }}$. Under proportional tax: $K_{ss}^{prop}={\left[\frac{\alpha \beta (1-\overline{\tau })}{1-\beta +\delta \beta (1-\overline{\tau }1}\right]}^{\frac{1}{1-\alpha }}and\frac{\partial K_{ss}^{prop}}{\partial \overline{\tau }}<0$.

Lemma 1. Under lump-sum tax, a government spending G is feasible at a zero debt steady state if and only if G < Ĝ_l, with:

Lemma 1 shows that under lump-sum tax, it is not feasible to finance more than Ĝ_l in a sustainab way even when the firm produces optimally and all the production is entirely devoted to finance government spending.

Assumption 1. $\hat{D}\le {\hat{D}}_l^{hi}\equiv \frac{1-\beta }{\beta }{\left(\frac{\alpha \beta }{1-\beta (1-\delta )}\right)}^{\frac{1}{1-\alpha }}$.

Proposition 4. Under tax is lump-sum, ${\hat{D}}_l^{hi}$ is the interest rate times the steady-state capital stock. If assumption 1 is violated, the firm will be constrained at steady state regardless of the size of the government.

Proposition 5. Let assumption 1 hold. The firm issues equity if and only if $G>\overline{G}{}_l\equiv {\hat{D}}_l^{hi}-\hat{D}$. $\overline{G}{}_l$ is the maximum government revenue that does not distort the firm investment.

Assumption 2. $\hat{D}<{\hat{D}}_l^{lo}\equiv -(1-\alpha ){\left(\frac{\alpha \beta }{1-\beta (1-\delta )}\right)}^{\frac{\alpha }{1-\alpha }}$