Friday, December 29, 2017

The Corporate Tax Rate, Part 2

John Cochrane posted a reply to my previous post on how changes in the corporate tax rate might affect investment. The key issues seem to relate to the specifics of what the tax code will allow as an expense. In my analysis, I treated investment spending by the firm as fully-expensed, which is not correct. However, the tax code does permit businesses to deduct interest on their debt, and depreciation, which I didn't include. What I'll do here follows - I think - a comment by Francois Gourio (Chicago Fed) on Twitter.

So, let's write down the firm's problem again, assuming a constant real interest rate r, which the firm's shareholders face (an important assumption - I'm neglecting taxes affecting the sharelholders). We'll assume that dividends are paid period-by-period to the firm's shareholders, with the firm maximizing the present value of dividends:
Here, K is the capital stock, N is the labor input, w is the wage rate, b is the firm's debt, and d is the depreciation rate. The firm's debt comes due in one period. Net proceeds for the firm in the current period consist of output minus the wage bill plus new debt issued, minus interest and principal on the debt issued in the previous period, minus investment, minus corporate taxes. The corporate tax rate t applies to output minus the wage bill, minus the interest payments on the debt, minus depreciation.

If the firm were to fund investment out of retained earnings (provided this does not violate a nonnegativity constraint) then a reduction in the corporate tax rate will indeed raise the after-tax marginal net payoff to investing. Alternatively, suppose that the firm always funds new investment by issuing debt, then pays the interest on the debt, retires debt as capital depreciates, and otherwise rolls the debt over. This implies that the firm's outstanding debt is always backed one-for-one by the firm's capital, or
Then, we can rewrite the first equation as
So, the firm's choice of labor input in each period, and its choice of capital in periods 1,2,3,... (equivalent to choosing investment) is independent of the tax rate t. Essentially, debt financing of investment permits full expensing of the investment expenditure - indirectly, through expensing of interest on the debt and depreciation.


1. We need to worry about how the household is taxed, which in this formulation determines what the objective function is for the firm.
2. To do a proper job here, we need to determine the optimal financial structure for the firm.

This is potentially quite complicated (not blog material), though I'm sure someone has addressed related problems in the taxation literature. To do the problem justice, we need a complete general equilibrium model. That said,

1. There's no presumption that the corporate tax rate reduction is going to matter much for intensive-margin decisions of the firm - decisions about labor input and investment.
2. Where the change in the corporate tax rate should matter is for entry decisions - here we need to start worrying about nonconvexities - e.g. fixed costs of entry. But some entry, relating to the treatment of pass-throughs, would just be a renaming of the productive unit - call yourself a business and you can be taxed at a lower rate. As well, firms may choose to relocate from other countries to the U.S., though as I mentioned in my previous post, those other countries won't give up without at fight.
3. There's a clear redistributive effect, as I mentioned in my previous post. Owners of stocks will benefit, and they tend to be richer people. Long-term, government transfers and expenditures on goods and services have to fall, and the burden of those reductions will be borne by the relatively poor.
4. If the Republican Congress actually wanted to increase investment spending, there are straightforward ways to do this through the tax code - an investment tax credit, for example.


  1. If the production function F of the firm is constant returns to scale and the firm faces a competitive market for labor, then the solution to the firm's maximization problem gives that the summand of V is zero in each period, so the corporate income tax doesn't distort any incentives, but it doesn't raise any money either. What am I missing here?

    The point about financial structure is of course important in the real world, as firms finance investment by equity and debt together. However, I think Cochrane would say that even if the corporate income tax rate reduction only alters the composition of financing towards more equity and less debt, that's still a good development. (Cochrane likes equity, as you know.)

    If Congress passed an investment tax credit, then they have to define what counts as investment and what doesn't, and that definition will probably end up distorting the investment choices that firms actually take in practice. Perhaps you have more faith in the ability of legislators and regulators to discern and define what is investment and what is not, but it shouldn't come as a surprise that Cochrane would be reluctant to go down that road.

    1. 1. The way I've modeled this, absent taxes the shareholders would be receiving the return to capital. Of course, given constant returns, output = wage bill + return to capital. Sometimes in models the households own the capital and receive the return to capital as rental payments. Here the households own the firm and receive the return to capital as dividends. So, what is being tax as corporate income is the return to capital minus interest payments on the debt, minus depreciation.

      2. "I think Cochrane would say that even if the corporate income tax rate reduction only alters the composition of financing towards more equity and less debt, that's still a good development." But, no idea whether that happens here. How would that work? Note that there are reasons to "like" debt. It's been with us for a long time, for good reasons.

      3. Last paragraph of your comment: No, I don't trust legislators and regulators to get it right. Particularly this lot. And I also wouldn't be defending this particular tax bill as resting on sound economic foundations.

    2. Thinking about 1. Looks like, when the firm is optimizing, V=0, so you're right, the government is collecting no revenue. But seems to be how the tax rules are specified. The corporate profits tax, with full expensing, implies the government collects zero from a corporate profits tax levied on constant returns firm. So, presumably, in practice the government is taxing monopoly profits. So looks like we need to think about how firms with monopoly power respond to the tax.

    3. 1. I don't understand how that makes any difference. The tax is levied on F(K_i, N_i) - w_i N_i - r b_(i-1) - d K_i, which is equal to F(K_i, N_i) - w_i N_i - (r + d) K_i given your assumption about the financing of investment, which is also the summand of V. If the firm maximizes V, then it sets this to zero in every period, so the government taxes nothing.

      I may need to see this in a general equilibrium context to understand how it's supposed to work, right now it's just not clicking for me.

      2. I also have no idea whether that happens here (I think you need a fully specified general equilibrium model with explicit financial frictions to say anything that makes sense about this issue), but my naive intuition is that the behavior of a firm trying to obtain the optimal financial structure should be similar to minimizing a quadratic loss function, with the weights being increasing functions of the marginal cost of obtaining each form of financing. If it's now cheaper to obtain equity financing because of the corporate tax rate reduction, that should induce the firm to use more equity and less debt financing.

    4. I don't know a lot about corporate finance, but we can get away from Modigliani-Miller, which implies financial structure is irrelevant, with taxation and financial frictions (costs of default, for example). I guess the approach would be to have the firm behave in the interests of the shareholders, then work out the optimal financial structure problem, then do everything in general equilibrium. I don't follow your argument about effects on financing - it's unclear to me what, if any, the effects would be on debt vs. equity financing.

    5. V is linear in the tax rate, t. Making no further assumptions, taking the first derivative with respect to the tax rate (i.e., DV/Dt), yields, DV/Dt = -F(K,N) +wN +rb +dK.

      The extent to which F(K,N) exceeds the sum of wN, rb, and dK, determines the extent to which the firm is advantaged by the reduction in the corporate tax rate, t.

      Corporate management will usually set the dividend rate per equity share at a fixed annual amount which will vary infrequently, or on a fixed schedule. It is not a given that the dividend rate will jump upward for all firms when the corporate tax rate is decreased. Whether the dividend or buy-back rate increases with the tax reduction depends on the firm’s circumstances, its competitive situation and the opportunities for growth in its or allied industries.

      The equation for V is also the equation for each incremental investment project the firm is considering in the next budget cycle. The decision to increase the dividend rate, increase the buy-back of outstanding common shares, retire outstanding debt, increase investment in R&D, or increase investment in real assets, or some combination of these actions, depends on the firm, the proclivities of management, shareholder opinion, and the value of growth opportunities (if any) facing the firm.

      For firms facing attractive future growth opportunities and/or increasing economies to scale, a reduction in the corporate tax rate will encourage investment, I, which will lead to an increase in K and N. Otherwise, the firm will payout the excess cash in the form of dividends and stock buy-backs, and the funds so released will be reinvested in firms with growth opportunities. If firms are without growth opportunities, then the excess cash is invested in government debt at i < r .

      To the extent that the quotient N/K is constant, greater investment leads to higher employment. In the case of automation that results in a decrease in the quotient, greater investment in K may or may not result in higher N. However, if N increases, some part of the tax rate reduction will be distributed to households in the form of w∙N rather than r∙b or v∙(K-b) (where v is the dividend rate or buy-back rate and K-b is the 'equity' stock component of fixed capital investment).

      The assumption of equilibrium is unlikely to hold in fact. But if equilibrium should hold, then for the all-debt-financed firm for which the initial capital investment is b0, V – b0 = 0 holds, not V=0.

  2. (responding to comment posted at 1:20 pm)

    There's certainly some of monopoly rent taxation going on, but I don't think that's the whole story. If firms raise any capital at all through equity, then the corporate income tax falls on the dividends, so with a naive intuition like the quadratic loss function I was talking about earlier (which represents a diversification bias) the firm's effective cost of capital at the margin is lowered by a corporate income tax rate reduction. That could lead to more investment, although to work it out properly you need to specify exactly why firms could prefer equity to debt, which probably requires stochastic variables in the model. What I'm saying on this subject is cocktail party speculation.

    If the firm can set prices, then you're right: the tax falls entirely on monopoly rents in this model, and permanent changes in the tax rate have no effect on the firm's pricing decisions, so no incentives are distorted in that case - a tax increase just raises more revenue. However, dividends in the real world also represent the return the firm has to pay to attract equity investors, and insofar as that's true and the firm tries to diversify its financing, higher corporate income tax rates are going to have an adverse effect on investment.

    From what I've seen, a lot of the debate about this has just been about the case where households own capital and collect rents on it, and that's the only source of financing the hypothetical firm in the model has. (Greg Mankiw's "corporate tax cuts raise wages more than one for one" model was based on this.) In practice, though, what happens is a lot more complicated.

  3. Imagine full expensing of investment, simply. So period profits are y-wl-I. The latter quantity has to be positive, actually rk-I, otherwise stockholders are not receiving the return on capital. So a tax t*(y-wl-I) should raise positive revenue, even under perfect competition. What am I missing?

    1. Yes, that's what I did in the other post. But y-wN-I need not be positive in every period.

  4. "Alternatively, suppose that the firm always funds new investment by issuing debt, then pays the interest on the debt, retires debt as capital depreciates, and otherwise rolls the debt over. This implies that the firm's outstanding debt is always backed one-for-one by the firm's capital..."

    Not a good assumption. The bill limits corporate net interest deduction to 30 percent of adjusted AGI (exceptions for small firms, floor inventory and real estate). See, (amended) IRC Section 163(j). Also, we have thin cap rules etc.

    Vivian Darkbloom

    1. "The bill limits corporate net interest deduction to 30 percent of adjusted AGI"
      For what fraction of firms do we think this would bind?

      "Also, we have thin cap rules etc."
      What's a "think cap rule?" What's "etc."?

    2. As far as how many companies I think it would apply to, probably the majority of the companies that would, under your unrealistic example, finance all of their investments through debt rather than equity.

      It appears to me that you are creating very stylized facts in order to save your initial argument. Why, if your initial argument were sound, would it be necessary to now make the unrealistic assumption that all investment is financed through debt? It might be time to just throw in the towel.

      If you have to's thin cap(italisation)--not "think cap"(!). See, e.g., section 385 and related rulings and case law. In addition to Sections 163(j) and 385, there are also common law rules (outside the scope of the tax code) regarding adequate capitalisation.


    3. Based on a quick analysis of IRS data, this limit would apply to approximately 13% of net interest deducted.

      Also, Vivian is correct that your debt financing assumption is not a good one. Furman has mistakenly made that argument as well. Treating investment as purely debt-financed is flat-out incorrect when evaluating aggregate investment effects. A 100% debt financing assumption is only relevant when comparing it with a 100% equity financing assumption, to evaluate the tax system's bias toward debt. When evaluating investment incentives, the important measure relies on "typically financed" investment, which in a steady state is equivalent to the firm's net debt-to-value ratio.

  5. Hi Steve
    The problem you wrote down only has dividend tax, not corporate tax (you have written it in such a way that they are equivalent). Firms get to deduct capital depreciation from corporate income taxes. This little detail changes your equations (you cannot factor out t anymore). If you write the euler equation for the firm and allow deduction for depreciation you will clearly see the distortion. Look at equation 10 here
    Under formulation that I am suggesting, lowering corporate tax rate has similar effect as investment subsidies that you suggested in 4.