Corporate tax policy, budget deficits and the capital stock in a neoclassical model of investment

Or, What would be the net effect on investment of the McCain tax plan?


Figure 1: Real nonresidential fixed investment (blue) and investment in equipment and software (red), SAAR. NBER defined recession dates shaded gray. Source: BEA GDP release of August 28, 2008, and NBER.As noted in a previous post, the McCain and Obama campaigns have many different components. The McCain tax plan involves a series of tax reductions aimed at lowering the cost of capital facing firms, with the aim at spurring investment; and as Jim pointed out, investment is a key determinant in our future prosperity. On the other hand, one particularly substantial difference with the Obama plan is that, as scored by the respective campaigns’ officials and tabulated by the nonpartisan Tax Policy Center, the McCain tax plan involves a $1.3 trillion dollar larger cumulative budget deficit over FY2009-2018. This suggests to me countervailing effects from implementing a McCain tax policy.

To examine the implications carefully, let’s resort to a formal model. Here I’ll use the workhorse neoclassical model of investment. Let Kt be the net capital stock at time t, Yt be real GDP, and rKt be the real rental cost of capital. Assume real output is a Cobb-Douglas function of capital (with exponent σ ) and other inputs. Then taking a partial differential of Y with respect to K yields the marginal product of capital, which in equilibrium should equal rKt. Re-arranging, one obtains:

K*t = σ Yt/rKt

Where * denotes the equilibrium value, and σ incorporates the Cobb-Douglas coefficient and the wage rate. Note that as rK rises, the optimal capital stock declines. This relates to investment in the following way:

It = Δ KtKt – K t-1

Note that with adjustment costs, the actual capital stock will not match the optimal at any given time. Investment will be a function of the gap between the optimal and actual capital stock in any number of partial adjustment mechanisms, as in the Jorgensen model.

What is rental cost of capital, rK? Conceptually, it is the amount that one needs to pay to rent one unit of capital. Of course, corporations don’t typically “rent” capital, but this is a useful artifice to think about the costs incurred in using capital in the production process. How does one define this parameter conceptually?

In order to answer this, define some terms. Let u be the tax rate on rental income (related to the corporate tax rate), and z is the investment incentive for each dollar of capital purchased (investment tax credit, accelerated write-offs or “depreciation for tax purposes”). Then firms equate:

(1-u)r Kt = (r t + d) (1-z t)P Kt

Where PKt is the price of a unit of capital goods (which is held constant; allowing for changes here alters the formula slightly but not importantly for our purposes), d is depreciation, and r is the real interest rate. Rearranging and solving for the rental cost of capital yields:

r Kt = [(r t + d) (1-z t)P Kt]/(1-u)

What this expression makes clear is that the rental cost of capital will decrease as the corporate tax rate is decreased, or investment incentives are increased. This makes clear the motivation for the tax reductions McCain has forwarded.

Of course, there is one additional term to consider in that expression: r — the real interest rate. The tax reductions have a direct implication for tax revenues and hence the budget deficit, which in turn will have an impact on the real interest rate, relative to the counterfactual of constant deficit.

How much of an increase? In Monday’s post, I argued that the extension of EGTRRA/JGTRRA alone induces a 2 ppts of GDP increase in the deficit relative to baseline. Since the McCain plan incorporates in addition the aforementioned investment inducements, multiply the amount by about 45% to obtain about a 3 ppts of GDP increase. Laubach (2003), using CBO projections in a standard bond pricing model, finds an effect of about 25 bps per 1 ppts of GDP increase in projected deficits. That implies a 0.75 increase in the real long term interest rate. So the determination whether investment rises or falls in the McCain plan hinges in part on the responsiveness of investment to changes in the corporate tax rate, capital gains rates, etc. The less responsive, holding all else constant, the more likely it is that the net effect will be negative.

One important observation: real interest rates fell in the wake of the 2001 and 2003 tax cuts (and concurrent recession and slow growth). Does this mean that all this is a silly worry? I think not, to the extent that in the past foreigners were willing to absorb all the new Treasuries issued at the going price. It is not clear to me this is still the case (and we know Agencies, which were a close substitute for Treasurys, are no longer absorbed by foreign demand with ease [0]). Hence, I think the case for higher interest rates [1] and hence crowding out in response to increasing budget deficits is much better now than in the 2001-05 period.

For more discussion of partial equilibrium analyses, crowding out, capital mobility, see these posts: [2], [3], [4]. See also the Economist‘s FreeExchange take on McCainomics.

Parting shot: I realize the preceding is an ad hoc, partial equilibrium analyses (even if it is more general than what people typically focus on). For a growth model perspective, see the US Treasury’s study from 2006, discussed here.

Originally published at Econbrowser and reproduced here with the author’s permission.