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.

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

  1. markg

    We all agree debt to gdp ratios can be too large. But at what level are they too large? Following WWII the ratio was 120%. Japan has had much larger ratios with no crowding out or higher interest rates. And can the debt to gdp ratio be too small? The PUBLIC (not total) debt to gdp ratio was under 40% from 1968 – 1982 and most would not consider those to be great economic times (the stock market started and ended that period at about the same level, interest rates trended higher, and unemployment increased). The ratio has been under 40% since 2000 and the economy has not been good for most Americans (again, the stock market has been essentually flat over that period).

  2. Robert Bell

    Thanks for the post, and an earlier recommendation of David Romer’s “Advanced Macro” which was very helpful in understanding the context.
    Just to clarify, Senator McCain and Senator Obama’s plan *both* imply an increase in the deficit, but from different sources. Would it be roughly correct to assume that, given that the McCain’s implied deficit is about 1.45x that of Obama, that the effect on the debt to GDP ratio would be roughly 2/3 of 3 pp, and the effect on the real interest rate (via the Laubach linear approximation) would be about 50 bp?

  3. Buzzcut

    Higher interest rates, lower taxes, higher oil prices, and a weaker dollar are all good things. More saving, less spending, export oriented growth. What’s not to like?

  4. GNP

    Robert Bell wrote: Thanks for the post, and an earlier recommendation of David Romer’s “Advanced Macro” which was very helpful in understanding the context.

    Romer’s Advanced Macroeconomics 2001 is a great textbook. Romer, however, dismisses the performance of inflation-targeting central banks on insufficient data and analysis. The dual-mandate-hamstrung US federal reserve has contributed to the current housing sector/financial sector crisis by holding overnight rates too low for too long earlier in the decade, even if US inflation has remained relatively tame over the decade.

    Many in the macro policy literature appear to believe that expansive fiscal policy makes the central bank’s job of price stability all that more difficult. Agents anticipating higher inflation would naturally bid up risk premiums and as a result the private cost of borrowing money.

    Thought I would add a rational-expectations-augmented Keynesian perspective to the neo-classical crowding-out argument for fun. But frankly, this discussion gets much more interesting and controversial when new classical supply side arguments are stirred into the mix.

    That requires looking in detail at the composition of deficit-financed public expenditures.

  5. Menzie Chinn

    markg: Yes, the optimal debt to GDP ratio is difficult to pin down. I would say that while the explicit liabilities of the Federal government were larger as a share of GDP in earlier periods (i.e., under Reagan, Bush the Elder), the contingent liabilities seem larger and more imminent now (from PBGC to Medicare Part D, etc.). See this post.

    Robert Bell: That’s correct. For a fuller, short to medium term perspective, one might wish to consider what the central bank’s reaction function is (assuming prices are sticky). Also, the neoclassical model is the simplest one, and the one I teach in intermediate macro. But there are arguments in favor of either a financial accelerator or q-theoretic approaches, which would modify further any conclusions; see this post.

    GNP: While I tend to agree that the Fed might have kept policy rates too low for too long, I don’t think we can lay the entire blame for this episode there. The deregulatory zeal of the Administration (where they sometimes stopped state regulatory authorities from slowing the “financial innovations” of subprime lending, alt-A primes, etc.) should also bear some of the blame, and in my view, perhaps even the majority. In that case, the housing boom is not a “rational stochastic bubble” driven by low interest rates, but rather “looting” a la Akerlof and Romer (2003). See this post for more.

    But you bring up a good point — to get at medium term effects, one would want something with short run rigidities and adjustment costs, like the dynamic stochastic general equilibrium models now being implemented at the Board, Fund, etc. But to my knowledge, they are not yet sufficiently detailed in corporate sector behavior to parse out the differential effects from the types of provisions being considered. If I’m wrong, I’d be obliged to hear from the readers.

  6. c thomson

    Like the Palin speech, this is all good fun. But, given that the predictive value of academic models and of economic forecasters has repeatedly proven to be somewhere in the neighborhood of zero, why bother with this stuff?
    Just bash Bush or whoever and skip the pseudo science. No one has the faintest idea of what McCain or Obama would be able to get through Congress, let alone what will happen at the same time in the real world.
    If anyone disagrees, please submit statistically meaningful evidence of successful bets on the one year out level of interest rates or of the stock market. Or the price of corn. Or whatever. Academic tenure or political appointments don’t count. Just cash bets won.
    Economics are a valuable tool for historical understanding and for winning bar arguments. Or, if you are Paul Krugman, for getting space to drivel in the NYT. (Wonder if the Times pay Krugman? Bet he would rant for free.)

  7. algernon

    Hear, hear c thomson! Well stated.
    Menzie, if you were more objective, you would have included the massive effect of foreign central banks buying US long-term debt (depressing those interest rates) & the constant complaint over the last decade from liberals about banks not lending to blacks & poor people as factors encouraging the housing bubble. I share your disdain for the Bush Administration, but their not telling banks to make sure they make loans that can be paid back seems like a minor point.

  8. Menzie Chinn

    c thomson: OK, I’ll fetch my dousing stick to help me out on my next post. I do wonder why you’re reading this blog if you are skeptical of the usefulness of economics.

    algernon: I do discuss the effect of China in this post (referenced in Monday’s post). I haven’t seen an analytical study regarding the quantitative magnitude of anti-redlining provisions in the “bubble”. But I disagree with your assessment assigning minimal effects from the Administration’s anti-regulatory crusade.

  9. Robert Bell

    Menzie: Thanks again
    GNP: I believe I have a 2006 edition of Romer. Menzie also recommended Fredric Mishkin – perhaps that will have more on inflation targeting.
    c thomson: “If anyone disagrees, please submit statistically meaningful evidence of successful bets on the one year out level of interest rates or of the stock market. Or the price of corn. Or whatever. Academic tenure or political appointments don’t count. Just cash bets won.”
    The problem with this line of thought is that you are prejudging what the science of economics should achieve. Specifically, you are saying, for example, that the inability to predict interest rates is a failure of economics.
    However, what if the point of economics is to characterize the world as it is, and one of the characteristics of that world is that certain things are *not* predictable. (
    I don’t see how this is materially different from the Heisenberg Uncertainty Principle, Godel’s Incompleteness Theorem, the Continuum Hypothesis ( or the Halting Problem ( all of which say that it is impossible to know or predict something, and all of which are considered great achievements in science or math.

  10. c thomson

    Professor Chin has misunderstood me. This is the best economics web site. As such, the economic opinions expressed range from the predictable, or Audacity of Hope, academic center to the deranged on both ends of the political spectrum. This makes the site useful, whatever the limited predictive value of conventional macro economics.
    Remember that that the stock market is a beauty contest in which the trick is to guess the verdict of the other judges.
    Economics is not a science, Robert Bell, but it is a useful tool for understanding what has happened in the past. Just like history itself. Despite the heroic efforts of Marx, history too has no predictive value. But wouldn’t a little knowledge of the British experience in Iraq been useful in Washington?
    Besides, economics keeps nice young mathematicians from having to teach Math 101 at South Dakota A & M for some humiliatingly low salary.
    Joe Biden has so wisely summed up the matter:
    “If economists could manage to get themselves thought of as humble, competent people on a level with dentists, that would be splendid.”

  11. GNP

    Menzie Chinn: Thank you. I’ll have to look up the Akerlof and Romer (1993) Brookings paper. From the abstract, I see interesting parallels with first-come, first-serve, boom ‘n bust resource exploitation policies. Privatize public natural wealth as quickly as possible and then shut down operations or exit completely. The welfare state then copes with the fall-out, or in less advanced nations, people starve.

    Robert Bell: From memory Frederic Miskin, is an advocate of inflation targeting, and in passing, yield curve forecasting.

    c thompson: You had me in absolute stitches. Thank you!

    The lack of predictive power of macroeconomic models–with the possible exception of the yield curve, and even that is weak–does not negate the scientific nature of economic inquiry. By this criteria, the biological and ecological natural sciences would be even less ‘scientific’ than economics! The saving grace of economists, particulary academic economists, is their penchant for publicly airing different view points, and emphasizing theoretical rigour. Moreover, the economics profession understands risk management better than any other profession.

    The problem with dentists is their penchant for performing fraudulent, expensive work. The field of information economics may not predict which dentist will attempt unnecessary work but does furnish a framework for understanding the phenomena.

  12. MarkS

    I would submit that the policy of extending governmental credit via lower tax rates or subsidies will disproportionately benefit those with a short time horizon over those with a longer point of view. More simply… The old want expenses securitized so that some of the costs are accrued beyond their lifetimes… The young want expenses paid as they occur, so that they are not burdened by prior profligacy.

    Whatever the mechanism, the piper has to be paid. America’s accumulated trade deficit is settled by the sale of industrial assets and currency devaluation. Alternatively, it could be paid by high interest rates and currency devaluation if protectionism were increased.

    The piper has to a large degree been avoided relative to the fiscal deficit, because the baby boom demographic bubble has been looted of their social security contributions. The piper is rapidly approaching however, and the economic and political ramifications will not be pretty.

    The piper is feeding in earnest on the financial system, having gobbled up about $4 trillion in real estate valuations, and $4 trillion in equities. The pain will continue with second tier bank failures, adjustable rate mortgage and commercial bond defaults, as well as illiquidity in derivative settlements.

  13. Arnold T

    The McCain tax cuts are across the board and supposedly equal for everyone. The net result will be the same as Bush’s Tax Cuts, huge deficits and a weaker Dollar. More of the SAME. Borrow and Spend economics.
    Obama’s Tax cuts have revenue producing Tax increases at the top income brackets. The result will be a zero balance change to the Deficit and a stable or strengthening Dollar due to fiscal restraint. A responsible Pay as You Go approach to economic relief and demand stimulation for the economy.
    Let me point out a very simple human factor in economics. Investors do NOT create jobs in a climate where confidence is shattered. This is not an equation, this is real world.
    Tax cuts to investors will NOT equate to a growing economy. A good businessman will see the lack of demand and NOT hire new employees and NOT build excess capacity. A hard to quantify Confidence Factor is needed before investors invest. During the mid 1930s and early 1970s this confidence was lacking.
    The Pitfalls of Capitalism is the Boom-Bust nature it exhibits periodically. Purist Capitalists hate to admit it, but the system will systematically almost self-destruct and in some cases has collapsed historically.
    Adam Smith was very Cynical about human nature. Our Government, as structured by Adam Smith, had Checks and Balances built-in to safe guard against abuses of power. A practical man, that Adam Smith, NOT an idealogical purist.
    The same is needed for Capitalism. Businessmen will lie, cheat and steal. Surprise. Regulations and Laws criminalize these abusive economic behaviors.
    More importantly, the top of an irrationally exuberant Long Tail curve can be cropped. The The Ponzi scheme collaspes, at the top of market blow-offs, can be mitigated. And the resulting destructive declines can be shortened. Really bad economic times are not required, they are an option we choose to have.
    So I warn all you NeoCon, Milton Freeman, FREE (Laissez-faire) Market economists. Your prideful rise in popularity for two decades will result in a vary hard fall. I just wish everyone did not have to suffer for the folly of a few.

  14. DickF

    Menzie wrote:
    What would be the net effect on investment of the McCain tax plan?
    …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.
    You do understand that people do not make investment decisions based on aggregate numbers don’t you?
    Your analysis shows what an economy may look like after the fact, but it does not at all demonstrate cause and effect. Let us look at two situations.
    Scenario 1. Let us say that in year 1 the tax code is changed so that all businesses are required to pay at least 1% of their sales in taxes as their cost of government services and those businesses with the highest corporate tax rates have their tax rates reduced slightly so that the entire impact of the tax has an increase of 2% of corporate income.
    Scenario 2. Let us say that in year 1 the tax code is changed so that no business in the bottom 75% of income pays any tax while the taxes on sales of the top 25% are imposed but only to the point where there is a net 2% decline in over-all taxation.
    What would you expect to happen to business investment in year 2 under each scenario?
    If you are a supply sider, an Austrian, or any other classical economist this question is a no brainer. Even with the slight tax increase, Scenario 1. will bring in more tax revenue in year 2 and with the larger businesses using their tax decrease to invest in capital the years beyond year 2 will also show increases.
    But in Scenario 2. there will be a decline in tax revenue in year 2 as the major businesses cut their cost of labor and capital by laying off workers and not investing in additional capital or maintenance of existing capital reducing their production and a decreasing their sales, moving a number of them into the bracket that pays no taxes. The number of businesses in the top 25% – those actually paying taxes – in year 2 will shrink significantly and this would be even more pronounced if the taxes were progressive.
    In the aggregate we would see a tax cut resulting in lower tax revenue while a tax increase would actually increase tax revenue. This is why I continue to say that if you believe that supply side economics is only about tax cuts you are seriously ignorant of what supply side is all about.
    Analysis in the aggregate has its place in preparing comparisons to what has actually happened and it can be very useful in measuring the impact of policy, if you know the details of policy differences, but to use aggregates to MAKE policy is total folly. To make policy you must analyze from the specific to the aggregate not from the top down.

  15. Anonymous

    Arnold T wrote:
    Obama’s Tax cuts have revenue producing Tax increases at the top income brackets.
    The balance of your economic statements are more fantasy than I have the time to deal with.

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