Appendix:  The trade-off between the level and variability of the tax rate

Introduction

This Appendix is concerned with the standard tax-smoothing model where the excess burden of taxation is convex in the tax rate but the other Ricardian assumptions hold. In particular, it is assumed that capital markets are perfect and complete and individuals and benevolent governments are rational decision makers. In this context, Bohn (1990, 1995) shows in a world of uncertainty that government policy should focus on managing the Crown’s aggregate balance sheet so as to minimise the variance of the tax rate. Bohn’s results are counter-intuitive, as at face value it would seem the government could choose a high return/high risk portfolio that reduces the expected tax level at the expense of higher variability of tax rates. The tax-smoothing literature appears to be at variance with standard financial theory.

This apparent contradiction has been something of a puzzle. Skilling (1997, p.14) suggested that it is “misleading to examine one of these factors [level and variability of the tax rate] in isolation, as it is the combination of these factors which generate the total deadweight loss.” Grimes (2001b) verifies Bohn’s conclusions by numerical simulation but does not explain why they hold. He also questioned their robustness to cases where the Ricardian assumptions have been relaxed.

The purpose of this Appendix is to explain the result in Bohn (1990, 1995). The analysis is not intended to imply that a trade-off could not exist in other models with different assumptions.

The Appendix is divided into four sections:

• a brief review of two special cases where the absence of a risk/return trade-off is clear;
• intuition for Bohn’s results with consumption-CAPM model;
• intuition for Bohn’s results with exogenous asset prices; and
• a comment on potential weaknesses in Bohn’s model.

Two special cases

Bohn has presented several special cases where it is intuitively clear that policy should focus on minimising the variability of the tax rate. Two particular cases occur where:

• citizens and/or investors are risk neutral (refer Bohn 1990); and
• fiscal risks are diversifiable (refer Bohn, 1995, p.35).[32]

In the first case, where parties are risk neutral, the price of systematic risk would be zero.[33] In this case, all securities would have equal expected returns. Thus, there would be no risk/return trade-off on portfolio returns and therefore no exploitable opportunity available to policy makers. Convexity of the deadweight loss function therefore implies that policy should focus on minimising the variability of the tax rate.

In the second case, where fiscal risks are diversifiable, all fiscal risks may be hedged without cost in terms of forgoing portfolio expected returns. This result occurs because the price of diversifiable risk is zero. The result applies irrespective of the risk tolerance of citizens and/or investors.

The general case with consumption-CAPM model

Bohn (1995) shows the policy conclusion in favour of minimising the variability of tax rates continues to hold in the general case where:

• citizens and investors may be risk averse or risk neutral;
• fiscal risks may include a systematic component; and
• asset prices may be determined in accord with an equilibrium pricing model or exogenously.

Under these conditions, Bohn’s result appears counter-intuitive since financial theory implies that hedging the systematic component would incur a cost in terms of lower portfolio expected return. It would appear that the risk/return trade-off applicable to portfolio returns should carry over to a trade-off between the level and variability of the tax rate.

To understand the fallacy of this intuition it is useful for the purposes of this Appendix to assume that asset prices are determined according to the consumption-CAPM model.[34] This could be relevant to a closed economy where optimality would imply citizens’ consumption choices and asset prices are jointly determined.

Two key factors are important for understanding the policy conclusion in favour of minimising the variability of tax rates:

• first, from the perspective of citizens, the Crown portfolio is one of several sub-portfolios making up their total portfolio. Frictions can place the managers of a sub-portfolio at a comparative disadvantage in providing the risk/return trade-off desired on the total portfolio, e.g. tax distortions may place the Crown sub-portfolio at a comparative disadvantage; and
• second, in an equilibrium model, deadweight losses should not be evaluated by applying constant discount rates in an NPV calculation. Instead, in a closed economy the equilibrium discount rate varies with the marginal utility of consumption. This means that any reduction in deadweight losses in states of nature where marginal utility is low (i.e. high consumption) is less valuable than in states where the marginal utility is high (i.e. low consumption). As a result, calculating the net present value (NPV) of losses at constant discount rates would not reflect accurately the economic value to consumers.

These two factors come together to produce the Bohn (1995) result. A convex deadweight loss function places the Crown sub-portfolio at a comparative disadvantage by increasing the spread of returns (net of tax): Bad returns on the Crown portfolio are very bad because they induce higher tax rates and therefore higher deadweight losses as well as lower after-tax returns on citizens’ other sub-portfolios; Good returns are very good because they induce lower taxes and therefore lower deadweight losses as well as higher after-tax returns on other sub-portfolios. Hence, except where assets are negatively correlated with the tax rate, this unfortunate distribution of tax rates means that switching an asset from the Crown’s portfolio to a portfolio held directly by individuals would reduce the variance of returns on citizens total wealth portfolios.

In addition, an inherent feature of equilibrium in the consumption-CAPM model is that the assets offering high expected returns are precisely those assets that produce high payoffs in states when consumption is high (low marginal utility) and low payoffs when consumption is low (high marginal utility). These assets must offer a higher expected return to compensate for their unfortunate distribution of payoffs (relative to the distribution of marginal utility). Hence, it is clear that an investment by the Crown in “high return” assets to reduce the expected NPV of deadweight losses (at constant discount rates) potentially could have negative economic value to consumers.

The proof that such investments necessarily would confer negative value to consumers derives from the Crown’s comparative disadvantage as portfolio manager. The unfortunate distribution of tax rates (as described above) exacerbates the unfortunate distribution of gross payoffs from “high return” assets. Thus, citizens’ would prefer to use one or more of their own sub-portfolios rather than the Crown sub-portfolio to optimise their holdings of risky assets. For these reasons, within this model, it is always best to assign the Crown the task of immunising its portfolio so as to eliminate citizens’ exposure to the Crown, i.e. policy should minimise the variance of the tax rate.

The general case with exogenous asset prices

The result derived for the consumption-CAPM model applies also when asset prices are determined by other equilibrium models (e.g. APT) or determined exogenously. Bohn (1995, Section 2) adopts the assumption of exogenous asset prices by writing his model in terms of prices for general state-contingent claims. This approach is relevant to a small open economy, such as New Zealand, where asset prices are determined by international capital flows.

Much of the intuition of the previous section still applies. Given asset prices, optimising citizens maximise utility by trading-off the level and variability of consumption, which they implement through their choice of asset holdings. To the extent that citizens’ income risks are diversifiable then their asset holdings would achieve smooth consumption at high average levels. However, to the extent that citizens’ income risks include a systematic component, then by definition the equilibrium profile of consumption chosen by rational consumers would result in the marginal utility of consumption being low when asset payoffs are high (and vice versa). Similarly, the Crown portfolio still faces a comparative disadvantage relative to other sub-portfolios.

An example[35]

The result above may be illustrated by a simple example involving one riskless asset earning r (assumed equal to zero) and one risky asset earning expected return of R. Suppose that in state s₁ (with probability 0.75) the risky asset has an excess return of 4 percent, and that in state s₂ (with probability 0.25) it has an excess return of -4 percent. The expected excess return is thus 2 percent.

Given this exogenous distribution of asset payoffs, a rational consumer would hold assets such that the distribution of marginal utility is uncorrelated with the distribution of asset returns, i.e. such that E[u'(c).(R_t+1-r_t+1)]=0, where u'(c) is the marginal utility of consumption and E is the expectations operator. Further, suppose that u'(c(s₁)) = 1 in state s₁. Then u'(c(s₂)) must equal 3 for this expectation to hold.

Now suppose that if the government makes an investment it can either reduce taxes by 1 percent in state 1 or increase them by 1 percent in state 2. Assume a quadratic deadweight loss function, h(τ) = τ². With an initial tax rate of 0.3, the marginal deadweight losses in the two states will be 0.58 and 0.62 respectively. When these are weighted by the respective marginal utility weights, the expected economic value to citizens is -0.03.

Grimes (2001b) obtains similar results in a range of numerical simulations.

Potential weaknesses

As noted above, the purpose of this Appendix is to clarify the rationale for Bohn’s theoretical results and why they are consistent with financial theory. No claim is made that the Bohn model is appropriate for analysing policy options for the New Zealand economy.

Bohn (1990, footnote 5) and Bohn (1995, p.28) take comfort that his results are robust to any form of equilibrium asset pricing model and indeed to the case where asset prices are exogenous. Bohn’s comments are in reaction to the Equity Risk Premium Puzzle, whereby the main theoretical asset pricing models, such as the consumption-CAPM and APT have received, at best, very limited empirical support.

However, irrespective of how asset prices are determined, Bohn’s results rely on citizens making fully rational consumption and investment decisions. In this respect, the empirical failure of the consumption-CAPM model is a failure of joint hypothesis about both consumption behaviour and asset pricing. From this perspective, the empirical failure potentially strikes at the heart of tax smoothing as Bohn’s results rely heavily on the relation between consumption and asset prices implied by equilibrium models.

Looking ahead, a crucial issue is to achieve a better theoretical understanding of observed behaviour of consumption and asset prices.