4.2  Convex loss function[13]

A convex deadweight loss function is the standard assumption in the literature. In this case, Bohn (1990, 1995) shows that Crown financial policy should minimise the variability of the tax rate. This conclusion is due to two factors:

• 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) while good returns are very good because they induce lower taxes (though convexity means the reduction in deadweight losses is proportionately smaller than the increase in bad states). An identical portfolio held directly by an individual would have lower variance of returns; and
• assets offering high expected returns have an unfortunate distribution of payoffs, in the sense that their payoffs tend to be high in states when consumption is high (low marginal utility) and low when consumption is low (high marginal utility). The high expected return is compensation for this unfortunate distribution of payoffs (relative to the distribution of marginal utility). Citizens wish to invest in such assets only up to the point where the marginal utility of higher average consumption equals the marginal disutility of higher variability of consumption.

The Crown’s comparative disadvantage exacerbates the unfortunate distribution of payoffs from “high return” assets. Thus, citizens would prefer to use one or more of their own sub-portfolios other than the Crown sub-portfolio to optimise their holdings of risky assets. For these reasons, the Crown is best assigned the task of immunising its portfolio to eliminate citizens exposure to the Crown. This means that the Crown should target the zero-variance portfolio,[14] i.e. zero diversifiable and systematic risk. The Appendix provides a more detailed intuitive explanation of Bohn’s result.

Instruments to achieve zero-variance portfolio

Consistent with the assumption of complete capital markets, most theoretical contributions favouring the zero-variance portfolio assume the government can issue and purchase state-contingent securities as desired to hedge all risks.[15]

An exception is Angeletos (2002), who shows in the context of a closed-economy equilibrium business cycle model that almost every risk can be hedged with non-contingent debt of different maturities. Angeletos presents a stylised example where a government implements the optimal portfolio policy by selling perpetuities and investing in short-term assets. Critical to his result is that shocks to government expenditure and/or the tax base affect the equilibrium interest rate, causing movements in the market value of long term debt greater than movements in the market value of short-term assets.

Summary for convex losses (with complete and perfect markets)

Economic objective

Minimise the expected economic value of the deadweight loss of taxation

CFP objectives

Tax policy

Minimise variance of the tax rate subject to IBC

Portfolio policy

Minimise variance of the Crown portfolio

Target

Zero variance Crown portfolio

Instruments

1. contingent securities, e.g. issue contingent debt with returns negatively indexed to public spending and positively indexed to productivity and other shocks to the tax base;
2. debt maturity structure, e.g. issue perpetuities and purchase short-term assets; and
3. any combination of market securities with appropriate covariances.

4.3  Incomplete capital markets

Thus far the analysis has assumed all Ricardian assumptions hold except those relating to tax distortions. This and the following subsections depart from the various Ricardian assumptions, while continuing to assume the deadweight loss function is convex.

The current subsection summarises the case where some fiscal risks cannot be hedged. The absence of hedging opportunities may be due to incomplete capital markets or simply that the Crown is unable to access certain markets or use particular instruments. Lack of access to particular markets or instruments may be due to asymmetric information issues and moral hazard incentives relating to government spending policy. These are discussed further in Sections 5 and 7. For the purposes of this section, the distinction between incompleteness and lack of access is irrelevant. The key underlying assumption is that the Crown faces a comparative disadvantage in managing citizens’ risk/return exposure.

Given the Crown’s comparative disadvantage, the optimal tax policy is the same policy objective as the previous section, i.e. minimise the variability of tax rates. Similarly, the optimal portfolio policy objective is to minimise the variance of the Crown portfolio.

Although the tax and portfolio objectives are unaltered, the assumption of incomplete markets restricts the instruments the Crown has available to achieve its policy objectives. Bohn (1995) shows that the optimal policy now has two parts:

• hedge to the fullest extent possible using available securities; and
• consider building and maintaining a positive balance of net worth as self-insurance against unhedged risks.[16]

Building a precautionary balance would require the tax rate to be held temporarily higher than otherwise. The cost of additional deadweight losses is worthwhile if and only if the unhedged risks would otherwise result in a negative correlation between tax rates and consumption. The rationale is similar to the previous section, and relates to the high cost in terms of forgone utility if tax rates have to be increased in states where consumption is already low (i.e. high marginal utility). Building up a buffer has value to the extent that the unfortunate timing of changes in tax rates can be avoided.

In contrast, if unhedged risks result in tax rates and consumption being positively correlated then tax adjustments facilitate consumption smoothing. Building a precautionary balance would remove this beneficial effect while also imposing deadweight losses due to temporarily higher tax rates. If the correlation between tax rates and consumption is zero or positive, the Crown should target CNW at zero, i.e. just satisfy the government inter-temporal budget constraint.

Whether the Crown should target a positive net worth buffer is an empirical issue. It depends on which risks cannot be hedged and how these shocks impact on tax policy and consumer behaviour.

Instruments to achieve minimum-variance portfolio

A substantive literature has developed analysing the hedging properties of marketable assets and observed debt instruments (in the absence of general state-contingent securities). Alternative debt denominations include nominal debt, price-indexed debt, and foreign-currency denominated debt. Short- versus long-term debt maturities have been analysed also. The key conclusion is that the appropriate instrument is an empirical question. The choice of debt denomination and maturity depends on the type of shocks hitting the economy and the serial and cross-correlations in macroeconomic variables within and across countries. The main results, summarised from Missale (1997), are provided in the box below.

Summary for incomplete markets

Economic objective

Minimise the expected economic value of the deadweight loss of taxation

CFP objectives

Tax policy

Minimise variance of tax rate

Portfolio policy

Minimise variance of the Crown portfolio

Targets

1. minimum-variance portfolio (i.e. zero diversifiable risk and minimum systematic risk consistent with risks that cannot be hedged);
2. positive CNW buffer (if unhedged risks cause negative correlation between tax and consumption, otherwise target net worth at zero)

Instruments

1. buy (short-sell) assets whose returns have positive (negative) correlation to public spending and negative (positive) correlation to tax base;
2. choice of debt instruments is an empirical issue (see Missale 1997):
   • nominal debt for government spending and productivity shocks;
   • price-indexed debt for monetary and real demand shocks causing inflation;
   • foreign currency debt when output and inflation shocks are correlated internationally;
   • maturity structure of debt to match structure of planned fiscal surpluses;
   • short maturity debt when positive correlation between output and real interest rates; and
3. tax rate, as instrument for building net worth buffer (if required)