I. The Model

Following the literature, the relation between inflation and unemployment is described by an expectations-augmented Phillips curve:²

where λ > 0; u, uⁿ and π are (respectively) the rates of unemployment, natural unemployment, and inflation; π^e is the public’s inflation forecast; and $\eta \sim N\left(0,{\sigma }_{\eta }^2\right)$ is a supply shock. The public is assumed to construct its expectations rationally:

where E is the expectations operator and I is the public’s information set.

The central banker affects π through a policy instrument. We can interpret this instrument as the rate of growth of a monetary aggregate or as a short-term nominal interest rate. The instrument is imperfect in the sense that in a stochastic world, it cannot determine inflation completely, as in:

where f(·) is a monotonic, continuous, and differentiate function, i is the policy instrument, and $\epsilon ~N\left(0,{\sigma }_{\epsilon }^2\right)$ is a control error uncorrelated with η. Since there is no private information in the model, the government’s and central banker’s information sets coincide with the public’s and are also given by I. Since i is chosen in the previous period, i ∈ I.³

The central banker is assumed to have additively separable preferences over inflation and unemployment. Preferences are described by the function:

where φ > 0, π^* and u^* are the targeted rates of inflation and unemployment, and α, γ are nonzero preference parameters. The components of (4) are described by the linex function g(x) = [exp (αx) - αx -1]/α² (Varian, 1974). This function has several important properties. First, it permits different weights for positive and negative deviations from the target. For example, in the case where α > 0, positive inflation deviations from the target are weighted more severely than negative ones in the central banker’s loss function, even if they are of the same magnitude. This means that both the size and sign of a deviation affect the central banker’s loss. Second, it relaxes certainty equivalence and allows a prudence motive on the part of the central banker. Then, moments of higher order than the mean might play a role in the formulation of monetary policy. Third, it is analytically tractable and yields a closed-form solution when shocks are normally distributed. Finally, it nests the quadratic function commonly used in previous literature as a special case when the preference parameter tends to zero.⁴ This result is important because it suggests that the hypothesis that the central banker’s preferences are quadratic over inflation (unemployment) could be evaluated by testing whether α(γ) is significantly different from zero.

The unemployment target is u^* = kuⁿ, where 0 < k < 1. This specification accommodates (i) the view that central bankers target the natural unemployment rate (that is, k = 1, see Blinder, 1998), and (ii) the usual assumption that labor market distortions make the natural unemployment rate higher than socially optimal and, consequently, 0 < k < 1. Analytical results obtained under both assumptions are compared below.

The problem of the central banker is to choose the value of the instrument that minimizes her expected loss. Formally,

subject to the expectations-augmented Phillips curve and taking π^e as given. The first-order condition of this problem defines implicitly the central banker’s reaction function in terms of the public’s inflation forecast, π^e:⁵

The terms ${\sigma }_{\pi }^2\mathrm{and}{\sigma }_u^2$ denote (respectively) the conditional variances of inflation and unemployment and are related to the variance of the structural disturbances as follows: ${\mathrm{\sigma }}_{\mathrm{\pi }}^2={\mathrm{\sigma }}_{\mathrm{\epsilon }}^2\mathrm{and}{\mathrm{\sigma }}_u^2={\mathrm{\sigma }}_{\mathrm{\epsilon }}^2+{\mathrm{\lambda }}^2{\mathrm{\sigma }}_{\mathrm{\eta }}^2$. In writing equation (5), I have used the fact that when shocks are normal, the distributions of inflation and unemployment conditional on I are normal. Hence, the distribution of exp(α(π − π^*)) and exp(γ(u − u^*)) are log normal.⁶

Following the literature, the rational expectations relation (equation 2) is interpreted as the public’s reaction function. The Nash equilibrium is the (expected) rate of inflation where equations (5) and (2) intersect. Conditions for the existence and uniqueness of the Nash equilibrium are presented in the following proposition:

Proposition 1. Provided that $\mathit{1}\mathit{+}\mathit{\left(}\mathit{\alpha \lambda \varphi }\mathit{/}\mathit{\gamma }\mathit{\right)}\mathit{\left[}\mathit{exp}\mathit{\left(}\mathit{\gamma }\mathit{\left(}\mathit{1}\mathit{-}\mathit{k}\mathit{\right)}{\mathit{u}}^{\mathit{n}}\mathit{+}{\mathit{\gamma }}^{\mathit{2}}{\mathit{\sigma }}_{\mathit{u}}^{\mathit{2}}\mathit{/}\mathit{2}\mathit{\right)}\mathit{-}\mathit{1}\mathit{\right]}\mathit{>}\mathit{0}$, there exists a unique π^e = E(π|I) such that h(E(π|I),π^e) = 0

Proof. To prove existence, construct a

Plugging (6) into (5) and using π^e = E(π|I) delivers h(E(π|I)π^e) = 0. To show uniqueness, assume there exists a second inflation forecast, say

that also lies on the 45^° line on the plane (π^e = E(π|I)) and satisfies h(E(π|I),π^e) = 0. Replace ${\hat{\pi }}^e$ in (5) and simplify to obtain

Since $1+(\mathrm{\alpha \lambda \varphi }/\mathrm{\gamma })(\exp (\mathrm{\gamma }(1-k)u^n+{\mathrm{\gamma }}^2{\mathrm{\sigma }}_u^2/2)-1)>0$ and α ≠ 0, then it must be the case that z = 0.

II. Theoretical Implications, Special Cases

General Case

These results indicate that prudence has two opposite effects on monetary policy under discretion. Prudence with respect to inflation (α > 0) reduces the inflation bias. Prudence with respect to unemployment (γ > 0) increases the inflation bias. The overall effect of prudence on equilibrium inflation is ambiguous. For given parameters values, the Nash equilibrium is unique (if it exists), but depending on the values of the preference and model parameters the equilibrium inflation rate will be different.

Consider the set of parameters values for which optimal monetary policy is an equilibrium outcome. A government that delegates monetary policy to a “prudent” central banker would obtain E(π|I) = π^* if and only if the preference parameters (α,γ) solve

$\begin{array}{cc}h\left(\alpha ,\mathrm{\gamma }\right)=-\alpha {\mathrm{\sigma }}_{\mathrm{\pi }}^2/2+\left(1/\alpha \right)ln\left[1+\left(\mathrm{\alpha \lambda \varphi }/\mathrm{\gamma }\right)\left(exp\left(\mathrm{\gamma }\left(1-k\right)u^n+{\mathrm{\gamma }}^2{\mathrm{\sigma }}_u^2/2\right)-1\right)\right]=0.& \left(7‴\right)\end{array}$

This equation has no closed-form solution, but it can be solved numerically for given parameter values. In order to develop some intuition, Figure 1 plots h(α,γ) = 0 for the case where $\mathrm{\lambda }=2,\varphi =0.8,\mathrm{\pi }*=0,k=0.75,u^n=5,{\mathrm{\sigma }}_{\mathrm{\pi }}^2=6$, and ${\mathrm{\sigma }}_u^2=2$. From the discussion above, points below (above) h(α,γ) = 0 correspond to Nash equilibria with an inflation (deflation) bias. Figure 1 also plots the Nash equilibrium predicted by the Barro-Gordon model and the set of values (α,γ) for which the Nash equilibrium does not exist (see Proposition 1, p. 460). Recall that the Barro-Gordon model corresponds to the case where (α,γ) → (0.0). Figure 1 indicates that the original inflation-bias result due to Barro and Gordon (1983b) is only one of the possible outcomes in a more general model with asymmetric preferences. No claim is made that the parameters used to construct this graph are realistic, but the basic point is that we cannot rule out a priori other Nash outcomes where monetary policy is optimal, there is a deflation bias, or there is an inflation bias that arises from different central bank objectives than in Barro and Gordon. In this sense, the econometric analysis of time series or cross-section data might be useful to obtain estimates of the central banker’s preference parameters and to assess the relative merits of different game-theoretical models.

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Possible Equilibrium Outcomes

Citation: IMF Staff Papers 2002, 004; 10.5089/9781589061224.024.A007

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III. Empirical Analysis

This section reports the results of an exploratory analysis of the model predictions using data from developed economies during the 1990s. Since the static nature of the model invites the cross-section analysis of the data, this paper uses observations from 21 OECD countries. The countries are Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Luxembourg, the Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, the United Kingdom, and the United States.⁹ The raw data were taken from OECD Main Economic Indicators. Average inflation was calculated using quarterly CPI inflation from 1990:1 to 1999:2. The conditional variances of inflation (unemployment) were computed from a regression of the first-difference of quarterly inflation (unemployment) on four of its lags. For reasons to be made clear below, I used two samples to compute the conditional variances: 1980:1 to 1989:4 and 1990:1 to 1999:2.

The simple game-theoretical model developed in Section I predicts that the conditional variances of inflation and unemployment help explain the rate of inflation. Linearize the Nash equilibrium (equation (6)) by means of a first-order Taylor series expansion and write it in reduced form as:¹⁰

where α = π^*+ λφ(l − k)uⁿ is a positive intercept; b = −α /2 and c = λφγ/2 are constant coefficients; j = 1, 2, …, 21, indexes the country; π_j is the average inflation rate in country j; ${\mathrm{\sigma }}_{\mathrm{\pi },j}^2\mathrm{and}{\mathrm{\sigma }}_{u,j}^2$ are the conditional variances of inflation and unemployment in country j; and ζ_j is a country-specific error term. In the absence of measurement error and heterogeneity, the Ordinary Least Square (OLS) projection of π on ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$ delivers consistent and unbiased estimates of a, b, and c. Recall that ${\mathrm{\sigma }}_{\pi }^2={\mathrm{\sigma }}_{\mathrm{\epsilon }}^2\mathrm{and}{\mathrm{\sigma }}_u^2={\mathrm{\sigma }}_{\mathrm{\epsilon }}^2+{\mathrm{\lambda }}^2{\mathrm{\sigma }}_{\mathrm{\eta }}^2$ are model parameters included in the information set I, and note that ζ can be interpreted as an average of rational-expectations forecast errors. It follows that ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$ are uncorrelated with ζ. From the estimate of b, one could recover an estimate of α as $\hat{\alpha }=-2\hat{b}$. From the estimate of c, it is not possible to recover an estimate of γ, but the sign of c is informative about the sign of γ because λ,φ > 0, by assumption. For example, a positive estimate of c implies γ > 0 and is consistent with the idea that the central banker weights more heavily positive than negative unemployment deviations from the natural rate.

Consider first the results for all countries in the sample, reported in column (1) of Table 1. The OLS estimate of b is numerically small and insignificantly different from zero at standard levels. This result suggests no systematic relation between the mean and the conditional variance of inflation during the period considered. An estimate of the (average) asymmetry preference parameter on inflation for OECD member countries is $\hat{\alpha }=-0.04\left(0.14\right)$. Thus, the null hypothesis of quadratic inflation preferences (α = 0) cannot be rejected at standard levels. On the other hand, the estimate of c is positive and significantly different from zero at the 10 percent level. Although the asymmetry preference parameter on unemployment is not identified, the finding that c > 0 supports the view that (on average) OECD central bankers weight more heavily positive than negative unemployment deviations from their target.

Table 1.

Results of Least-Squares Regressions

Notes: The figures in parentheses are White heteroskedasticity-consistent standard errors. The superscripts * and † denote the rejection of the null hypothesis that the coefficient is zero at the 5 percent and 10 percent significance levels, respectively, using the exact t distribution.

Table 1.

Results of Least-Squares Regressions

	All Countries								EU Only
${\sigma }_{\pi ,}^2{\sigma }_u^2$ Constructed Using Data	Post-1990				Pre-1990				Post-1990				Pre-1990
	(1)		(2)		(3)		(4)		(5)		(6)		(7)		(8)
Intercept	2.07	*	2.11		2.15	*	2.17	*	2.17	*	2.25	*	1.97	*	2.19	*
	(0.21)		(0.20)		(0.33)		(0.27)		(0.29)		(0.25)		(0.27)		(0.25)
${\sigma }_{\pi }^2$	0.02		0		0.07		0		0.05		0		0.88		0
	(0.07)				(0.48)				(0.09)				(0.66)
${\sigma }_u^2$	6.86	†	7.29	*	4.53		5.01		6.86	*	7.94	*	4.72		9.63	†
	(3.30)		(2.97)		(4.27)		(4.57)		(3.02)		(3.03)		(5.85)		(4.86)
R2	0.21		0.20		0.13		0.13		0.26		0.24		0.49		0.38

Notes: The figures in parentheses are White heteroskedasticity-consistent standard errors. The superscripts * and † denote the rejection of the null hypothesis that the coefficient is zero at the 5 percent and 10 percent significance levels, respectively, using the exact t distribution.

View Table

Table 1.

Results of Least-Squares Regressions

	All Countries								EU Only
${\sigma }_{\pi ,}^2{\sigma }_u^2$ Constructed Using Data	Post-1990				Pre-1990				Post-1990				Pre-1990
	(1)		(2)		(3)		(4)		(5)		(6)		(7)		(8)
Intercept	2.07	*	2.11		2.15	*	2.17	*	2.17	*	2.25	*	1.97	*	2.19	*
	(0.21)		(0.20)		(0.33)		(0.27)		(0.29)		(0.25)		(0.27)		(0.25)
${\sigma }_{\pi }^2$	0.02		0		0.07		0		0.05		0		0.88		0
	(0.07)				(0.48)				(0.09)				(0.66)
${\sigma }_u^2$	6.86	†	7.29	*	4.53		5.01		6.86	*	7.94	*	4.72		9.63	†
	(3.30)		(2.97)		(4.27)		(4.57)		(3.02)		(3.03)		(5.85)		(4.86)
R2	0.21		0.20		0.13		0.13		0.26		0.24		0.49		0.38

Notes: The figures in parentheses are White heteroskedasticity-consistent standard errors. The superscripts * and † denote the rejection of the null hypothesis that the coefficient is zero at the 5 percent and 10 percent significance levels, respectively, using the exact t distribution.

There are at least three difficulties in interpreting the above results. First, despite similarities in the economies in the sample, there might well be substantial heterogeneity in the natural rate of unemployment and preference parameters across countries. As it is well known, pooled least-squares estimates can be biased when the intercepts and slopes are heterogeneous. Second, since the conditional variances are estimated using inflation and unemployment data, ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$ are generated regressors. Pagan (1984) and Pagan and Ullah (1988) examine the implications of generated regressors in estimation and inference. In many cases, generated regressors can be problematic because they measure with noise the true, but unobserved, regressor. The problem is that measurement error in ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$ forms part of the residual ζ, and orthogonality conditions required for the consistency of OLS might not hold. Third, using estimates of ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$ based on the same sample period as π might lead to biased estimates of the slope parameters if the inflation distribution is skewed (see Flood and Marion, 2001). If the inflation distribution is skewed positively (negatively), there will be an upward (downward) bias. In this case, the problem is that measurement error in π is correlated with ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$. Hence the OLS estimate of the slope not only reflects the correlation between (the true) π and the conditional variances, but also includes a term capturing the correlation between measurement error and the explanatory variables. The latter term is proportional to the skewness of the independent variable and is only zero in the special case where π is not skewed.

In order to address the first issue, I examine a subsample that consists of members of the European Union (EU). Although there might still be substantial heterogeneity, it can be argued that these countries share a number of labor market features (e.g., EU work legislation) and have attempted in recent years to harmonize monetary policy through explicit institutional arrangements like the Exchange Rate Mechanism and Monetary Union. Hence, the data points could be regarded as separate realizations of the same data-generating process. Estimates using this subsample are reported in column (5) in Table 1. Results are essentially the same as those obtained for the full sample. The estimate of b is not significantly different from zero at standard levels. The estimate of the (average) asymmetry preference parameter on inflation for EU member countries is $\hat{\alpha }$ only −0.10 (0.18). The null hypothesis of quadratic inflation preferences cannot be rejected. The estimate of c is positive and significantly different from zero at the 5 percent level.

The possible bias that could be caused by measurement error in the explanatory variables could be addressed by using an instrumental variable procedure. Unfortunately, it is not easy to find instruments for the conditional variances of inflation and unemployment in the cross-section dimension. In preliminary work, I considered using the conditional variances of energy prices and detrended Solow residuals as instruments, but they proved weak in the sense that their correlation with the variables they were meant to instrument for was too low to yield reliable results.

A strategy to address the third issue is to construct estimates of the conditional variances using pre-1990 data.¹¹ In particular, I use quarterly data from 1980:1 to 1989:4. Provided that measurement error in π is not serially uncorrelated and the measure of ${\mathrm{\sigma }}_{\pi }^2\mathrm{and}{\mathrm{\sigma }}_u^2$ is perfect, OLS delivers consistent estimates of b and c. Results are reported in column (3) for the full sample and column (7) for the EU subsample. Results are similar to the ones previously obtained. The null hypothesis that b = 0 cannot be rejected at standard levels. The estimate of c is positive in both cases but is not statistically different from zero. Since the number of data points is limited, more precise estimates of c could be obtained if the restriction that inflation preferences are quadratic (α = 0) is imposed. All specifications were reestimated and results reported in columns (2), (4), (6), and (8) in Table 1. Estimates are consistent with the ones previously obtained but, as expected, standard errors are smaller. The null hypothesis c = 0 cannot be rejected at the 5 percent level in two cases, at the 10 percent level in one case, and is rejected in only one case.

In summary, the empirical analysis of the game-theoretical model indicates that the hypothesis that preferences are quadratic in inflation cannot be rejected. On the other hand, estimates of the coefficient on ${\sigma }_u^2$ are always positive and in most cases significantly different from zero. Although the preference parameter cannot be identified, this result is consistent with a positive value of γ meaning that the (average) central banker attaches a larger loss to positive than negative unemployment deviations from its target. For the reasons discussed below, however, these empirical results should be interpreted with caution and are best regarded as exploratory.

IV. Conclusion

This paper presents initial results in a larger research project that examines monetary policy under asymmetric preferences. The relevance of this generalization of the central banker’s loss function is illustrated here in the simplest possible setup where the central banker and the public play a one-shot game without private information. It is shown that although the central banker’s reaction function is nonlinear on the public’s inflation forecast, there are conditions under which the Nash equilibrium exists and is unique. More importantly, relaxing certainty equivalence means that uncertainty can induce a prudence motive on the part of the monetary authority, which has two distinct and contradictory effects on equilibrium inflation. Prudence on inflation moderates the incentive to create surprise inflation, while, in unemployment, prudence increases the expected benefit of surprise inflation. The overall effect depends on the relative magnitude of the preference parameters and the conditional variances of inflation and unemployment.

While the empirical results are suggestive of asymmetric unemployment preferences, they are best regarded as exploratory for at least two reasons. First, results depend very obviously on the estimation procedure and on the countries included in the sample. Second, although this paper allows for the strategic interaction of the central banker and the public, equilibrium concepts other than Nash might be empirically important. Current and future research by the author seeks to address these observations. For example, Ruge-Murcía (2001b) focuses on inflation targeting regimes and estimates the central banker preferences parameters for Canada, Sweden, and the United Kingdom using the time series of inflation and unemployment. Ruge-Murcía (2001c) examines time series data from G-7 economies to test whether an inflation bias could arise even when central bankers target the natural unemployment rate. Still, given our limited understanding of central bankers’ behavior and preferences, it is probably premature to dismiss the notion that prudence can play a role in modern monetary policymaking.