On Impatience and Policy Effectiveness

Silvia Sgherri; Tamim Bayoumi

doi:10.5089/9781451871654.001.A001

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On Impatience and Policy Effectiveness

Author: Ms. Silvia Sgherri¹ and Mr. Tamim Bayoumi

¹ 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Author’s E-Mail Address: tbayoumi@imf.org; ssgherri@imf.org

Publication Date:: 01 Jan 2009

eISBN:: 9781451871654

Language:: English

Keywords:: WP; real interest rate; income hypothesis; multiplier rise; interest rate equation; net tax rate equation

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An increasing body of evidence suggests that the behavior of the economy has changed in many fundamental ways over the last decades. In particular, greater financial deregulation, larger wealth accumulation, and better policies might have helped lower uncertainty about future income and lengthen private sectors' planning horizon. In an overlapping-generations model, in which individuals discount the future more rapidly than implied by the market rate of interest, we find indeed evidence of a falling degree of impatience, providing empirical support for this hypothesis. The degree of persistence of "windfall" shocks to disposable income also appears to have varied over time. Shifts of this kind are shown to have a key impact on the average marginal propensity to consume and on the size of policy multipliers.

Abstract

I. Introduction

An increasing body of evidence suggests that the behavior of the economy has changed in substantial and fundamental ways over the last decades.¹Inter alia, our previous work has shown that changes in the efficacy of monetary policy are linked to shifts in the degree of forward-lookingness of price/wage setters and that these improvements in supply-side flexibility have been instrumental in reducing the volatility of inflation and, even more importantly, output.² Have households also become more far-sighted over time?

Greater financial deregulation, larger wealth accumulation, and better policies might indeed have helped lower uncertainty about future income and lengthen consumers’ planning horizon. In an overlapping-generations model, in which individuals discount the future more rapidly than implied by the market rate of interest, evidence of a falling rate of time preference would provide further support for this hypothesis. Intuitively, this kind of shifts in private-sector behavior is likely to have a divergent impact on the effectiveness of monetary and fiscal policy: while forward-looking price setters tend to increase the effectiveness of interest rate changes in stabilizing the economy, tax and transfer policies tend to have less impact on the consumption plans of far-sighted households.

To look into changes in the transmission mechanism of monetary and fiscal policy, we need to pin down the structural parameters describing the private-sector behavior as well as those describing the behavior of the policymaker. Extending Bayoumi and Sgherri (2006), this is done within an overlapping-generations framework which dispenses with the assumption of the infinite-lived representative agent. The model assumes that consumers have finite planning horizons and, therefore, discount the future more rapidly than implied by the government’s budget constraint. As a result, households value tax cuts today more highly than the implied future tax increases, allowing expansionary tax and transfer policies to have real effects on consumption, even though they are optimizing their lifetime consumption plans subject to intertemporal budget constraints. Adding a life-cycle dimension to consumption provides more realistic consumption dynamics, with spending responding less to a temporary fiscal policy shocks than to a long-term one, as predicted by the permanent income hypothesis. In such a theoretical model, the impact of any policy shock on consumption depends on three characteristics—the persistence of the shock, whether it is anticipated or not, and the discount wedge, i.e., the consumers’ excess of discount with respect to the prevailing market interest rate.

A very important feature of this model is its ability to generate testable econometric relationships between the coefficients of the consumption function and the underlying structural parameters of the model. In this way, we succeed in identifying and estimating the parameter that measures the life horizon, thereby nesting the infinite-horizon model (and Ricardian equivalence) as a special parameter configuration. Cross-equation restrictions also permit to investigate the data admissibility of alternative consumption theories within a unique modeling framework. Given our focus on changes in behavior over time we look at results from rolling regressions. These results permit to identify meaningful sub-periods over which structural parameters have been relatively stable. Estimates for these samples will be subsequently used to analyze the impact of shifts in agents’ behavior on the evolution of the propagation mechanisms of monetary and fiscal shocks.

To anticipate our conclusions, estimation reveals that the finite-horizon model fits post-war US data reasonably well, entailing a discount wedge of about 3 percent a year. The discount wedge appears to have fallen over time, meaning that the time horizon relevant for individual decision making has lengthened. At the same time, the degree of persistence of “windfall” shocks to disposable income has increased, whereas shocks to the real interest rate seem to have become shorter-lived than they used to. The analysis of the combined effect of these shifts on the size of income and policy multipliers suggests that changes in the persistence of shocks have been more important than those affecting the length of consumers’ horizon.

The plan of the paper is as follows. Section II provides the theoretical framework for the analysis. The intertemporal model is estimated in section III. As mentioned before, implied fiscal and monetary multipliers depend on three characteristics—the persistence of the shock itself, whether it is anticipated or not, and the rate to which consumers discount the future. Section IV explores these interactions in more detail as well as time variation in the coefficient estimates. Broader implications for policy analysis are discussed in the concluding section.

II. Theoretical Model

The model used in this paper is a discrete-time, closed-economy version of the overlapping-generations framework in Blanchard (1985) and Yaari (1965), in which Ricardian equivalence is broken through the assumption that consumers face a constant probability of dying in each period. More precisely, while households make their consumption plans on the basis of a finite horizon, the society (and, thereby, the government) has an infinite horizon due to the continuous entry of new generations. The difference between the time horizons relevant for individual and public decision making results in discrepancies between the private and public sectors’ discount rate, meaning that individuals discount the future at a faster rate than implied by the government’s budget constraint. In this context, a tax cut (or a rise in spending) boosts current consumption because the wedge between the real interest rate and the discount rate implies that the net present value of the tax cut (or the rise in spending) exceeds that of subsequent increases in taxes (or spending cuts) needed to keep the government solvent.

Monetary policy

In a closed economy, the real interest rate (r_t) is endogenously determined by the monetary authority. The central bank follows an interest rate rule, according to which the real interest rate moves each period to adjust for past deviations from its steady state level $(\bar{r}) :$

\begin{matrix} Δ (r_{t + i} - \bar{r}) = θ^{r} (r_{t + i - 1} - \bar{r}) + ε_{t + i}^{r} & (1) \end{matrix}

where the parameter θ^r determines the degree of interest rate smoothing and $ε_{t}^{r}$ is an unexpected monetary policy shock.³ Given equation (1), it should be clear that at each period t, the expected level of the real interest rate is equal to: $E_{t} r_{t + 1} = - \sum_{i = 0}^{\infty} {(1 + θ^{r})}^{i} θ^{r} \bar{r} = \bar{r}$ .

Consumption

Crucially, in addition to the usual discount rate (assumed equal to the expected real interest rate, $\bar{r}$ ), consumers of all ages face an additional discount wedge, λ, reflecting their constant probability p of dying in the next period, such that $p = \frac{λ}{(1 + λ)}$ . The size of each age cohort at birth is also normalized to p and is assumed to decline deterministically over time, at rate p. In this way, total population at each time t is constant and equal to 1.⁴ Individuals are assumed to be born with zero financial wealth. The assets/liabilities of the dead are transferred to outside life insurance companies operating under zero-profit conditions and able to borrow/lend freely from the government to service their interest costs.⁵ To simplify the modeling, we assume that utility is quadratic, which ensures certainty equivalence, and that labor income follows an exogenous and quite general stochastic process, encompassing both the hypotheses of unit root behavior and trend stationarity.⁶

The maximization problem of each consumer born at time s is thus:

\begin{array}{l} Max E_{t - 1} \sum_{i = 0}^{\infty} \frac{U (c_{t + i})}{{[(1 + r_{t}) (1 + λ)]}^{i}} \\ s . t . \sum_{i = 0}^{\infty} \frac{c_{t + i}}{{[(1 + r_{t}) (1 + λ)]}^{i}} = \sum_{i = 0}^{\infty} \frac{y_{t + i}}{{[(1 + r_{t}) (1 + λ)]}^{i}} \\ Δ (r_{t + i} - \bar{r}) = θ^{r} (r_{t + i - 1} - \bar{r}) + ε_{t}^{r} & (2) \\ Δ (y_{t + i} - y_{t + i}^{*}) = θ^{y} (y_{t + i - 1} - y_{t + i - 1}^{*}) + μ^{y} + ε_{t}^{y} \\ y_{t + i}^{*} = κ^{y} + ζ^{y} time \\ U (c_{t + i}) = c_{t + i} - \frac{1}{2} c_{t + i}^{2} \end{array}

where U(·) is the utility function, c is the individual’s consumption level, and y is the individual’s disposable income, which may either follow a random walk with drift (ζ^y, θ^y = 0) or a stationary autoregressive process (ζ^y ≠ 0, θ^y <0) around a deterministic trend (time). In addition, λ indicates the discount wedge, Δ is the first difference operator, and remaining Greek letters reflect underlying parameters.

It should be stressed from the start that by “death” we mean economic death rather than its physical counterpart. Limited planning horizons (or “impatience”) in economic decision making can be due to disconnectness of current households from future generations or lack of an altruistic bequest motive.⁷ They can also occur through more subtle factors, for example imperfect access to financial markets. We regard the probability of “death” as an unknown parameter to be estimated.

It should also be noted that, for each individual consumer, the first order condition of the dynamic optimizing problem (1) implies E_t-1Δc_t = 0, regardless of the stochastic properties of labor income. The implication that changes in individual consumption are unpredictable—as they are in the case of a representative consumer model with infinite lifetime—stems from the fact that the discount rate and the return on financial wealth are the same, that is $(1 + \bar{r}) (1 + λ)$ . That’s not the case in the aggregate. In the aggregate, the discount rate is still $(1 + \bar{r}) (1 + λ)$ , but the return on financial wealth is just $(1 + \bar{r})$ , as insurance premia and wealth transfers among consumers disappear. This implies that, as long as λ > 0, changes in aggregate consumption become predictable, while aggregate consumption and aggregate labor income share a common trend.

To see this, notice that the expected path for aggregate consumption is defined as:

\begin{matrix} E_{t - 1} c_{t} = p E_{t - 1} c_{t, t} + (1 - p) c_{t - 1} & (3) \end{matrix}

where the first component reflects the expected level of consumption of the new cohort born at time t, and the second component denotes the one-period-ahead consumption of those who were alive both at time t-1 and at time t.

Given the assumption that individuals are born with no financial wealth, for the newly born cohort the expected level of consumption will only depend on the discounted sum of current and future labor income. Using the expression for individual human wealth:

\begin{matrix} H_{t} = \frac{1}{(1 + r_{t}) (1 + λ)} \sum_{j = 0}^{\infty} {[\frac{1 + θ^{y}}{(1 + r_{t}) (1 + λ)}]}^{- j} E_{t - 1} y_{t + j} & (4) \end{matrix}

and decomposing expected labor income into its current level (y_t-1) and the discounted sum of anticipated future changes (E_t-1Δy_t), we can derive the following expression:

\begin{matrix} E_{t - 1} c_{t, t} = (\bar{r} + λ) E_{t - 1} H_{t} = \frac{\bar{r} + λ}{\bar{r} + λ - θ^{y}} y_{t - 1} + \frac{(1 + \bar{r}) (1 + λ)}{\bar{r} + λ - θ^{y}} [Δ y_{t} + \frac{\bar{r}}{λ} ε_{t}^{y}] & (5) \end{matrix}

where $ε_{t}^{y} = Δ y_{t} - E_{t - 1} Δ y_{t}$ denotes unexpected changes in labor income, regarded as wealth. Weighting equation (5) appropriately and substituting for into equation (3) produces the following aggregate consumption function:

\begin{matrix} Δ c_{t} = - \frac{λ}{1 + λ} (c_{t - 1} - \frac{\bar{r} + λ}{\bar{r} + λ - θ^{y}} y_{t - 1}) + \frac{λ (1 + \bar{r})}{\bar{r} + λ - θ^{y}} Δ y_{t} + \frac{\bar{r} (1 + \bar{r})}{\bar{r} + λ - θ^{y}} ε_{t}^{y} + ε_{t}^{c} & (6) \end{matrix}

where $ε_{t}^{c} = c_{t} - E_{t - 1} c_{t}$ indicates idiosyncratic shocks to aggregate consumption.

It should thus be clear that changes in consumption depend on three factors, namely: (i) an “error correction” mechanism on the difference between the level of lagged consumption and income due to the “birth” of new individuals, (ii) predictable changes in labor income reflecting a positive wedge in the discount rate, and (iii) the familiar “random walk” effect from unanticipated changes in income.

Importantly, the model nests the infinite-horizon model as a special case: if the discount wedge (and, thereby, the probability of dying) is zero, aggregate consumption collapses to a pure random walk and changes in aggregate consumption become unpredictable. Conversely, as long as the discount wedge is positive, aggregate consumption and aggregate labor income share a common trend. Thus, if y_t is a unit root process with drift (θ^y, ζ^y = 0), then c_t will also be; if y_t is trend stationary (θ^y <0, ζ^y ≠ 0), both series will comprise a deterministic trend. Average changes in consumption will also become proportional to expected changes in labor income, implying $E_{t - 1} Δ c_{t} = \frac{\bar{r} + λ}{\bar{r} + λ - θ^{y}} (μ^{y} + ζ^{y})$ .

Fiscal policy

In this model, real public spending (expressed as a ratio to aggregate income and denoted as g) is financed by lump-sum tax payments net of lump-sum transfers (tt) and government debt (B), both expressed as ratios to aggregate income.⁸ The expected real return on one-period government bonds is equal to the market real rate, $\bar{r} .$ Under the usual transversality condition, $\lim_{i \to \infty} \frac{B_{t + i}}{{(1 + r_{t})}^{i}} = 0$ , the intertemporal government budget constraint is of the form:

\begin{matrix} \sum_{i = 0}^{\infty} \frac{g_{t + i}}{{(1 + r_{t})}^{i}} = \sum_{i = 0}^{\infty} \frac{{tt}_{t + i}}{{(1 + r_{t})}^{i}} & (7) \end{matrix}

As the government’s budget constraint needs to be satisfied, any cut in net taxes (or a rise in public spending) has, at some point, to be counterbalanced by a future increase in net taxes (or a cut in public spending). We model this by assuming that the long-term rate of taxes less transfers (denoted as tt^*) moves each period, reflecting a deterministic trend (time) and the long-term costs of this period’s innovation to the net tax rate itself (denoted as ε^tt). Hence, an unexpected fall in taxes (rise in transfers) is simultaneously accompanied by an increase in the expected long-term rate of taxes less transfers. Futhermore, we allow the net tax rate to adjust for past deviations from its long-run rate and to vary over the cycle, mimicking a progressive tax and transfer system. Akin to labor income, the net tax rate may either follow a random walk with drift (θ^tt = 0) or a stationary autoregressive process (θ^tt <0) around its deterministic trend. Specifically:

E_{t - 1} Δ ({tt}_{t + i} - {tt}_{t + i}^{*}) = θ^{tt} ({tt}_{t + i - 1} - {tt}_{t + i - 1}^{*}) + ϕ Δ y_{t + i} + μ^{t t} where {tt}_{t + i}^{*} = κ^{t t} + ζ^{t t} time - \frac{\bar{r}}{\bar{r} - θ^{tt}} ε_{t + i}^{tt} .

Symmetrically, the long-term level of public expenditure (denoted as g^*) moves each period, reflecting a deterministic trend (time) and the long-term costs of this period’s innovation to public spending (denoted as ε^g). Hence, an unexpected rise in public spending is simultaneously accompanied by a cut in g^*. Once again, we generalize the government spending process by letting it follow either a random walk with drift (θ^g = 0) or a stationary autoregressive process (θ^g <0) around its deterministic trend. Specifically:

E_{t - 1} Δ (g_{t + i} - g_{t + i}^{*}) = θ^{g} (g_{t + i - 1} - g_{t + i - 1}^{*}) + μ^{g} where g_{t + i}^{*} = κ^{g} + ζ^{g} time - \frac{\bar{r}}{\bar{r} - θ^{t t}} ε_{t + i}^{g}

The consumer’s problem is now modified by the presence of public expenditure in the utility function and the stream of tax payments (net of lump-sum transfers) altering the notion of disposable labor income:

\begin{array}{l} Max E_{t - 1} \sum_{i = 0}^{\infty} \frac{U (c_{t + i}, g_{t + i})}{{[(1 + r_{t + i}) (1 + λ)]}^{i}} \\ s . t . \sum_{i = 0}^{\infty} \frac{c_{t + i}}{{[(1 + r_{t + i}) (1 + λ)]}^{i}} = \sum_{i = 0}^{\infty} \frac{(y_{t + i} - t_{t + i})}{{[(1 + r_{t + i}) (1 + λ)]}^{i}} \\ Δ (r_{t + i} - \bar{r}) = θ^{r} (r_{t + i - 1} - \bar{r}) + ε_{t + i}^{r} \\ Δ (y_{t + i} - y_{t + i}^{*}) = θ^{y} (y_{t + i - 1} - y_{t + i - 1}^{*}) + ϕ Δ y_{t + i} + μ^{y} + ε_{t + i}^{y} \\ y_{t + i}^{*} = κ^{y} + ζ^{y} time & (8) \\ Δ ({tt}_{t + i} - t t_{t + i}^{*}) = θ^{tt} ({tt}_{t + i - 1} - {tt}_{t + i - 1}^{*}) + μ^{t t} + ε_{t + i}^{tt} \\ t t_{t + i}^{*} = κ^{t t} + ζ^{t t} time - \frac{\bar{r}}{\bar{r} - θ^{tt}} ε_{t + i}^{tt} \\ Δ (g_{t + i} - g_{t + i}^{*}) = θ^{tt} (g_{t + i - 1} - g_{t + i - 1}^{*}) + μ^{g} + ε_{t + i}^{g} \\ g_{t + i}^{*} = κ^{g} + ζ^{g} time - \frac{\bar{r}}{\bar{r} - θ^{g}} ε_{t + i}^{g} \\ U (c_{t + i}, g_{t + i}) = c_{t + i} - Γ c_{t + i}^{2} - γ g_{t} c_{t} \end{array}

The resulting consumption function looks very much like the earlier one except that unanticipated cuts in taxes (ε^tt) and unanticipated increases in government spending (ε^g) lower consumption through a Ricardian offset on tt^* and g^*, respectively, whereas unexpected increases in income (ε^y) raise consumption through higher saving. In addition, the presence of a progressive tax and transfer system (φ>0) is likely to lower the impact of changes in income on consumption. There may also be subtle differences in the coefficients on income and net taxes in the “error correction” mechanism due to the specific speed of adjustment of the two stochastic processes:

\begin{array}{l} Δ c_{t} & = \frac{λ (1 + \bar{r})}{\bar{r} + λ - θ^{y}} (1 - \frac{ϕ λ (1 + \bar{r})}{\bar{r} + λ - θ^{tt}}) (Δ y_{t} + \frac{\bar{r}}{λ} ε_{t}^{y}) - \frac{λ (1 + \bar{r})}{\bar{r} + λ - θ^{tt}} (Δ {tt}_{t} - \frac{\bar{r}}{\bar{r} - θ^{tt}} ε_{t}^{tt}) \\ - \frac{ξ^{r}}{\bar{r} + λ - θ^{r}} Δ r_{t} - \frac{ξ^{g}}{\bar{r} + λ - θ^{g}} (Δ g_{t} - \frac{\bar{r}}{\bar{r} - θ^{g}} ε_{t}^{g}) & (9) \\ - \frac{λ}{1 + λ} [c_{t - 1} - \frac{\bar{r} + λ}{\bar{r} + λ - θ^{y}} y_{t - 1} + \frac{\bar{r} + λ}{\bar{r} + λ - θ^{tt}} {tt}_{t - 1}] + ε_{t}^{c} \end{array}

III. Empirical Estimates

The model was estimated from 1955 using annual data on (the logarithm of) real consumption (c) and real income (y), the ratio of public expenditure to income (g), (the logarithm of one minus) the net tax rate (tt, direct taxes net of transfers as a ratio to income), and the real interest rate (r). Corresponding series are plotted in Figure 1. Annual data were used because taxes are levied on yearly income and it simplifies the time series characterization of the data, while 1955 was chosen to have as long a time series as possible without including the large shocks experienced by the economy over the great depression, Second World War, and immediate postwar period. While indirect taxes are not included specifically, they affect real income and consumption through the deflator. In any case, most of the active fiscal policy in the United States has occurred through the federal government, whose main tax base is personal income.

A preliminary analysis of the time series properties of aggregate consumption and aggregate income suggests that these series are nonstationary, and better characterized by the presence of a stochastic trend with positive drift than by a deterministic one. In contrast, corresponding evidence for the net tax rate seems to imply that g is trend-stationary, while tt might even be a mean reverting process. Table 1 reports results of Augmented Dickey-Fuller (1979), Phillips-Perron (1988), and Ng-Perron tests for unit roots allowing for appropriate deterministic components in the data generating process. All test statistics corresponding to c and y fall within the 95 percent confidence region and are thus consistent with the presence of a unit root in those series, whereas in the case of tt the statistical support for this hypothesis is mixed.

Table 1.

United States: Unit Root Tests

^aDickey and Fuller (1979) t-statistic. MacKinnon (1996) one-sided p-values.

^bPhillips and Perron (1988) t-statistic. MacKinnon (1996) one-sided p-values.

^cNg-Perron (2001) t-statistic and corresponding critical values (see Ng-Perron (2001), Table 1).

^dCritical values correspond to the null of a unit root against a trend-stationary alternative.

^eCritical values correspond to the null of a unit root against a mean-reverting alternative.

Table 2 reports the estimated cointegrating regression between c, y, and tt, as well as two statistics corresponding to the trace and the maximum eigenvalue cointegration tests. The null of no cointegration between consumption, aggregate income, and net taxes is systematically rejected by the data. This result provides initial support for both the “finite-horizon” consumption model (equation 9), that predicts the existence of a cointegrating relation between these three series.

Table 2.

United States: Cointegration Tests^a

^aJohanssen cointegration rank test between (c_t, y_t, tt_t) allowing for a restricted constant (κ).

^bMacKinnon-Haug-Michelis (1999) p-values.

^{^*}Denotes rejection of the corresponding hypothesis at the 5% level.

Table 2.

United States: Cointegration Tests^a

# Cointegr. Relations	Eigenvalue	Statistic	5% Critical Value	P-values^b
		Trace Statistic
None^*	0.41	48.77	35.19	0.00
At most 1	0.19	20.20	20.26	0.05
At most 2	0.15	8.93	9.16	0.06
		Maximum Eigenvalue
None^*	0.41	28.57	22.30	0.01
At most 1	0.19	11.27	15.89	0.23
At most 2	0.15	8.93	9.16	0.06
Implied Cointegrating Relation
Normalized cointegrating coefficients (standard error in parentheses)
c_t	y_t	tt_t	κ
1	-1.00 (0.01)	0.98 (0.53)	0.52 (0.16)
Adjustment coefficients (standard error in parentheses)
Δc_t	-0.12 (0.05)
Δy_t	0.05 (0.09)
Δtt_t	-0.10 (0.05)

^aJohanssen cointegration rank test between (c_t, y_t, tt_t) allowing for a restricted constant (κ).

^bMacKinnon-Haug-Michelis (1999) p-values.

^{^*}Denotes rejection of the corresponding hypothesis at the 5% level.

Table 2.

United States: Cointegration Tests^a

# Cointegr. Relations	Eigenvalue	Statistic	5% Critical Value	P-values^b
		Trace Statistic
None^*	0.41	48.77	35.19	0.00
At most 1	0.19	20.20	20.26	0.05
At most 2	0.15	8.93	9.16	0.06
		Maximum Eigenvalue
None^*	0.41	28.57	22.30	0.01
At most 1	0.19	11.27	15.89	0.23
At most 2	0.15	8.93	9.16	0.06
Implied Cointegrating Relation
Normalized cointegrating coefficients (standard error in parentheses)
c_t	y_t	tt_t	κ
1	-1.00 (0.01)	0.98 (0.53)	0.52 (0.16)
Adjustment coefficients (standard error in parentheses)
Δc_t	-0.12 (0.05)
Δy_t	0.05 (0.09)
Δtt_t	-0.10 (0.05)

^aJohanssen cointegration rank test between (c_t, y_t, tt_t) allowing for a restricted constant (κ).

^bMacKinnon-Haug-Michelis (1999) p-values.

^{^*}Denotes rejection of the corresponding hypothesis at the 5% level.

Basic Model

The unrestricted system we estimate comprises:

\begin{array}{l} Δ c_{t} = α^{c} + β^{y} Δ y_{t} + β^{tt} Δ {tt}_{t} + β^{tt} Δ g_{t} + β^{t t} Δ r_{t} + β^{ecm} (c_{t - 1} + γ^{y} y_{t - 1} + γ^{tt} t t_{t - 1}) + ε_{t}^{c} \\ Δ r_{t} = α^{r} + θ^{r} r_{t - 1} + ε_{t}^{r} \\ Δ y_{t} = α^{y} + θ^{y} y_{t - 1} + ε_{t}^{y} & (10) \\ Δ {tt}_{t} = α^{t t} + θ^{t t} t t_{t - 1} + ϕ Δ y_{t} + ε_{t}^{t t} \\ Δ g_{t} = α^{g} + ζ^{g} t i m e + θ^{g} g_{t - 1} + ε_{t}^{g} \end{array}

where c, y, r, g, and tt correspond to previous definitions. Assuming rational expectations, these equations reflect the specification derived in the theoretical section, except that the underlying data-generating process for changes in income, net taxes and public spending is now explicitly consistent with time series evidence. We hence assume that average changes in consumption and net taxes are proportional to average labor income growth, while allowing for a deterministic trend in the public spending equation.⁹

Results from estimating this unrestricted model are reported in Table 3. The model was estimated using both Seemingly Unrelated Regressions (SUR) over the whole sample (55-05).¹⁰

Table 3

United States: Seemingly Unrelated Estimates of Unrestricted Model (Eq. 10)

Note: One and two asterisks denote that the coefficient is different from zero at 5 and 1 percent significance level, respectively.

Table 3

United States: Seemingly Unrelated Estimates of Unrestricted Model (Eq. 10)

Sample: 1955-2005
Consumption equation
α^c	-.05 (.05)
β^y	.46 (.05)^**
β^tt	-.07 (.18)
β^g	.05 (.08)
β^r	-.13 (.05)^*
β^ecm	.10 (.05)^*
γ^y	-1.03 (.03)^**
γ^tt	1.07 (.66)
R²	.76
DW	2.01
Interest rate equation
α^r	.00 (.00)
θ^r	-.26 (.09)^**
R²	.13
DW	1.80
Income equation
α^y	.78 (.47)
θ^y	-.10 (.06)
R²	.09
DW	1.70
Net tax rate equation
α^tt	-.02 (.02)
Φ	.31 (.07)^**
θ^tt	-.13 (.08)
R²	.39
DW	1.84
Government spending equation
α^g	.04 (.02)^*
ζ^g	-.00 (.00)
θ^g	-.17 (.08)^*
R²	.11
DW	1.43
*Finite-horizon model restrictions:*χ² (4)	0.10

Note: One and two asterisks denote that the coefficient is different from zero at 5 and 1 percent significance level, respectively.

Table 3

United States: Seemingly Unrelated Estimates of Unrestricted Model (Eq. 10)

Sample: 1955-2005
Consumption equation
α^c	-.05 (.05)
β^y	.46 (.05)^**
β^tt	-.07 (.18)
β^g	.05 (.08)
β^r	-.13 (.05)^*
β^ecm	.10 (.05)^*
γ^y	-1.03 (.03)^**
γ^tt	1.07 (.66)
R²	.76
DW	2.01
Interest rate equation
α^r	.00 (.00)
θ^r	-.26 (.09)^**
R²	.13
DW	1.80
Income equation
α^y	.78 (.47)
θ^y	-.10 (.06)
R²	.09
DW	1.70
Net tax rate equation
α^tt	-.02 (.02)
Φ	.31 (.07)^**
θ^tt	-.13 (.08)
R²	.39
DW	1.84
Government spending equation
α^g	.04 (.02)^*
ζ^g	-.00 (.00)
θ^g	-.17 (.08)^*
R²	.11
DW	1.43
*Finite-horizon model restrictions:*χ² (4)	0.10

Note: One and two asterisks denote that the coefficient is different from zero at 5 and 1 percent significance level, respectively.

Reported estimates imply that consumers spend almost one-half of the change in their income, but a small and statistically insignificant proportion of any change in net taxes. It also implies that any deviation between the underlying level of consumption and disposable income is reversed at a rate of about 10 percent a year. In addition, the hypothesis of a unit marginal propensity to consume out of disposable income appears congruent with the data. The equations for changes in income and net tax rate indicate that unexpected disturbances to these aggregates are also reversed at a rate close to 10 percent a year, although these estimates are subject to larger standard errors. Expected (average) income growth is about 3 percent a year. In the net tax rate equation, revenues rise by about one-third of a percent for every one percent change in income—suggesting the personal tax and transfer system is reasonably progressive. The interest rate equation is consistent with a significant degree of interest smoothing (about 75 percent), while only 15 percent of any deviation of public expenditure from trend a year is corrected in the same year. The consumption equation fits relatively well, with R-squares of 0.76 and no evidence of correlation in the residuals.

Wald tests of the coefficient restriction implied by the finite horizon model are also reported in Table 1 (assuming a real interest rate of 3 percent a year, in line with the corresponding sample mean). Assuming rational expectations, the restrictions are:

\begin{matrix} \begin{array}{l} β^{y} = \frac{λ (1 + r)}{r + λ - θ^{y}} (1 - \frac{ϕ λ (1 + r)}{r + λ - θ^{t t}}) \\ β^{tt} = - \frac{λ (1 + r)}{r + λ - θ^{tt}} (1 - \frac{r}{r - θ^{tt}}) \\ β^{ecm} = - \frac{λ}{1 + λ} \\ γ^{y} = - \frac{r + λ}{r + λ - θ^{y}} \\ γ^{t t} = \frac{r + λ}{r + λ - θ^{tt}} \end{array} & (11) \end{matrix}

The finite-horizon model can be accepted at conventional levels. This is not surprising as the estimated coefficients—a larger coefficient on income than on taxes and an even smaller value on the error correction mechanism—are in line with the predictions of the model. Time-varying p-values corresponding to the joint and individual Wald restrictions are plotted in Figure 2, confirming the validity of the model over time.

Table 4 reports results from estimating the deep parameter of the finite-horizons model—the wedge on the discount rate—using SUR estimates. The specification for consumption—which excludes innovations to income under the assumption of rational expectations—is as follows:

\begin{array}{l} Δ c_{t} & = \frac{λ (1 + \bar{r})}{\bar{r} + λ - θ^{y}} (1 - \frac{ϕ λ (1 + \bar{r})}{\bar{r} + λ - θ^{tt}}) (Δ y_{t} + \frac{\bar{r}}{λ} ε_{t}^{y}) - \frac{λ (1 + \bar{r})}{\bar{r} + λ - θ^{tt}} (Δ {tt}_{t} - \frac{\bar{r}}{\bar{r} - θ^{tt}} ε_{t}^{tt}) \\ - \frac{ξ^{r}}{\bar{r} + λ - θ^{r}} Δ r_{t} - \frac{ξ^{g}}{\bar{r} + λ - θ^{g}} (Δ g_{t} - \frac{\bar{r}}{\bar{r} - θ^{g}} ε_{t}^{g}) \\ - \frac{λ}{1 + λ} [c_{t - 1} - \frac{\bar{r} + λ}{\bar{r} + λ - θ^{y}} y_{t - 1} + \frac{\bar{r} + λ}{\bar{r} + λ - θ^{tt}} t t_{t - 1}] + ε_{t}^{c} & (12) \end{array}

Table 4.

United States: Estimates of Restricted Model with Impatient Consumers (Eq. 9)

Notes: See ^{Table 3}. Restrictions on implied coefficients are provided in equation (13)

Table 4.

United States: Estimates of Restricted Model with Impatient Consumers (Eq. 9)

Sample: 1955–2005
Consumption equation
α^c	.04 (.01)^**
ξ^g	.01 (.05)
ξ^r	-.05 (.02)^*
λ	.02 (.00)^**
R²	.72
DW	1.96
Interest rate equation
α^r	.00 (.00)
θ^r	-.26 (.09)^**
R²	.13
DW	1.80
Income equation
α^y	.12 (.03)^**
θ^y	-.01 (.00)^**
R²	.09
DW	1.70
Net tax rate equation
α^tt	-.02 (.02)
Φ	.34 (.07)^**
θ^tt	-.13 (.08)
R²	.39
DW	1.84
Government spending equation
α^g	.04 (.02)^*
ζ^g	-.00 (.00)
θ^g	-.18 (.07)^**
R²	.11
DW	1.43
*Implied Coefficients*
β^y	0.40
β^tt	-0.12
β^g	-0.08
β^r	-0.16

Notes: See ^{Table 3}. Restrictions on implied coefficients are provided in equation (13)

Table 4.

United States: Estimates of Restricted Model with Impatient Consumers (Eq. 9)

Sample: 1955–2005
Consumption equation
α^c	.04 (.01)^**
ξ^g	.01 (.05)
ξ^r	-.05 (.02)^*
λ	.02 (.00)^**
R²	.72
DW	1.96
Interest rate equation
α^r	.00 (.00)
θ^r	-.26 (.09)^**
R²	.13
DW	1.80
Income equation
α^y	.12 (.03)^**
θ^y	-.01 (.00)^**
R²	.09
DW	1.70
Net tax rate equation
α^tt	-.02 (.02)
Φ	.34 (.07)^**
θ^tt	-.13 (.08)
R²	.39
DW	1.84
Government spending equation
α^g	.04 (.02)^*
ζ^g	-.00 (.00)
θ^g	-.18 (.07)^**
R²	.11
DW	1.43
*Implied Coefficients*
β^y	0.40
β^tt	-0.12
β^g	-0.08
β^r	-0.16

Notes: See ^{Table 3}. Restrictions on implied coefficients are provided in equation (13)

To compare these results with the unrestricted coefficient estimates reported in Table 3, the implied coefficients on the change in income (β^y), change in net tax rate (β^tt), change in public spending (β^g), and change in real interest rate (β^r) are reported using the restrictions from equation (11).

The SUR results imply an excess private sector discount rate just below 2½ percent, which is significantly different from zero at the 1 percent level, hence rejecting the fully Ricardian model. The implied coefficients for the restricted regressions are all reasonably close to the freely estimated values, consistent with the results from the Wald test, and the fit of the model is largely unaffected. The dynamics of labor income, tax rate, public expenditure and interest rate are virtually unchanged, with restrictions helping pinning down a very slow but significant adjustment in labor income changes. Coherently with the results from the unrestricted model, the finite-horizon model implies that consumers spend 40 percent of their changes in income, but only 12 percent of their changes in net taxes. Interestingly, one extra dollar of public spending is estimated to crowd out 8 cents of private consumption, while a 1 percent increase in real interest rate is estimated to trim down private consumption by about one-sixth of 1 percent.

IV. Analysis and Discussion

A fundamental feature of the intertemporal model used in this paper is that the impact of a change in income/net taxes on consumption depends on several characteristics—its persistence, whether it is anticipated or not, and the average length of consumers’ planning horizon. This section explores these interactions in more detail.

In our previous study (Bayoumi and Sgherri, 2006) we showed how the impact of unanticipated changes in income and net taxes rises as the level of impatience increases. We also showed that the effectiveness of changes in income and net taxes depends on the degree of persistence of the shock: whereas the effect of an unanticipated change in underlying income rises steadily from around 5 cents in the dollar for a temporary change to a one-for-one impact if the change is permanent, the net tax multiplier rises from around 3 cents in the dollar to peak at just over 19 cents for a shock that converges at 5 percent a year. At convergence rates below 5 percent, the net tax multiplier starts to fall as the Ricardian offset increases rapidly. Indeed, it falls to zero for a “permanent” shock to net taxes, as this violates the intertemporal budget constraint and hence the “change” in taxes is fully offset by the opposite movement in the long-term tax rate.

Here we investigate these interactions along a time dimension, using rolling estimates of the finite-horizons model (Table 4) with windows of 26 years. Figure 3 shows that the degree of consumers’ impatience has fallen from 3 to 2 percent over the last 20 years, thereby lengthening households’ planning horizon from 33 to 50 years. At the same time, though, the persistence of both income and fiscal shocks has increased (Figure 4). In particular, we notice the following: (i) shocks to the income level do not show any tendency to mean revert by the end of the sample; (ii) the rate of convergence (e.g., 1+θ) of changes in net taxes has slowed down from 80 to 90 percent over the last 20 years; changes in public spending have also become more persistent over the same period, with convergence delayed from 65 to 90 percent. In contrast, real interest rate shocks have become shorter-lived, as their convergence rate rose from about 85 percent in 1982 to less than 70 percent in 2005.¹¹

Changes of this kind are expected to have an important impact on the average MPC out of windfall income shocks as well as on the size of policy multipliers. However, the direction of these changes is not clear a priori.

Time variation in the effectiveness of income shocks and policy levers on consumption is visualized in Figure 5. An extra dollar of income raises consumption by 42 cents in 2005 and only by 32 cents in 1982. Analogously, a one-dollar cut in public spending produces larger increases in private consumption today than 20 years ago (9 versus 5 cents on a dollar). On the contrary, a one-dollar rise in real interest rate induces a smaller decline in consumption in 2005 (14 cents) than it used to in 1982 (25 cents). As for tax cuts, their effectiveness has been quite volatile over the sample, and related changes are thereby negligible, despite a statistically significant upward trend in the persistence of shocks to the tax rate.

Based on counterfactual analysis, Figure 6 plots the combined effect of changes over time in consumers’ impatience and in shocks’ persistence on the magnitude of income and policy multipliers. It seems clear that the increase in the persistence of income and fiscal shocks has been predominant over the fall in consumers’ impatience, bringing about bigger (rather than smaller) income and fiscal multipliers. Had the persistence in these shocks always been at the current higher level, the effectiveness of changes in income and public spending would have fallen (rather than increased) over the sample, and so would have fallen the effectiveness of any tax cut. In the hypothetical case of no change in shocks’ persistence, any change in the effectiveness of income and fiscal shocks on consumption would have been due to the fall in the discount wedge, implying smaller multipliers in 2005 than in 1982. In the case of real interest shocks, persistence has fallen drastically over the sample, implying much smaller multipliers in 2005 than in 1982. Also in this event, the impact of changes in shocks persistence on the size of the multiplier has been a lot more important than any lengthening of the households’ planning horizon.

Interestingly, time variation in the effects of income changes on consumption is found to be nonlinear: as income approaches a random walk behavior, changes in the effectiveness of any extra dollar of income on consumption only occur through further drops in the degree of consumers’ impatience. This seems to imply that there is only limited scope to increase the effectiveness of “windfall” shocks to disposable income, once the consumers’ planning horizon becomes longer and longer.

V. Conclusions and Policy Implications

Identifying the size and impact of policy changes is a tricky business. In this paper we have developed empirical estimating equations for a structural model with rational consumers who discount the future more rapidly than the rate implied by the government’s budget constraint. In this way, we attempt to reconcile theory and empirical evidence regarding the effect of income and policy shocks on private consumption.

Estimation reveals that the finite-horizon model fits US postwar data rather well. The implied excess rate of discount is of the order of 2-3 percent—a value which seems broadly in line with macroeconomic estimates of the excess volatility of consumption with regard to income and with predictions of previous life-cycle models with calibrated consumers’ expected lifetimes.¹² It also appears compatible with the prevailing level of interest rate on credit card debt, the main form of unsecured borrowing available to consumers—10-15 percent (in nominal terms) over the sample—and with mortgage lengths up to 40 years.

The great advantage of our framework is that it brings the intertemporal nature of disturbances to disposable income back to the fore of analysis. Indeed, thinking of changes in the effectiveness of policy levers within an intertemporal setting provides a range of insights. First, if the marginal propensity to consume out of “windfall” shocks to disposable income is directly proportionate to the “average degree of impatience”, a progressive lengthening of the households’ planning horizon also implies a progressive drop in the average MPC over time. Hence, over time, consumers seem to have become more far-sighted and better able to smooth consumption over their lifetime, possibly thanks to larger wealth accumulation, greater financial deregulation, and a more extensive global financial integration.¹³

Have these developments limited the effectiveness of fiscal policy over time? Not yet, as the longevity of changes in net taxes and public spending has also lengthened, indeed offsetting the impact of having to cope with “more Ricardian” consumers. In other words, our results seem to make the case that fiscal stimuli have so far retained their effectiveness. Notably though, as households become less and less impatient, the scope to further increase the effectiveness of “windfall” shocks to disposable income is likely to become more and more limited.

At the same time, the real effects of real interest rate changes appear to have fallen dramatically over time, supporting the wide-spread view that the sacrifice ratio has ultimately come down. Consistently with our previous work, this result seems to be due to a fall in the persistence of real interest rate shocks, likely owing to the drop in inflation inertia.¹⁴ Even in this case then, a greater degree of forward-lookingness in private sector behavior seems to be an essential factor behind the substantial changes in (monetary) policy effectiveness.

Linking policy effectiveness and shifts in private sector behavior has also a number of important implications. First, it calls into question the large body of work that assesses monetary and fiscal rules by assuming that such rules has no impact on underlying private sector behavior. While such analysis may be useful for the short-term impact of changes in policy effectiveness, the analysis in this paper suggests that it is fraught with difficulty as a guide to the longer-term consequences of a policy shift. Second, it implies that there is a direct connection between some of the more recent improvements in the U.S. economy, such as the fall in output volatility, and the conduct of fiscal and monetary policy. At the same time though, reductions in the variance of output are general phenomena across a wide range of countries, suggesting that this analysis may have wider implications than the United States.

References

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See, among many others, the contributions of Blanchard and Simon (2001), Boivin and Giannoni (2006), Cogley and Sargent (2001), Primiceri (2006), Stock and Watson (2002).

Bayoumi and Sgherri (2004a, 2004b).

This simple reaction function is consistent with a pure inflation targeting policy, in which the nominal interest rate moves one-to-one with deviations of inflation from its equilibrium value.

Using s to denote the age cohort born in period s, total population at time t (where t≥s) will is hence equal to $\sum_{s = - \infty}^{t} p {(1 - p)}^{t - s} = 1$ .

The presence of life insurance companies operating under perfect competition can be seen as an effective transfer of wealth (denoted with W) within members of the same age cohort, s. In each period t, a total wealth of $(p N_{s, t - 1}) W_{s, t - 1} (1 + \bar{r})$ is transferred to the (1-p)N_s,t-1 individuals who are still alive, each hence receiving $[p / (1 - p)] W_{s, t - 1} (1 + \bar{r})$ . Added to the usual return on invested wealth, this yields an effective return of $(1 + \bar{r}) {(1 - p)}^{- 1}$ , or equally $(1 + \bar{r}) (1 + λ)$ , where $(1 + \bar{r})$ reflects the market rate and (1+λ) the additional insurance premium due to life uncertainty.

Individual labor supply is assumed to be time invariant, so that no intergenerational redistribution of labor income occurs. However, as the (exogenous) technology level may follow a (stochastic or deterministic) trend, aggregate labor income grows over time.

On this point, see also Barro (1974) and Evans (1993).

Monetization of deficits is treated as an inflationary tax and its proceeds are included in tt.

Importantly, assuming trend stationarity for the labor income process does not seem to affect our estimates in any significant way. Needless to say, though, the two assumptions bear divergent implications in terms of “smoothness” of consumption with respect to labor income. See Deaton (1988) for details.

The model has been also estimated using GMM. The results are very similar and available upon request.

The statistical significance of these changes can be checked by simply looking at t-statistics for the linear trends fitted through corresponding rolling estimates. In all cases, the hypothesis of no change over time is strongly rejected at the usual 95% confidence level (results are also available upon request).

See, for instance, Galì (1990).

On this point, see also the recent contribution by Cecchetti, Flores-Lagunes, and Krause (2006).

Bayoumi and Sgherri (2004a, 2004b).

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On Impatience and Policy Effectiveness

Author: Ms. Silvia Sgherri and Mr. Tamim Bayoumi

Volume/Issue:: Volume 2009: Issue 018

Publisher:: International Monetary Fund

ISBN:: 9781451871654

ISSN:: 1018-5941

Pages:: 28

DOI:: https://doi.org/10.5089/9781451871654.001

Keywords
I. Introduction
II. Theoretical Model
III. Empirical Estimates
- Basic Model
IV. Analysis and Discussion
V. Conclusions and Policy Implications
References

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United States: The Data, 1955-2005

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United States: Validity of Model Restrictions over Time

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United States: Time Variation in the Discount Wedge

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United States: Time Variation in Persistence of Income/Policy Shocks

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United States: Time Variation in Income/Policy Multipliers

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United States: Counterfactual Analysis

Foresee

Table 4.

United States: Estimates of Restricted Model with Impatient Consumers (Eq. 9)

Sample: 1955–2005
Consumption equation
α^c	.04 (.01)^**
ξ^g	.01 (.05)
ξ^r	-.05 (.02)^*
λ	.02 (.00)^**
R²	.72
DW	1.96
Interest rate equation
α^r	.00 (.00)
θ^r	-.26 (.09)^**
R²	.13
DW	1.80
Income equation
α^y	.12 (.03)^**
θ^y	-.01 (.00)^**
R²	.09
DW	1.70
Net tax rate equation
α^tt	-.02 (.02)
Φ	.34 (.07)^**
θ^tt	-.13 (.08)
R²	.39
DW	1.84
Government spending equation
α^g	.04 (.02)^*
ζ^g	-.00 (.00)
θ^g	-.18 (.07)^**
R²	.11
DW	1.43
*Implied Coefficients*
β^y	0.40
β^tt	-0.12
β^g	-0.08
β^r	-0.16

Notes: See ^{Table 3}. Restrictions on implied coefficients are provided in equation (13)

	ADF^a	PP^b	NP^c
c_t	-1.95	-1.71	-1.38
y_t	-2.25	-2.28	-1.85
Critical Values^d 5% level	-3.51	-3.51	-2.91
tt_t	-3.42	-2.76	-2.01
Critical Values^e 5% level	-2.92	-2.92	-1.98

	ADF^a	PP^b	NP^c
c_t	-1.95	-1.71	-1.38
y_t	-2.25	-2.28	-1.85
Critical Values^d 5% level	-3.51	-3.51	-2.91
tt_t	-3.42	-2.76	-2.01
Critical Values^e 5% level	-2.92	-2.92	-1.98

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Abstract

I. Introduction

II. Theoretical Model

Monetary policy

Consumption

Fiscal policy

III. Empirical Estimates

Basic Model

IV. Analysis and Discussion

V. Conclusions and Policy Implications

References

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