U.S. OUTPUT AND INFLATION: SOME FACTS

This section presents key features of the behavior of U.S. output and inflation that I seek to match in this paper.

The subsequent analysis is based on U.S. GDP and GDP inflation for the period Q1:1959 to Q4:2002.⁷

Detrended U.S. data.

Figure 2 depicts the auto- and cross correlations of the detrended data. The panels on the main diagonal of the figure show the autocorrelation for output and inflation, respectively. Output and inflation are positively autocorrelated for about 1 year, which shows that there is considerable persistence in these variables. Thereafter, the autocorrelations turn negative, showing that above-average output (inflation) tends to be followed by below-average output (inflation) circa 1 to 4 years down the road. These findings are consistent with those reported by Stock and Watson (1999).

Auto- and cross correlations of U.S. output and inflation.

The lower-left graph in Figure 2 reveals that inflation is positively correlated with lagged output for about

years, with the maximum correlation being attained at around 1 year. This suggests that inflation responds sluggishly to output deviations. Correspondingly, the upper-right graph shows that output is negatively correlated with lagged inflation for about the first 3 years: Inflation is an indicator of future output losses. Taylor (1999) has called these features the “reverse dynamic cross correlation” of output and inflation. Qualitatively similar results have been reported by Fuhrer and Moore (1995) who estimated autocorrelation functions for output and inflation using a vector autoregression that included a short-term nominal interest rate.

Although an analysis of the dynamic correlations is informative about the comovements present in the data, it remains uninformative about the causes underlying these movements. Impulse response functions provide answers about potential causal links but require identifying assumptions.

Figure 3 depicts the impulse response to a nominal demand shock obtained from fitting a vector autoregression (VAR) with two lags to yearly output and inflation data.⁹

years to a monetary policy shock whereas output reacts well within the first year.

Impulse responses to a demand shock with 2 standard error Monte Carlo error bands.

Figure 3 shows the impulse response for a positive demand shock of a magnitude of 1 standard deviation. Output remains about 0.3 standard deviation above average in the year after the shock, illustrating that there is considerable output persistence. Inflation increases by about 0.4 standard deviation in both years after the shock, indicating that the price response is rather persistent, as suggested by the lower left panel in Figure 2.

In summary, the preceding analysis indicates that demand shocks should generate a persistent increase in output and a sluggish and persistent increase in inflation. Moreover, the rise in inflation should be followed by a persistent output decrease later in the cycle. The simple model discussed in the remaining part of this paper can replicate all of these facts using white-noise nominal demand disturbances as driving forces.

A SIMPLE BUSINESS-CYCLE MODEL

This section outlines a highly stylized business-cycle model with monopolistic competition [Dixit and Stiglitz (1977)] and sticky prices where firms set prices one period in advance.¹⁰

In slight deviation from a standard setup there are two kinds of agents: entrepreneurs who own monopolistically competitive firms and finance consumption using the monopolistic profits, and workers who finance consumption by offering their services in a competitive labor market. To make the distinction between workers and entrepreneurs economically relevant, contingent-claim markets that would allow for risk sharing between workers and entrepreneurs are assumed to be unavailable. This ensures that workers' labor supply decisions do not depend on current and expected future profits and simplifies the analysis.

Entrepreneur i∈[0, 1] maximizes a standard utility function of the form

where cⁱ_t denotes consumption, mⁱ_t the entrepreneur's real money holdings at the end of period t, τ_t the real value of a possibly negative government cash transfer, Π_t the inflation factor from period t−1 to t, and Φⁱ_t the monopoly rents from ownership of firm i. The first constraint forces entrepreneurs to use money to pay for consumption goods. The second constraint is the flow budget constraint.

Each firm i produces an intermediate consumption good qⁱ that is an imperfect substitute in the production of the aggregate consumption good c:

Profit maximization implies that each firm sets its price Pⁱ_t as a fixed markup over expected production costs. When the production technology is linear in labor, this implies that

where P_t denotes the price index of the final consumption good and w_t the real wage. Dividing the preceding equation by P_t−1 and assuming that all entrepreneurs and firms have identical expectations delivers

Equation (1) summarizes optimal behavior by firms.

Next, consider workers. Each worker j∈[0, 1] chooses consumption c^j_t and labor supply n^j_t to maximize

When u, v∈C², u′>0, u″<0, v′>0, v″[ges ]0,

for all c[ges ]0, and the cash-in-advance constraint is binding, utility maximization implies a labor supply function of the form¹¹

Inverting this labor supply function with respect to the first argument delivers an expression for the real wage,

where the linearity of the production function has been used to substitute n_t by y_t. Given the specified utility functions, the real wage increases in the demand for labor and in the expected inflation tax.

Finally, consider the government that issues money via lump-sum transfers.

where τ_t is a mean zero white-noise shock with small bounded support and is the only source of randomness in the model. When prices are preset and the cash-in-advance constraint is binding for all agents, output is demand determined in the short run and the previous equation is a specification of the demand side of the economy.¹³

Using (1), (2), and (3), one obtains the temporary equilibrium equations that describe current output and inflation as a function of past output and expectations about future inflation rates:

The remaining part of the paper considers the linearizations of equations (4) and (5) around the deterministic steady-state equilibrium and makes different assumptions about how agents forecast inflation.

RATIONAL EXPECTATIONS EQUILIBRIA

Linearizing the model—equations (4) and (5)—around the deterministic monetary steady state yields the following linear approximation for the stochastic system¹⁴

where ε denotes the (real wage) elasticity of labor supply and

the steady-state output level. In Appendix Section A.2, we prove that the rational expectations solutions to (6) have a minimum state variable representation as a two-dimensional AR(1) process,

and that there are two rational expectations solutions of this form. The first solution is stationary and given by

Output in this equilibrium is white noise, and inflation responds with a delay of one period to output deviations. The second solution is nonstationary,

and diverging from the steady state (almost surely). The diverging paths have either increasing inflation rates and decreasing output levels or decreasing inflation rates and increasing output levels.¹⁶

LEARNING AGENTS

The preceding section considered rational inflation forecasts. This section considers agents who have to learn how to formulate their expectations.

In particular, agents are assumed to consider a limited set of econometric forecast models, to fit the considered forecast models to the data, and to choose the model with the best past prediction performance to forecast the future.

Although such behavior seems intuitively plausible, it requires that the modeler impose the set of forecasting models considered by agents. The next section supposes that agents consider a particularly simple set of forecasting models. The generality of the results obtained with this particular set are discussed in Section 7.

Learning to Forecast

Suppose agents consider the following set of simple regression models to forecast inflation rates:

Here, x denotes an explanatory variable that is believed to predict future inflation and (α_t−1, β_t−1) agents' least-squares estimates of the relationship between x_t−1 and Π_t based on information up to time t−1.

With the economy being described by two state variables (real output and inflation), there exist only two forecasting models of form (10)¹⁷

For simplicity the time subscripts for α and β will be dropped from now on.

:

Model Y predicts inflation using past output whereas Model Π uses past inflation. Importantly, Model Y can generate expectations that are consistent with the rational expectations equilibria (8) and (9), provided it is suitably parameterized. Model Π, however, never generates rational expectations since inflation never depends on lagged inflation in a rational expectations equilibrium.

Admittedly, the assumption regarding the set of forecasting models is ad hoc and in several ways quite special.¹⁸

Experimental evidence has given considerable support to the notion that agents consider simple forecasting rules of form (10); see Adam (2004).

Note, however, that it generalizes the standard setup in the learning literature, which assumes that agents consider only a single model that is assumed consistent with rational expectations. Here, the agents do not possess a priori knowledge about the structure of the economy's rational expectations solution.

Several economic interpretations exist as to why agents may consider a restricted set of forecasting models only. First, the prediction technology, that is, agents' knowledge about preparing and evaluating forecasts, may simply impose such a restriction upon them. Given the relative simplicity of the underlying economic model, restriction (10) may be interpreted as representing such a technological restriction. Second, a restricted set of forecast models may be the result of costs associated with the formulation of forecasts. Such costs cause agents to trade off forecasting performance with the cost of considering smaller or larger classes of models. Finally, the restriction may describe a situation in which agents perform a specification search for suitable forecast models and initially consider a certain simple class of models. Unsatisfactory prediction performance may then lead to changes in the considered class.¹⁹

The experimental evidence in Adam (2004) suggests that this might be the economically most relevant interpretation.

It remains to be explained how agents choose between competing forecast models. Given that agents use least-squares estimation, it seems natural to assume that the agents' model choice depends on the models' mean-squared forecast error (MSFE). In particular, agents are assumed to choose the model with the lowest past MSFE to predict future inflation. Use of the MSFE criterion can be justified on the grounds that it constitutes a second-order approximation to the correct utility-based choice criterion.

The previous assumptions imply that agents asymptotically forecast with minimal MSFE, provided learning converges and generates a stationary economic environment. Therefore, agents' forecasts will be constrained rational in any stationary equilibrium, where the equilibrium notion is made precise in the next section.

Equilibrium with Learning Agents

The economy now evolves as explained in the following. Each period, agents estimate both Model Y and Π by least squares and choose the model with the lowest past MSFE to forecast inflation. Agents then maximize utility, assuming that the economy evolves according to the forecasts from the selected forecasting model. This generates a new inflation rate and output level according to equation (6), where the operator E[·] might now denote the potentially nonrational expectations generated by the chosen forecast model. Using the new data point, agents adapt their least-squares estimates and their model choices, and the process repeats itself.

Informally, an equilibrium is a situation in which the new inflation rate and output level “reconfirm” the previous parameter estimates and the previous model choice and where a learning process would generate convergence to such a situation from starting points sufficiently close to it. The resulting equilibrium is called a model equilibrium, below.

To formally define a model equilibrium, one can build on the concept of a restricted perceptions equilibrium (RPE), developed by Evans and Honkapohja (2001), which I give here for convenience.²⁰

The RPE has been called rational expectations equilibrium with econometric models by Anderson and Sonnenschein (1985). The RPE is related to the limited-information rational expectations equilibrium and the reduced-order limited-information rational expectations equilibrium developed by Marcet and Sargent (1989) and Sargent (1991), the consistent-expectations equilibrium of Hommes and Sorger (1997), and the self-confirming equilibrium of Fudenberg and Levine (1993) recently used by Sargent (1999) in a macroeconomic context.

DEFINITION 1. A restricted perceptions equilibrium in Model M (M=Y, Π) is a stationary sequence {y_t, Π_t}^∞_t=0 generated by equation (6) where agents use Model M with parameters (α*_M, β*_M) to forecast future inflation rates and where (α*_M, β*_M) is the orthogonal projection of Π_t on

Note that the RPE takes the choice of forecast model as given and only requires optimality (in a mean-squared error sense) of the forecast model's parameterization. Building on this definition, one can then define a model equilibrium as follows.

DEFINITION 2. A Model M equilibrium (M=Y, Π) is a restricted perceptions equilibrium in Model M, where

1. Model M generates a lower mean-squared forecast error than model M′={Y, Π}\{M}.
2. The restricted perceptions equilibrium in Model M is expectationally stable (E-stable).

A model equilibrium requires optimality of agents' model choice and E-stability of the associated RPE. The former implies that agents forecast optimally, subject to the constraints that have been imposed. The latter ensures that agents can learn the equilibrium, since expectational stability usually governs the stability of equilibria under least-squares learning and related learning procedures [Evans and Honkapohja (2001)].²¹

The results of Evans and Honkapohja (2001, Chap. 13.1.2) suggest that this is the case even if the resulting equilibrium is not a rational expectations equilibrium.

CALCULATING MODEL EQUILIBRIA

This section derives conditions under which Model Y and Model Π equilibria exist and compares their properties with U.S. data.

Model Y Equilibria

Suppose that agents use Model Y to forecast inflation rates. Substituting the forecasts of Model Y for the expectations in the temporary equilibrium equations (6) implies that the actual law of motion for inflation will be given by

where the coefficients a and b depend on agents' least-squares estimates α_Y and β_Y.²²

See Appendix Section A.3, for how to express E_t−1[Π_t+1] as a function of lagged output.

Equation (11) reveals that the actual law of motion for inflation coincides with the structural assumption of Model Y. This implies that in a Model Y equilibrium,

Otherwise the parameter estimates would not remain stable. Consequently, a Model Y equilibrium is a rational expectations equilibrium. The converse, however, is not necessarily true because a rational expectations equilibrium might fail to be stationary or expectationally stable.

Section A.3 of the Appendix shows that the stationary rational expectations equilibrium (8) is expectationally stable.²³

Section A.3 also shows that the nonstationary rational expectations equilibrium (9) fails to be E-stable so that one could rule it out based on expectational stability alone.

Output and inflation in Model Y equilibrium are thus given by

A 1% money shock increases output and expected inflation by the same amount.²⁴

Agents' Model Y estimate is asymptotically identical to equation (12).

As a result, entrepreneurs increase prices by 1% [see equation (1) and note that the expected wage is equal to 1−σ], which implies that the excess money stock will not persist into the next period.

Consequently, with rational expectations output and inflation are white-noise processes and inflation reacts with a delay of one period due to the stickiness of prices. The rational expectations equilibrium thus performs poorly in matching the features of U.S. output and inflation data shown in Section 2.

Model Π Equilibria

This section considers equilibria in which agents use Model Π to predict inflation rates. As argued before, agents' expectations are then only constrained rational.

Preliminaries

Suppose agents use Model Π to forecast inflation rates. Since least-squares estimates deliver forecasts that are (on average) unbiased, average inflation in a Model Π equilibrium will coincide with average expected inflation. Given that such a relation holds only at a (rational expectations) steady state, average output and average inflation in a stationary Model Π equilibrium will be same as in a Model Y equilibrium.

As a result, only the second moments of output and inflation will be affected by the use of Model Π. Moreover, on a more technical level, it implies that one can linearize the model around the same steady-state values as one usually does when calculating the linearized rational expectations solutions.

Substituting the predictions of Model Π for the inflation expectations in (6) delivers an equation that describes current inflation and output as a function of the past values of these variables, the Model Π parameters (α_Π, β_Π), and the labor supply elasticity ε²⁵

The relevant features of this process are unaffected by the average level of output

.

:

The preceding equation reveals that in a potential Model Π equilibrium, inflation depends (generically) on past inflation and past output. The law of motion for inflation thus lies outside the class of forecast models that agents consider. With all forecast models being misspecified, Model Π may deliver a better fit to (14) than Model Y.

Existence of Model Π Equilibria

This section discusses conditions under which a Model Π equilibrium exists.

The least-squares estimate of β_Π in Model Π equilibrium is given by

where Π_t is generated by the actual law of motion (14). Since β_Π also enters equation (14), determining a Model Π equilibrium involves solving a fixed-point problem, as when calculating a standard rational expectations equilibrium. Section A.4 of the Appendix shows how to solve for the fixed-point β*_Π of equation (15).

Figure 4 graphs β*_Π as a function of the labor supply elasticity ε. It is shown that β*_Π increases with ε and is equal to zero for ε=1. The case ε=1 is special because a 1% demand shock then generates a 1% increase in expected costs. Markup pricing by firms then causes inflation to increase by 1%. This drives the excess money balances back to zero. As a result, there is no persistence in excess demands and inflation as in the rational expectations equilibrium. As labor supply becomes more elastic, however, a 1% demand shock generates a less than proportionate labor cost and inflation increase. Excess money balances are then not devaluated in a single period but persist into future periods where they again cause above-average costs and inflation. Inflation is then positively autocorrelated, which explains the positive slope in Figure 4.

β*Π as a function of ε.

Now, substitute the solution for β*_Π together with α*_Π=1−β*_Π into equation (14).²⁶

The relation α*_Π=1−β*_Π follows from the unbiasedness of least-squares forecasts and

.

The resulting stochastic process denotes an RPE in Model Π, provided it is stationary. As is easily verified, the autocorrelations of this process depend solely on the labor supply elasticity ε, and numerical calculations show that the RPE is stationary for 0.35 [les ] ε [les ] 2.15 which is the range of elasticity values considered from now on.²⁷

These and the subsequent boundaries are only approximate.

For the RPE to be a Model Π equilibrium, one has to verify that use of Model Π is indeed optimal and that the RPE in Model Π is expectationally stable (E-stable).

Expectational stability of the RPE is shown to hold in Section A.5 of the Appendix. This leaves the question of whether Model Π can outperform Model Y in terms of its mean-squared forecast error. Clearly, for values of ε around 1, this cannot be expected because past inflation has almost no predictive power. However, for larger and smaller values, inflation is increasingly autocorrelated; see Figure 4.

Figure 5 depicts the mean-squared forecast errors of the two forecast models for various values of ε, assuming that the economy is in an RPE in Model Π. The graph reveals that for a sufficiently elastic labor supply (ε_n,w[ges ]1.75), Model Π will indeed generate better predictions than Model Y, which establishes the existence of Model Π equilibria.²⁸

Admittedly, the required supply elasticity lies on the high end of plausible values. However, similar elasticity levels are not uncommon in the macroeconomic literature. Christiano et al. (1997), for example, report satisfactory performance of a limited participation model for a labor supply elasticity of 2.

Mean-squared forecast error, Model Y $(\smash{\raise0.6pt\hbox{&ndash;}}\ddiam\smash{\raise0.6pt\hbox{&ndash;}})$ vs. Model Π $(\smash{\raise0.6pt\hbox{&ndash;}}\drect\smash{\raise0.6pt\hbox{&ndash;}})$.

Output and Inflation in Model Π Equilibrium

This section presents the output and inflation dynamics in Model Π equilibrium and compares these with the behavior of U.S. data. For illustrative purposes the section assumes a labor-supply elasticity of ε=1.8, which is at the lower end of values for which Model Π equilibria exist. The effects of larger elasticity values are discussed at the end of the section.

Figure 6 depicts the auto- and cross correlations of output and inflation in Model Π equilibrium for six model periods. This corresponds to the 6 years of U.S. data shown in Figure 2, provided that a model period is interpreted as a year.²⁹

Figure 6 depicts the correlations for the unfiltered data. Transforming these “yearly” data into four equal quarterly values and applying the bandpass filter that has been used for detrending the data lead to a very similar graph. Unfiltered data are shown because they allow for a clear interpretation in terms of the model's mechanisms.

Such an interpretation seems reasonable, given the degree of price stickiness in the model.

Auto- and cross correlations in Model Πequilibrium.

The auto- and cross-correlations shown in Figure 6 match those of Figure 2 remarkably well: Output and inflation are persistent; both variables are positively correlated for short lags and negatively for longer lags; output is a positive leading indicator for inflation, and inflation is a leading indicator of future output losses. However, none of these features shows up in a rational expectations equilibrium.

Figure 7 depicts the impulse responses to a demand shock in a Model Π equilibrium. The shock hits the economy in period 1. Inflation in period 2 reacts only sluggishly, causing output to remain above its steady-state level. Inflation increases further in period 3 and remains positive until period 4, thereby driving real balances and output below their steady-state values. Output and inflation then slowly return to their equilibrium values. Again, the shock responses qualitatively match the estimated responses for U.S. data shown in Figure 3.

Impulse responses to a demand shock in Model Π equilibrium.

The sluggish and persistent reaction of inflation in response to a demand shock can be explained as follows. In general, inflation increases because firms expect either inflation or real wages to increase. When a demand shock hits the economy, current prices are preset. This together with the use of Model Π implies that inflation expectations are “preset” and that wage expectations initially drive inflation. Since labor supply is relatively elastic, wages are expected to increase only slightly and the demand shock generates a rather weak initial inflation reaction. Once inflation has picked up, however, inflation expectations pick up and start to drive actual inflation, explaining why inflation in period 3 is even higher than in period 2. Since inflation expectations have been underreacting initially and are unbiased on average, they now overreact. This causes inflation to remain positive and drives output below its steady value.

I now briefly discuss the effects of a more elastic labor supply for the impulse responses shown in Figure 7. Clearly, the larger ε, the lower the expected cost increase. This causes the initial inflation response to be even more sluggish. Once inflation has picked up, there are two opposing effects. On the one hand, inflation shows higher persistence; see Figure 4. This causes expectations to increase more strongly in response to any given initial inflation increase. On the other hand the initial inflation increase has been less pronounced. As it turns out, the net effect is positive and the peak of inflation is higher than the one shown in Figure 7. Higher elasticity values thus generate a more sluggish initial inflation reaction, but a stronger reaction thereafter. This results in a shorter length of the cycle, and the correlations shown in Figure 6 cross the axis at an earlier date.

Note that the Model Π equilibrium, unlike the Model Y equilibrium, also generates empirically plausible real wage behavior. In particular, real wages are lagging over the output cycle in Model Π equilibrium, a feature documented by Galí and Gertler (1999) for the United States. Figure 8 displays the output–real wage correlation in Model Y and Model Π equilibria, respectively.³⁰

The labor-supply elasticity is assumed to be given by ε=1.8. Higher elasticity values lead to similar pictures. The correlations have been obtained numerically from 10,000 simulated data points.

In both cases, output is positively correlated with current wages due to an upward-sloping labor supply function. Yet, in Model Π equilibrium output shows a higher correlation with the wage in the subsequent model period. Since inflation and inflation expectations are lagging over the cycle, workers expect a higher inflation tax after the peak of the cycle, thereby shifting the labor supply curve upward and resulting in higher real wages despite decreasing output levels.

Correlation between output (t) and wage (t + i).

OUTLOOK

Some questions related to the findings in this paper might deserve further attention. First, it seems desirable to evaluate the existence and empirical performance of model equilibria in a less stylized and more realistic model of the business cycle. If model equilibria can replicate other comovements of the data, in particular, investment and consumption dynamics, this would certainly increase confidence in the empirical relevance of such equilibria. Second, it seems important to evaluate the plausibility of forecasting restrictions directly, that is, not only via the implied model predictions but also through the use of inflation survey data or experimental evidence. The results in Ádam (2004) show that the inflation expectations reported by subjects participating in an experimental version of the presented economy are described surprisingly well by the expectations implied by the Model Π equilibrium.