Let Us Assume That a Social Planner Is Trying to Minimize Social Loss Function of the Form

Government policy and the minimization of the social loss function

Nissim Ben David

Abstract

The social loss function used in the paper is quadratic in deviations of actual from optimal values of the objectives. In order to activate an optimal policy suggested in this paper, the planner should estimate an econometric system of equations that relate between the values of exogenous policy variables and targeted endogenous variables. Minimizing social loss, subject to estimated equations constraints, the policy planner can determine the optimal level of policy exogenous variables.

Assuming a policy planner in the U.S. is trying to minimize social damage, I estimated the optimal policy of relevant exogenous variables, according to the suggested model.

Key Words: Government policy; social loss function

Ben-David Nissim, Department of Economics and Management, Emek Yezreel Academic College, 19300 Emek Yezreel, Israel. Email: Phone 972-4-6398847, Fax 972-4-6291452

I. Introduction

The recent literature has seen a growth of interest in the interactions between fiscal

and monetary policies, after a long period in which each had been analyzed in isolation as if there was only one policymaker who was able to act alone and in his/her own self-interest. The idea that policies might interact is an old one, going back to Tinbergen (1954), Mundell (1962) and Cooper (1969), as well as a recent one, for example, Dixit and Lambertini (2003), Persson et al. (2006), Hughes Hallett et al. (2005).

Monetary policies cannot be committed if fiscal policies are not committed at the same time. However, the public must be aware that they are committed. This can be achieved through monitoring, or because the institutional structure locks the policymakers (see Hughes Hallett and Weymark (2007)).

The literature on discretion and commitment since the works of Kydland and

Prescott (1977) and Barro and Gordon (1983) assumes that policymakers have

preferences over inflation and employment that correspond to a quadratic loss

function.

How do and should authorities minimize the loss function? Most of the literature discusses this in terms of a commitment to a simple reaction function, a simple “instrument rule”, where the central bank mechanically sets its instrument rate (usually a short interest rate like a one or two-week repurchase rate) as a given simple function of a small subset of the information available to the central bank. Although several different simple instrument rules have been discussed since the 1970s, the best known and most discussed is the Taylor (1993) rule, where the interest rate is determined as a function of the gap between actual and optimal rate of inflation and the gap between actual and potential output. A large volume of research, for instance in Taylor (1999), has examined the properties of the Taylor (1993) rule and its variants in different models, with respect to determinacy of equilibria, performance measured by social loss, robustness of different models, etc. Several papers have also estimated empirical reaction functions of this type.

Inflation targeting obviously involves an attempt to minimize deviations

from the explicit inflation target. Whereas there may previously have been some controversy about whether inflation targeting involves concern about output gap variability, there is agreement in the literature that this is indeed the case, for instance, Fischer (1996), King (1996), Taylor (1996) and Svensson (1996) in Federal Reserve Bank of Kansas City (1996) all discuss inflation targeting with reference to a loss function.

In practice, no central bank follows an instrument rule, either explicit or implicit. Every central bank uses more information than the suggested simple rules rely on,

especially in open economies. In particular, no central bank responds in a prescribed

mechanical way to a prescribed information set. As is known by every

student of modern central banking, the bank's Board or Monetary Policy

Committee reconsiders its monetary policy decisions more or less from scratch

at frequent intervals, by taking all the relevant information into account. The bank frequently reconsiders (and, at best, re-optimizes); once and for all, and then simply applies the resulting reaction function forever after. This reconsideration of the bank's decisions means that the situation is best described as decision-making under discretion rather than commitment; there will inevitably be reconsiderations and new decisions in the future, and, in practice, there is no commitment mechanism for preventing this.

Therefore, the role of simple or complex instrument rules is, in practice, never

to commit the banks. Instead, they serve as base-lines, that is, as comparisons

and frames of reference, for the actual policy and its evaluation. In contrast,

targeting rules have the potential to serve as a kind of commitment (namely a commitment to a loss function, although it is still minimized under discretion), and are potentially closer to the actual practice and decision framework of (at least) inflation targeting central banks.

Under such policy circumstances, with the banks and fiscal authorities not committed to instrument rules, actual policy achievement should be compared to potential achievements, given that authorities would have been committed. Such a comparison can be made by calculating deviations from targets under discretion (actual deviations) and under commitment.

In this paper, I present a model with a policy planner that tries to minimize a social loss function. The social loss is determined as a sum of square deviations of actual from optimal levels of targeted economic variables. In the first step, by using historical data, the social planner estimates a simultaneous system of equations that connect between endogenous (target) and exogenous (instrument) variables[1]. In the second step, the planner minimizes the social loss function subject to the constraints determined by the simultaneous estimated equations. The optimization creates a set of equations that determine the optimal level of the exogenous variables (instruments).

Given actual and optimal levels of the exogenous variables, the optimal and actual social loss are calculated. In each time period ,the gap between actual and optimal loss is calculated. The results point into times with bad economic policy and times with good economic policy.

This paper is organized in the following manner: The model structure and an example are laid down in section II. Estimation of the model by using U.S. data is presented in Section III, and Section IV presents the summary.

II. The model

Let us assume that a social planner is trying to minimize social loss function by taking optimal policy measures. The social loss function is of the form: , where is the optimal value of economic variable i, while is its actual value. might be a macro economic variable such as inflation level, unemployment rate, production rate of growth etc.

The planner can use N different economic policy measures, where .

is the amount of policy measure he can take. Notice that might be the quantity of money, or amount of government expenditure etc.

There is a connection between policy measures taken and the value of targeted economic variables as well as simultaneous connections between the levels of the economic targets as presented by equation (1).

Assuming a linear connection, we can define (1) as:

If we can estimate (1a) as a simultaneous system of equations (by using a proper data set) and identify all k equations, each of the Yi variables can be defined as a function of the Xi exogenous variables.

We would get:

Where ei is estimation error.

The planner problem can be defined as:

We should notice that estimation errors are included at the constraints and will affect policy instruments levels.

Substituting the constraint into the minimization function we get:

Differentiating (2a) in respect to Xi and equalizing to zero we get:

Example

Let us assume that the social planner is trying to optimize the levels of two economic variables Y1 and Y2. Their optimal value is and , while the planners' loss function is: .

The following equations define the connection between policy measures and policy targets:

(We should notice that given a data set of Y1t, Y2t, X1t and X2t these two equations can be estimated as a simultaneous system with exact identification).

The planner problem is:

Using (4a) we isolate Y1 and Y2 and get:

Defining

We get:

Substituting into, we get:

Differentiating in respect to X1 and X2 we get:

Isolating X1 and X2 from (a) and (b) we get X1 and X2 as function of the A's parameters and of e1, e2, and .

III. Optimal levels of inflation, unemployment and trade balance deficit in the U.S.

Let us assume that policy planners in the U.S. are trying to achieve a certain optimal decrease in inflation rate, unemployment rate, and trade balance deficit.

The optimal target can be defined by - targeted decrease of inflation rate, - targeted decrease of unemployment rate and - targeted decrease in trade of balance deficit.

The planner problem is:

Estimation of constraints

First, we should estimate a simultaneous system of equations that would define the connections between the above endogenous variables and exogenous economic policy measures. The exogenous variables are controlled by the planner and we assume them to be Dprime - the rate of change of prime interest rate, Dgovexp – rate of change in government expenses, Dgovreceipt – rate of change in government receipts, Dm2

- the rate of change in M2 stock of money and D1, D2 and D3 - quarterly dummy variables.

Data

I used U.S. quarterly data for the period 1990.1-2008.3 of the consumer price index, government expenses, government receipts, M2 – money supply, net exports, net government savings, prime rate and unemployment rate[2].

First, in Table 1 we examine the stationarity of various series by applying the Augmented Dickey-Fuller Test.

Table 1

Augmented Dickey-Fuller Test

*MacKinnon (1996) one-sided p-values

The only non-stationary series is Dm2. I replaced it by using the difference series D(dm2), which is stationary.

Several simultaneous versions were estimated by using the following instrumental variables: . After removing non-significant variables, I arrived at the following system of forecasted equations (see regression results in appendix 1).

As we can see, there is no simultaneous relations between d(dcpi) – the change in inflation rate and d(unemp) – the change in unemployment rate. The change in net export (dnetexport) is negatively effected by the change in unemployment rate.

The endogenous variables are effected only by dgovexp and dprime, which are determined by the policy planner.

At this stage, the planner should minimize the social damage subject to estimated constraints. The planner problem becomes:

Substituting (6b) into (6c) we get:

Substituting (6a), (6b) and (6c)' into (5) we get:

Differentiating in respect to , and we get:

Solving (a') and (b') we get:

Given d3, e1, D(dcpi)*, D(unemp)*, d2, e3 and D(netexports)*, we get an optimal level of Dgovexp and dprime – which are determined exogenously by a planner (these optimal policy levels would minimize social damage presented in equation (5)).

We should notice that the level of policy measures taken are defined as function of constants (parameters and targeted values for the objectives), and of the error terms of the objectives. The authority observes the estimation errors (e1, e2, e3) and then set policy instruments levels. Such a policy takes in consideration the fact that econometric models are not accurate. By using estimated errors the policy maker is improving his ability to reach closer to the targets and to minimize social loss.

Simulation

Let us assume that the planner determines the following targeted levels:

D(dcpi)*=-0.5%, D(unemp)*=-0.5% and D(netexports)*=+2%.

Using (6a), (6b) and (6c) I calculated the errors e1, e2 and e3.

Using (a'') and (b''), I calculated the optimal levels of dgovexp and dprime for the period 1990.1 – 2008.3.

Figure 1 present actual and optimal levels of the rate of change in government expenditure.

Figure 1

As we can see, actual change in government expenditure is much less volatile in comparison to the optimal desired change in government expenditure.

Figure 2 presents actual and optimal levels of the rate of change in the prime interest rate.

Figure 2

As we can see, the actual rate of change in the prime rate is moving in the same direction as the optimal change of the prime rate.

Optimal forecasted levels of targeted economic variables versus actual levels

Given that government authorities would have used the model, presented in this paper, to activated the optimal rate of change in government expenditures and prime rate during the period 1990.1 – 2008.3, I forecasted the rate of change in inflation rate, unemployment rate and trade balance deficit.

Figures 3, 4, and 5 present actual versus optimal levels.

Figure 3

Figure 4