THE OPTIMAL DEGREE OF COMMITMENT where r is the level of the nominal interest rate and Er(p,+1)-pr represents the rate of inflation expected by investors, based complete period t information. The serially uncorrelated goods market demand disturbance is ur N(O, 04; u may be viewed as a transitory shift in intertemporal consumption preferences The demand for real money balances is a decreasing function of the nominal interest rate and an increasing function of output where m is the logarithm of the nominal money supply and v is a shift in portfolio preferences between money and bonds U-N(O, 02). To simplify exposition, the disturbances, U, u, and z, are assumed to be independent and serially uncorrelated 3. The social loss function The principal differences between the present paper and I vious rational expectations cum wage contracting analyses of monetary stabilization policy derive from the specification of the social objective function. Because most models embody the nat. ural rate hypothesis, the issue of whether or not the central bank wishes it could lower the average level of employment is com monly ignored. But this potential source of tension is fundamental le conduct of stabilization policy. Indeed, if there does not exist any temptation for the monetary authorities to inflate sys tematically, then there is no reason to consider any regime other than fully discretionary monetary policy. (Note that the central bank's incentives to inflate need not be motivated by employment considerations, but can also arise due to the presence of nominal government debt or short-term rigidities in the tax system. Here we shall assume that some factor such as income taxa tion or unemployment insurance distorts the labor-leisure deci sion and causes the market-determined level of employment n and Gordon [1983a). We shall further assume that a ee barro to lie below the socially optimal level of employment nl(se constant and equal to n -n The social loss function A depends on deviations of employ ment and infation from their optimal(socially desired)levels n19rnodified tion. the main 学此比品 in Kydland and Prescott multiperiod objective fu
THE OPTIMAL DEGREE OF COMMITMENT 1173 where r is the level of the nominal interest rate and E,(p,, 1) - Pt represents the rate of inflation expected by investors, based on complete period t information. The serially uncorrelated goods market demand disturbance is ut - N(0,a2); u may be viewed as a transitory shift in intertemporal consumption preferences. The demand for real money balances is a decreasing function of the nominal interest rate and an increasing function of output: (9) mt - Pt - Xrt + 'PYt + Vt, where m is the logarithm of the nominal money supply and v is a shift in portfolio preferences between money and bonds; v N(O, U). To simplify exposition, the disturbances, v, u, and z, are assumed to be independent and serially uncorrelated. 3. The Social Loss Function The principal differences between the present paper and previous rational expectations cum wage contracting analyses of monetary stabilization policy derive from the specification of the social objective function. Because most models embody the natural rate hypothesis, the issue of whether or not the central bank wishes it could lower the average level of employment is commonly ignored. But this potential source of tension is fundamental to the conduct of stabilization policy. Indeed, if there does not exist any temptation for the monetary authorities to inflate systematically, then there is no reason to consider any regime other than fully discretionary monetary policy. (Note that the central bank's incentives to inflate need not be motivated by employment considerations, but can also arise due to the presence of nominal government debt or short-term rigidities in the tax system.) Here we shall assume that some factor such as income taxation or unemployment insurance distorts the labor-leisure decision and causes the market-determined level of employment nt to lie below the socially optimal level of employment h' (see Barro and Gordon [1983a]). We shall further assume that h' - ni is constant and equal to h - n. The social loss function A depends on deviations of employment and inflation from their optimal (socially desired) levels:5 5. A similar social objective function is employed in Kydland and Prescott [1977] and Barro and Gordon [1983a]. Although the analysis below would have to be modified substantially if the central bank had a multiperiod objective function, the main points would still obtain
1174 QUARTERLY JOURNAL OF ECONOMICS (10) A2=(n-n1)2+X(m1-亓)}, where T,=p,- Pr-1, t is the socially desired trend inflation rate. and x is the relative weight society places on inflation stabili zation versus employment stabilization It is somewhat difficult, in the context of a rational expec- tations model, to argue that the level of the inflation rate has much direct weight in the social loss function b(However, the analysis below does not depend on x being particularly large The costs of inflation include the administrative costs of posting new prices and the costs of adjusting the tax system to be fully neutral with respect to inflation. And, of course, high rates of inflation force agents to economize on their holdings of non-in- terest-bearing money-the so-called shoe leather cost of infa tion. Despite the foregoing considerations, i may be nonzero if through distortions(see Phelps [1973/ erate deadweight costs alternative taxes to seignorage also ger III TIME-CONSISTENT EQUILIBRIUM UNDER FULLY DISCRETIONARY MONETARY POLICY Here, stochastic equilibrium is derived under the assumption that the monetary authorities attempt to minimize the social loss function A, given by equation(10)above Expectations about the future path of the money supply are not exogenously given in this model, but depend endogenously on agents' expectations about the monetary authorities future short run stabilization objectives. Wage setters will not believe prom sed future paths for the money supply that are not time-con sistent. Instead, equilibrium nominal wage increases are set at a sufficiently high level so that, in the absence of disturbances, the central bank will not choose to inflate the money supply beyond the point consistent with wage setters desired real wage At this high level of inflation, the central bank finds that the marginal gain from trying to raise employment above the natural rate is fully offset by the marginal cost of still higher inflation. Note also that no individual group of wage setters has any incentive to change their wage bargain in the time-consistent equilibrium Even though individual wage setters are concerned about infla 6. Unanticipated infl rough its effect on employment. Fischer and M nomic costs of both anticipated and unanticipated inflation
1174 QUARTERLY JOURNAL OF ECONOMICS (10) A, = (n, - t')2 + X(TIrt - where Irt Pt - Pt- 1, r is the socially desired trend inflation rate, and X is the relative weight society places on inflation stabilization versus employment stabilization. It is somewhat difficult, in the context of a rational expectations model, to argue that the level of the inflation rate has much direct weight in the social loss function.6 (However, the analysis below does not depend on X being particularly large.) The costs of inflation include the administrative costs of posting new prices and the costs of adjusting the tax system to be fully neutral with respect to inflation. And, of course, high rates of inflation force agents to economize on their holdings of non-interest-bearing money-the so-called "shoe leather cost of inflation." Despite the foregoing considerations, fr may be nonzero if alternative taxes to seignorage also generate deadweight costs through distortions (see Phelps [1973]). III. TIME-CONSISTENT EQUILIBRIUM UNDER FULLY DISCRETIONARY MONETARY POLICY Here, stochastic equilibrium is derived under the assumption that the monetary authorities attempt to minimize the social loss function A, given by equation (10) above. Expectations about the future path of the money supply are not exogenously given in this model, but depend endogenously on agents' expectations about the monetary authorities' future shortrun stabilization objectives. Wage setters will not believe promised future paths for the money supply that are not time-consistent. Instead, equilibrium nominal wage increases are set at a sufficiently high level so that, in the absence of disturbances, the central bank will not choose to inflate the money supply beyond the point consistent with wage setters' desired real wage. At this high level of inflation, the central bank finds that the marginal gain from trying to raise employment above the natural rate is fully offset by the marginal cost of still higher inflation. Note also, that no individual group of wage setters has any incentive to change their wage bargain in the time-consistent equilibrium. Even though individual wage setters are concerned about infla- 6. Unanticipated inflation enters indirectly into the social loss function (10) through its effect on employment. Fischer and Modigliani [1978] catalog the economic costs of both anticipated and unanticipated inflation
THE OPTIMAL DEGREE OF COMMITMENT 1175 tion, the contract at their firm has only a small impact on the aggregate inflation rate By substituting equation(6)into equation(10), and recalling that n'-n'=n-n, the central bank's objective function under fully discretionary monetary policy may be written as D,=A=[z/n +(p- wP)/a-(n-n)]2 where superscript D stand for fully discretionary regime. "The central bank maximizes social welfare by choosing a level of the money supply consistent with pp, the period t price level that minimizes A: (12) p +X(pt-1+亓) Recall that (the logarithm of) wage setters'target real wage is zero. Thus, wage setters select wp by taking expectations across (12)and setting wP= Et-1(p?): 8 (13)?=E4-1m)=p2-1+亓+(h-n)xQ=p1-1+m By choosing w? according to(13), wage setters assure themselves that the monetary authorities will not systematically drive down the real wage. Thus, as Kydland and Prescott [1977] point out, the time-consistent rate of inflation is too high when n>n We are now prepared to evaluate social welfare under fully discretionary monetary policy. But to facilitate exposition in later sections, we shall first develop a notation for evaluating the ex- pected value of the social welfare function under any arbitrary monetary policy regime"A", AA 7. pl is found by setting aD/ ap,=0. The second-order conditions for a min (81)E1-1(m2+=(n-n)s+1)xa+(s+1)亓+p s≥0. (For simplicity, we treat the monetary authorities' objective function as con trolled it directly, ignoring the fact that the central bank directly controls onl he assumption of saddl stability. (For microec ion in monetary models
THE OPTIMAL DEGREE OF COMMITMENT 1175 tion, the contract at their firm has only a small impact on the aggregate inflation rate. By substituting equation (6) into equation (10), and recalling that h' - n' = h - n, the central bank's objective function under fully discretionary monetary policy may be written as (11) Dt = At = [zWbq + (Pt - )/o- (nf - n + X[Pt - Pt 1 -*], where superscript D stand for "fully discretionary regime." The central bank maximizes social welfare by choosing a level of the money supply consistent with pr, the period t price level that minimizes At:7 (12) pD [t h - n - zt/ -*]/[+(1 2 Recall that (the logarithm of) wage setters' target real wage is zero. Thus, wage setters select u' by taking expectations across (12) and setting e =-Et -ID):8 I (13) O = Et-1(pD) = Pt-i + * + (h - n)/Xot = Pt-i + ir1D. By choosing t according to (13), wage setters assure themselves that the monetary authorities will not systematically drive down the real wage. Thus, as Kydland and Prescott [1977] point out, the time-consistent rate of inflation is too high when h > n. We are now prepared to evaluate social welfare under fully discretionary monetary policy. But to facilitate exposition in later sections, we shall first develop a notation for evaluating the expected value of the social welfare function under any arbitrary monetary policy regime "A", AA: 7. pt is found by setting 8DtlIpt = 0. The second-order conditions for a minimum are met; given the quadratic form of D, the minimum is global. 8. Investors can apply the same algorithm repeatedly to derive a time-consistent path for all future prices: (8.1) Et-1 (pP?8) = (h - n)(s + l)/xo + (s + 1)f* + Pt+ , S _ 0. (For simplicity, we treat the monetary authorities' objective function as constant.) Note that we are treating the price level as if the monetary authorities controlled it directly, ignoring the fact that the central bank directly controls only the money supply. The anticipated future path of the money supply consistent with (8.1) may be found using the macro model of equations (7)-(9), together with the assumption of saddlepath stability. (For macroeconomic justification of the saddlepath assumption in monetary models, see Obstfeld and Rogoff [1983].)