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The dynamic behaviour of a simple macroeconomic model with a tax based incomes policy
Economics Letters 3 (1979) 139-143 0 North-Holland Publishing Company
THE DYNAMIC BEHAVIOUR
OF A SIMPLE MACROECONOMIC
MODEL WITH
A TAX BASED INCOMES POLICY
D.A. PEEL * University of Liverpool, Liverpool L69 3BX, UK
Received 18 July 1979
Tax based incomes policies are studied in a mark-up pricing framework with adaptive expectations. Stability may be endangered by such policies. Implications of these stability effects for policy decisions are drawn.
One policy which continues to generate interest 1 as a possible adjunct to other policies in the control of inflation is to tax profits at a rate which varies with the extent to which wage increases in a firm exceed the wage increase norm. Such a policy is known as a tax based incomes policy or T.I.P. Whilst a cost push interpretation of inflation would seem to be the most reasonable rationale for such a policy there has been some limited discussion of the likely impact of the policy in monetarist models of an economy. In particular Seidman (1978) argues that T.I.P. will reduce the natural rate of unemployment. His argument is based on manipulation of the simple augmented ‘Phillips’ curve framework first presented by Tobin (1968) and Smith (1970). The framework is described most simply by the following equations. First the neoclassical wage equation w=WD-S)/S)+P~,
(1)
where w is the rate of change of money wages, D, S are the demand and supply of labour services, pe is the expected rate of change of prices, and h is a positive constant. Second the assumption that excess demand for labour services can be made a function of the unemployment rate, (D-S)/S=F(U).
(2)
r I am grateful, as always, to Stan Metcalfe for helpful comments. See e.g. Brookings Papers on Economic Activity 2, 1978, the whole issue relates to the subject, and American Economic Review Papers and Proceedings, May 1979, ‘Controlling inflation: Incentives for wage and price stability’. 139
140
D.A. Peel / Tax based income policies
Third, abstracting equation
from normal productivity
growth, the simple mark-up price
P=W,
(3)
where p is the actual rate of inflation. When unanticipated inflation is zero juxtaposition
of(l),
(2) and (3) yields
F(U)=0.
(4)
The solution of (4) generates the natural rate of unemployment. Seidman suggests that the introduction of the T.I.P. will lead to a uniform reduction (Y) in the rate of wage increase for any level of excess demand or expected price change. Consequently, (1) is modified to yield w=F(U)tpe-y. Juxtaposition
(5) of (3) and (5), when there is no unanticipated
inflation
yields
F(II)=Y . The solution of this equation yields a lower rate of unemployment than (4). Though attractive, Seidman’s analysis is questionable. Introduction of the tax and its consequent influence on the demand for labour services will be dependent upon the particular objective function which it is assumed firms have. [See for example Latham and Peel (1977).] In any event it is not self evident that the introduction of the tax might not reduce the natural rate of employment through a leftward shift in the demand function, and hence output. Why it should influence the natural rate of unemployment is unclear. [Also argued by Wachter (1978).] The probIem with Seidman’s analysis is ln deducting the tax in (5). The appropriate methodology, within the framework, is to realise that the tax will at best influence the excess demand (supply) for labour services and via this route, ceteris paribus, wage inflation. However, if the impact of the tax on the natural rate of unemployment is questionable it seems likely that the introduction of the tax could change both the stability and cyclical characteristics of the economy. Since this issue has, as yet, not been formally examined it seems of some interest to examine the various possibilities in a variant of the simple stylized model of a monetary economy outlined in, for example, Black (1975), Laidler-Parkin (1975), and Peel (1979). This model consists of a wage and price equation, a demand for money function, the assumption that price expectations are formed adaptively and for simplicity the assumption that there is instantaneous adjustment between the demand and supply of money. The supply of money is assumed exogenous. The minor modification which we make to the model is to modify the mark-up price equation to allow for the possibility that firms will pass a proportion of the tax forward. In particular we assume that the price equation is given by p=w
t(3tri,,
(6)
141
D.A. Peel / Tax based income policies
where t is the tax rate, fl is a positive constant, and I.$ is (d/dt)(w). This type of notation is adopted in the remainder of the note. This equation is derived by assuming that the level of prices is a fixed mark-up (k) on unit wage costs and a proportion (z) of the tax per unit of output. Assuming further that normal output per man is fixed and the fixed wage norm (which does not influence the argument) is zero we obtain after differentiating to a first-order approximation that p=wtz(k-
l)*ti,
(7)
letting fl= z(k - 1) we obtain (6). For simplicity we assume (in common with the papers cited above) that our augmented Phillips curve is given by w =h
WQ/Q*)
+ tf ,
@I
where h is a positive constant, log is natural logs, and Q” is the trend level of real output assumed fixed for simplicity. The demand for money in proportionate rates of change is given by md=ptbq-c$,
(9)
where q is the proportionate rate of change of output, and b and c are positive constants. Eq. (9) is a standard form of demand function in the monetary literature. The Fisher hypothesis is assumed where either the real rate of interest is a constant or a logarithmic function of real output [as in e.g. Taylor (1977)]. The adaptive price expectations mechanism is given by P’=C$(p-pe).
(10)
Finally, assuming instantaneous
adjustment
in the money market so that
md LmS 9
01)
our model is given by (6), (8), (9), (10) and (11). These equations following second-order differential equation (i = 0):
reduce to the
$[b + &h(l - c$) - b$$pt] + ph(l - c@ + @r) + $hp = @hm t h(1 + q@f)liz + hprii. Stability of this second-order that
equation
(12) for constant monetary
expansion
requires
b t /3th(l - c$) - b&Q 2 0, 1 -cf#Jt$l@>O.
(13)
Cycles occur if h(1 - c$ + &3t)z < 4$[b t Pth(l -c@) - bW]
.
(14)
142
D.A. Peel / Tax based income policies
From (13) we notice that if t = 0, a necessary and sufficient condition for stability is the Cagan condition that 1 > c#. In these circumstances we notice that if h(l - c$) - b@ < 0, the introduction of the tax introduces an element of instability into a previously stable situation. Conversely, if 1 2(1 - c@)lM - 4@V&$ -
(15)
Consequently, if 1 < c@, the introduction of the tax both introduces some stability into a previously unstable situation and reduces the possibility of cycles. By contrast, if 1 > c@ but h(l - c#) - b# < 0, the tax introduces an element of instability into a previously stable economy but reduces the possibility of cycles [from (15)]. Finally, we notice from (12) that in equilibrium, the rate of inflation is equal to the rate of monetary expansion. (Recall Q* is assumed constant.) Consequently the rate of inflation is independent of the tax rate. (Assuming, as clearly must be the case, that proponents of T.I.P. are not simply looking for an extra source of tax receipts.) However, it follows from the mark-up price equation that the introduction of the tax will lead, ceteris paribus, to a one shot increase in the price level. This is a result also obtained in some micro-considerations of the tax [e.g. Latham and Peel (1977)]. We might note that if the tax is levied on the excess of the rate of price increase over the norm (rather than the wage increase), the price equation assuming some passing forward is given by p =w +pti,.
(16)
Assuming the other structures of the model given above, we obtain a third-order differential equation which is always unstable for constant monetary expansion. Conclusions. The purpose of this note has been to consider the effects on stability and the cyclical transition path of a monetary economy of the introduction of a T.I.P. when firms practise some tax shifting. We showed how the tax could introduce an element of instability into a previously stable situation but could also reduce the possibility of cycles. To conclude, whilst the analysis in this paper could be repeated for more complicated models the basic point of the analysis remains. That is that the implications of the tax for stability and cycles must be considered before implementation of the tax in the real economy.
References Black, J., 1975, A dynamic model of the quantity theory, in: J. Parkin and A. Nobay, Current economic problems (Cambridge University Press, Manchester).
eds.,
D.A. Peel / Tax based income policies
143
Laidler, D.E.W. and J.M. Parkin, 1975, Inflation - A survey, Economic Journal. Latham, R.W. and D.A. Peel, 1977, The tax on wage increases when the firm is a monopsonist, Journal of Public Economics. Peel, D.A., 1979, The dynamic behaviour of a simple macro model with endogenous demand supply of labour, International Economic Review, forthcoming. Seidman, L.S., 1978, Tax-based incomes policies, in: Brookings Papers on Economic Activity 2, Special issue - Innovative policies to slow inflation. Smith, W., 1970, On some current issues in monetary economics: An interpretation, Journal of Economic Literature. Taylor, D., 1977, A simple model of monetary dynamics, Journal of Money Credit and Banking. Tobin, J., 1968, Inflation - its causes, consequences and control, in: Proceedings of a Symposium at New York University. Wachter, M., 1978, Comments on Seidman, in: Brookings Papers on Economic Activity 2.
THE DYNAMIC BEHAVIOUR
OF A SIMPLE MACROECONOMIC
MODEL WITH
A TAX BASED INCOMES POLICY
D.A. PEEL * University of Liverpool, Liverpool L69 3BX, UK
Received 18 July 1979
Tax based incomes policies are studied in a mark-up pricing framework with adaptive expectations. Stability may be endangered by such policies. Implications of these stability effects for policy decisions are drawn.
One policy which continues to generate interest 1 as a possible adjunct to other policies in the control of inflation is to tax profits at a rate which varies with the extent to which wage increases in a firm exceed the wage increase norm. Such a policy is known as a tax based incomes policy or T.I.P. Whilst a cost push interpretation of inflation would seem to be the most reasonable rationale for such a policy there has been some limited discussion of the likely impact of the policy in monetarist models of an economy. In particular Seidman (1978) argues that T.I.P. will reduce the natural rate of unemployment. His argument is based on manipulation of the simple augmented ‘Phillips’ curve framework first presented by Tobin (1968) and Smith (1970). The framework is described most simply by the following equations. First the neoclassical wage equation w=WD-S)/S)+P~,
(1)
where w is the rate of change of money wages, D, S are the demand and supply of labour services, pe is the expected rate of change of prices, and h is a positive constant. Second the assumption that excess demand for labour services can be made a function of the unemployment rate, (D-S)/S=F(U).
(2)
r I am grateful, as always, to Stan Metcalfe for helpful comments. See e.g. Brookings Papers on Economic Activity 2, 1978, the whole issue relates to the subject, and American Economic Review Papers and Proceedings, May 1979, ‘Controlling inflation: Incentives for wage and price stability’. 139
140
D.A. Peel / Tax based income policies
Third, abstracting equation
from normal productivity
growth, the simple mark-up price
P=W,
(3)
where p is the actual rate of inflation. When unanticipated inflation is zero juxtaposition
of(l),
(2) and (3) yields
F(U)=0.
(4)
The solution of (4) generates the natural rate of unemployment. Seidman suggests that the introduction of the T.I.P. will lead to a uniform reduction (Y) in the rate of wage increase for any level of excess demand or expected price change. Consequently, (1) is modified to yield w=F(U)tpe-y. Juxtaposition
(5) of (3) and (5), when there is no unanticipated
inflation
yields
F(II)=Y . The solution of this equation yields a lower rate of unemployment than (4). Though attractive, Seidman’s analysis is questionable. Introduction of the tax and its consequent influence on the demand for labour services will be dependent upon the particular objective function which it is assumed firms have. [See for example Latham and Peel (1977).] In any event it is not self evident that the introduction of the tax might not reduce the natural rate of employment through a leftward shift in the demand function, and hence output. Why it should influence the natural rate of unemployment is unclear. [Also argued by Wachter (1978).] The probIem with Seidman’s analysis is ln deducting the tax in (5). The appropriate methodology, within the framework, is to realise that the tax will at best influence the excess demand (supply) for labour services and via this route, ceteris paribus, wage inflation. However, if the impact of the tax on the natural rate of unemployment is questionable it seems likely that the introduction of the tax could change both the stability and cyclical characteristics of the economy. Since this issue has, as yet, not been formally examined it seems of some interest to examine the various possibilities in a variant of the simple stylized model of a monetary economy outlined in, for example, Black (1975), Laidler-Parkin (1975), and Peel (1979). This model consists of a wage and price equation, a demand for money function, the assumption that price expectations are formed adaptively and for simplicity the assumption that there is instantaneous adjustment between the demand and supply of money. The supply of money is assumed exogenous. The minor modification which we make to the model is to modify the mark-up price equation to allow for the possibility that firms will pass a proportion of the tax forward. In particular we assume that the price equation is given by p=w
t(3tri,,
(6)
141
D.A. Peel / Tax based income policies
where t is the tax rate, fl is a positive constant, and I.$ is (d/dt)(w). This type of notation is adopted in the remainder of the note. This equation is derived by assuming that the level of prices is a fixed mark-up (k) on unit wage costs and a proportion (z) of the tax per unit of output. Assuming further that normal output per man is fixed and the fixed wage norm (which does not influence the argument) is zero we obtain after differentiating to a first-order approximation that p=wtz(k-
l)*ti,
(7)
letting fl= z(k - 1) we obtain (6). For simplicity we assume (in common with the papers cited above) that our augmented Phillips curve is given by w =h
WQ/Q*)
+ tf ,
@I
where h is a positive constant, log is natural logs, and Q” is the trend level of real output assumed fixed for simplicity. The demand for money in proportionate rates of change is given by md=ptbq-c$,
(9)
where q is the proportionate rate of change of output, and b and c are positive constants. Eq. (9) is a standard form of demand function in the monetary literature. The Fisher hypothesis is assumed where either the real rate of interest is a constant or a logarithmic function of real output [as in e.g. Taylor (1977)]. The adaptive price expectations mechanism is given by P’=C$(p-pe).
(10)
Finally, assuming instantaneous
adjustment
in the money market so that
md LmS 9
01)
our model is given by (6), (8), (9), (10) and (11). These equations following second-order differential equation (i = 0):
reduce to the
$[b + &h(l - c$) - b$$pt] + ph(l - c@ + @r) + $hp = @hm t h(1 + q@f)liz + hprii. Stability of this second-order that
equation
(12) for constant monetary
expansion
requires
b t /3th(l - c$) - b&Q 2 0, 1 -cf#Jt$l@>O.
(13)
Cycles occur if h(1 - c$ + &3t)z < 4$[b t Pth(l -c@) - bW]
.
(14)
142
D.A. Peel / Tax based income policies
From (13) we notice that if t = 0, a necessary and sufficient condition for stability is the Cagan condition that 1 > c#. In these circumstances we notice that if h(l - c$) - b@ < 0, the introduction of the tax introduces an element of instability into a previously stable situation. Conversely, if 1
(15)
Consequently, if 1 < c@, the introduction of the tax both introduces some stability into a previously unstable situation and reduces the possibility of cycles. By contrast, if 1 > c@ but h(l - c#) - b# < 0, the tax introduces an element of instability into a previously stable economy but reduces the possibility of cycles [from (15)]. Finally, we notice from (12) that in equilibrium, the rate of inflation is equal to the rate of monetary expansion. (Recall Q* is assumed constant.) Consequently the rate of inflation is independent of the tax rate. (Assuming, as clearly must be the case, that proponents of T.I.P. are not simply looking for an extra source of tax receipts.) However, it follows from the mark-up price equation that the introduction of the tax will lead, ceteris paribus, to a one shot increase in the price level. This is a result also obtained in some micro-considerations of the tax [e.g. Latham and Peel (1977)]. We might note that if the tax is levied on the excess of the rate of price increase over the norm (rather than the wage increase), the price equation assuming some passing forward is given by p =w +pti,.
(16)
Assuming the other structures of the model given above, we obtain a third-order differential equation which is always unstable for constant monetary expansion. Conclusions. The purpose of this note has been to consider the effects on stability and the cyclical transition path of a monetary economy of the introduction of a T.I.P. when firms practise some tax shifting. We showed how the tax could introduce an element of instability into a previously stable situation but could also reduce the possibility of cycles. To conclude, whilst the analysis in this paper could be repeated for more complicated models the basic point of the analysis remains. That is that the implications of the tax for stability and cycles must be considered before implementation of the tax in the real economy.
References Black, J., 1975, A dynamic model of the quantity theory, in: J. Parkin and A. Nobay, Current economic problems (Cambridge University Press, Manchester).
eds.,
D.A. Peel / Tax based income policies
143
Laidler, D.E.W. and J.M. Parkin, 1975, Inflation - A survey, Economic Journal. Latham, R.W. and D.A. Peel, 1977, The tax on wage increases when the firm is a monopsonist, Journal of Public Economics. Peel, D.A., 1979, The dynamic behaviour of a simple macro model with endogenous demand supply of labour, International Economic Review, forthcoming. Seidman, L.S., 1978, Tax-based incomes policies, in: Brookings Papers on Economic Activity 2, Special issue - Innovative policies to slow inflation. Smith, W., 1970, On some current issues in monetary economics: An interpretation, Journal of Economic Literature. Taylor, D., 1977, A simple model of monetary dynamics, Journal of Money Credit and Banking. Tobin, J., 1968, Inflation - its causes, consequences and control, in: Proceedings of a Symposium at New York University. Wachter, M., 1978, Comments on Seidman, in: Brookings Papers on Economic Activity 2.