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Inflation: A simple Friedman theory with a Phillips twist
Journal of Monetary Economics 1 (1975) 117-122. ONorth-Holland
Publishing Company
John VANDERKAMP* Umkersity of Guelph, Glelph, Ont ., Canada
e
Introduction
This paper constructs a simple ‘textbook’ model of inflation which incorporates some of Friedman’s notions about the demand for money, the basic idea of the Phillips curve, and the effects of inflationary expectations.’ The simple model has three basic components which are discussed in turn: (i) a simple demand for money relation; (ii) a Phillips curve incorporating price change expectations in the short-run these expectations are constant and in the long-run they fully reflect actual inflation; and (iii) a mechanism which equilibrates this system, with real output acting as the equilibrator through its effect on the unemployment rate. The demand for money is assumed to be M = kPy;
IL1is the demand for money (as well as the supply, since there is assumed to be continuous market clearing), P is the overall price level and y the real output level. It is assumed that k, which is the reciprocal of velocity, is constant.2 Since we are interested in the rate of inflation, we differentiate eq. (1) logarithmicdly, ti
(2)
*Thanks are due to my Chdpl~ CXGX~UC~, to Brian Scarfc and lo an anonymous refci*cc. ‘See Friedman (1971) and Vanderkamp (1972). For rcccnt literature see various issues of the Brookings Papers on Economic Activity, in particular by Gordon (1971a, 1971b, 1972) and the important work of the Manchester pr cct, starting publication with Parkin and Sunlner (197,“). *It may be argued that velocity is positively related to the nominal interest rate which is in turn affected by inflationary expectatiotis [see Sargent (19731. Incorporating this would modify cq. (2) by adding a term representing the rate at which inflationary expectations are changing over time. Since ’ zero in short- and long-run equilibrium conditions it has for simplicity been o e re ere owever, that the rate of expecta&ns adjustment will not only affect the position of the Phillips curve but also the demand for money schedule.
118
.I. Vanderkdimp, “Textbook’ model of i@¶ation
Variables with dots over them represent percentage changes; thus P is the inflation rate and j is the rate of growth of real output. In eq. (2) the change in money supply determines the change in nominal income levels. [See Friedman (1971).] However, we do not know from this how much of this money supply change will go into prices and how much into output. To determine the actual split between price and output changes we need to add the Phillips curve relation, P = a()+qU-‘+P*.
(3)
The unemployment rate is represented by U and it appears in inverse form to reflect the non-linearity of the Phillips curve. P* is the expected rate of inflation which has a coefficient of unity, reflecting the presumption that price change expectations are fully incorporated in wage (and thus price) changes. 3 Coefficient a, is positive and a0 is assumed to be negative; these two parameters determine the shape of the Phillips curve. The expected rate of inflation is, in general, influenced by past rates of inflation. We shall distinguish between the shot+run, in which inflationary expectations are constant (for simplicity at P* = 0), c;nd the long-run, in which inflationary expectations fully catch up with actual inflation experience (9” = 9). The third component of this model is the equilibrating mechanism, which works by adjustment of the rate of change in output and the unemployment rate. jt* is the (constant) growth rate of real output consistent with the long-term growth rate of the labour force and of labour productivity. When actual output growth exceeds y* unemployment declines, and when $ < j* the unemployment rate increases, in accord with
ir = b(j-j*); 0 is the rate of change in unemployment and
(4) b
is negative.
peration of the model The past history of the inflation rate (which affects P*) is a predetermined variable and hi is an exogenous variable in this system. Graphically this system of equations, (2), (3) and (4), is represented in fig. 1. The vertical axis represents the rate of inflation t, and the right-hand side of the horizontal axis shows the unemployment U. The Phillips relation (3) is shown in this P-U space as Dpo and the expected rate of inflation PyI’is shown as a horizontal line (initially assumed to be at P* = 0). The left-hand side of the horizontal axis represents Jo with larger rates of output increase further toward the left side. e constant “See Yanderkamp (1972) and Gordon (1972).
J. Vmderkamp,
‘Textbook’ model of inflation
119
long-term growth rate in output is shown by the vertical line p*. In the left quadrant two different rates of money supply increase are shown by lines hi, and A&. It will be noted that because of our formulation of eq. (2), the A&lines are at 45”.angles to the axes; a money supply increase can potentially all be used for inflation, or all for real output increases, or for linear combinations of the two. The equilibrium solution corresponding to money s .pply increase ti, is j = 3*,2) = 0, and W = U*. For hj, the equilibrium position is f = $*, P = p1 and U = U1 (under the assumption that initially p* = 0 and remains there). To see how this equilibrium is established let us start from situation 0 and see how we move to situation 1 when & is increased from A?, to Ml . When the rate of monetary expansion is raised to I\;r,, the rate of output increase will initially go to j,, because at a moment of time U is given at U = U*. But now $ is greater than j.*, and the rate of unemployment is lowered. As unemployment goes down we are travelling up the Phillips curve, implying that 9 > 0. When we arrive at
Y
Fig. 1
the PI--U, combination this process stops and equilibrium is re-established with j = j,*, p s-. p, and U zz U1. As long as i)* stays at zero (and other things are the same) this will remain the equilibrium situation; we label this ~ITOPGI’UIZ equilibrium because P* is constant. For different rates of monetary expansion we will trace out a set of shcrt-run equilibrium positions (and adjustment paths) along the DP, Phillips curve. At the same time, looking ut the left-hand uadrant, we will trace a series of equilibrium points along the f* line with P = hi--J*. Thus looking at one side of the picture we obtain a perfect Phillips curve explanation of inflation, while the other siale gives us a fine quantity theory explanation of inflation. While clearly tl,t ultimate cause of inflation stems from the rate of monetary expansion, we require knowledge about the Phillips curve as well to determine the complete equilibrium configuration. But the situation with P = P, and P* = 0 carmot maintain itself forever. To ibrium shifts it will be assumed that p* catches up with actual P. In fig. 2 we again start from situation 0 of fig. 1 and observe what
120
J. Vanderkamp,“Textbook’model of inflation
happens wher? the rate of monetary expansion is raised from fi, to fi, . The initial long-vu?? equilibrium position is at (p*, p = P* = 0, U*). After the shift from n;f, to &Z1 the system will start to kavel in the direction of short-run equilibrium (j*, P, , U,), but while this happens P* will begin to rise and eventually catch up with the actual inflation experience. Long-run equilibrium will be re-established at (j7*,P, and U*), with p* having shifted up to P,. Thus for comparison of long-run equilibrium situations the quantity theory relation holds On the other hand, the Phillips curve loses the ability to with P = a-j*. explain differences in long-run equilibrium positions. At the same time, the Phillips curve is important in determining the adjustment path traced out between long-run equilibria, and the Phillips curve determines U*, the ‘natural’ manployment rate.
0
Fig. 2
3. Afijustment path Ultimately the rate of monetary expansion determines the rate of inflation in this model, but the Phillips curve plays a very important role in the adjustment process. This role is the more important the less rapidly inflationa.:y expectations adjust to actual experience. If p* is constant and does not adjust at all, then monetary policy can be employed to move along the PhiHips curve to the most desired P-U combination. While monetary expansion is the moving force, the other side of the picture is of great importance and only ti good estimate of the Phillips curve will give us an accurate measure of the trade-off costs between 8’ and U. If price change expectations do adjust then there is a natural unemployment rate U* to which the system will tend in \the long-run. But this does not mean that illips curve is of no importance. Firstly, the Phillips curve shape determines e of this natural unemployment rate, which will not only reflect structural antes and downward rigidities, but also spontaneous or cost-push types of n pressures. Secondly, the Phillips curve is an important determkant of
J. Vanderkamp, ‘Textbook’ model of inflation
121
the adjustment path between long-run equilibria. This may be seen m fig. 3 which shows the response of the system to an increase in the rate of monetary expansion from ti, to hi, . The adjustment path between the two equilibrium situations 0 and 1 is shown in fig. 3. If expectations adjust quite slowly (a) will be the more probable path of adjustment, while rapid expectations adjustment will produce a path such as (a’). Going in the opposite direction, in response to a reduced rate of monetary expansion, we will travel along an adjustment path such as (b) or (b’). Path (b) corresponds to the case of slow adjustment of inflationary expectations. It will be obvious that in the downward direction the path of adjustment is likely to be tortuous. The reason for this is the non-linear shape of the Phillips curve. To make a significant reduction in the r:rte of inflation below the expected rate requires a large increase in unemployment beyond U*. And even a large
Fig. 3
increase in unemployment will only reduce the rate of price increase very little, which means that inflationary expectations will only moderate slightly. If price change expectations are slow to adjust, then the unemployment experience along the adjustment path [such as (b) in fig. 3j is likely to be very prolonged and severe. If P* reacts slowly to experience, then one can use monetary policy very effectively to reduce unemployment at least for a while. Ironically, the more attractive such a policy is the more expensive it becomes to reduce the rate of inflation by reduced monetary expansion; path (a) in fig. 3 provides a possible trade-off for a time, but the corresponding downward path (b) is very expensive. Thus slow expectational reactions make the Phillips curve trade-off a more attractive trap, but they also greatly increase the costs of extrication if one wishes to reduce a long established high inflation rate. Assuming that governe temptations of increasing employment in the short-run, we would predict a trend of ever-increasing rates of inflation.
122
J. Vanderkamp, ‘Textbook’ model of inflation
The short-run policy objective of increasing employment may be even more attractive when an incomes policy can be used to shift the Phillips curve to the left. An inco,i;enes policy will have the effect of reducing the rate of spontaneous wage and price inflation and lowering inflationary expectations. But these effects are only temporary, since expectations are bound to reflect actual experience in the end. Thus when we are pursuing an expansionary monetary policy, an incomes policy will only add to our sense of illusion that we are making permanent employment gains. On the other hand, an incomes policy may well be a useful tool when it is employed together with a restrictive monetary policy to aid in the downward adjustment from an established inflation rate. The incomes policy would then reduce the unemployment burden resulting from the transition to a lower inflation rate.
This model suggests that the Phillips curve is not dead, but that it is in fact a more dangerous device than the simple trade-off approach implies. The danger lies in the costs of reducing inflation, which may not be realized when the economy is moving to higher inflation rates. This problem is, of course, of no importance if one is not concerned about the rate of inflation. In that case there is presumably some optimal rate of increase in the rate of monetary expansion (and an optimal rate of increase of tne inflation rate) which depends on the shape of the Phillips curve, the effect of output changes on employment, and on the rate at which inflationary expectations adapt to experience. References Friedman, M., 1970, A theoretical framework for monetary analysis, Journal of Political Economy 78, March-April, 193-2.38. Friedman, M., 1971, A monetary theory of nominal income, Journal of Political Economy 79, March-April. Gordon, R.J., 1971a, Inflation in recession and recovery, Brookings Papers on Economic Activity I. Gordon, R.J., 197lb, Steady anticipated inflation: Mirage or oasis, Brookings Papers on Economic Activity Il. Gordon, W.J., 1972, Wage-price control and the shifting Phillips curve, Brookings Papers on Economic ActivitJl II. Parkin, M. and M.T. Sumner, eds., 1972, Incomes policy and inflation (Manchester University Press, Manchester). Sargent, T.S., 1972, Anticipated inlqation and the nominal rate of interest, Quarterly Journal of Economics 86, .MC%y, 212-225. Vanderkamp, J., 1968, The Phillips relation: A theoretical explanation - a comment, Economica 35, May. Vanderkamp, J., 1972, Wage adjustment, productivity and price change expectations, The Review (.>fEconomic Studies 39. Vanderkamp, J., nd., Excess demand, unemployment, vacancies and wage adjustment, Discussion paper (Department of Economics: University of Guelph, Guelph, Ont.).
Publishing Company
John VANDERKAMP* Umkersity of Guelph, Glelph, Ont ., Canada
e
Introduction
This paper constructs a simple ‘textbook’ model of inflation which incorporates some of Friedman’s notions about the demand for money, the basic idea of the Phillips curve, and the effects of inflationary expectations.’ The simple model has three basic components which are discussed in turn: (i) a simple demand for money relation; (ii) a Phillips curve incorporating price change expectations in the short-run these expectations are constant and in the long-run they fully reflect actual inflation; and (iii) a mechanism which equilibrates this system, with real output acting as the equilibrator through its effect on the unemployment rate. The demand for money is assumed to be M = kPy;
IL1is the demand for money (as well as the supply, since there is assumed to be continuous market clearing), P is the overall price level and y the real output level. It is assumed that k, which is the reciprocal of velocity, is constant.2 Since we are interested in the rate of inflation, we differentiate eq. (1) logarithmicdly, ti
(2)
*Thanks are due to my Chdpl~ CXGX~UC~, to Brian Scarfc and lo an anonymous refci*cc. ‘See Friedman (1971) and Vanderkamp (1972). For rcccnt literature see various issues of the Brookings Papers on Economic Activity, in particular by Gordon (1971a, 1971b, 1972) and the important work of the Manchester pr cct, starting publication with Parkin and Sunlner (197,“). *It may be argued that velocity is positively related to the nominal interest rate which is in turn affected by inflationary expectatiotis [see Sargent (19731. Incorporating this would modify cq. (2) by adding a term representing the rate at which inflationary expectations are changing over time. Since ’ zero in short- and long-run equilibrium conditions it has for simplicity been o e re ere owever, that the rate of expecta&ns adjustment will not only affect the position of the Phillips curve but also the demand for money schedule.
118
.I. Vanderkdimp, “Textbook’ model of i@¶ation
Variables with dots over them represent percentage changes; thus P is the inflation rate and j is the rate of growth of real output. In eq. (2) the change in money supply determines the change in nominal income levels. [See Friedman (1971).] However, we do not know from this how much of this money supply change will go into prices and how much into output. To determine the actual split between price and output changes we need to add the Phillips curve relation, P = a()+qU-‘+P*.
(3)
The unemployment rate is represented by U and it appears in inverse form to reflect the non-linearity of the Phillips curve. P* is the expected rate of inflation which has a coefficient of unity, reflecting the presumption that price change expectations are fully incorporated in wage (and thus price) changes. 3 Coefficient a, is positive and a0 is assumed to be negative; these two parameters determine the shape of the Phillips curve. The expected rate of inflation is, in general, influenced by past rates of inflation. We shall distinguish between the shot+run, in which inflationary expectations are constant (for simplicity at P* = 0), c;nd the long-run, in which inflationary expectations fully catch up with actual inflation experience (9” = 9). The third component of this model is the equilibrating mechanism, which works by adjustment of the rate of change in output and the unemployment rate. jt* is the (constant) growth rate of real output consistent with the long-term growth rate of the labour force and of labour productivity. When actual output growth exceeds y* unemployment declines, and when $ < j* the unemployment rate increases, in accord with
ir = b(j-j*); 0 is the rate of change in unemployment and
(4) b
is negative.
peration of the model The past history of the inflation rate (which affects P*) is a predetermined variable and hi is an exogenous variable in this system. Graphically this system of equations, (2), (3) and (4), is represented in fig. 1. The vertical axis represents the rate of inflation t, and the right-hand side of the horizontal axis shows the unemployment U. The Phillips relation (3) is shown in this P-U space as Dpo and the expected rate of inflation PyI’is shown as a horizontal line (initially assumed to be at P* = 0). The left-hand side of the horizontal axis represents Jo with larger rates of output increase further toward the left side. e constant “See Yanderkamp (1972) and Gordon (1972).
J. Vmderkamp,
‘Textbook’ model of inflation
119
long-term growth rate in output is shown by the vertical line p*. In the left quadrant two different rates of money supply increase are shown by lines hi, and A&. It will be noted that because of our formulation of eq. (2), the A&lines are at 45”.angles to the axes; a money supply increase can potentially all be used for inflation, or all for real output increases, or for linear combinations of the two. The equilibrium solution corresponding to money s .pply increase ti, is j = 3*,2) = 0, and W = U*. For hj, the equilibrium position is f = $*, P = p1 and U = U1 (under the assumption that initially p* = 0 and remains there). To see how this equilibrium is established let us start from situation 0 and see how we move to situation 1 when & is increased from A?, to Ml . When the rate of monetary expansion is raised to I\;r,, the rate of output increase will initially go to j,, because at a moment of time U is given at U = U*. But now $ is greater than j.*, and the rate of unemployment is lowered. As unemployment goes down we are travelling up the Phillips curve, implying that 9 > 0. When we arrive at
Y
Fig. 1
the PI--U, combination this process stops and equilibrium is re-established with j = j,*, p s-. p, and U zz U1. As long as i)* stays at zero (and other things are the same) this will remain the equilibrium situation; we label this ~ITOPGI’UIZ equilibrium because P* is constant. For different rates of monetary expansion we will trace out a set of shcrt-run equilibrium positions (and adjustment paths) along the DP, Phillips curve. At the same time, looking ut the left-hand uadrant, we will trace a series of equilibrium points along the f* line with P = hi--J*. Thus looking at one side of the picture we obtain a perfect Phillips curve explanation of inflation, while the other siale gives us a fine quantity theory explanation of inflation. While clearly tl,t ultimate cause of inflation stems from the rate of monetary expansion, we require knowledge about the Phillips curve as well to determine the complete equilibrium configuration. But the situation with P = P, and P* = 0 carmot maintain itself forever. To ibrium shifts it will be assumed that p* catches up with actual P. In fig. 2 we again start from situation 0 of fig. 1 and observe what
120
J. Vanderkamp,“Textbook’model of inflation
happens wher? the rate of monetary expansion is raised from fi, to fi, . The initial long-vu?? equilibrium position is at (p*, p = P* = 0, U*). After the shift from n;f, to &Z1 the system will start to kavel in the direction of short-run equilibrium (j*, P, , U,), but while this happens P* will begin to rise and eventually catch up with the actual inflation experience. Long-run equilibrium will be re-established at (j7*,P, and U*), with p* having shifted up to P,. Thus for comparison of long-run equilibrium situations the quantity theory relation holds On the other hand, the Phillips curve loses the ability to with P = a-j*. explain differences in long-run equilibrium positions. At the same time, the Phillips curve is important in determining the adjustment path traced out between long-run equilibria, and the Phillips curve determines U*, the ‘natural’ manployment rate.
0
Fig. 2
3. Afijustment path Ultimately the rate of monetary expansion determines the rate of inflation in this model, but the Phillips curve plays a very important role in the adjustment process. This role is the more important the less rapidly inflationa.:y expectations adjust to actual experience. If p* is constant and does not adjust at all, then monetary policy can be employed to move along the PhiHips curve to the most desired P-U combination. While monetary expansion is the moving force, the other side of the picture is of great importance and only ti good estimate of the Phillips curve will give us an accurate measure of the trade-off costs between 8’ and U. If price change expectations do adjust then there is a natural unemployment rate U* to which the system will tend in \the long-run. But this does not mean that illips curve is of no importance. Firstly, the Phillips curve shape determines e of this natural unemployment rate, which will not only reflect structural antes and downward rigidities, but also spontaneous or cost-push types of n pressures. Secondly, the Phillips curve is an important determkant of
J. Vanderkamp, ‘Textbook’ model of inflation
121
the adjustment path between long-run equilibria. This may be seen m fig. 3 which shows the response of the system to an increase in the rate of monetary expansion from ti, to hi, . The adjustment path between the two equilibrium situations 0 and 1 is shown in fig. 3. If expectations adjust quite slowly (a) will be the more probable path of adjustment, while rapid expectations adjustment will produce a path such as (a’). Going in the opposite direction, in response to a reduced rate of monetary expansion, we will travel along an adjustment path such as (b) or (b’). Path (b) corresponds to the case of slow adjustment of inflationary expectations. It will be obvious that in the downward direction the path of adjustment is likely to be tortuous. The reason for this is the non-linear shape of the Phillips curve. To make a significant reduction in the r:rte of inflation below the expected rate requires a large increase in unemployment beyond U*. And even a large
Fig. 3
increase in unemployment will only reduce the rate of price increase very little, which means that inflationary expectations will only moderate slightly. If price change expectations are slow to adjust, then the unemployment experience along the adjustment path [such as (b) in fig. 3j is likely to be very prolonged and severe. If P* reacts slowly to experience, then one can use monetary policy very effectively to reduce unemployment at least for a while. Ironically, the more attractive such a policy is the more expensive it becomes to reduce the rate of inflation by reduced monetary expansion; path (a) in fig. 3 provides a possible trade-off for a time, but the corresponding downward path (b) is very expensive. Thus slow expectational reactions make the Phillips curve trade-off a more attractive trap, but they also greatly increase the costs of extrication if one wishes to reduce a long established high inflation rate. Assuming that governe temptations of increasing employment in the short-run, we would predict a trend of ever-increasing rates of inflation.
122
J. Vanderkamp, ‘Textbook’ model of inflation
The short-run policy objective of increasing employment may be even more attractive when an incomes policy can be used to shift the Phillips curve to the left. An inco,i;enes policy will have the effect of reducing the rate of spontaneous wage and price inflation and lowering inflationary expectations. But these effects are only temporary, since expectations are bound to reflect actual experience in the end. Thus when we are pursuing an expansionary monetary policy, an incomes policy will only add to our sense of illusion that we are making permanent employment gains. On the other hand, an incomes policy may well be a useful tool when it is employed together with a restrictive monetary policy to aid in the downward adjustment from an established inflation rate. The incomes policy would then reduce the unemployment burden resulting from the transition to a lower inflation rate.
This model suggests that the Phillips curve is not dead, but that it is in fact a more dangerous device than the simple trade-off approach implies. The danger lies in the costs of reducing inflation, which may not be realized when the economy is moving to higher inflation rates. This problem is, of course, of no importance if one is not concerned about the rate of inflation. In that case there is presumably some optimal rate of increase in the rate of monetary expansion (and an optimal rate of increase of tne inflation rate) which depends on the shape of the Phillips curve, the effect of output changes on employment, and on the rate at which inflationary expectations adapt to experience. References Friedman, M., 1970, A theoretical framework for monetary analysis, Journal of Political Economy 78, March-April, 193-2.38. Friedman, M., 1971, A monetary theory of nominal income, Journal of Political Economy 79, March-April. Gordon, R.J., 1971a, Inflation in recession and recovery, Brookings Papers on Economic Activity I. Gordon, R.J., 197lb, Steady anticipated inflation: Mirage or oasis, Brookings Papers on Economic Activity Il. Gordon, W.J., 1972, Wage-price control and the shifting Phillips curve, Brookings Papers on Economic ActivitJl II. Parkin, M. and M.T. Sumner, eds., 1972, Incomes policy and inflation (Manchester University Press, Manchester). Sargent, T.S., 1972, Anticipated inlqation and the nominal rate of interest, Quarterly Journal of Economics 86, .MC%y, 212-225. Vanderkamp, J., 1968, The Phillips relation: A theoretical explanation - a comment, Economica 35, May. Vanderkamp, J., 1972, Wage adjustment, productivity and price change expectations, The Review (.>fEconomic Studies 39. Vanderkamp, J., nd., Excess demand, unemployment, vacancies and wage adjustment, Discussion paper (Department of Economics: University of Guelph, Guelph, Ont.).