Inflation, Unemployment and Unemployment Benefits

Hagen, Kurt von dem Article Inflation, Unemployment and Unemployment Benefits Zeitschrift für Wirtschaftsund Sozialwissenschaften (ZWS) - Vierteljahresschrift der Gesellschaft für Wirtschaftsund Sozialwissenschaften, Verein für Socialpolitik Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Hagen, Kurt von dem (1976) : Inflation, Unemployment and Unemployment Benefits, Zeitschrift für Wirtschaftsund Sozialwissenschaften (ZWS) - Vierteljahresschrift der Gesellschaft für Wirtschaftsund Sozialwissenschaften, Verein für Socialpolitik, ISSN 0342-1783, Duncker & Humblot, Berlin, Vol. 96, Iss. 3, pp. 235-260, https://doi.org/10.3790/schm.96.3.235 This Version is available at: https://hdl.handle.net/10419/291373 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Inflation, Unemployment and Unemployment Benefits By Kurt von dem Hagen* An unemployment compensation system is integrated into a short-run macroeconomic model of prices and employment. The dynamic properties are discussed and some qualifications are made with respect to the stabilizing effects of unemployment benefits. I. Introduction The unemployment cum inflation phenomenon or "stagflation", as politicians like to describe it, has gained much attention during the last decade from theorists and practitioners alike. Various remedies have been tried without significant success, and the gap of communication and understanding seems to widen betweens those who are in charge of the measures and those who believe they know what sohuld be done. In this paper, we shall not try to add to these recommendations, but rather point out one aspect of unemployment which seems to have been pushed aside during large parts of the discussion: the inflationary effects of unemployment benefits. Usually, the unemployment compensation system is regarded as some kind of built-in stabilizer (Musgrave, 1959, pp. 505 - 6). Authorities install income redistribution programs by which, during periods of "normal" employment, those who are employed have to support the jobless part of the labour force. If the unemployment ratio rises beyond its "normal" level, available funds fall short of the amounts needed and government steps in by extending credit to the employment agencies. These loans are to be repaid during the next recovery, when inflows exceed disbursements. It remains questionable, however, whether we really experience such countercyclical switches from deficit spending to surplus saving, and even if this were the case, no straightforward conclusions should be drawn as to the stabilizing effects on employment. An extension of * This article was written while I was at the University of Mannheim. Thanks for comments on a first draft are due to Horst Herberg, Hans Jürgen Jaksch, Michael Schmid and Ann Schwarz-Miller. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 236 Kurt von dem Hagen government transfers to households which is geared to a rise in the unemployment ratio may raise effective demand, but this does not necessarily mean that employment will be raised too. Moreover, the fact that unemployed workers are to receive transfers should encourage trade unions not to bother too much with the employment problem when bargaining over new wage rates. It is not surprising that during inflationary recessions only employers tend to stress the job situation, while unions concentrate on the income losses due to inflation. In what follows we shall set up a short run macroeconomic model of prices and employment which is dynamically stable and we shall describe the adjustment processes which may or may not be supported by employment benefit programs. From the assumptions about the wage determination it follows that, except at the price of a running inflation, no monetary policy will be able to raise employment beyond what is considered as "normal" by the bargaining groups in the labour market. It also is to be shown that, if unions disregard the job situation during recessions (because members are "insured" against unemployment), a stable unemployment ratio above the normal level may exist which is accompanied by a positive rate of inflation. II. Assumptions and Notations There are three sectors in the economy: firms, private households, and a central bank and government sector. Relative prices of goods and services produced by firms remain constant so that total real output (X) can be conceived of as one conglomerate good, which may be either consumed or invested. Total output is a function of capital (K) and labour (N) employed, and is subject to diminishing marginal returns. We shall further assume that for the time period considered labour is the only variable factor, i. e. (1) X = X (N), XN > 0, XNN < 0 .i Firms are price takers in the labour market, i. e. given the present output price (P), at which entrepreneurs expect to be able to sell their products, and given the current nominal wage rate (W), the demand for labour is determined by the equality of this money wage rate and the value of the marginal product of labour, (2) W = PXN . 1 Throughout this paper the (partial) derivate of a function f with respect to a variable x is denoted by fx. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 237 Using w for the real wage rate, W (3) w: = — , from (1) and (2) we obtain a demand function for labour which falls with a rising real wage, (4) N* = N (w), Nw< 0 . On the supply side of the labour market it is assumed that trade unions follow the concept of a "fair" real wage rate, w*, from which they deduce the demanded money wage rate (5) W = w*Pe , where Pe is the price level expected to be valid during the near future. At this wage rate supply is perfectly elastic, which means that labour demanded by firms is always equal to labour employed (6) Nd = N . If, however, there is a deviation of the amount of labour demanded (and employed) from some "natural" or "normal" level, N, unions correct their desired real wage rate according to the following differential equation:2 Dw* . N — N (7) — = X ———, X > 0 and constant. w*N From (5) and (7) we can derive a corresponding adjustment equation for the money wage rate, DW (8) —— =71 a u , where (9) and (10) 7i: = - DP* p7~ u : = N-N N 2 D is the differential operator d/dt. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 238 Kurt von dem Hagen The unemployment ratio (10), which is based on normal employment rather than on full employment of the whole labour force, is taken to be the relevant figure for union policy. Trade unions do not feel responsible for any kind of unemployment regarded as "structural" or "frictional", but if the ratio defined by (10) becomes positive, i. e. if employment is lower than regarded as normal, they would prefer to gain some more employment rather than to offset the expected rate of inflation completely. On the other hand, if the demand for labour is higher than N (u < 0), they raise the money wage rate by more than would be necessary to compensate for the expected real income losses due to rising output prices.3 Once the level of employment is determined, firms produce output according to production function (1) and pay out real income X to wage earners and shareholders. If total real sales fall short of X, unsold goods are stored. On the other hand, if excess demand for goods and services is positive, firms cut down inventories of finished products accumulated in the past.4 To keep the argument simple, we shall assume that demand is always satisfied, i. e. any excess demand for goods and services will lead to passive disinvestment of exactly that amount. The general idea is that in oligopolistic markets producers first of all try to keep profits in a certain relation to costs, i. e. they raise output prices whenever the money wage rates have gone up. Unintended investment, however, indicates that markets have not been cleared and prices should be checked and/or corrected. So any excess demand for goods together with the corresponding change in inventories should lead to a further correction of the prices. A price setting function which cointains these two elements may be represented by the following differential equation DP DW ^ (11) ——— = ———b p v, ¡x > 0 and constant,5 3 The fixed normal employment N could also be conceived of as a classical supply curve for labour which is completely inelastic with respect to the real wage rate. It should be mentioned, however, that (7) and (8) allow for a more general interpretation, if N is not taken as a target figure but simply indicates that level of employment at which the bargaining forces of employers and workers counterbalance each other, thus leaving the distribution of real income unchanged. See Stein (1974) for a similar version of this PhillipsLipsey type of wage rate adjustment. In this connection, reference should also be made th the approach taken by Michael Mussa (1975), which was brought to my attention after the present paper had been completed. 4 The concept of inventories seems a bit strange if reference is made to services produced by firms. Here one would have to think of producers who are able to switch between "routine" and "project" work, i. e. if they face a positive excess demand for their services, they delay any routine work and concentrate their employees on the projects ordered. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 239 where v is the (relative) excess of planned investment (I) and consumption (C) as explained further below over output produced: I + C-X (12) v = - . It should be noted that Walras' Law has been expelled partly from the model economy. First, we excluded the labour market by determining realized employment and, consequently, realized income. Second, the goods and services market was settled in a similar way and we had a spillover to the financial markets. We now assume that there are only two financial assets in the economy: equities or ownership of real capital (K) and fiat money (M). On the markets for these assets we shall abandon the above non-tâtonnement process and allow for any kind of recontracting until a (portfolio) stock equilibrium is reached.6 A necessary assumption for this outcome would be that financial markets are perfectly organized and immediately react to disturbances created by shifts in demand and/or supply. The demand for money in real terms may be given by a conventional liquidity preference function7 (13) L = g (r, X), gr < 0, gn < 0, gx > 0 where r is the expected real rate of return on capital (equities), n again is the expected rate of inflation (the negative rate of return on real balances), and X is the level of income, which may be taken as determining the transactions demand for money. Permanent stock (portfolio) equilibrium in the money and equities market insures that we always have (14) where M is the amount of fiat money existing in the economy. Another feature of the model is that we shall allow the valuation of existing capital goods (P/c) to differ in the short run from their reproduction cost (P), i. e. although real capital is to be conceived of as one 5 The adjustment coefficient jti is taken as a constant in (11). Usually, it should be expected to vary with the market rate of interest since passive investment enlarges the need for new funds. We shall omit this aspects for convenience. * In the standard IS-LM context this means that we are always on the LM curve, which is a bond (equity) and money market equilibrium curve. 7 Wealth does not appear as an explicit variable in (13). Hence, it must be assumed implicitly that along a moving equilibrium households would be willing to put all their savings into equities, i. e. they are not diversifiers on the margin although, according to (13), they diversify their average portfolio. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 240 Kurt von dem Hagen homogeneous good (which, for convenience, may be undestructable), the relative price PK (= PK/P) may be different from one.8 If, for example, the demand of households for real balances rises relative to the demand for equities, sales on the stock market will force share prices down and at the same time raise the imputed rate of return on these shares. Because newly created equities will also have to be served at an interest rate equal to this higher rate of return, investment projects which seemed profitable up to this point will have to be cancelled and the level of investment will go down in the next period (next instant). For the moment, i. e. until the slowing down of investment has raised the price of existing capital goods to its portfolio equilibrium level, PK will stay below one or, equivalently, the marginal efficiency of capital will be lower than the market rate of interest. There will be an inverse relation between the market valuation, PK, of existing capital and the real rate of return upon it, r, i. e. if the real marginal efficiency of capital, X/c, is expected to remain constant in the future, the market return on equities follows from the present value formula (15) r = XK/pK It should be noticed that (15) is a definition of either r or PK• Once PK is determined in the market, the value of r follows and vice versa. We assume that a difference between the two rates XR and r is the major force governing investment decisions, i. e. taking R as an exogenous variable the short run investment function is (16) I = I (r), Ir< 0 . The demand for consumption goods (C) is assumed to depend on current disposable income which, as long as government activities are not considered, is simply the factor income created during the production process: (17) C = c Y, 1 > c > 0 and constant. Finally, expectations about the rate of inflation are assumed to be changed according to past experience: 8 By assumption, financial markets are cleared at any moment in time, while this is not necessarily the case in the goods markets. Consequently, the valuation of capital should only in very special cases be the same as the output price. Cf. Brainard and Tobin (1968), Tobin (1969), and Stein (1971, 1974). Frenkel and Rodriguez (1975) arrive at essentially the same point by arguing in terms of adjustment costs. The basic reference for this second approach would be Gould (1968). OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 241 (18) D n = e nj , e > 0 and constant The above formula has two limiting cases for e 0 and e ->- oo, respectively. The first one implies D n = 0 and is often referred to as "static" expectations, because economic agents do not learn from or react to their own experience. The second one implies "perfect myopic foresight", i. e. the observed rate of inflation is expected to remain the same in the near future. For reasons, which will become clear in the next section, we shall assume that e is fairly small, i. e. people react slowly to any discrepancies between realized and expected rates of inflation. III. Stability and Adjustment Processes The model to be discussed consists of three differential equations (8), (11) and (18), one stock demand function (13), and a market clearing condition (14), which is continuously maintained by appropriate changes in the market rate of interest or, equivalently, in the price of equities. Assuming that (13) can be solved explicitly for r, we can insert (14) and derive the following equilibrium condition for the money and equities markets (19) r = G (m, n, X) , 1 gr r where (20) m : = -y- . From equations (1) - (4) and (6) it follows that output is a function of the real wage rate only, (21) X = h(w), hw = XNNw<0 . We then may insert (16), (17) and (21) into (12) and, substituting r by equation (19), rewrite the relative excess demand for goods and services, v, as a function of m, ny and w: (22) v =v (m, n,w), vm > 0, v„ >0, vw > 0 , where the signs of the partial derivatives follow directly from the functions involved. 16 Zeitschrift fiir Wirtschaftsund Sozialwissenschaften 1976/3 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 242 Kurt von dem Hagen It should be remembered that, since planned saving always equals realized saving, there are no influences on v from the consumption side. The above variables work on v only via realized output, X, and planned investment, I. Planned investment (and, consequently, u) shrinks whenever the real rate of interest rises, which is the case if either m or n (or both) decline. A decline in the real wage rate also leads to a decline in v, because output, and therefore supply, is increased while at the same time a higher demand for transaction balances will raise the required rate of return on equities, thereby reducing planned investment demand. By means of (4) and (6) we can interpret the unemployment ratio (10) as a function of the real wage rate, (23) u = u (w), uw > 0 . Next we insert (23) and (22) into equations (8) and (11) and, together with the adaptive expectations function (18), arrive at the following differential equations: (24) DW/W = 7i - Xu(w) , (25) DP/P = DW/W + fiv(m, nt w) , (26) D n = s [DP/P - n\ . At first sight one might have the impression that this system is capable of producing a steady state inflation, where DP/P = DW/W = 7i = some positive constant at full employment and zero excess demand for goods. As long as the government remains passive, however, the amount of fiat money does not change, and a positive rate of inflation will drive real balances down. Lower real balances in turn via the interest rate mechanism lead to a lower rate of investment which again slows down the rate of price inflation. In other words: regardless of the forces, which may have set an inflationary process into motion, it will peter out, if it is not fed by an appropriate monetary expansion. The foregoing argument becomes clear, if we replace (24) and (25) with two differential equations in the real wage rate and real balances, respectively. Taking account of relationships (3) and (20) we may derive from (24) - (26) (27) Dm = — m [n + p v (m,7T,w) — ku (w)] . (28) Dw = — w ¡i v (m, n, w) , (29) D N = S [¡IV (m, TI, W) — X u (W)] . OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 249 dogenous forces (as, perhaps, experienced in the past), deviations from N should not only be considered transitory but also "symmetrical" in the sense that periods of low employment (N > N) will be followed by periods of high employment (N < N), during which loans are to be refunded. To simplify matters further, it is assumed that unemployment transfers are based on potential rather than on past income, i. e. every unemployed worker is to receive a certain percentage a, of the wage income, which he would be able to earn. Consequently, total unemployment benefits, measured in real terms, will amount to a w (N — N), where N stands for full employment of the whole labour force. On the other hand, if insurance premiums are to be certain fraction, /?, of actual income, real payments to the insurance agencies will be ¡3 wN leading to a net cash flow of (34) ¡3 w N — a w (N — N) . Because of the above assumption that during "normal" employment the compensation program simply consists of a redistribution of wage income, expression (34) must vanish for N = N, i.e. a and ft have to be fixed by the government in such a way as to ensure the following balance equation: (35) pwN - ocw(N -N) = 0 . Using (35) we may substitute for (N) in (34) and rewrite the net cash flow to insurance agencies as (36) (a + p) w (N - N) , which will be positive (negative) whenever employment rises beyond (falls below) its "average" level, N. By assumption, these net cash inflows or outflows are part of the government budget. Disregarding, as before, all other government activities, we may set (36) equal to the change in the supply of money (in real terms): DM (37) (a + p) w (N - N) — . Let us further assume that government sets N equal to N, i. e. the insurance program is based on exactly that level of employment which indicates a standstill in the real wage bargaining process.11 From (10), 11 From what has been said at the beginning of this section it should be clear that unemployment benefits are not considered a policy instrument. The central authorities just try to find an average level of employment such that they will be able to stay out the insurance business in the long run. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 250 Kurt von dem Hagen (6) and (4) we may then substitute for the unemployment ratio, u (w), and rewrite (37) as DM (38) -p- = B (w) , where (39) B(w) = (a + fi)Nwu(w) , Bw > 0 . The real flow supply of money, being identical to a real income transfer from the government to private households, is an increasing function of the real wage rate, and it is positive (negative) whenever u is positive (negative). There are two channels through which unemployment benefits affect excess demand. They are represented by the left and right hand side of (39), respectively. First, a change in the money supply influences the demand for commodities via the interest rate mechanism. Second, government transfers are part of disposable income and therefore affect demand via the consumption function. Let v indicate, as before, the relative excess demand for goods and services, when government remains absent. The general expression for excess demand, ~ i + c(X + B)-X (40) v = may then be written (41) v = v + c b , where B(w) (42) b = and bw > 0 . It should be understood that the relevant figure for price decisions of firms is now given by v instead of v, i. e. instead of differential equation (11) we have DP DW (43) = — + [xlv + cb] . From (43) and (38) together with (24) and (266 we may derive the following system of differential equations: (44) Dw = — w [a, (v + c b) , (45) D n = s [u (v + c b) - X u] , (46) Dm = B — m [n — X u + p (v + c b)7 . OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 251 The stationary solution to these equations is the same as that of (27) - (29). From Dw = 0 = D it it follows that u would have to be zero, and if u vanishes, so will both B and b, and therefore, the difference between systems (27) - (29) and (44) - (46). As to the dynamic behaviour of the economy, we shall again disregard all but small disturbances and linearize around the stationary solution: (47) Dw D n Dm - w* ¿i(vw + c bw) -w*/*vn £ {M (vw + cbw) - Xuw } eiivn £f*vm Bw - m* {ju (vw + cbw) - luw} - m* (1 + ¡i vn) - m* juvn w — w* n Til — 771* Since (47) and (30) are both evaluated at the same point in the phase space, the values of partial derivatives must be the same. The only difference then between the two systems consists of bw and Bw terms in the first colums of (47). It is easily shown that conditions (31) and (32), while still being sufficient, are no longer necessary for stability, i. e. after introduction of unemployment benefits weaker restrictions would suffice.12 The unemployment compensation program outlined above thus may be termed "stabilizing" in the general sense that it enlarges the set of parameter distributions, which ensure dynamic stability of the system. This does not necessarily mean, however, that the adjustment process will be speeded up. If Xi and (i = 1, 2, 3) are the roots of characteristic equations corresponding to (30) and (47), respectively, the following results may be obtained: (48) | X! + x9 + | < | y, + Vq + V3 I and (49) X! = yjy2y.3 , i. e. at least one but at most two of the terms on the right hand side will be greater in absolute value than the corresponding x% roots. Therefore, it cannot be stated that unemployment benefits always increase stability in the sense that a certain neighbourhood around the equilibrium point will be reached in shorter time. 12 This and the following results are obtained in Appendix 2. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 252 Kurt von dem Hagen So far, we have described the economy by means of m, w and n. In terms of these variables, stationary solutions to (27) - (29) and (44) - (46) are identical, and disequilibrium dynamics, as given by (30) and (47), do not exhibit characteristic differences. However, there remains one important modification with respect to the money supply. While M was fixed in the model of section III, in the present framework it is being changed throughout the adjustment process. As a result, it is not possible to determine the final price level in the economy without first having evaluated the change of M along the time path. A graphic exposition may be given by means of standard macroeconomic "demand" and "supply" schedules. In Figure 3, all curves are drawn under the assumption of zero inflationary expectations. Let A represent an initial equilibrium identical to the starting point of Figure 2, and let the autonomous shift in the demand for goods be of the same size in both diagrams. We then may identify two conceptual differences. First, the "demand" curves in Figure 3, being calculated from v + cb = 0 instead of v = 0, are flatter than the corresponding ones in Figure 2, while each respective pair has one point in common at X = X* (i. e. b = 0). Accordingly, a short run "Keynesian" solution under a fixed money supply would represent both a higher output and price level than in the case of Figure 2. (For comparison, demand curves from Figure 2 repeated and appear as dotted lines in Figure 3.) Second, in the present model the pull which carries income away from X* will create a government surplus and thus, via a decline in M, shift the demand schedule downwards. Without further quantitative specification it cannot be told whether this shift fully or partly compensates the upward move in the demand curve due to rising inflationary expectations. As long as the time path shows a cyclical pattern, there will be at least one period of unemployment accompanied by monetary expansion and a corresponding (partial) upward shift in the demand curve. In Figure 3, the overall net effect on M is assumed to be negative, i. e. the final equilibrium point, D, represents lower nominal money balances than at the outset and, consequently, a lower output price level.13 13 Taking into account the assumption about stability and linearity as well as the direction of the initial demand shift, the above solution seems most plausible. However, a rigorous proof has not yet been carried out. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 253 P Figur 3 V. Unemployment plus Inflation In this last section consider the case where labour unions are reluctant to accept any wage rates that would lower their members' real income, i. e. they restrict the wage bargain by (50) DW/W > 7i . There may be times during which unions are neither willing nor able to follow the rule given by (50). The aftermath of the oil crisis provides a fairly good example of their not being certain as to which wage raise they should demand (and which might be feasible). During the course of a normal trade cycle, however, restriction (50) seems quite plausible OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 254 Kurt von dem Hagên with respect both to the trade unions' target of defending what they have achieved in the past and the general asymmetry of the bargaining process. In terms of the wage adjustment equation (24) it follows from (50) that A should become zero whenever employment falls below Nf i. e. (51) > if L>J The model now consists of two distinct parts. Up to the full employment real wage rate, which may be termed w°, equations (44) - (46) are still effective while for real wage rates higher than w° the coefficient X would vanish: (52) Dw = — w jit (v + cb) , (53) D jz = en (u + cb) , (54) Dm = B — m [JLI (v + c b) + n] . At the full employment wage rate, where u, B and b all become zero, both systems of differential equations coincide, and we shall argue that w = w° as a limiting case is contained in either part of the model. As to equations (44) - (46), we know from the last section that a stationary solution would require stable prices as well as full employment, i. e. the equilibrium point would lie on the common border just mentioned. A solution of (52) - (54) would require zero excess demand for goods (55) v (m, TT,W) + cb (to) = 0 and (56) B (w) - m n = 0 . These equations hold not only if both B and n vanish. If employment is normal and prices are stable14, they are also compatible with a whole array of pairs of positive B and n15 or, less formally, with many different combinations of unemployment and inflation. Let us inspect system (52) - (54) more closely. From the first two of these equations we may derive (57) Dw/w = — D JZ / E . 14 A zero inflation rate would follow from D = 0 = n. 15 Since (54) - (56) are only valid for w > to0, net transfers B cannot be negative, and the same follows for n in (56), since m should be positive for obvious reasons. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 255 After integrating, this becomes (58) Inw = - n/ s + A , where A is a constant. In that part of the phase space, which is governed by (52)-(54), variables w and n obviously have to stay in a certain relation. Moreover, time does not explicitly appear in (58), so the relation must be entirely determined by the values of w and n which happen to be realized when the phase path "crosses the border", i. e. when the unemployment ratio becomes positive. The reason for this is quite simple. From the definition of the real wage rate it follows that Dw = w [DW/W - DP/P] , while the adaptive expectations rule is D 7i — e [DP/P - n] . Both of the bracketed terms above are equal in absolute value if DW/W = Jt , i. e. if the change in the money wage rate is equal to inflationary expectations, as assumed in (51) for employment levels lower than N. As to the determination of the constant of integration in (58), once the value of n is known (call it n0) at which the phase path goes through the full employment coordinate, w = w°, it follows from (58) that InwO = _ n**/e + A and, consequently, for all values of w > w°: (59) n = — [In w — In tu°7 . Because of the negative slope of (59) the initial value JI° is the maximum rate of inflation that may be realized under system (52) - (54). On the other hand, equation (56) requires stationary values of n to be nonnegative. Therefore, if happens to be negative, an equilibrium solution for (52) - (54) does not exist within the range w° < w <C oo. This can be demonstrated by means of Figure 4, which is a projection of parallel cuts through the phase space at different levels of m. Equilibrium condition (55) is represented16 by the set of lines running from the w° coordinate to the southeast, while (56) is a bunch of rays from a common origin at w = w° and n = 0. Note that the w° coordinate and the abszissa itself are the limiting cases for m 0 and m oo, 16 For convenience, both (55) and (56) are linearized. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 256 Kurt von dem Hagen respectively. Intersections of corresponding pairs of these curves are connected by the dotted line, which starts from w = w° and n = 0 with a positive slope and later bends back towards the w° axis. Points on this curve are candidates for a stationary equilibrium with both unemployment and inflation. Finally, two graphs of equation (59) are plotted down in Figure 4, each starting at a different initial level of n. If = n\ > 0, a unique equilibrium is given by intersection point E, while in the second case, where = n\ < 0, no stationary solution will exist. This last outcome (as well as the uniqueness of the above equilibrium) is mainly due to the OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 Inflation, Unemployment and Unemployment Benefits 257 negative slope of (59). At present, however, we shall not discuss this phenomenon any further and shall confine ourselves to the statement that the partitioned system as given by (44) - (46) and (52) - (54) has at least one and at most two stationary solutions, where the first is always the equilibrium developed in previous sections. Stability conditions for the partial system (52) - (54) are given in Appendix 3. Assuming that they are met we still cannot say very much about the actual time path. If the economy starts at a high employment level (i. e. N> N) and follows a cyclical path towards the stationary equilibrium, unemployment will become positive at some point in time and thus set behavioural system (52) - (54) into effect. If on the common boundary inflationary expectations are positive (i. e. = 0), there may be a stable path towards a stationary solution characterized by unemployment cum inflation, but if this part of the model also produces cycles, the time path might swing back to the high employment region, and the above process would start all over anew, with a different value of n0 and a different unemployment equilibrium. These remarks may suffice for the present until an analysis of joint stability of the two partial systems has been carried through. What seems remarkable at this point is the tendency towards a path of selfsustained inflation and unemployment. Of course, no government would tolerate such a constellation in the long run; new policy tools might be activated and, more important, the old tools would have to be checked and corrected. In the present model, the principal source of inflation is the growth in the money supply, which could be stopped by an appropriate raising of unemployment insurance premiums (as it was done in Germany at the beginning of 1976, when premiums jumped by fifty percent). However, it does not become clear from the model at which employment level authorities should fix N. If the economy has reached a stable position in the unemployment region, a decline of N to the prevailing level of employment brings the growth of money down to zero immediately, but a bit later the fact that N exceeds N may create new difficulties of its own kind. The problem does not seem little enough to justify its being neglected. At least the traditional fiction of the stabilizing forces of unemployment benefits should deserve a more critical interpretation. 17 Zeitschrift fur Wirtschaftsund Sozialwissenschaften 1976/3 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36 258 Kurt von dem Hagen Appendix A.l Stability of (30) If the characteristic equation of the linearized system (30) is written as (A.l) x3 + ax x2 + ag x + a3 = 0 , the Routh-Hurwitz theorem states that necessary and sufficient conditions for all roots Xj (i = 1, 2,3) to have negative real parts are (A.2) <n > 0 , (A.3) ata2as>0 , (A.4) a3 > 0 , where c^ is the negative of the trace of the system matrix in (30), 0$ is the sum of all second order principal minors, and is the negative of the determinant. Inequality (A.2) is the same as (31) in the text and is not repeated here. Expansion of the determinant gives us (A.5) ctg = e I [a, m* w* uw vm > 0 , i. e. condition (A.4) will be fulfilled. The sum of the second order principal minors is (A.6) a-2 = ^ [m* vm (s + X w* uw) — s X w* uw vj . This expression is not necessarily positive, but it must be if (A.3) holds. Inserting the proper expressions for alf a.2 and a3 in (A.3) we get (A.7) (a! - E) I m* w* uw vm > a^ e (k w* uw vn - m* vm) , which is the equivalent of condition (32). A.2 Stability of (47) Write the characteristic equation of (47) as y3 + bty2 + b2y + b3 = 0 and obtain the following results: bt = ju [w* (vw + c bW) - E v„ + m* vm] (A.8) = at + /¿w* cbw> ai , (A.9) b2 = 02 + Bw w* ii vm > 02 , (A. 10) b3 = og . If ax and a3 are positive, so will be b^ and b3. OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/schm.96.3.235 | Generated on 2023-04-04 11:52:36