The Importance of the Retention Ratio in a Kaleckian Model. with Debt Accumulation

Transcription

1 Kyoto University, Graduate Shool o Eonomis Researh Projet Center Disussion Paper Series The Importane o the Retention Ratio in a Kalekian Model with Debt Aumulation Hiroaki Sasaki and Shinya Fujita Disussion Paper No. E Researh Projet Center Graduate Shool o Eonomis Kyoto University Yoshida-Hommahi, Sakyo-ku Kyoto City, , Japan November 2010

2 The Importane o the Retention Ratio in a Kalekian Model with Debt Aumulation Hiroaki SASAKI Ɨ and Shinya FUJITA Abstrat Hein (2007) investigates the eets o debt o irms and interest rates on output and growth, and obtains the ollowing three results. First, the long-run equilibrium is stable only i the short-run equilibrium is debt-led growth. Seond, to obtain a positive long-run equilibrium value o the debt-apital ratio under the debt-led growth regime, extremely high interest rates are neessary. Third, the long-run equilibrium value o the rate o apital aumulation is inreasing in interest rates and is independent o inome distribution. However, these onlusions depend ruially on the assumption that the retention ratio o irms is equal to unity, and hene, that there is no dividend to shareholders. By relaxing the above assumption, we show in this paper that even the long-run equilibrium under the debt-burdened growth regime an be stable and that the long-run equilibrium value o the debt-apital ratio will be positive with plausible interest rates irrespetive o whether the long-run equilibrium is debt-led growth or debt-burdened growth. Moreover, the eets o interest rates and inome distribution on apital aumulation dier rom regime to regime. 1 Introdution While many post-keynesian researhers debate the relationship between inome distribution and eonomi growth, some Kalekian researhers have investigated the link between debt aumulation and eonomi growth. The eonomy is termed debt-led i inreases in the debt-apital ratio and the interest rate raise the rate o apaity utilization and/or the rate o apital aumulation. The eonomy is orrespondingly termed debt-burdened i inreases in the debt-apital ratio and the Ɨ Graduate Shool o Eonomis, Kyoto University. sasaki@eon.kyoto-u.a.jp Graduate Shool o Eonomis, Nagoya University. ujita@soe.nagoya-u.a.jp 1

3 interest rate lower the rate o apaity utilization and/or the rate o apital aumulation. Models that onsider the eet o debt-aumulation on eonomi growth inlude Taylor (2004, h. 8), Lima and Meirelles (2007), and Charles (2008a, 2008b). 1 Taylor (2004, h. 8) introdues the eet o interest payments into the investment untion and shows the possibility o debt-led and debt-burdened regimes. In addition, he shows that the eet o isal expenditure depends on the regime. Both Lima and Meirelles (2007) and Charles (2008a, 2008b) introdue a Minskyan taxonomy with regard to the inanial ragility o the eonomy into Kalekian models that onsider debt-aumulation. However, Lima and Meirelles s model assumes that the savings rate o apitalists is equal to the retention ratio o irms. Aordingly, their model eonomy is neither debt-led nor debt-burdened. Charles s model assumes that irms investment depends only on their retained proits, whereas in the usual Kalekian models, investment depends on the rate o apaity utilization and on interest payments. Consequently, apital aumulation in Charles s model neessarily leads to debt-burdened. Lavoie (1995) and Hein (2007) an be haraterized as benhmark models o the monetary theory o prodution in that they learly investigate the eets o monetary variables suh as interest rates and the debt-apital ratio on real variables suh as apaity utilization and apital aumulation. While Lavoie (1995) assumes that investment is exogenously given, Hein (2007) onsiders a Kalekian investment untion with a negative eet o interest payments, whih is also adopted in Taylor (2004). In the short run o these models, disequilibrium o the goods market is adjusted through hanges in apaity utilization with the debt-apital ratio remaining onstant. 1 The ollowing are Kalekian models that onsider inanial ators. Asada (2006) presents a kind o Kalekian model that onsiders debt-aumulation and analytially shows the existene o ylial lutuations. Charles (2010) presents a Kalekian model in whih irms propensity to invest hanges endogenously and shows the ourrene o haoti dynamis in a relatively simple ramework. However, his result depends ruially on the assumption that time is disrete. Hein (2010) presents a model in whih the outside inane-apital ratio is an endogenous variable; the undamental struture o the model is the same as that o Hein (2007). For Hein (2010), see also the empirial study o Hein and van Treek (2010). Setterield (2009) introdues a monetary poliy termed the Pasinetti rule and investigates the eet o the poliy. 2

4 Lavoie and Hein analyze how inreases in the debt-apital ratio and interest rates aet the rates o apaity utilization and apital aumulation. In the long run, the goods market always lears and the debt-apital ratio beomes an endogenous variable. Hein (2007) inds that, i the short-run equilibrium rate o apital aumulation is debt-led, the long-run dynamis o the debt-apital ratio are always stable. In ontrast, i the short-run equilibrium rate o apital aumulation is debt-burdened, the long-run dynamis are always unstable. However, Hein s result that the long-run dynamis o the debt-apital ratio are stable only i the short-run equilibrium is debt-led growth is questionable rom an empirial standpoint. For example, Hein and Shoder (2009) empirially study the eet o the debt-apital ratio on the rate o apaity utilization in the US and Germany, and show that both ountries are debt-burdened. It is implausible that the debt-apital ratio o these debt-burdened eonomies diverges to ininity. In addition, in Hein (2007), extremely high interest rates are needed to obtain a positive long-run equilibrium value o the debt-apital ratio under the stable debt-led growth regime. Hein and Shoder (2009) estimate parameters o the investment untion, rom whih we an dedue that unless interest rates are kept at 8% in the US and 20% in Germany, the equilibrium debt-apital ratio o Hein (2007) will be negative. Moreover, in Hein (2007), the long-run equilibrium rate o apital aumulation leads to the produt o the savings rate o inanial apitalists and the rate o interest. From this, it ollows that the paradox o thrit does not hold and that an inrease in the rate o interest leads to a rise in the rate o apital aumulation, whih ontradits the usual Keynesian intuition. In this paper, we show that Hein s results depend ruially on the assumption that the retention ratio o irms is equal to unity, and hene, that there are no dividend payments to shareholders. Indeed, i this assumption is relaxed that is, i the retention ratio o irms is less than unity the long-run equilibrium will be stable even i the short-run equilibrium is debt-burdened. In addition, under this relaxed assumption, the long-run equilibrium value o the debt-apital ratio will be positive with plausible 3

5 interest rates irrespetive o whether the eonomy is debt-led or debt-burdened growth. Moreover, the eets o interest rates and inome distribution and interest rates on apital aumulation dier rom regime to regime. The remainder o the paper is organized as ollows. Setion 2 briely explains the struture o Hein s (2007) model and shows its drawbaks. Setion 3 presents our modiied model, in whih the retention ratio o irms is stritly less than unity and shows some analytial results. Setion 4 presents some numerial examples to asertain the results o the previous setion and to lariy those aspets that annot be known analytially. Setion 5 onludes the paper. 2 The basi struture o Hein s (2007) model This setion surveys the basi struture o Hein s (2007) model and shows some speial results that arise rom the assumption that the retention ratio is equal to unity. We assume a losed eonomy without government. There exists a single good, whih an be used or both prodution and onsumption. Tehnologial progress is not expliitly onsidered: both the potential output-apital ratio and the output-labor ratio are assumed to be onstant. There are three agents in our eonomy: irms with exess apaity, workers, and inanial apitalists. Firms invest without issuing new shares, using a part o proits and external unds that are inaned by apitalists via banks, whih in turn set the nominal loan rate (interest rate). The proit rate (r ) is deined as the nominal proit (Π ) divided by the prie level ( p ) and the existing apital stok ( K ). r Π Π Y Y h = u pk py Y K = v, ( 0,1) h, (1) where Y denotes real output and Y denotes potential output. The rate o proit is omposed o three ators: the proit share ( h ), the rate o apaity utilization ( u = Y / Y ), and the potential output-apital ratio ( v = Y / K ). 2 In the ollowing 2 Hein (2007) onsiders the ase where the mark-up rate in the prie equation elastially hanges aording to the interest rate. However, we abstrat rom this ase beause suh an equation does not hange our results very muh. 4

6 disussion, we assume v = 1 or the sake o simpliity. Aording to the post-keynesian horizontalist view (Moore (1988), Rohon (1999), and Fontana (2009)), we assume that irms an make use o loans, whih are inaned by apitalists, under a given nominal interest rate (i ). Beause irms hold the nominal sok o debt ( L ), they have to pay interest (il ) with a dividend to apitalists in eah period. Thereore, irms nominal proits are split into their retained proits ( Π ) and the apitalists inome ( Π ), whih is omposed o dividend and interest reeipts. Π pk = s Π pk i L pk = s ( hu i), ( 0,1) s, i > 0, (2) Π pk = ( s )( hu i) + i 1, (3) where ( L / pk ) s denotes the retention ratio, whih is assumed to be onstant; and denotes the debt-apital ratio. The irst term o the RHS in Equation (3) represents dividends, and the seond term represents interest inome. We assume that apitalists save a onstant ration ( s ) o their inome, while workers do not save. Total saving ( S ) is omposed o retained proits and savings rom apitalists inomes. Using Equations (2) and (3), we obtain the aggregate saving untion. S K [ ] = s ( hu i) + s ( 1 s )( hu i) i, ( 0,1) + s. (4) Firms make an investment plan ( I ), whih is given by the ollowing desired investment untion: I g = α + βu + τh θi, α > 0, β > 0, τ > 0, θ > 0. (5) K Equation (5) shows that the rate o apital aumulation ( g ) responds positively to the proit share and the rate o apaity utilization (Bhaduri and Marglin (1990)) and negatively to interest payments per existing apital, i (Taylor (2004), and Taylor 5

7 and Arnim (2008)) Short-run equilibrium In the short run, disequilibrium between investment and saving is adjusted through hanges in apaity utilization under given apital and debt stoks. Hein (2007) assumes that s = 1 or the sake o simpliity: irms do not pay a dividend to apitalists. To see what happens in this ase, this setion aepts the assumption that s = 1. The equilibrium rate o apaity utilization is obtained rom Equations (4) and (5). ( 1 s θ ) α + τh + u = h β i. (6) This equilibrium will be stable i saving responds more elastially to variations in apaity utilization than investment, that is, i h β > 0, whih we assume in the ollowing disussion. I the reation oeiient o investment to interest payments and apitalists propensity to save are suiiently small, that is, i 1 s θ > 0, inreases in the debt-apital ratio and the interest rate raise the rate o apaity utilization. This ase is alled the debt-led apaity utilization (DLCU, hereater) regime in the reent Kalekian literature. I 1 s θ < 0, then inreases in the debt-apital ratio and the interest rate redue the rate o apital aumulation; this ase is alled the debt-burdened apaity utilization (DBCU, hereater) regime. Substituting Equation (6) in Equation (5), we get the rate o apital aumulation in the short-run equilibrium. h g = ( α + τh) + [ β ( 1 s ) θh] i h β. (7) 3 Post-Keynesian models eature many variants o the desired investment untion. Hein (2006) uses a normal Kalekian-type investment untion that positively relates investment per apital to the proit rate and the rate o apaity utilization. Charles (2008a, 2008b, 2008) assumes that investment is an inreasing untion o retained proits. Under this assumption, apital aumulation is debt-burdened in the short-run equilibrium. 6

8 Whether apital aumulation is debt-led or debt-burdened depends on the proit share and the reation oeiient o investment to the rate o apaity utilization, in addition to the oeiient o interest payment and to the apitalists savings propensity. I β ( 1 ) θh > 0 s, then the short-run equilibrium is debt-led growth (DLG, hereater). I, on the other hand, β ( 1 ) θh < 0 s, then the short-run equilibrium is debt-burdened growth (DBG, hereater). As we have seen, the ondition or DLCU (DBCU) diers rom the ondition or DLG (DBG). Note that DBCU is not ompatible with DLG. 4 Thereore, in Hein (2007), there exist the ollowing three ases or the debt regimes. Case (a): the short-run equilibrium is both DLCU and DLG. Case (b): the short-run equilibrium is both DBCU and DBG. Case (): the short-run equilibrium is both DLCU and DBG. 2.2 Long-run equilibrium In the long run, the debt-apital ratio beomes an endogenous variable, while the goods market always lears. The growth rate o the debt-apital ratio is given by L L = g = g L, (8) pk where inlation is assumed to be zero. Firms loans in eah period are equal to the dierene between total investment and their internal proits. Sine the goods market always lears, loans are inally idential with the savings rom apitalists inomes. 5 4 The ondition or DBCU is given by x 1 θ < 0. s The ondition or DLG is given by y β ( 1 s ) θh > 0. Using these two equations, we obtain y β ( x + θ ) θh = βx + θ ( β h). Here, x < 0 and β h < 0 imply y < 0. Thereore, DBCU neessarily leads to DBG. 5 I the agents who save out o inome are not apitalists but workers, irms loans are 7

9 L pk = I K ( hu i) = s i. (9) Substituting Equation (9) in Equation (8) yields = ( i g). (10) s The long-run equilibrium debt-apital ratio ( ) is derived rom = 0 and rom Equations (7) and (10). si = ( h β ) h( α + τh) i[ β ( 1 s ) θh]. (11) Here, we abstrat a trivial solution, = 0, beause it is eonomially meaningless. In addition, empirial studies by Taylor and Arnim (2008) and Hein and Shoder (2009) show that the debt-apital ratio is positive in advaned apitalisti ountries; hene, we assume that > 0. satisied. The long-run equilibrium will be loally stable i the ollowing ondition is g = = [ β ( s ) θh] i 1 h β < 0. (12) Beause h β > 0 and > 0, the neessary and suiient ondition or loal stability is given by β ( 1 ) θh > 0 s ondition or DLG. Thereore, we obtain the ollowing result.. Note that this ondition oinides with the Result 1. (Hein, 2007) Suppose that the long-run equilibrium debt-apital ratio is positive. When s = 1, the long-run equilibrium is stable (unstable) i the short-run equilibrium is DLG (DBG). Now, onsider the ase where the long-run equilibrium is loally stable, in other words, where the short-run equilibrium is DLG (Case (a)). For the positive equilibrium debt-apital ratio, the numerator o the RHS in Equation (11) must be positive (that is, s i ( h β ) h( α + τh) > 0 ) beause its denominator is also positive. This yields equivalent to workers savings (Dutt and Amadeo (1993)). 8

10 i > h( α + τh) /[ s ( h β )]. Thereore, we obtain the ollowing result. Result 2. (Hein, 2007) To obtain a positive debt-apital ratio in the long-run equilibrium, extremely high interest rates are needed. For example, suppose that α = 0, s = 1, and h = 0. 3, and substitute them in the inequality. Then, the ondition o > 0 leads to i > 0.09τ /( 0. 3 β ). On the basis o the empirial study o Hein and Shoder (2009) that inds that β = and τ = 0.14 in the US and that β = and τ = in Germany, a positive requires i > or the US and i > or Germany. Furthermore, as s gets smaller or α gets larger, a positive β requires higher interest rates. Again, onsider the ase or the stable equilibrium. Provided that ( 1 ) θh > 0 s debt-apital ratio is given by, the eet o the interest rate on the long-run equilibrium = h( α + τh) [ β ( 1 s ) h] 2 i i θ > 0. (13) Equation (13) ontradits the statement o Hein (2007, pp ) that an inrease in the interest rate an either raise or redue the debt-apital ratio. By substituting Equation (11) in Equation (6), we obtain the long-run equilibrium rate o the apaity utilization ( u ). Dierentiating it with respet to i yields u i = s ( 1 s θ ) [ β ( 1 s ) θh]. (14) The denominator o the RHS o Equation (14) is positive beause o the loal stability ondition. Moreover, based on Footnote 4, its denominator is negative when its numerator is negative, whih ontradits the stability ondition. Thereore, both the numerator and denominator must be positive, 6 whih leads to the result that an inrease in the interest rate raises the rate o apaity utilization in the long run. 6 As mentioned above, the ondition or the positive numerator o the RHS o Equation (14), that is, 1 s θ > 0, implies that the short-run equilibrium is DLCU. As a result, the stable long-run equilibrium an be ompatible only with Case (a): the short-run equilibrium is both DLCU and DLG. 9

11 Finally, we obtain the long-run equilibrium rate o apital aumulation From this, we obtain the ollowing result. g = s i. Result 3. (Hein, 2007) A rise in the interest rate inreases the rate o apital aumulation irrespetive o the debt regime. This result seems strange rom the Keynesian perspetive that holds that lower interest rates produe higher growth. 7 Furthermore, apital aumulation in the long run is independent o the proit share, whih is ontrary to the typial Kalekian proposition onerning the relation between growth and distribution. 3 Modiied model: the ase in whih dividends are distributed In this setion, we explain how some analytial results o Hein (2007) are modiied when we take into aount the ase where irms distribute dividends to apitalists. Assuming s < 1 instead o s = 1 and using Equations (4) and (5), we obtain the short-run equilibrium rate o apaity utilization. α + τh + [ s (1 s ) θ ] i u =, σ s + s ( 1 s ) > 0. (15) σh β Hereater, we onsider only the ase where the short-run equilibrium is stable, whih is equivalent to supposing that σh β > 0. We immediately ind that the short-run equilibrium is DLCU i s ( 1 ) θ > 0, while it is DBCU i ( 1 ) θ < 0 s The short-run apital aumulation is given by σh( α + τh) [ βs g = A + B, A > 0, B σh β The short-run equilibrium is DLG i s ( 1 s ) σθh > 0 ( 1 s ) σθh < 0 β s ( B < 0 ). s. s (1 s ) σθh i. (16) σh β ] β ( B > 0 ) and DBG i We also ind, using the same method as in Footnote 4, that the ondition or DBCU ontradits the ondition or DLG. Thereore, as in Hein (2007), there exist 7 In addition, g = si denies the paradox o thrit that implies that an inrease in the savings propensity o apitalists leads to a derease in the rate o apital aumulation. 10

12 three ases o debt regimes. Case 1: the short-run equilibrium is both DLCU and DLG. 8 Case 2: the short-run equilibrium is both DBCU and DBG. Case 3: the short-run equilibrium is both DLCU and DBG. Firms loans are given by the ollowing equation. L I = s 1 pk K [ i] ( hu i) = s ( s )( hu i) +. (17) Substituting Equations (16) and (17) in Equation (8) and rearranging, we obtain the dynamial equation o the debt-apital ratio or the ase where the retention ratio is below unity. G ( ; i) s (1 s = σ s ss = 1 σ g + i σ. (18) ) A s (1 s ) B ss i 2 + A + B σ σ Needless to say, substituting s = 1 in Equation (18) yields Equation (10): our modiied model boils down to Hein s (2007) model. 3.1 Stability analysis in the long run: The DLG ase We here assume that B > 0. This orresponds to Case 1: the short-run equilibrium is both DLCU and DLG. Then, the RHS o Equation (18) shows a parabola with its vertex oriented upwards in the (,) plane. The inlexion axis o the urve is given by s ( 1 s ) B σa + ss i =. (19) 2σB I we assume that s = 1, the parabola passes through the origin o the plane, whih is identiied with Hein (2007). Otherwise, an interept o the parabola is inevitably positive. Figure 1 shows the ase where 0 s < 1 and > 0; this ase orresponds to < 8 Case 1 is divided into Case 1-1 and Case 1-2 in the ollowing analysis. 11

13 Case 1-1 o Table 1 in the next setion. Figure 2 shows the ase where 0 s < 1 and < 0 ; this ase implies Case 1-2 o Table 1. Figure 3 shows the speial ase where s = 1 and > 0. 9 From Figures 1 and 2, we an immediately understand that when the retention rate is lower than unity, Equation (18) always intersets the horizontal axis one within > 0 regardless o the sign o the inlexion axis. This result is important beause the equilibrium debt-apital ratio is always positive under the DLG regime without any partiular ondition or the interest rate, whih we impose in Setion 2.2. One again, the inlexion axis must be positive in the ase o s = 1 or the positive equilibrium value (Figure 3). In other words, the interest rate must be high enough to satisy i( h ) h( α + τh) > 0 s β. < Proposition 1. The long-run equilibrium is stable i the short-run equilibrium is DLG. Moreover, to obtain a positive debt-apital ratio in the long-run equilibrium, extremely high interest rates are not needed. We represent the long-run equilibrium debt-apital ratio ( ) in Figure 1. Even i an external shok moves the debt-apital ratio to the point 0 on the horizontal axis, this ratio ontinues to derease in the range o < 0 until it onverges to the equilibrium. In ontrast, even i the debt-apital ratio happens to be below its equilibrium, this ratio ontinues to inrease beause o > 0 and inally onverges to the equilibrium. This reasoning explains why a unique positive equilibrium under the debt regime is always loally stable. (Figures 1, 2, and 3 to be inserted here) 3.2 Stability analysis in the long run: the DBG ase Next, we onsider the ase where B < 0, whih means that the short-run equilibrium 9 We exlude the ase where s = 1 and < 0 beause this ase does not have a positive equilibrium value. For the same reason, we also abstrat this ase under the DBG regime. 12

14 is DBG (either Case 2 or 3). In this ase, the RHS o Equation (18) depits a parabola with its vertex oriented downwards in the (,) plane. The neessary and suiient ondition or the existene o the positive equilibrium value, whih is equivalent to the ondition that the urve rosses the horizontal axis, is that the disriminant o ( ; i) = 0 G is positive. However, it is diiult to speiy this ondition beause our 0 < s < 1 assumption requires very ompliated alulations. Here, we assume that the disriminant is positive. 10 Figure 4 shows the ase where 0 s < 1 and > 0 ; Figure 4 orresponds to < either Case 2 or Case 3 in Table 1. Figure 5 shows the ase where 0 s < 1 and < 0, while Figure 6 shows the speial ase where s = 1 and > 0. There exists at least one positive equilibrium debt-apital ratio i > 0 <, that is, s ( 1 s ) B σ A + ss i < 0 is satisied (Figures 4 and 6). However, in ontrast to Hein (2007), ulilling this inequality does not require a higher interest rate. This inequality will be satisied i parameters α, τ, and h are suiiently large. Proposition 2. The long-run equilibrium an be stable even i the short-run equilibrium is DBG. Moreover, to obtain a positive debt-apital ratio in the long-run equilibrium, extremely high interest rates are not needed. Figure 4 shows that the ase where 0 s < 1 and > 0 has multiple equilibria ( 1, 2 ). The smaller equilibrium, < 1, is loally stable, while the larger one, 2, is loally unstable. One more, there exists a unique positive but unstable equilibrium in the ase where s = 1, as we mentioned in Setion (Figures 4, 5, and 6 to be inserted here) 3.3 The eet o the interest rate on apital aumulation in the long run Although our modiied model is somewhat ompliated, it is possible to investigate the 10 Numerial examples in the next setion show the existene o the positive equilibrium under plausible parameter settings. 13

15 eet o the interest rate on the long-run equilibrium. In the ollowing disussion, we onsider only the stable and positive equilibrium. The equilibrium in the long run is deined by = G( ; i) = 0 ( i) G, and its stability ondition is given by ) B ss i A + 2B < 0. (20) σ ; = = s (1 s σ Note that the stable equilibrium always satisies Equation (20). Totally dierentiating ( ; i) = 0 G yields d di G i ( ; i) G( ; i) / =, (21) is given by where the denominator is negative and G( d ; i) / i G( ; i) [ s ( 1 s ) σ ][ βs ( 1 s ) σθh] + ss ( σh β ) i = σ σh β. (22) Using g and, we an rewrite Equation (18) with = 0 as ollows: g ss i = σ s ( 1 s ). (23) Equation (23) implies that the positive equilibrium values or the debt-apital ratio and σ s s or the rate o apital aumulation impose the ondition that ( 1 ) 0 >. We suppose a situation where this inequality is always satisied. Then, i the short-run equilibrium is DLG (Cases 1-1 and 1-2), the sign o G( ; i) / i beause βs ( 1 s ) h > 0 is ambiguous σθ, whih leads to the ambiguous sign o d / di. However, i we onsider the situation where the short-run equilibrium is DBG (Cases 2 and 3), we obtain ( ; i) / i > 0 G beause βs ( 1 s ) h < 0 σθ, whih leads to d / di > 0. Furthermore, dierentiating the long-run equilibrium rate o apital aumulation g + = A B with respet to the interest rate yields d i [ βs (1 s ) σθh d = + i. (24) σh β di ] Proposition 3. A rise in the interest rate dereases the long-run equilibrium rate o apital aumulation i the short-run equilibrium is DBG. 14

16 In the ase o βs ( 1 s ) h < 0 σθ, we obtain / di < 0 dg beause d / di > 0. This result is ontrary to the result o Setion 2.2. However, in the ase o ( 1 s ) h > 0 βs σθ, this eet is not lear beause the sign o d / di is ambiguous. 4 Numerial examples This setion presents numerial examples to show that the eonomially meaningul long-run equilibrium atually exists under plausible parameter settings and that the results o omparative statis analysis dier depending on the irumstanes. In this setion, we investigate how inreases in the interest rate and the proit share aet the long-run equilibrium values o the debt-apital ratio, the rate o apital aumulation, and the rate o proit. Reall that in Hein (2007), the long-run equilibrium value o apital aumulation is given by g = s i. Thereore, an inrease in the interest rate neessarily inreases the rate o apital aumulation. In addition, an inrease in the proit share never aets the rate o apital aumulation. However, in our model, as will be shown below, an inrease in the proit share raises the rate o apital aumulation in some ases and lowers it in others. From the above analysis, we know that there exist situations where the short-run equilibrium is DLCU or DBCU and DLG or DBG. Note that the ombination o DBCU and DLG is impossible. Thereore, our ases Cases 1-1, 1-2, 2, and 3 are easible; these are listed in Table 1. (Table 1 to be inserted here) Reall that Cases 1-1 and 1-2 orrespond to Figures 1 and 2, respetively, and that Cases 2 and 3 orrespond to Figure 4. Note that Figure 4 alone annot distinguish whether the short-run equilibrium is DBCU or DLCU. We ondut the numerial analysis as ollows. First, we ind a benhmark or eah ase. Seond, starting rom eah benhmark, we inrease the interest rate (or proit 15

17 share) slightly and then ompute the new equilibrium values. When onduting the numerial analysis, we must onsider that the short-run equilibrium belongs to the orresponding ase even ater the interest rate (or proit share) hanges. The parameters or eah benhmark are listed in Table 2. Naturally, the long-run equilibrium is stable in every ase. (Table 2 to be inserted here) 4.1 Case 1-1 Table 3 shows an example that orresponds to Case 1-1. Here, we ompute the long-run equilibrium values by inreasing the interest rate rom 0.1 to and the proit share rom 0.3 to When the interest rate inreases, all the long-run equilibrium values inrease. This result is similar to the result in Hein (2007). In Hein (2007), or the long-run equilibrium to be stable, the short-run equilibrium must be DLG, whih neessarily makes the short-run equilibrium DLCU. In this ase, an inrease in the interest rate inreases all long-run equilibrium values. When the proit share inreases, the debt-apital ratio also inreases. In addition, the rate o apital aumulation dereases, and aordingly, the long-run equilibrium is wage-led growth. Moreover, the rate o apaity utilization dereases, and aordingly, the long-run equilibrium is stagnationist. These two results are typial o the Kalekian model. 11 However, the rate o proit inreases, that is, we never obtain the paradox o ost in whih an inrease in the proit share lowers the rate o proit. In this respet, our model is dierent rom the typial Kalekian model. (Table 3 to be inserted here) 11 The eonomy is termed wage-led (proit-led) growth i an inrease in the proit share lowers (raises) the rate o apital aumulation. In addition, the eonomy is alled stagnationist (exhilarationist) i an inrease in the proit share lowers (raises) the rate o apaity utilization. For more on this issue, see Bhaduri and Marglin (1990) and Bleker (2002). 16

18 4.2 Case 1-2 Table 4 shows an example that orresponds to Case 1-2. Here, we ompute the long-run equilibrium values by inreasing the interest rate rom 0.02 to and the proit share rom 0.4 to Note that in this ase, the rate o interest is small ompared to the rate in Case 1-1. The eets o a rise in the interest rate on the long-run equilibrium values are the same as those in Case 1-1; that is, all equilibrium values inrease. The result that when the interest rate inreases, the rates o apital aumulation and apaity utilization inrease seems to be paradoxial. In the usual post-keynesian theory, an inrease in the interest rate is expeted to have a negative eet on output and output growth. The eets o a rise in the proit share are the same as in Case 1-1, exept or the eet on the rate o proit. When the proit share inreases, the rate o proit dereases, implying the paradox o ost. (Table 4 to be inserted here) 4.3 Case 2 Table 5 shows an example that orresponds to Case 2. Here, we ompute the long-run equilibrium values by inreasing the interest rate rom 0.02 to and the proit share rom 0.4 to When the interest rate inreases, the debt-apital ratio also inreases, but the rate o apital aumulation, the rate o apaity utilization, and the rate o proit derease. The ase in whih a rise in the interest rate leads to a deline in the rates o apital aumulation and apaity utilization is onsidered to be a normal ase. When the proit share inreases, the debt-apital ratio dereases. In Cases 1-1 and 1-2 where the short-run equilibrium is DLG, the debt-apital ratio inreases as a result o a rise in the proit share. Thereore, in Case 2, the result we obtain is opposite to the 12 To learly disern the hanges in the equilibrium values, we use instead o

19 results in Cases 1-1 and 1-2. In Case 2, the rate o apital aumulation inreases, whih means that the long-run equilibrium is proit-led growth. Considering that we have wage-led growth in Cases 1-1 and 1-2 where the short-run equilibrium is DLG, we might say that there is a relationship between DLG and wage-led growth and that there is a relationship between DBG and proit-led growth. However, note that this inerene is based on numerial examples and not on rigorous analytial solutions. In Case 2, the rate o apaity utilization dereases, implying that the long-run equilibrium is stagnationist, whih is the same as in Cases 1-1 and 1-2. A rise in the proit share inreases the rate o proit. Consequently, we never obtain the paradox o ost. (Table 5 to be inserted here) 4.4 Case 3 Table 6 shows an example that orresponds to Case 3. Here, we ompute the long-run equilibrium values by inreasing the interest rate rom 0.02 to and the proit share rom 0.4 to When the interest rate inreases, the debt-apital ratio, the rate o apaity utilization, and the rate o proit inrease, whereas the rate o apital aumulation dereases. The eets o a rise in the proit share are the same as in Case 2. (Table 6 to be inserted here) 5 Conlusions Reent Kalekian studies, taking into aount irms debt aumulation, analyze how the stability ondition and the long-run eet o the interest rate on the real eonomy dier aording to debt regime (debt-led or debt-burdened). Hein (2007), one o the representatives among those researh works, abstrats the distribution o dividends to apitalists and assumes that the retention ratio is equal to unity. However, this assumption leads to the ollowing extreme results. First, the long-run equilibrium 18

20 debt-apital ratio is stable only i the short-run equilibrium is debt-led growth. Seond, the positive equilibrium debt-apital ratio under a debt-led growth regime requires an extremely high interest rate. Third, an inrease in the interest rate always raises apital aumulation in the long run, while the inome distribution never aets apital aumulation. This paper investigates the stability ondition and the long-run eet o the interest rate when irms distribute dividends to inanial apitalists, that is, when the retention rate is lower than unity. Our results are summarized as ollows. The long-run positive equilibrium debt-apital ratio an be stable not only under the debt-led growth regime but also under the debt-burdened growth regime. Regardless o the debt regime, this equilibrium does not impose any speial ondition on the size o the interest rate. In addition, i the short-run equilibrium rate o apital aumulation is debt-burdened, an inrease in the interest rate redues the equilibrium rate o apital aumulation in the long run. Although the long-run eet o the interest rate on apital aumulation under the debt-led regime is ambiguous, our numerial examples using plausible parameters show that this eet seems to be positive. Furthermore, the numerial examples show that a rise in the proit share dereases long-run apital aumulation under the debt-led growth regime, while it inreases long-run apital aumulation under the debt-burdened regime. The paradoxial eet o inome distribution on eonomi growth (that is, wage-led growth à la Kaleki) seems to orrespond to the puzzling relation between debt aumulation and eonomi growth, that is, debt-led growth. These onlusions teah us that setting the retention rate to unity or the sake o simpliity, unlike the zero saving propensity hypothesis o workers, produes biased analytial results. Thereore, we must pay due attention to this assumption when we onstrut a theoretial model. 19

21 Reerenes Asada, T. (2006) Stabilization poliy in a Keynes-Goodwin model with debt aumulation, Strutural Change and Eonomi Dynamis, Vol. 17 (4), pp Bhaduri, A. and S. Marglin (1990) Unemployment and the real wage: the eonomi basis or ontesting politial ideologies, Cambridge Journal o Eonomis, Vol. 14, pp Bleker, R. A. (2002) Distribution, demand and growth in neo-kalekian maro-models, in Setterield, M. (ed.) The Eonomis o Demand-led Growth, Challenging the Supply-side Vision o the Long Run, Cheltenham: Edward Elgar. Charles, S. (2008a) Corporate debt, variable retention rate and the appearane o inanial ragility, Cambridge Journal o Eonomis, Vol. 32 (5), pp Charles, S. (2008b) A post-keynesian model o aumulation with a Minskyan inanial struture, Review o Politial Eonomy, Vol. 20 (3), pp Charles, S. (2008) Teahing Minsky s inanial instability hypothesis: a manageable suggestion, Journal o Post Keynesian Eonomis, Vol. 31 (1), pp Charles, S. (2010) Explaining persistent yles in a short-run ontext: irms propensity to invest and omnipotent shareholders, Journal o Post Keynesian Eonomis, Vol. 32 (3), pp Dutt, A. K. and E. J. Amadeo (1993) A post-keynesian theory o growth, interest, and money, in Baranzini M. and J. Harourt (eds.) The Dynamis o the Wealth o Nations: Growth, Distribution and Strutural Change, New York: Mamillan. Fontana, G. (2009) Money, Unertainty and Time, Abingdon: Routledge. Hein, E. (2006) Interest, debt and apital aumulation: a Kalekian approah, International Review o Applied Eonomis, 20 (3), pp Hein, E. (2007) Interest rate, debt, distribution and apital aumulation in a post Kalekian model, Metroeonomia, Vol. 56 (2), pp Hein, E. (2010) Shareholder value orientation, distribution and growth short- and medium-run eets in a Kalekian model, Metroeonomia, orthoming. Hein, E. and C. Shoder (2009) Interest rates, distribution and apital aumulation a 20

22 post-kalekian perspetive on the US and Germany, Institute or International Politial Eonomy Working Paper No. 7/2010. Hein, E. and T. van Treek (2010) Finanialisation and rising shareholder power in Kalekian/post-Kalekian models o distribution and growth, Review o Politial Eonomy, Vol. 22 (2), pp Lavoie, M. (1995) Interest rates in post-keynesian models o growth and distribution, Metroeonomia, Vol. 46 (2), pp Lavoie, M. and W. Godley (2001-2) Kalekian models o growth in a oherent stok-low monetary ramework: a Kaldorian view, Journal o Post Keynesian Eonomis, Vol. 24 (2), pp Lavoie, M. and M. Seareia (2001) Minsky s inanial ragility hypothesis: a missing maroeonomi link, in Belloiore, R. and P. Ferri (eds.) Finanial Fragility and Investment in the Capitalist Eonomy: The Eonomi Legay o Hyman Minsky, Volume II, Cheltenham: Edward Elgar. Lima, G. T. and Meirelles, A. J. A. (2007) Marodynamis o debt regimes, inanial instability and growth, Cambridge Journal o Eonomis, Vol. 31 (4), pp Marglin, S. and Bhaduri, A. (1990) Proit- squeeze and Keynesian theory, in Marglin, S. and J. Shor (eds.) The Golden Age o Capitalism: Reinterpreting the Postwar Experiene, Oxord: Clarendon Press. Moore, B. J. (1988) Horizontalists and Vertialists: The Maroeonomis o Credit Money, Cambridge: Cambridge University Press. Rohon, L. P. (1999) Credit, Money and Prodution: An Alternative Post-Keynesian Approah, Cheltenham: Edward Elgar. Setterield, M. (2009) Maroeonomis without the LM urve: an alternative view, Cambridge Journal o Eonomis, Vol. 33 (2), pp Taylor, L. (2004) Reonstruting Maroeonomis: Struturalist Proposals and Critiques o the Mainstream, Massahusetts: Harvard University Press. Taylor, L. and C. R. V. Arnim (2008) Debt-equity yles in the twentieth entury: empirial evidene and a dynami Keynesian model, in Flashel, P. and M. 21

23 Landesmann (eds.) Mathematial Eonomis and the Dynamis o Capitalism: Goodwin s Legay Continued, Abingdon: Routledge. 22

24 0 Figure 1: 0 s < 1 and > 0 under the debt-led ase < Figure 2: 0 s < 1 and < 0 under the debt-led ase < Figure 3: s = 1 and > 0 under the debt-led ase 23

25 1 2 Figure 4: 0 s < 1 and > 0 under the debt-burdened ase < Figure 5: 0 s < 1 and < 0 under the debt-burdened ase < Figure 6: s = 1 and > 0 under the debt-burdened ase 24

26 Table 1: Four ases in whih the long-run equilibrium is meaningul and stable Utilization Aumulation Shape o parabola Case 1-1 DL DL Case 1-2 DL DL Case 2 DB DB Case 3 DL DB Convex upward, positive axis Convex upward, negative axis Convex downward, positive axis Convex downward, positive axis Table 2: Benhmark parameters or our ases s s h i α β τ θ Case Case Case Case Table 3: Results or Case 1-1 Benhmark Rise in interest rate Rise in proit share Debt-apital ratio Aumulation rate Utilization rate Proit rate

27 Table 4: Results or Case 1-2 Benhmark Rise in interest rate Rise in proit share Debt-apital ratio Aumulation rate Utilization rate Proit rate Table 5: Results or Case 2 Benhmark Rise in interest rate Rise in proit share Debt-apital ratio Aumulation rate Utilization rate Proit rate Table 6: Results or Case 3 Benhmark Rise in interest rate Rise in proit share Debt-apital ratio Aumulation rate Utilization rate Proit rate