IVR Working Paper WP 2015/01. Can Public Debt Be Sustainable? - A Contribution to the Theory of the Sustainability of Public Debt

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1 IVR Workin Paper WP 2015/01 Can Public Debt Be Sustainable? - A Contribution to the Theory o the Sustainability o Public Debt Frank C. Enlmann February 2015 Abstract The sustainability o overnment s inancial situation has two aspects: solvency and liquidity. The overnment is solvent i it satisies its intertemporal budet constraint. In the steady state o a rowin economy, the overnment is solvent i the public-debt-to-gdp ratio is constant and the real interest rate on overnment debt exceeds the rowth rate o real GDP. The overnment is liquid i its instantaneous budet constraint is satisied. Accordin to Gärtner 2009, or a constant imarydeicit-to-gdp ratio, this steady state debt-to-gdp ratio is unstable i the real interest rate on overnment debt exceeds the rowth rate o real GDP. Hence, there seems to be a tension between sustainability concernin overnment s solvency and sustainability concernin overnment s liquidity i the latter is deined as stability o the steady state debt-to-gdp ratio. This leads to the main research question o this paper: Is it true that in a rowin economy a constant debt-to-gdp ratio can only be stable i the overnment s solvency constraint is violated? Or put dierently: Is it true that public debt cannot be sustainable? Obviously, one counterexample is enouh to show that a oposition does not hold in eneral. Such a counterexample based on the Solow rowth model is esented in the paper. Hence, in eneral, sustainability o public debt is possible in the sense, that both the solvency constraint and the liquidity constraint with a stable steady state debt-to-gdp ratio are satisied or a set o parameter constellations. JEL Classiication Numbers: E62, H60, O40; keywords: Public debt, sustainability, stability, public capital, economic rowth. Institute o Economics and Law, University o Stuttart, Keplerstrasse 17, Stuttart, Germany ( enlmann@ivr.uni-stuttart.de). I want to thank Lars Feld, David Greenlaw, James Hamilton, Peter Hooper, Stean Hombur, Gebhard Kirchässner, and Carl Christian von Weizsäcker or helpul comments and Tobias Stamm, Markus Till Berkeeld, Konstanze Hülße, and Klaus Hauber or excellent research assistance. The usual disclaimer applies.

2 1. Introduction Public debt and its sustainability have become an acute oblem since the beinnin o the Great Recession in various countries, especially in the Euro Area. As, amon others, Kindleberer and Aliber 2011 or Reinhart and Roo 2009 have pointed out, this is by no means a new phenomenon in economic history. Yet, there seem to exist dierent views on what are the conditions or the sustainability o public debt or more speciically the public debt-to-gdp ratio. The most undamental concept o the sustainability o public debt seems to be that the overnment remains solvent, i.e. that it satisies its intertemporal budet constraint, named in the ollowin: solvency constraint. As ollows e.. rom Romer 2012 p. 586, this is the case i the real interest rate on real overnment debt is always positive and larer than the rate o rowth o real overnment debt. In a rowin economy, this means that the overnment s solvency constraint is satisied i the public-debt-to-gdp ratio is constant and the real interest rate on overnment debt exceeds the rowth rate o real GDP. Thus, with respect to solvency, a condition or sustainable public debt is a suiciently hih real rate o interest compared to the rowth rate o real public debt which, or a constant debt-to-gdp ratio, just equals the rowth rate o real GDP. Yet, durin the Euro crisis hih rates o interest on overnment bonds, especially hih seads relative to the German Bund were considered a sin o iscal weakness, as overnments o countries like Greece, Ireland, Italy, Portual, and Spain had to oer hih yields in order to stay liquid by sellin their bonds in inancial markets. A overnment is liquid i its instantaneous budet constraint 1, named in the ollowin liquidity constraint, is satisied. In case o public deicits, satisyin the overnment s liquidity constraint requires a suicient demand or newly issued overnment bonds i monetization o public debt is excluded. The overnment s instantaneous budet constraint translates into a dierential equation or the debt-to-gdp ratio rom which an equilibrium debt-to-gdp ratio can be computed (see e.. Gärtner 2009 p. 394 and de Grauwe 2014 annex; in the ollowin: G2). Accordin to Gärtner 2009 p. 395, or a constant imary-deicit-to-gdp ratio, this steady state debt-to-gdp ratio is unstable 2 i the real interest rate on overnment debt exceeds the rowth rate o real GDP. In this case a shock, that makes the actual debt-to-gdp ratio larer than its equilibrium value, leads to an ever increasin debt-to- GDP ratio and an ever increasin deicit-to-gdp ratio. In a closed economy, this increasin deicit-to- GDP ratio cannot be inanced out o ivate aents savins i it apoaches unity, and public debt becomes unsustainable, as the overnment becomes illiquid. Hence, iscal sustainability concernin overnment s liquidity in the sense o resilience to shocks requires the stability o the equilibrium debt-to- GDP ratio which in turn requires the rowth rate o real GDP to exceed the real rate o interest on overnment debt or a iven imary-deicit-to-gdp ratio. Hence, there seems to be a tension between sustainability concernin overnment s solvency and sustainability concernin overnment s liquidity i the latter is deined as stability o the steady state debt-to- GDP ratio. This leads to the main research question o this paper: Is it true that in a rowin economy a constant debt-to-gdp ratio can only be stable i the overnment s solvency constraint is violated? An airmative answer would imply that, by its very nature, public debt is unsustainable and, hence, has to be avoided. This would amount to a theoretical justiication o a public debt brake. Obviously, one counterexample is enouh to show that a oposition does not hold in eneral. Such a counterexample is esented here. To this end the simple Solow rowth model (Solow 1956) is eneralized in order to include public capital and public debt as well as the overnment s liquidity and solvency constraint. A rowth model o the Solow type is chosen because its relevant steady state is asymptotically stable. Hence, the conditions under which the oriinally stable steady state becomes unstable can 1 In the ollowin time is assumed to be continuous. 2 Now and in the ollowin, by stable we mean asymptotically stable in the sense that ater a shock a system returns to its equilibrium (see e.. Gandolo 2010 pp. 331). Contrary to e.. European Central Bank 2011 p. 64, by stable we do not mean constant. 2

3 be studied. This research stratey obviously could not be ollowed i the startin point were a model whose steady state is either asymptotically unstable as a rowth model o the Harrod-Domar type 3 or a saddle point as an optimal rowth model o the Ramsey type. Whereas G2 treat the interest rate on public debt, the rowth rate o GDP, and the imary-deicit-to- GDP ratio as exoenous variables, in the model esented below they will be endoenous. Here the stability o the (economically relevant) steady state and hence the equilibrium public debt ratio does not depend on whether the equilibrium rowth rate o GDP exceeds the equilibrium rate o interest. From an economic point o view the main dierence between the model esented below and the model esented e.. by G2 consists in the ollowin: Whereas in G2 s model the debt dynamics are driven by the supply side (which miht be called Say s Law o Public Finance ), namely the overnment s need to inance a budet deicit, in the model esented below the debt dynamics are driven by the demand side, namely the demand o ivate households or overnment bonds, just as in the oriinal Solow rowth model where ivate capital accumulation is driven by the savins o ivate households and hence the demand o ivate households or stock or bonds issued by ivate irms in order to inance ivate investments in real capital. In almost all countries in the world there has been public debt, at least durin the last decades. In some countries soverein debt crises were experienced, but luckily not in all. Hence in a realistic model or countries without soverein debt crisis, that can describe and explain several stylized acts, there should exist steady states with a positive and stable debt-to-gdp ratio without violatin the overnment s solvency constraint. The latter requires the real rate o interest on public debt to exceed the rowth rate o real GDP, which in turn requires a imary public budet surplus or a positive steady state debt-to-gdp ratio. In the model esented below such steady states exist, as lon as ivate households want to hold a part o their savins in overnment bonds as an asset with (under normal conditions) a positive nominal rate o return and comparatively low risk. As in the oriinal Solow model money is not taken into account. Hence there cannot be any liquidity eerence in the sense o a demand or money. But the demand or overnment bonds can be viewed as stemmin rom a sort o liquidity or saety eerence which induces ivate households to accept a lower rate o return, i.e. a saety discount or investments in overnment bonds compared to investments in bonds or equities issued by ivate irms. How stron this saety eerence can become one could observe durin the Euro crisis, when the nominal yield on bonds issued by the Federal Government o Germany sometimes became neative. Presentin a model with a positive steady state debt-to-gdp ratio, where the overnment is both solvent and liquid in the sense o resilience to shocks and, hence, where its public debt is sustainable, does not mean to deny the possibility o unsustainable public debt. The model esented can thus orm the base or indin out deeper reasons or the occurrence o iscal crises. 4 In the model ivate households opensity to invest in overnment bonds is assumed to be an exoenous parameter just as the saety discount actor. These rather stron assumptions can be justiied by three considerations: (i) they are in line with the oriinal Solow rowth model, where the savin rate is assumed to be exoenous; (ii) this is a paper on soverein debt sustainability, not on soverein debt crises includin runs on overnment bonds; (iii) in another paper by the author both ivate households opensity to invest in overnment bonds and the saety discount actor will be endoenized. The remainder o the paper is oranized as ollows: In section 2 the Solow rowth model with public debt is esented. In section 3 we derive the system o dierential equations, the steady state, and carry out various comparative dynamic analyses. In section 4 we will turn to the question whether it is really true that in a rowin economy a constant debt-to-gdp ratio can only be stable and hence sustainable i the real rate o interest on overnment debt is lower than the rowth rate o real GDP which in turn 3 Domar 1944 is a seminal paper or the discussion o the sustainability o public debt in the context o economic rowth. 4 First steps in this direction are undertaken in an accompanyin paper by the author. 3

4 leads to a violation o the overnment s solvency constraint. To answer this question we will irst describe Romer s apoach (Romer 2012) and how it its into the model esented here. Then we describe the apoach by Gärtner 2009 and de Grauwe 2014 (G2), and inally we compare their apoach with the one esented here. In the model esented here the steady state and hence the steady state ratio o public debt to national income turn out to be stable under rather eneral conditions, namely, whenever the sum o the rate o technical oress and the rate o chane o labor supply, i.e. the equilibrium rowth rate o real GDP is positive and the partial oduction elasticity o labor in eiciency units lies within a rane between zero and unity. Hence, stability does not depend on the sin o the dierence between the rowth rate and the rate o interest on overnment bonds. What really matters or the sustainability o public debt is not any more or less arbitrary level o the public-deicit-to-gdp ratio or o the public-debt-to-gdp ratio, but whether there is a imary surplus or deicit in the steady state. Section 5 concludes. 2. A Simple Solovian Growth Model with Public Debt, Exoenous Propensity to Invest in Government Bonds, and Exoenous Saety Discount Factor Followin Aschauer 1989 we take account o public capital S 1 K in the oduction unction Y A L K K ; 1 (1), S where Y denotes areate supply o oods and services oduced in the economy, A technical eiciency, L labor input, K ivate capital. 5 We assume a Cobb-Doulas oduction unction with constant returns to scale in order to be able to carry out numerical analyses. The main results o the analysis do not depend on the speciic orm o the oduction unction as lon as constant returns are eserved and the oduction unction has a reesentation o the orm Y F(AL,K,K ), i.e. S technical oress can be reesented as purely labor-aumentin (see Acemolu 2009, pp. 58.). For the sake o simplicity, the dependence o the variables on time is omitted in eneral. Total actor oductivity rows at the rate o technical oress, labor supply at the natural rate n. Labor supply is inelastic with respect to the real net wae. Capital and labor are ully employed: t A A0e ; 0 (2), nt L L0e ; n 0 Areate demand or oods and services consists o ivate well as o net ivate (I ) and net public investment (I ). D (3). (C ) and public consumption (C ) as Y C I C I (4). As in the Solow model, the market or oods and services is in equilibrium: S D Y Y Y (5), 5 This oduction unction diers rom the one e.. in Barro 1990 and Minnea and Villieu These authors take oductive public expenditures into account, not public capital. 4

5 where Y denotes net domestic oduct which equals net national income, as we consider a closed economy. 6 Private consumption depends on available income o households (Y T r B) C c (YT rbb) (6), where c denotes households opensity to consume with respect to disposable income, T net transer payments rom households to overnment (taxes plus social contributions rom ivate households minus overnment s transer payments to ivate households), and r B the rate o interest on overnment bonds B. For households opensity to save obviously holds: s 1 c (7), and hence or ivate savins S S s (YT rbb) (8). In the oriinal Solow model neither inancial markets nor a bankin sector are explicitly modelled. It is simply assumed that the inancial market or irms (newly issued) bonds or equities is in equilibrium. Hence, (in the end) households savins are just invested by ivate irms. Now we have two inancial markets: one or irms (newly issued) bonds or equities and one or overnment bonds. Accordinly, one part o savins S is invested in bonds/equities issued by ivate irms B another in overnment bonds where B S s (YT r B) (9), B S s (YT r B) (10), 7 s s s (11). Thus, in this as in Solow s model bonds and equities are held directly by households, whereas in reality overnment bonds are held to a lare extent by intermediaries (see or recent data e.. Andritzky 2012 and Broner et al. 2014). In the ollowin, or brevity s sake T will be called taxes, and the correspondin rate tax rate. Furthermore, as in Solow s oriinal contribution we just stipulate consumption, savin, and investment unctions without derivin them explicitly throuh a utility maximization exercise. 8 In any case, implicitly we assume that households instantaneous utility depends on both consumption o oods and wealth which is invested in irm equities or bonds and overnment bonds accordin to the respective savin rates. Taxes depend on the tax rate and ross household income (Y rb B), which lows rom ivate irms as labor and capital income and rom overnment as interest payments on overnment bonds T (Y rb B) (12). By postulatin the savin unctions (9) and (10), we assume that households eerences are such that they want to diversiy their investments by investin their savins both in ivate equities/bonds and 6 In the ollowin, or brevity s sake the term net will be omitted in eneral. Just as in Solow s oriinal contribution to the theory o economic rowth (Solow 1956) we do not treat deeciation explicitly in the model. But we will consider deeciation when we come to the Maastricht criteria concernin the debt-to-gdp and the public-deicit-to-gdp ratio. 7 This assumption is in line with von Weizsäcker See or a theoretical justiication Akerlo s Presidential Address at the Annual Meetin o the American Economic Association 2007 (Akerlo 2007, especially pp 13.). 5

6 overnment bonds. Furthermore, we assume that ivate households just reinvest their inancial means in irm and overnment bonds, whenever outstandin irm and overnment bonds become due. Alternatively we can assume that irms and overnments issue perpetual bonds. Secondary markets or equities and overnment bonds are not explicitly modeled. Savins that low to ivate irms real capital K Savins that low to overnment overnment s budet deicit B : S are used to inance ivate irms net investments I in ivate I K S (13). 9 S are used to buy new overnment bonds B D which inance the D (14). B B S Here and in the ollowin a dot above a variable denotes the irst derivative with respect to time. Eq. (14) has the ollowin economic implication: the overnment supplies additional overnment bonds that ivate households demand accordin to their disposable income and risk eerences. Government budet deicits respond to the portolio choices o ivate households just as irms investments, i.e. irms deicits. Both are determined by the demand o ivate households or new assets. Government budet deicits are an equilibrium phenomenon just as ivate irms deicits inanced by issuin stocks or bonds. For the correspondin public-deicit-to-national-income ratio we et by simple division o eq. (14) by national income: The overnment s liquidity constraint is: B Y S Y TB C I rbb (15). (16). Excludin monetization o public debt, the overnment s total revenue consists o taxes plus net increase in overnment debt B. This revenue is used to inance public expenditures on consumption C and investment I plus interest payments on outstandin overnment debt r B B. The overnment s imary deicit D is deined as ollows: D C I T (17), and the correspondin ratio o the imary deicit to national income d as: D C I T d (18). Y Y From eqs (14) and (17) we obtain: With the public-debt-to-national-income ratio b (19). B D rb B B b (20) Y 9 Here and in the ollowin we use the ollowin abbreviations: ˆ 6 x dx/dt and x x/x.

7 we et rom eqs (18) - (20): d (Bˆ r )b B (21). For b 0 there is a imary deicit ( d 0) i the rate o chane o public debt exceeds the real rate o interest on public debt, and there is a imary surplus i the real rate o interest on public debt exceeds the rate o chane o public debt. 10 From eqs (19), (20), (14), (12), and (10) we can also obtain another exession or the imary deicit ratio, namely: d s (1 ) rb 1s (1 ) b (22). Hence, the imary debt ratio does not only depend on households opensity to invest in overnment bonds and the tax rate, but on the ratio o overnment s interest payments to national income as well, as the latter inluence households disposable income. The budet deicit can be used to inance overnment s net investments in public real capital K or a part o them as well as to inance a part o public consumption or interest payments on outstandin overnment debt. We introduce the public-investment-to-budet-deicit ratio that is supposed to be set by the parliament when it decides on the budet: I (23). B In case o 1 the so-called Golden Rule o Public Finance is ollowed (see e.. Bassetto and Sarent and 2006) 11. For public capital accumulation we obtain: (24). K I B; 0 From eqs (16) and (24) we can deduce: T C rb B (11/ )I (25). I the Golden Rule o Public Finance is ollowed, taxes on national income and interest payments on public debt just serve to inance public consumption and interest payments on public debt. From eqs (12) and (25) we et or the share o public consumption in national income: C B I (1 )r B (11 / ) (26), Y Y Y and rom eq. (12) or the share o taxes in national income: T B (1 r B ) (27). Y Y From eqs (4) - (6), (8) - (10), (13), (20), (24) and (27) the ollowin exession can be derived or the share o public consumption in national income: C 11 (1 )s (1 )(1r B b) (28). 12 Y 10 Contrary to Gärtner 2009 and de Grauwe 2014 the ratio o the imary deicit to national income is endoenous. 11 See in this context also Buiter 2001, Blanchard and Giavazzi 2004 and Sachverständienrat 2007 pp It should be noted that eqs (26) and (28) are not independent rom each other. 7

8 The share o public consumption in national income is an endoenous variable in this model which is determined by the overnment s liquidity constraint, and hence (i) by the public-deicit-to-nationalincome ratio which in turn is determined by households opensity to invest in overnment bonds, (ii) by the share o taxes in national income which in turn is determined by the tax rate and the share o interest payments on public debt in national income, and (iii) by the share o public investment in national income which in turn is determined by the public-investment-to-budet-deicit ratio and, aain, by the public-deicit-to-national-income ratio. As in Solow s rowth model we assume perect competition in the markets or oods and services, labor and ivate capital. The ice level is assumed to be unity. As all incomes low to households only ivate households pay income tax. Hence, the rental ice o ivate capital R equals the marinal oductivity o ivate capital and the wae rate equals the marinal oductivity o labor: Y R YK A L K K (29), K 1 1 Y w YL A L KK (30). L Furthermore, we assume that the use o public capital is ree o chare. This leads to oits ( ) even with perect competition: For the respective rate o oit r we can compute rom eq. (29): and hence, or the overall rate o return on ivate capital r K From eqs (29), (33), and (34) ollows: (1 )Y (31). r (32) K Y r (1 ) (33), K rk R r (34). Y (35). r K (1 ) K Even assumin perect markets, we allow that the rate o return on ivate capital may exceed the one on overnment bonds due to risk and liquidity considerations o ivate households. This implies that households consider an investment in overnment bonds less risky and more liquid than an investment in ivate enterises. Private households risk and liquidity considerations are taken into account by the exoenously iven saety discount actor. Hence, we postulate r B (1 )r K; 0 1 (36). By assumption, the saety discount actor does not exceed unity. Hence, we consider it a disequilibrium phenomenon when even the nominal rate o interest on overnment bonds is neative like the German Bund durin certain periods o the Euro crisis. Finally, we deine ivate and public capital-labor ratios in eiciency units: K AL (37),

9 K AL (38), as well as net domestic oduct per capita in eiciency units: Y y (39). AL The oduction unction can be rewritten by usin eqs (39), (37) and (38): From eq. (24) we obtain 1 Y (AL) K K 1 y 1 y, (40). AL (AL) (AL) (AL) K B (41) i remains unchaned over time, and hence accordin to eq. (20) or the debt ratio b: B B AL b Y Y AL y For the interest rate on public debt we can derive rom eqs (35) and (36) B K (42). Y r (1 )(1 ) (1 )(1 ) (43), and hence with eqs (43) and (42) or the share o overnment s interest payments in national income: 1 r b (1 )(1 ) (44), B and hence with eq. (27) or the ratio o households disposable income to national income: YTr B T B B 1 1 rb 1 (1 )rbb (1 ) 1 (1 )(1 ) (45). Y Y Y From eqs (45), (9), and (10) we can deduce the shares o households investment in ivate and public bonds in national income: S 1 s (1 ) 1 (1 )(1 ) (46), Y S 1 s (1 ) 1 (1 )(1 ) Y Finally, rom eqs (35) and (40) we derive or the rate o return on ivate capital: 1 1 K K (47). Y r (1 ) (1 ) (48). 9

10 3. Steady State and Comparative Dynamics In this section we deal with rowth dynamics. Here we derive the system o dierential equations or the ivate and the public capital-labor ratio in eiciency units 13 and the steady state. With ^ denotin the percentae rate o chane, we et rom eq. (13) and eqs (37) - (39), and (45): ˆK s (1 ) (1 )(1 ) (49) From eq. (24) we et with eq. (10) and eqs (37) - (39), and (45): ˆK s (1 ) (1 )(1 ) (50). From eqs (37), (2) and (3) we can derive the percentae rate o chane o the ivate capital-labor ratio in eiciency units: ˆ ˆ K n (51), and rom eqs (38), (2) and (3) the percentae rate o chane o the public capital-labor ratio in eiciency units: ˆ Kˆ n (52). We obtain the system o dierential equations by insertin eq. (49) into eq. (51): and eq. (50) into eq. (52): s (1 ) (1 )(1 ) ( n) (53), s (1 ) (1 )(1 ) ( n), (54). The steady state ( ) is derived by settin both eqs (53) and (54) equal to zero. By dividin the two resultin equations, we can see that the ollowin relationship holds in the steady state: We deine the ratio o the opensities to save: s s (55). s s s (56). By insertin eq. (55) into eqs (46) and (47) we can derive the steady state values o the shares o households investment in ivate and public bonds in national income: and 1 s s (1 ) 1 (1 )(1 ) s ; 0, 0, 0, 0, 0 s s s s s s s (57) 13 For simplicity s sake, in the ollowin text we will oten omit in eiciency units. 10

11 1 s s (1 ) 1 (1 )(1 ) s ; s s s s s 0, 0, 0, 0, 0 s s Finally, in the steady state, we can rewrite eqs (53) and (54) in the ollowin way: and s y, ( n) (58). (59), s y, ( n) (60). Eqs (59) and (60) are the equivalents to the well-known equilibrium condition in the Solow model: sy( ) ( n). Households opensity to save and invest in ivate capital with respect to national income s decreases with the tax rate and the saety discount actor, because in both cases households disposable income shrinks. The same holds or the households opensity to save and invest in overnment bonds with respect to national income s which, accordin to eq. (15), equals the steady state public-deicit-to-national-income ratio: 1 s s (1 ) 1 (1 )(1 ) s B (61). Y From eq. (61) we see that the steady state public deicit ratio is constant, just as in the oriinal Solow rowth model the ratio o the chane o capital to national income is equal to the savins rate which is constant ( K / Y s ). 14 We obtain the steady state values o ivate and public capital per capita by insertin eq. (55) into eq. (53): 15 and inally, aain usin eq. (55): 1/ 1 s s ; ( n) 0, 0, 0, 0, 0, 0, 0 ( n) s s (62), 14 With a more eneral oduction unction than the Cobb-Doulas oduction unction the sum o the oduction elasticities o ivate and public capital 1 and hence the public deicit ratio would not be exoenously iven. 15 As in the oriinal Solow model there is a second steady state, namely 0. This steady state is an unstable node. 11

12 1/ 1 s s ; ( n) 0, 0, 0, 0, 0, 0, 0 ( n) s s Fiure 1 shows how the steady state values o the ivate and public capital-labor ratios can be determined raphically in the three-dimensional Solow Diaram in the intersection point o the two dierential equations and the 0 -plain. (63). (, ) 12 Fiure 1: Steady state values o the ivate and public capital-labor in eiciency units ratios in the three-dimensional Solow Diaram ( s 0.01;s 0.1; 1; 0.7; 0.2; 0.25;n 0; 0.02; 0.5; 1 ) 16 By insertin eqs (62) and (63) into eq. (40) and takin account o eq. (58) the steady state value o national income per capita in eiciency units ollows: (1 ) 1/ (1 ) s y s ; n y y y y y y y 0, 0, 0, 0, 0, 0, 0 ( n) s s In analoy to the simple Solow model (Solow 1956), accordin to eq. (64), the steady state national income per capita increases with a oportional increase in the opensities to save and a decrease in the equilibrium rowth rate ( n). Furthermore, the steady state national income per capita in eiciency units increases with increases in the public-investment-to-budet-deicit ratio and with decreases in the tax rate ( ) and the saety discount actor ( ). Obviously, in the steady state the rowth rates o national income, ivate and public capital as well as public debt coincide with the rowth rate o labor supply in eiciency units: (64). Y ˆ K ˆ K ˆ B ˆ n (65). 16 The parameter values in the various iures are based on rouh estimates or Germany.

13 By insertin eqs (62) and (63) into eq. (42), and usin eq. (58), we et or the steady state value o the ratio o public debt to national income: s (1 ) 1 s b 1 (1 )(1 ) s ; n n b b b b b b b 0, 0, 0, 0, 0, 0, 0 ( n) s s Eq. (66) shows the well-known result that the steady state value o the debt ratio is equal to the steady state public deicit ratio divided by the steady state rowth rate o national income. Under the additional assumptions s 0 and n 0 the steady state debt ratio b is also positive. Accordin to eq. (66) the steady state debt ratio is the larer, the larer the households opensity to save and invest in overnment bonds (s ) and the larer the sum o the partial oduction elasticities o ivate and public capital (1 ). The steady state debt ratio is the smaller, the larer the equilibrium rowth rate ( n), the tax rate ( ), the saety discount actor ( ), and the households opensity to save and invest in shares o ivate irms (s ). 17 What may be surisin at irst siht is the act that the steady state debt ratio does not depend on the public-investment-to-budet-deicit ratio. But this act is in line with the overnment budet deicit bein determined by ivate households demand or overnment bonds as an asset or their savins. Government can inluence the steady state public deicit ratio and hence the steady state debt ratio only by varyin the tax rate, thus inluencin disposable income and hence households demand or newly issued overnment bonds. (66). Fiure 2: Steady state public deicit ratio and the Maastricht criterion ( s 0.1; 1; 0.7; 0.2;n 0; 0.02; 0.5; 1 ) Fiure 2 shows how the steady state public deicit ratio varies mainly with households opensity to invest in overnment bonds. The respective slope is close to unity. It is the larer the smaller is the tax rate. The Maastricht criterion o 3 % reers to the debt-to-gdp ratio. The correspondin criterion or the r r public-deicit -to-ndp ratio b would be about % hiher because o B/Y (B/ Y) (Y / Y), where r Y denotes GDP and r (B/Y) the public-deicit-to-gdp ratio. The same reasonin applies to 17 The sins o the impacts o variations o the rowth rate and the tax rate on the steady state public debt ratio are the same as in the simple public debt dynamics model o Gärtner 2009 in the case o a positive stable equilibrium public debt ratio. 13

14 the debt ratio. Yet, or simplicity s sake, in the iures we nelect the distinction between public-deicitto-ndp ratio and public-deicit-to-gdp ratio. By multiplyin eqs (64) and (66) or by dividin eq. (62) by we can derive the steady state value o public debt per capita: / 1/ 1/ (1 )/ 1/ y b ( n) s s yb yb yb yb yb yb yb (67). 0, 0, 0, 0, 0, 0, 0 ( n) s s Even i, as pointed out above, the steady state value o the ratio o public debt to national income remains unaected by chanes in, this does not hold or the steady state value o public debt per capita. This increases with as does per capita income. By insertin eq. (55) into eq. (44) we derive or the steady state value o the ratio o public interest payments to national income: s (1 )(1 ) rb b ; s r B b r B b r B b r B b r B b r B b r B b 0, 0, 0, 0, 0, 0, 0 ( n) s s For the steady state value o the ratio o public interest payments to national income to be smaller than unity it is suicient that households invest at least as much in ivate irms as in overnment bonds. Accordin to eq. (68) there is a neative relationship between the steady state values o the interest rate on overnment bonds and the public debt ratio. The lower the public debt ratio, the hiher is the rate o interest on overnment bonds, iven the saety discount actor, labor s partial oduction elasticity, and the ratio o savin rates. Remarkably, overnment cannot inluence the ratio o public interest payments to national income by chanin the tax rate. In act, in the esented model the steady state value o the ratio o public interest payments to national income is determined by eerences and risk considerations o ivate households and by technoloy, not by overnment. 18 I we divide eq. (68) by eq. (66) we inally obtain the steady state value o the interest rate on public debt: ( n)(1 )(1 ) ( n)(1 )(1 ) r B ; (1 ) s s (1 )(1 ) s s rb rb rb rb rb rb r 0, 0, 0, 0, 0, 0, B 0 ( n) s s Eq. (69) is the analoous exession o the steady state value o the real rental ice o (ivate) capital in the oriinal Solow rowth model, namely r ( n)(1 ) /s. 19 Accordin to eq. (69) the steady K (68). (69). 18 This result depends on the oduction unction bein o the Cobb-Doulas-type. 19 From this equation ollows that the real rate o interest in Solow s model exceeds the steady state rowth rate as lon as the savin rate exceeds the partial oduction elasticity o capital 1. An analoous result was derived by Pasinetti 1962, where he shows that the steady state real rate o interest equals the ratio o the steady state rowth rate to the savin rate o capitalists under the additional condition that the distribution o wealth between (savin) workers and capitalists remains 14

15 state rate o interest on overnment bonds decreases with an increase in the opensities to save and the saety discount actor ( ) as well as with a decrease in the equilibrium rowth rate ( n) and the tax rate ( ). Neither the steady state value o the ratio o public interest payments to national income nor the steady state rate o interest on overnment bonds depend on the ratio o public investment to the budet deicit. Fiure 3: Steady state rate o interest on overnment bonds ( s 0.01;s 0.1; 1; 0.7; 0.2;n 0; 0.02; 1 ) Fiure 3 shows that the steady state rowth rate exceeds the steady state rate o interest only or rather hih values o the saety discount actor. By usin eqs (18), (24), (27), (28), (66) - (69) the equilibrium value o the ratio o the imary deicit to national income d can be computed as: 1 d s (1 ) (1 )(1 ) 1s (1 ) s ; d d d d d d 0, 0, 0, 0, 0, 0 s s It can be positive or neative or zero. From eqs (21) and (65) we directly et the ollowin relationship between the steady state values o the imary deicit ratio, the debt ratio, and the dierence between the rowth rate o real national income and the real rate o interest on overnment bonds: For b 0 we can derive: B (70). d Yˆ r b (71). d 0 Yˆ r B (72). Hence, or b 0, in the steady state there is a imary deicit i the rowth rate o real national income exceeds the real rate o interest on public debt, and there is a imary surplus i the rowth rate o real national income is smaller than the real rate o interest on public debt. Fiure 4 shows that the steady state values o the imary deicit ratio and o the dierence between the rowth rate o real national income and the real interest rate on public debt are more sensitive with respect to chanes in the saety discount actor than to chanes in the tax rate. I the overnment increases the tax rate in order to turn the steady state value o a imary deicit into a imary surplus this unchaned. This clearly shows that Piketty s claim that the wealth distribution becomes more and more unequal i the interest rate exceeds the rowth rate does not apply to models with reesentative aents (see Piketty 2014). 15

16 increases the steady state value o the real interest rate on public debt in such a way that the dierence between the steady state values o the rowth rate o real national income and the real interest rate on public debt turns neative, too. In this sense, or iven values o the other parameters, overnment cannot only inluence the sin o the imary deicit, but the sin o the dierence between the rowth rate o real national income and the real rate o interest as well. 16 Fiure 4: The steady state values o the imary deicit ratio and o the dierence between the rowth rate o real national income and the real interest rate on public debt ( s 0.01;s 0.1; 1; 0.7; 0.2;n 0; 0.02; 1 ) The reason or (72) to hold is the overnment s liquidity constraint (see eq. (19)). I rb Yˆ in the steady state then overnment s interest payments exceed the budet deicit. Accordin to eq. (19) this ap has to be illed by a imary surplus: B B rb Yˆ r ˆ D Bb rb Y b d 0 Y Y Y (73). I a debt brake is introduced as in Germany (see Feld 2010), where the debt-brake-public-deicit-to-gdp ratio is restricted to 0.35 % in the constitution: DB r B DB Y GER s.0035 (74), Y GER Y GER GER with DB reerrin to debt brake and GER to Germany, the correspondin debt-brake-tax rate DB can be computed rom eq. (58): DB s DB 1 ; 1 s 1 (1 )(1 ) s (75). DB DB DB DB DB 0, 0, 0, 0, 0 s s From eq. (75) we can see that the debt brake tax rate increases with households opensity to invest in overnment bonds. Thus, in order to satisy the debt brake, the overnment has to increase the tax rate when ivate households increase their opensity to invest in overnment bonds, thus directly actin aainst ivate households eerences. The debt brake tax rate decreases with an increase o the saety discount actor, because an increase in the saety discount actor reduces households interest income rom overnment bonds and this lowers households opensity to save and invest in overnment bonds with respect to national income. By insertin the debt-brake-public-deicit ratio into eq. (66) we can compute the correspondin steady state debt-brake-public-debt ratio:

17 b DB DB s n (76). 20 Accordin to eq. (76) a decrease o the public deicit ratio due to the introduction o a debt brake does not only reduce the steady state public debt ratio, but accordin to eq. (64) also per capita income unless overnment increases the public-investment-to-budet-deicit ratio suiciently in order to counterbalance the adverse impact. By dierentiatin eqs (64) and (58) we can compute the percentae chane o the public-investment-to-budet-deicit ratio which is necessary in order to keep national income per capita unchaned i the tax rate is increased in order to reduce the public deicit ratio: d (1 ) ds (1 ) d (1 ) s (1 )(1 ) (77). 4. Sustainability o Public Debt Dynamics We now turn to the question posed in the introduction, namely: Is it really true that in a rowin economy a constant debt-to-gdp ratio can only be stable i the real rate o interest on overnment debt is lower than the rowth rate o real GDP which in turn leads to a violation o the overnment s solvency constraint? To answer this question we will irst describe Romer s apoach (Romer 2012) and how it its into this model. Then we describe the apoach by Gärtner 2009 and de Grauwe 2014 (G2), and inally we compare their apoach with the one esented here. Followin Romer 2012, p. 586 we write the overnment s solvency constraint in the ollowin orm: R B (t) lim PVB(t) lim B(t)e 0 (78), 21 t t with t R B (t) r( )d (79), 0 where PVB denotes the esent value o public debt in t = 0 and R B(t) the compound interest on overnment bonds rom t = 0 until t. As public debt is positive as lon as households opensity to invest in overnment bonds is positive, the esent value o public debt has to convere to zero in order to satisy the overnment s solvency constraint in our model. Hence, the rate o chane o PVB has to be neative when time oes to ininity: ^ B t t t lim PVB(t) lim B(t) lim r (t) 0 (80). Obviously, i the model economy is not in the steady state riht rom the start, the steady state values are only apoached i the steady state is asymptotically stable. Only under this condition holds: ^ ^ lim PVB(t) B r B PVB 0 (81). t 20 I we assume an averae rowth rate o nominal GDP o about 3 % or Germany the resultin lon run debt brake public debt ratio is about 10 %. 21 This ormulation is more eneral than the one used by Blanchard et al p. 12, who assume a constant rate o rowth o real GDP and a constant real rate o interest. 17

18 Hence, the stability o the steady state also matters or the answer to the question o whether the overnment s solvency constraint is satisied or not. In the steady state the rate o chane o public debt equals the steady state rowth rate o national income ( n ) accordin to eq. (65). Thus rom (81) ollows the minimum threshold o the steady state rate o interest on overnment bonds r BMin PVB that has to be exceeded in order to satisy the solvency constraint: PVB rbmin Takin into account eq. (69) condition (81) can be rewritten as: ^ s (1 )(1 ) PVB ( n) 1 0 ss n (82) ^ ^ ^ ^ ^ PVB PVB PVB PVB PVB 0, 0, 0, 0, 0 s s Accordin to eq. (83), or ( n) 0 the sin o the steady state rate o chane o the esent value o public debt does not depend on the size o the steady state rowth rate o national income ( n ), as the latter inluences the steady state values o both the rate o chane o public debt and the rate o interest on overnment bonds in a multiplicative manner. Instead, the sin o the steady state rate o chane o the esent value o public debt depends on all the other parameters that inluence the steady state rate o interest on overnment bonds alone (see the raction in eq. (83) and takin into account eq. (69)). Considerin the partial derivatives o the steady state rate o chane o the esent value o public debt, it is certainly not surisin that a hiher tax rate helps to satisy the overnment s solvency constraint. From eq. (83) and eq. (61) ollows as the minimum threshold o the tax rate PVB Min that has to be exceeded or the overnment s solvency constraint to be satisied: PVB PVB PVB PVB PVB (1 )(1 ) Min Min Min Min Min 1 ; 0, 0, 0; 0 (84). 22 s s (1 )(1 ) s s I the tax rate exceeds the minimum threshold there is a imary surplus in the steady state accordin to (72). Hence, in order to ulil the solvency constraint it is suicient or the overnment to realize a imary surplus in the steady state, as lon as the steady state is stable. The minimum threshold o the tax rate increases with increases in the saety discount actor, the oduction elasticity o labor, and households savin rates (see Fiure 5). Alternatively, or a iven tax rate, rom eq. (83) and eq. (61) ollows as maximum threshold o the saety PVB discount actor Max that is not to be exceeded or the overnment s solvency constraint to be satisied: (83). PVB PVB PVB PVB PVB s (1 ) Max Max Max Max Max (1 ) 1s (1 ) s s 1 ; 0, 0, 0; 0 (85) Since the outbreak o the Great Recession in many countries the central banks have lowered the level o interest rates at which banks can reinance themselves. In the majority o these countries this has led to a decline o the rate o interest or overnment bonds. As a irst apoximation, such a decline can be modeled by an increase in the saety discount actor. 18

19 At irst siht it may seem counterintuitive that a hiher saety discount actor and hence a lower steady state rate o interest rate on overnment bonds, a lower steady state ratio o overnment s interest payments to national income, and a lower steady state level o public debt makes it more diicult to satisy the overnment s solvency constraint in the lon run. But this result is due to the act that the steady state rowth rate o overnment debt is not aected by a chane in the saety discount actor, whereas the steady state rate o interest on overnment bonds is lowered. Fiure 5 shows how stronly the minimum threshold o the tax rate increases with the saety discount actor. For small values o the saety discount actor the minimum threshold o the tax rate is neative and hence smaller than the actual tax rate. But or lare values o the saety discount actor a urther increase may require a substantial increase o the tax rate in order to satisy the overnment s solvency constraint. Fiure 5: Minimum threshold or the tax rate as a unction o the opensity to invest in overnment bonds and the saety discount actor ( s 0.1; 1; 0.7; 0.2;n 0; 0.02; 1 ) For ( n) 0 inequality (83) is satisied i s (1 )(1 ) s r Bb s B B (86), Y Y i. e. i the steady state public-deicit-to-national-income ratio is smaller than the steady state overnment-interest-payments-to-national-income ratio. Accordin to inequality (86) the steady state overnment-interest-payments-to-national-income ratio is the upper threshold o the steady state public-deicitto-national-income ratio (B/Y) that is not to be reached i the overnment s solvency constraint is to be satisied. This upper threshold is the lower, the lower is the opensity to invest in public bonds compared to the opensity to invest in ivate bonds/shares, the lower is capital s share in national income, and the hiher is the saety discount actor. 23 Since the outbreak o the Great Recession in many countries the central banks have lowered the level o interest rates at which banks can reinance themselves. In the majority o these countries this has led to a decline o the rate o interest or overnment bonds. As a irst apoximation, such a decline can be modeled by an increase in the saety discount actor. 19

20 Fiure 6: The steady state public deicit ratio, its upper threshold, and the Maastricht criterion ( s 0.04;s 0.1; 1; 0.7; 0.2;n 0; 0.02; 1 But, as Fiure 6 shows, by settin a tax rate that exceeds the minimum tax rate accordin to inequality (84) the overnment has an instrument to make sure that its solvency constraint is satisied, even i the saety discount is lare. 24 For the correspondin upper threshold o the steady state debt ratio ollows rom inequality (86): From (73), (81), (86), and (87) ollows b s s (1 )(1 ) r Bb b n s ( n) n (87). ^ ˆ B B rb Y rbb 0 d 0 PVB b b (88) Y Y From eqs (80) and (83) we observe that satisyin the overnment s solvency constraint as a condition or the sustainability o public debt requires the steady state rate o interest on overnment bonds to exceed the steady state rowth rate o national income. But, accordin to Gärtner 2009 p. 395 and de Grauwe 2014 annex (G2), this condition leads to instability o the steady state debt ratio and hence nonsustainability o public debt. Thus let us have a closer look at their apoach. 25 By dierentiatin the debt ratio with respect to time (see eq. (42)), we can compute BY YB B b Yb ˆ 2 Y Y (89). By insertin eq. (16) into eq. (89) and takin into account the deinition o the imary budet deicit the ollowin dierential equation or the debt-to-national-income ratio can be derived: C I T b (Yˆ r ˆ B)b d (Yr B)b Y (90), rom which by settin b 0 26 the equilibrium value o b can be directly computed as: d b (91), Ŷ r B 24 Assumin the labor supply to depend on the tax rate miht chane this conclusion. 25 See in this context Greenlaw et al. 2013, too. 26 de Grauwe calls this the necessary condition or maintainin solvency (de Grauwe 2014). Hence, his use o the term solvency is dierent rom mine. 20

21 which is just the same equation as eq. (71) i d, Ŷ, and r B are equal to their steady state values. In inciple, d, Ŷ, and r B could be unctions o b. Hence, by linearizin eq. (90) around the equilibrium by means o a Taylor series o the irst order we et: ˆ B b ˆ b B b (92), b T d(b ) Y(b ) r (b ) b d ' (b )(bb ) b Y ' (b ) (r ) ' (b ) (bb ) where T reers to Taylor and d(b ) d, Y(b ˆ ) Y ˆ, r B(b ) rb. G2 treat d, Ŷ, and r B as exoenous parameters. Hence they (implicitly) assume: ' ' ' d b(b ) Y ˆ b(b ) (r B) b(b ) 0 (93). Accordinly, G2 s (linear) dierential equation or the debt ratio can be written as ollows: (94), G G b d (Yˆ r B )b ' b where G reers to G2. As the rowth rate and the rate o interest on overnment bonds do not directly depend on the debt ratio, we maintain the simpliyin assumption Ŷ ' (b ) (r ) ' (b ) 0 as a irst apoximation, but not d (b ) 0, as the imary deicit ratio directly depends on the debt ratio accordin to eq. (22), hence: b d b ' (b ) 1s (1 ) rb (95). By inspectin eq. (94) we see that the dierential equation stipulated by G2 is linear, and intersects the G vertical axis ( b ) at d G and the horizontal axis ( b ) at b b G 0. I d 0 and hence Ŷ r B G the resultin dierential equation b is positively sloped and, thus, the equilibrium seems to be unstable even i it is stable, as will be shown in the ollowin. In the model esented all the markets are assumed to be in equilibrium even outside the steady state. Hence, we are dealin with stability in the restricted sense o equilibrium dynamics, not disequilibrium dynamics (Hahn and Matthews 1964, p.782 and pp. 804.). By linearizin the dierential equations (53) and (54) in the relevant steady state we can compute the discriminant : 2 2 ( n) (1 ) 0 ( n) 0; 1 (96). Hence, we have two real roots. For the trace tr o the Jacobian matrix we et: tr ( n)(1 ) 0 ( n) 0; 0 (97), and or the determinant det o the Jacobian matrix: 2 det ( n) 0 ( n) 0; 0 (98). Because o eqs (96) - (98) the relevant steady state is a stable node (see Gandolo 2010, p.358), as lon as ( n) 0 and It is interestin to note that the stability o the economically relevant steady state just depends on the sum o the rowth rates o technoloical oress and labor supply as well as on the partial oduction elasticity o labor in eiciency units, not on households opensities to save, the tax rate, the saety discount actor or the ratio o public investment to budet deicit. B b 27 At irst siht, one miht even arue that, accordin to the inequalities (96) - (98), the equilibrium is a stable node or 1, too. But this would not take into account the act that the dierential equations were derived under the assumption that the areate oduction unction is linear homoenous. 21

22 Remarkably, the stability conditions are essentially the same as in the oriinal Solow model. Hence, or stability it does not make a dierence whether households savins are invested in one or several types o capital, ully or partly, as lon as the savin rates and correspondin investment rates are exoenous. 28 In order to compare the apoach o G2 more directly with ours we now derive the dierential equation or the debt-to-national-income ratio rom the model esented above where the number o new overnment bonds traded is determined by the demand o ivate households (see eqs (14) and (10)): 22 D B B S s (1 )(YrB B) (99). By insertin eq. (99) into eq. (90) the ollowin dierential equation or the debt to national income ratio can be derived: ˆ ˆ b s (1 ) 1s (1 ) rb b YrB b s (1 ) Ys (1 )rb b (100). Thus, by usin the steady state values o Ŷ and r B and assumin ' ' Ŷ b(b ) (r B) b(b ) 0, rom eqs (69), (95) and (100) we obtain the ollowin linearized (hence superscript l) dierential equation or the debt ratio which is equivalent to the Taylor apoximation accordin to eq. (92) or d ' (b ) 0 and ' ' Ŷ b(b ) (r B) b(b ) 0: l ( n)s l b s (1 ) b s (1 )(1 ) (101). Obviously, or 0,,,s,s 1, accordin to eq. (101) the resultin equilibrium debt ratio is stable, as lon as ( n) is positive, the same result as derived above (see inequalities (96) - (98)). From eq. (100) we see that or the equilibrium debt ratio to be stable the rowth rate o national income does not have to exceed the rate o interest on overnment debt, but just the tiny raction s (1 ) o it. In a stable steady state the rate o interest on overnment debt may exceed the rowth rate o national income and hence the overnment s solvency constraint may be satisied. Thus the answer to the question whether in eneral a constant debt-to-gdp ratio can only be stable in a rowin economy i the rowth rate o real GDP and hence real overnment debt exceeds the real rate o interest on overnment debt, which in turn leads to a violation o the overnment s solvency constraint, is no. From eqs (22), (65), and (69) we can inally derive the ollowin linearized relation o the imary deicit l ratio ( d ) and the debt ratio: 1 s (1 ) ( n)(1 )(1 ) l l d s (1 ) b (1 ) s s (1 )(1 ) b (102). Accordin to eq. (102), or a constant rate o interest on overnment debt at its steady state level ( r ), B an increase in the debt ratio implies a all in the imary deicit ratio in order to satisy the overnment s liquidity constraint. 28 How stability is aected i investment unctions, that are independent rom savin, are introduced in the Solow model is treated in Nikaido However, this is a study o disequilibrium dynamics in the sense o Hahn and Matthews 1964.