An equilibrium is in principle defined as in Sect.2.1but it differs a bit due to the fact that wages are not flexible now. Definition5determines an equilibrium.
Definition 5 An equilibrium is a sequence of variablesfC.t/;K.t/;B.t/g1tD0 and a sequence of pricesfw.t/;r.t/g1tD0such that
(a) equations (6) and (20) hold,
(b) equations (22), (23) and (27) withLd <Lhold and (c) equations (12) and (13) hold.
As regards long-run growth we define a balanced growth path analogously to the last section, that is a BGP is given if all variables grow at the same rate except for public debt possibly that may be constant or grow less than the other economic variables.
The growth rate of consumption is now obtained as CP
C D.1/.1˛/˛˛=.1˛/A˛=.1˛/.w=K/˛=.1˛/.Cı/ (29) where we used the optimality conditions (22) and (23).
The economy wide resource constraint is derived by combining the budget constraint of the household with that of the government as
KP K D Y
K C K Cˇ B
K C./ Y
K C.LLd/w
K ı ; (30)
Hence, the economy is completely described by Eqs. (13), (27), (29) and (30), with the return to capital,r, given by (22) and with Ld given by (23) and output determined by (21).
To get further insight into our model economy we define the new variablescWD C=K,xWDw=KandbWDB=K. Differentiating these variables with respect to time gives,6
P cDc
.Y=K/..1˛/.1/1.//Ccˇb.LLd/x
(31) xPDx
.1/Ld=LC.LLd/=L
.1/ .1.1˛// ˇb./.Y=K/Y=KC
6Again, we setAD1in the following analysis without loss of generality.
cCı.LLd/x
(32) bPDb
Y K
.1˛/.1/
b 1./
ˇCcˇb.LLd/x
(33) withY=KD.Ld/˛andLdD˛1=.1˛/x1=.˛1/.
4 Analysis of the Model
A rest point of (31)–(33) gives a situation where all endogenous economic variables grow at the same rate, except public debt in case the government runs a balanced budget or a slight deficit. The next proposition clarifies the question of existence of a BGP.
Theorem 4 Assume that the rate of time preference and the depreciation rate are sufficiently small and that Ld < L. Then, there exists a unique balanced growth path for
Cı
< 1
.1/ .1.1˛//: For
Cı
> 1
.1/ .1.1˛//
there exists no balanced growth path or there are two balanced growth paths.
Proof See Appendix.
Proposition 4 gives conditions such that a unique BGP exists. It can also be seen that the model may produce two BGPs or no BGP at all. Whether there is a unique BGP or whether there exist two BGPs depends on the rate of time preference and on labour market conditions. The government affects the outcome indirectly by its income tax policy and by determining unemployment payments since these variables influence the reservation wage of trade unions. However, public debt and deficit policies have no effect on that outcome. The latter also holds for the long-run allocation of resources which is the contents of Proposition5.
Theorem 5 Assume that there exists at least one BGP for the model with wage rigidity. Then, the debt policy of the government does not affect the balanced growth rate.
Proof See Appendix.
Proposition5demonstrates that public debt in our model is neutral in the sense that it does not affect the allocation of resources in the long-run and, thus, has no effect on economic growth. The reason for this result is that the marginal product of capital that determines the incentive to invest depends on labour demand in the case of rigid wages which, for its part, is a negative function of the wage rate relative to the capital stock,w=K D x. Since public debt policy does not affect that variable, it has no effect on the balanced growth rate. The wage rate is determined by trade unions maximizing the wage sum relative to a reference wage sum according to Eq. (24). Hence, the wage policy of trade unions does not only determine labour income but also economic growth.
Next, we study stability of the model with wage rigidity. As in the case of wage flexibility s stronger reaction of the primary surplus to higher public debt tends to stabilize the economy. Proposition6 gives the result assuming that the BGP is unique.
Theorem 6 In the case of permanent deficits, there exists one negative real eigenvalue forˇ sufficiently small. Forˇsufficiently large, there are two negative eigenvalues (or two eigenvalues with negative real parts).
Proof See Appendix.
This proposition shows that for a balanced government budget or for a slight deficit the economy is saddle point stable. Given a fixed initial ratio of public debt to capital and also a fixed value of the wage rate relative to capital, there exists a uniquely determined value for initial consumption such that the economy converges to the BGP in the long-run.
If the government runs permanent deficits that also holds provided the reaction of the government to public debt is sufficiently large, where sufficiently large means that it exceeds at least the rate of time preference of the household. In fact, numerical examples suggest that two negative eigenvalues occur if and only ifˇ > holds, although that cannot be proven for the analytical model. If that does not hold, i.e. if the reaction of the government to public debt is very small, there is only one negative eigenvalue. In this case, convergence to the balanced growth path is not given for a fixed initial public debt and wage rate.
To illustrate Proposition6 we compute a numerical example. To do so we set the labour share to60percent, i.e.˛ D 0:6, andA D 0:5. Total labour supply is LD1:25and unemployment benefits are50percent, D0:5. The income tax rate is10percent, D 0:1. The parameter reflecting the weight given to cumulated past income in the determination of the reference wage is set to1. The inverse of the labour supply elasticity is set to D0:1, the rate of time preference isD0:08 and the depreciation rate of capital is set toı D 0:1. Interpreting one time period as 2 years implies that the annual rate of time preference is about3:9percent and the annual depreciation rate about4:9percent. The balanced growth rate associated
with these parameter values isg D 0:0199which gives an annual growth rate of about1percent. The long-run unemployment rate amounts to4:7percent.
From Propositions5and6we know that the balanced growth rate is independent of public debt policy in the case of wage rigidity but public debt policy affects the dynamics. To see this we set D 0:01andˇ D 0:12andˇ D 0:0801which is larger than the rate of time preference which isD0:08. For these parameters there are two negative real eigenvalues. If we setD0:01andˇD 0:07andˇD 0:01 there is only one negative real eigenvalue such that this examples suggests that for values ofˇlarger (smaller) than the time preferencethere are two (one) negative eigenvalues.7
5 Conclusion
In this paper we have analyzed how public debt influences the long-run allocation of resources in a basic endogenous growth model with income taxation and unproductive public spending. Assuming that the inter-temporal budget constraint of the government must hold which implies that higher deficits must go along with higher primary surpluses, we could show that the long-run growth rate is the smaller the higher the debt to GDP ratio when wages are flexible. This holds for a fixed income tax rate and with non-distortionary public spending. The reason is that higher government debt implies a smaller shadow price of wealth that makes the household reduce its saving and labour supply.
In the case of rigid wages that give rise to unemployment public debt turns out to be neutral. This means that the long-run growth rate is independent of the debt and deficit policy of the government. The economic mechanism behind this outcome is that labour demand crucially determines the marginal product of private capital and, thus, the incentive to invest. If public spending is unproductive, variations in public debt that imply shifts in public spending have no effect on labour demand and on the return to private capital and, therefore, they do not influence long-run growth.
Further, we have seen that the economy is stable only if the government puts a sufficiently high weight on stabilizing public debt. If that does not hold the process of debt accumulation becomes explosive and, thus, the whole economy.
This underlines the importance of the government sector as regards the stability of market economies.
Acknowledgements I thank two referees for valuable comments on an earlier version that helped to improve the paper.
7Forˇ >.</ we must set < .>/ 0to getb> 0on the BGP.
Appendix
Proof of Proposition 1
To prove this proposition we setC=CP D PB=B;which must hold on a BGP in the case of permanent public deficits, giving
cˇ=.1ˇC/b1Dˇ .A:1/
Substituting this relation inb=bP gives,
b=bP Dc cˇ=.1ˇC/..1/.1˛/C =.ˇ//
From.A:1/we know thatb> 0implies thatandˇhave the same sign so that
=.ˇ/ > 0holds. With this, it is easily seen that the following relations hold,
c!lim0.b=b/P D 1; lim
c!1.b=b/P D C1; @.b=b/=@cP > 0:
This proves the existence of a uniquec?which solvesb=bP D0.
In case of a balanced budget or a slight deficit we haveb?D0since public debt is constant while the capital stock grows over time or grows less than capital and, in addition, we also haveD0(see the proof of Lemma7). Using this, the equation c=cP can be written as
c=cP D
cc˛=..1˛/C/ .1/.1.1˛//
;
It is easily seen that the following relations hold,
c!lim0.c=c/P D 1; lim
c!1.c=c/P D C1; @.c=c/=@cP > 0:
This proves the existence of a uniquec?which solvesc=cP D0.
Proof of Proposition 2
SettingCP=CD PB=Bimplies
cˇ=.1ˇC/ˇbD b Substituting in cˇ=.1ˇC/ˇbbybin (17) gives,
cc˛=..1˛/C/ .1/.1.1˛//bD0
From this we get by implicit differentiation:
dc
db D
1C.˛=..1˛/C//c1˛=..1˛/C/ .1/.1.1˛// > 0 Since a highercimplies a lower balanced growth rate the proposition is proven.
Proof of Lemma 7
To prove that lemma we see from (10) that setting D 0andˇ D .1/gives BP D 0(balanced budget scenario) and setting D 0and < ˇ < .1/gives 0 < B=BP < C=CP D g (slight deficit). Both of these debt/deficit policies imply b? D 0so thatbP D 0always holds. The balanced growth rate, then, is determined by the solution ofPcD0with respect toc. Sinceb?D0andD0hold both in the balanced budget scenario and for the slight deficit scenario, both scenarios imply the samec?and, therefore, the same balanced growth rate.
Proof of Proposition 3
To prove Proposition3, we first note that the balanced budget rule (rule (i)) implies b?D0since public debt is constant while the capital stock grows over time. Further, D0andˇD.1/rhold in this case.
The Jacobian of the dynamic system (17)–(18) is given by JD a11c.1/ ˛ c˛=.1˛C/
0 g
;
withcandbevaluated at the rest pointfc?; 0ganda11given by a11 DcC.1/.1˛/ .˛=.1˛C//c˛=.1˛C/:
The eigenvalues area11> 0;andg< 0so that the BGP is saddle point stable.
For slight deficits (rule (ii)) we also have b? D 0 and D 0. The reaction coefficient now is < ˇ < .1/rand the Jacobian J is obtained as,
JD a11 ˇc 0 ˇ
;
withcandbevaluated at the rest pointfc?; 0g,a11as above and where we used that C=CP D PK=Kholds on the BGP. Since < ˇholds for rule (ii) one eigenvalue is negative and one is positive implying saddle point stability, which proves the first part of the proposition.
In the case of permanent deficits (rule (iii)), the Jacobian matrix evaluated at the rest point of (17)–(18). The Jacobian is given by
JD a11 ˇc
a21 c˛=.1˛C/b1ˇb
;
withcandbevaluated at the rest pointfc?;b?ganda11anda21given by a11Dc
1C.˛=.1˛C// c1˛=.1˛C/.C.1/.1˛/
a21Db
1C.˛=.1˛C// c1˛=.1˛C/..1Cb1/C.1/.1˛//
The determinant of the Jacobian matrix can be computed as detJD.ˇ/
cC.˛=.1˛C// c˛=.1˛C/.1/.1˛/
C .˛=.1˛C// c˛=.1˛C/
FromCP=C D PB=B;which must hold on a BGP, we have c˛=.1˛C/b1 D ˇ. Using this we can rewrite the determinant as follows,
detJD.ˇ/
cC.˛=.1˛C// c˛=.1˛C/.1/.1˛/
C .˛=.1˛C//.ˇ/b
Forˇ > the determinant is negative sinceb> 0holds.
Forˇ < the determinant is positive. To show that the BGP is unstable we have to compute the trace of the Jacobian, trJ;which is given by,
tr JDcˇbC.˛=.1˛C// c˛=.1˛C/.C.1/.1˛//
C b1c˛=.1˛C/:
To see that trJis positive we first note that a positive value ofbimplies > 0for ˇ < . Further, fromc=cP D0we getcˇbDCc˛=.1˛C/.C.1/.1
˛// > 0;so that the trace of the Jacobian is positive, too. Since the trace and the determinant are both positive, the BGP is unstable forˇ < .
Finally, we note that on the BGP we have from (14) the relationD.1/rg.
Thus, Proposition3is proven.
Proof of Proposition 4
To prove that proposition we setPx=xD 0and solve that equation with respect toc giving
cD .1/Ld=LC.LLd/=L
.1/ .1.1˛// CˇbC./.Ld/˛C.Ld/˛ıC.LLd/x;
withLd D˛1=.1˛/x1=.˛1/. Inserting thatcincPleads to the following equation that we denote byf:
f D.1˛/.1/ ˛˛=.1˛/x˛=.˛1/
.1/ .1/.1.1˛//
˛x1=.˛1/=L.Cı/C .1=..1/.1.1˛////:
A solutionf D0with respect toxgives a rest point of (31)–(33). As regardsf we have
x!lim0f D 1; lim
x!1f D .Cı/C
1
.1/.1.1˛//
:
The first derivative off is given by
@f
@x D ˛ 1˛
x1C˛=.˛1/˛˛=.1˛/.1˛/.1/C 1
1˛
x1C1=.˛1/
.1/ .1/.1.1˛//
˛=L:
The second derivative off is
@2f
@x2 D ˛
.1˛/2 x2C˛=.˛1/˛˛=.1˛/.1˛/.1/C .1/ .2˛/
.1˛/2 x2C1=.˛1/
.1/ .1/.1.1˛//
˛=L:
Setting@f=@xD0gives
xDxmD ˛
˛L˛˛=.1˛/.1˛/.1/
.1/ .1/.1.1˛//
:
Insertingxmin@2f=@x2 shows that the sign of the resulting expression is equiva- lent to
˛ ˛˛=.1˛/.1˛/.1/ < 0:
This demonstrates that the functionf reaches a maximum forx D xmand it has a unique turning point given by
xDxwDxm .1C.1˛// :
Thus, the functionfis concave-convex, starts at1, reaches a maximum atxDxm, has a turning point atxDxwand converges to.Cı/C.1=..1/.1.1
˛////forx! 1. This implies that for.Cı/C.1=..1/.1.1˛//// > 0 there exists a unique rest point of (31)–(33) and for.Cı/C.1=..1/.1 .1˛//// < 0there exist two rest points for Eqs. (31)–(33) or no rest point iff
does not intersect the horizontal axis.
Proof of Proposition 5
We know that a BGP is given for a value of x such that the function f in Proposition4 equals zero. Looking at f it is immediately seen that this function does neither depend on the ratio of public debt to capital,b, nor on the parametersˇ
and.
Proof of Proposition 6
We compute the Jacobian matrix evaluated at the rest point of (31)–(33). For the balanced budget (rule (i)) and for the slight deficit (rule (ii)), the Jacobian is given by
JD 2 4c c
@.CP=C/=@x@.KP=K/=@x ˇc x x
@.w=w/=@xP @.K=K/=@xP ˇx
0 0 a33
3 5;
wherecandxevaluated at the BGP and where we have used thatbD0holds on the BGP with a balanced government budget or a slight deficit. In case of a balanced budget we havea33 D PK=Kand for the slight deficita33 Dˇ < 0, since the slight deficit is obtained for0 < < ˇ < .1/r. The eigenvalues of that matrix are given by PK=KD g< 0(balanced budget) and byˇ < 0(slight deficit)
and by the eigenvalues of the matrixJ1which is J1 D c c
@.C=C/=@xP @.K=K/=@xP x x
@.w=w/=@xP @.K=K/=@xP The determinant of that matrix is obtained as
detJ1Dcx
@.wP=w/=@x@.CP=C/=@x
D.1/c x.@f=@x/;
withffrom the proof of Proposition5. With a unique rest point of (31)–(33),fhas a positive derivative atf D0implying that the determinant is negative so thatJ1has one negative and one positive eigenvalue.
In case of permanent deficits (rule (iii)) we have
J2D 2 4c c
@.C=C/=@xP @.K=K/=@xP
ˇc x x
@.w=w/=@xP @.K=K/=@xP
ˇx b b
@.BP=B/=@x@.KP=K/=@x
ˇbC.ˇ/
3 5;
The determinant is given by
detJ2D.1/c x.ˇ/ .@f=@x/
and the trace, trJ2, is trJ2DcCx
@.w=w/=@xP @.K=K/=@xP
C.1/.ˇb.ˇ//DC1.ˇ/; .A:1/
withC1containing terms that are independent ofˇand independent ofcandbthat are determined byˇon the BGP.
SettingˇD0we can explicitly compute the eigenvaluesevi,iD1; 2; 3as ev1D; ev2;3D.1=2/
trJ1˙q
.trJ1/24detJ1
;
withJ1as for the balanced budget that has one positive and one negative eigenvalues.
The determinant of the JacobianJ2 is negative in this case. This shows that two eigenvalues ofJ2are positive and one is negative. For reasons of continuity this also holds in anenvironment aroundˇD0.
Ifˇis sufficiently large, i.e. at least larger than, the determinant of the Jacobian J2is positive, since.@f=@x/ > 0. Because of detJ2Dev1 ev2 ev3we know that in this case there is no negative eigenvalue or two negative eigenvalues (or eigenvalues with negative real parts in case of complex conjugate eigenvalues). Further, we know that we have for the trace, trJ2 D ev1Cev2 Cev3. From.A:1/we see that trJ2 monotonically declines withˇso that it becomes negative for a sufficiently largeˇ implying that there is at least one negative eigenvalue. But, because of detJ2 > 0 there must be two negative eigenvalues in that case.
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Change
Elmar Hillebrand
Abstract This paper develops an endogenous directed technical change growth model with financial intermediation. Technical change is driven by R&D invest- ments of private agents in response to market incentives and can take different directions. Key feature is that innovators are capital constrained and need external funds to finance R&D effort. Financial intermediaries finance these ventures.
The main theoretical result shows that credit interest rates—a “risk effect”—add to the determinants of directed technical change: Beside the price and the market size effect, the risk effect encourages innovations in those sectors, where the risk of innovation failure is lower. The degree of substitutability regulates the power of these different effects and determines how innovations respond to changes in relative factor supply, given that the risk effect is an additional determinant of directed technical change.
1 Introduction
There is little doubt that technical change is a key driving force of economic growth. Less clear, however, is the role of financial intermediation in the process of innovations and technical change: “pre-crisis” macroeconomics with regard to financial intermediation and economic growth offer deviating positions. Somewhat oversimplifying existing approaches, we can summarize these as follows. First, Some economists suggest that financial markets have at the most a minor relevance in the sense that the development in the financial sector follows“real”-sector activity (Robinson1952) or that macroeconomic research overestimates the role of financial development when exploring the determinants of economic growth (Lucas1988).
Second, other economic researchers go one step further and proclaimed their view on financial matters with respect to economic growth by simply ignoring it (Chan- davarkar1992; Meier and Seers1984; Stern1989). Third and last, some researchers assign financial intermediation an active role that cannot be ignored when exploring
E. Hillebrand
EEFA Research Institute, Muenster, Germany e-mail:e.hillebrand@eefa.de
© Springer International Publishing AG 2017
B. Bửkemeier, A. Greiner (eds.),Inequality and Finance in Macrodynamics, Dynamic Modeling and Econometrics in Economics and Finance 23, DOI 10.1007/978-3-319-54690-2_6
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