- Email: [email protected]
Indeterminacy under non-separability of public consumption and leisure in the utility function
Economic Modelling 21 (2004) 409–428
Indeterminacy under non-separability of public consumption and leisure in the utility function ´ ´ Ruiz Esther Fernandez, Alfonso Novales*, Jesus Departamento de Economia Cuantitativa, Universidad Complutense de Madrid, Campus de Somosaguas, Madrid 28223, Spain Accepted 18 June 2003
Abstract In a one sector growth model with public consumption in the utility function, the competitive equilibrium can be indeterminate for plausible values of the elasticity of intertemporal substitution of consumption, under constant returns to scale and endogenous government expenditures. Non-separability between public consumption and leisure in the utility function is crucial for this result. 䊚 2003 Elsevier B.V. All rights reserved. JEL classifications: E32; E62; D90 Keywords: Indeterminacy; Multiple equilibria; Externalities; Non-separable utility; Public consumption
1. Introduction We characterize conditions on preferences under which competitive equilibrium can be indeterminate in a single sector growth model with public consumption in the utility function, constant returns to scale in production and endogenous public expenditures financed through a constant rate income tax. General equilibrium models that display indeterminacy have been the focus of attention in recent years. Indeterminacy implies that there will be multiple paths converging to a given steady state. Hence, indeterminacy guarantees existence of a continuum of sunspot stationary equilibria, i.e. stochastic rational expectations *Corresponding author. Fax: q34-91-394-26-13. E-mail address: [email protected] (A. Novales). 0264-9993/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 0 3 . 0 0 0 3 8 - 5
410
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
equilibria determined by perturbations unrelated to the uncertainty in economic fundamentals. The interest of sunspot equilibria is that they provide a theoretical justification for ‘animal spirits’ underlying economic instability. Characteristics that produce indeterminacy of equilibria in one- or multi-sector real business cycle models or in endogenous growth models have been widely studied. Initially, it was shown that when these models are extended to include either productivity externalities or some market imperfection, indeterminacy can arise if social returns to scale in production are sufficiently high, so that the labor demand curve has a slope which is not only positive, but also greater than that of the labor supply curve (see Benhabib and Farmer (1994) and Farmer and Guo (1994), for one-sector models). These models have been widely criticized because to produce multiple equilibria, they require larger returns to scale than observed in actual data (see Aiyagari (1995)). More recent work has described two situations in which increasing returns needed to produce indeterminacy are lower than initially thought, sometimes allowing for negatively sloped labor demand curves: (i) two sector economies with externalities in one sector (Benhabib and Farmer (1996), Peril (1998), Benhabib and Nishimura (1998) and Weder (1998), which introduces a durable consumption sector in addition to the investment and non-durable consumption sectors, or Barinci and ´ Cheron (2001) in an environment with financial constraints); (ii) one sector models with non-standard characteristics, such as non-separable preferences (Bennett and Farmer (2000)), or capacity utilization affecting production (Wen, 1998). These models are able to produce indeterminacy of equilibria with relatively low increasing returns to scale, but the elasticity of intertemporal substitution needed is often too large, relative to values considered in the real business cycle literature (Weder (1998) and Bennett and Farmer (2000), among others). This will generally lead to rather volatile consumption paths, against the observation in most actual economies. The same criticism applies to existing endogenous growth models producing indeterminacy (see Benhabib and Perli (1994) and Xie (1994)). We show that if preferences of the representative agent are non-separable in public consumption and leisure, there are conditions under which competitive equilibrium is indeterminate for standard values of the elasticity of intertemporal substitution and constant returns to scale in production. If, in addition, public and private consumption enter as a composite good in the utility function, indeterminacy is compatible with labor supply (defined as in Benhabib and Farmer (1994), i.e. holding constant consumption) and demand curves having the standard slopes. As for the most part of the literature on indeterminacy, our model is subject to an externality, public consumption entering into the utility function. This extension can be justified from the point of view that to design fiscal policy, it is important to bear in mind the effect that public consumption may have on private consumption and work-leisure decisions. In this respect, there is some empirical work that provides evidence on complementary or substitutability between private and public consumption (Ni (1995), among others). Schmitt-Grohe´ and Uribe (1997) also characterize conditions for indeterminacy of equilibria under constant returns to scale without including public consumption
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
411
in the utility function. These authors show that under a constant level of public expenditures and endogenous proportional income taxes that balance the budget every period, indeterminacy arises for a range of values of the tax rate and the elasticity of intertemporal substitution consistent with those in the real business cycle literature. However, indeterminacy in their economy would be precluded under fixed income tax rates and endogenous government expenditures. Cazzavillan (1996) studies indeterminacy in a simple one sector model with public expenditures entering into the utility and production functions. The presence of both externalities is crucial to get indeterminacy in his economy, which displays endogenous growth. Hence, the contribution of our work is to show that there can be indeterminacy in a simple neoclassical growth model with only a source of externality: public consumption entering in the utility function non-separably with leisure. Relative to fiscal policy, we assume that (i) public consumption is financed through distorting taxes (a proportional tax on labor and capital income); and (ii) the government budget balances every period. Our economy can display indeterminacy for fixed income tax rates, plausible values of the intertemporal elasticity of substitution of consumption, constant returns to scale and endogenous government expenditures. Furthermore, (a) the region of the parameter space producing indeterminacy is independent of the income tax rate, and (b) the elasticity of the constant-consumption labor supply function with respect to the level of public expenditures can have either sign. Finally, our indeterminacy condition can be interpreted in a similar manner to Bennett and Farmer (2000): the economy displays indeterminacy if the Frisch labor supply curve (holding constant marginal utility of consumption) crosses the labor demand curve with the ‘wrong slope’. The model is described in Section 2. In Section 3 we characterize the transitional dynamics of the model and the conditions for indeterminacy. In Section 4 we provide an economic interpretation of the condition that generates indeterminacy. We provide a numerical example in Section 5, and the paper closes with some remarks in Section 6. 2. The economy 2.1. The consumer’s problem There is a continuum of identical consumers, all with the same preferences defined on private and public consumption and leisure, according to the utility function:
UŽc,g,n.s
Žcqfg.1yr 1yr
yg1yrn1qx, r)0, r/1,
(1)
where c is private consumption, g public consumption and n is the supply of
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
412
labor.1 To simplify notation, time dependence of all variables is suppressed. The parameter fG0 determines the degree of substitution between private and public consumption (as in Ni (1995)), xG0 defines the elasticity of labor supply (as in Benhabib and Farmer (1994)) and r is the inverse of the elasticity of intertemporal substitution of consumption, defined in a broad sense (Cscqfg). The possibility that rc1 allows for public consumption to either be complement or substitute for leisure. The single period utility function in Eq. (1) is homogeneous of degree 1yr in c and g. In our model, this condition allows for the existence of a balanced growth path although, for simplicity, we do not consider growth in the economy.2 Each consumer receives income on labor and capital, which can be used to consume, save, and pay taxes at a constant rate tg(0,1) on the two sources of income: ˙ cqkqdks Ž1yt.Žvnqrk.,
(2)
where v is the real wage and r is the return on capital (k). Hence, the representative consumer faces the optimization problem: Max
|
`
eybtU(c,g,n) dt
(3)
0
subject to Eq. (2) and given k0, with b)0 being the time discount. First order conditions for this problem are:
Žcqfg.yrsl,
(4)
Ž1qx.g1yrnxslŽ1yt.v,
(5)
l˙ y sŽ1yt.ryŽdqb., l
(6)
lim eybtls0,
(7)
t™`
l being the multiplier associated to the consumer’s budget constraint. Combining Eqs. (4) and (5): vs
1qx Žcqfg.rg1yrnx, 1yt
(8)
1 Farmer (1997) considers a similar class of utility functions, which includes real balances instead of public consumption. 2 Farmer (1997) showed that, in the case of a monetary model with real balances, MyP, in the utility function, the condition that allows for the possibility of balanced growth is that U(c,MyP,n) be homogenous in c and MyP.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
413
so that the marginal rate of substitution between private consumption and leisure must be equal to the after tax real wage. Combining Eqs. (4) and (6) and using Cscqfg: ˙ 1w C s Ž1yt.ryŽdqb.z~, C ry x
|
(9)
so that consumption growth is every period proportional to the return on capital, net of taxes and depreciation. 2.2. Firms There is a continuum of identical firms operating in a competitive environment. For simplicity, we normalise their number to one. Aggregate production exhibits constant returns to scale: ysAkan1ya, agŽ0,1.
(10)
y being aggregate output and k, n the demand for the two production factors. Under perfectly competitive markets for the two factors, profit maximizing conditions are: y rsa , k
(11)
y vsŽ1ya. . n
(12)
2.3. Government The government chooses a tax rate t and balances its budget every period. Hence, the instantaneous government budget constraint is: gstŽvnqrk.,
(13)
where g represents government spending on goods and services that contribute to household’s utility. 2.4. Equilibrium Given t and k0, a competitive equilibrium is a set of trajectories {c,n,k,g,v,r} such that: (i) given the paths for {v,r}, {c,n,k} solve consumer’s problem wEqs. (2), (7)–(9)x, with Cscqfg; (ii) given {v,r}, {n,k} solve the firm’s problem
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
414
wEqs. (11) and (12)x; (iii) government budget constraint Eq. (13) holds every period; and (iv) markets clear, in particular, from Eqs. (2), (10) and (13): ˙ Akan1yascqŽkqdk .qg,
(14)
where gstAkan1ya.
(15)
3. Local dynamics Let us now discuss the local properties of the equilibrium dynamics of the system. A steady state is a vector (css kss, nss, lss, gss) satisfying the equations for competitive equilibrium such that if it is ever reached, the system will stay at that ˙ sssc˙ sssk˙ sssl˙ sss0.: point forever ŽC w
z
1
Ž1yt.Aa 1ya kss | , sx nss y dqb ~
(16)
css dŽ1ya.qb s , kss a
(17)
css dŽ1ya.qb 1yt s , gss dqb t
(18)
w
1
B
Eyrz 1qx
dŽ1ya.qb 1yt 1yt 1ya Cfq F nsssx 1qx D dqb t G y t lsssŽcssqfgss.yr,
| ~
,
(19) (20)
where Eq. (16) comes from Eqs. (6), (10) and (11); Eq. (17) refers to Eqs. (14)– (16); Eq. (18) is obtained from Eqs. (15)–(17) and (19) comes from Eq. (8) together with Eqs. (10), (12), (15), (16) and (18); and Eq. (20) is obtained from Eq. (4). All these equations are evaluated at steady-state. To characterize the local dynamics of the system, it is enough to analyze the transition of the state and co-state variables (k,l). That requires to write the control variables (c,n) as a function of (k,l). w
1 Ž1ya.Ž1yt.Ar arz 1qxyr Ž1ya. ns lk , tŽ1yr.Ž1qx. y ~
x
|
(21)
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428 wB
xC
csly1yryf
Ž1ya.Ž1yt. E
Ft
1qx
yD
xqa
1qx
aŽ1qx.
1ya
1ya
Ž1ya.
G
A
k
z
|
l
415
1ya 1qxyrŽ1ya.
(22)
,
~
where Eq. (21) comes from Eqs. (5), (10) and (12); and Eq. (22) is obtained from Eq. (4) together with Eqs. (15) and (21). The first order approximation to the dynamic system made up by Eqs. (6) and (14) is:
CF C F B˙ E
Bkyk
k
E
,
sG
˙G Dl
ss
DlylssG
with transition matrix G (see Appendix A):
C B
Gs
w
x
B
Ždqb.Ž1qx.C1qf
Ždqb.Ž1qx. B t E C1qf Fyd 1qxyrŽ1ya. D 1yt G
Bk
Bl
Ždqb.Ž1ya. y 1qxyrŽ1ya.
E
C
E
D
F
arŽ1qxyrŽ1ya..
D lss Gy
1qxyr C FŽ1ya.Ždqb. D G k 1qxyr Ž1ya. D ss ss
ss
F
zE
t E F 1yt G y
|
d r~
.
G
(23) Since one of the variables in the system is predetermined and the other is free, the system will have a unique, locally determined equilibrium when the steady state is a saddle point, which requires two characteristic roots of matrix G to be of opposite sign. Indeterminacy of equilibria arises only when the two roots of G are negative. This means that trace of G be negative, and its determinant be positive. For any given values of the vector (a,d,t,f,b), let w1(r), w2 (r) denote the linear functions:
w1Žr.sy1qŽ1ya.r, w
dyŽdqb.xaqf y
w2Žr.s
t z | 1yt ~
t bqŽdqb.f 1yt
y
dŽ1ya. t bqŽdqb.f 1yt
r.
Proposition 1. Det(G))0 if and only if x-w1(r). Proof. See Appendix A.
Proposition 2. Tr(G)-0 if and only if
xyw2Žr. xyw1Žr.
-0.
416
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
Proof. See Appendix A. Let r*,r** be the zeroes of w1(Ø) and w2(Ø). We have: 1 1 w dqb B t Ez Caqf F| , r**s x1y 1ya 1ya y d D 1yt G~ and r*)r** for all a,d,t,f,b. r*s
Proposition 3. Let (x0,r0) be a point in the (x,r)-plane with 0Fx0-w1(r0). x0yw2Žr0. Then -0. x0yw1Žr0. Proof. Since r* is the single root of w1(r)s0, which is an increasing linear function of r, 0Fx0-w1(r0)´r0)r*)r** and hence, w2(r0)-0, since r** is the single root of w2(r)s0, w2(Ø) being a decreasing linear function of r. This implies that x0yw2(r0))0, and, hence x0yw2Žr0. -0. x0yw1Žr0. Proposition 4. The competitive equilibrium is indeterminate in economy Eqs. (1)– (15), if and only if 0Fx0-w1(r0). Proof. Let us assume the competitive equilibrium is indeterminate. Then, Det(G)) 0 and Tr(G)-0. By Proposition 1, x0-w1(r0), while x0 is constrained to be nonnegative in Eq. (1). Conversely, let us assume that 0Fx0-w1(r0) Proposition 1, Proposition 2 and Proposition 3 then imply that Det(G))0 and Tr(G)-0 and hence, the competitive equilibrium is indeterminate.
Corollary 5. A necessary condition for indeterminacy of the competitive equilibrium 1 is r) 1ya Proof. This condition is needed for w1(r))0, and only then condition in Proposition 4 can hold. The previous results show that, in this economy: i. For values of a as usually calibrated in the real business cycle literature (approx. 0.35), indeterminacy can arise for values of the relative risk aversion parameter r)1.54, (i.e. intertemporal elasticity of substitution smaller than 0.65), a range of values consistent with existing empirical estimates. ii. fiscal policy parameters do not affect the conditions guaranteeing indeterminacy of equilibria, at a difference of Schmitt-Grohe´ and Uribe (1997). In their model, indeterminacy is possible in a range of intermediate steady-state tax rates. In particular, indeterminacy is not possible for tax rates in the decreasing part of
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
417
the Laffer curve and, as the authors point out, it is crucial for their results that labor income taxes be endogenous. In the economy, we consider, indeterminacy can arise for any value of the income tax rate and hence, for any size of the public sector, defined as the ratio of government expenditures to output. The goverment cannot do anything to stabilize the economy. Guo and Lansing (1998) show indeterminacy in a one-sector real business cycle under increasing returns to scale with a tax policy similar to ours, but without public consumption in the utility function. They consider alternative parameterizations allowing for regressive, proportional and progressive taxes, showing that progressive taxes may act as an economic stabilizer, precluding indeterminacy of equilibria. These same fiscal rules could be used in our framework to discuss whether they could be used to stabilize the economy, in the same sense of these authors. Non-separability of public expenditures and leisure in the utility function is crucial for the previous results, as the next proposition shows. Žcqfg.1yr 1qx yn , r)0, r/1, xG0, fG0, be an utility funcLet V(c,g,n)s 1yr tion similar to Eq. (1) except for the fact that it is separable in public consumption and leisure. Proposition 6. If preferences of the representative agent in the economy are represented by utility function V(c,g,n), there is no point in the parameter space for which equilibrium is indeterminate. Proof. See Appendix A.
4. Indeterminacy and the labor market Most work discussing indeterminacy of equilibria in one sector models shows that it is associated with upward sloping demand curves, downward sloping supply curves, or both. The main result of Bennett and Farmer (2000) is that the Benhabib– Farmer condition wBenhabib and Farmer (1994)x, that labor demand and supply curves cross with the ‘wrong slopes’ generalizes to the case when preferences are non-separables in private consumption and leisure. However, Bennett–Farmer show that it is the Frisch labor supply curve (i.e. holding constant the marginal utility of consumption3) rather than the constant–consumption labor supply curve, which should be considered in this analysis. In this section, we relate our conditions for indeterminacy to the slopes of labor demand and both labor supply curves. In our model, under the utility function U(c,g,n), the Frisch labor supply curve and the constant–consumption labor supply curve differ. The Frisch labor supply 3
The use of tins term follows Bennett and Farmer (2000) and Browning et al. (1985).
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
418
curve (from Eq. (5)) is: vs
1qx
Ž1yt.l
g1yrnx,
(24)
and using again Eq. (15) to eliminate g from Eq. (24), we get the reduced-form Frisch labor supply curve: vs
1qx ŽtAka.1yrnxqŽ1yr.Ž1ya., Ž1yt.l
(25)
having a slope in the neighborhood of the steady state: B dv Es
C D
F s
dn GF
vss w xqŽ1yr.Ž1ya.z~, nss y x
|
which will be negative, when r)
(26) 1 (which implies r)1) and (ry1)(1ya)) 1ya
x, being positive otherwise. From Eq. (12), the slope of the labor demand curve in the neighborhood of steady state is: B dv Ed
C D
v F sya ss , dn G nss
which is always negative. Proposition 7. The competitive equilibrium is indeterminate in our economy if and only if the reduced-form Frisch labor supply curve is downward-sloping, and steeper than the labor demand curve. Proof. From Proposition 4 and its corollary, if equilibrium is indeterminate we have 1 r) and x-y1qr(1ya). Since (ry1)(1ya))y1qr(1ya) for all a, r, 1ya the labor supply curve slopes downward. It will also be steeper than the labor demand curve, since the former condition also implies: (ry1)(1ya))yx)a. Conversely, if the latter condition holds, then x-y1qr(1ya) and equilibrium is indeterminate, by Proposition 4. Hence, the appropriate condition for indeterminacy of the competitive equilibrium to arise in the economy, is that the Frisch labor supply curve crosses the labor demand curve with the ‘wrong slope’, the former being steeper (i.e. the Benhabib– Farmer condition holds).
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
419
The intuition behind the existence of stationary sunspot equilibria are as follows. If the private agents expect future public consumption to be above average, then they will also expect higher future income taxes. Future hours worked for any given capital stock will then be expected to increase because of Eq. (25), raising the expected marginal product of capital. The increase in the expected return to capital leads to an increase in current labor supply and output. Since the income tax rate is fixed, current public consumption will also be higher. Thus, expectations of aboveaverage public consumption of next period lead to a higher current public consumption. Under the V(c,g,n) utility function, it is easy to see that the Frisch labor supply curve does not depend on the public consumption, and the previous argument breaks down. This is consistent with Proposition 6, which shows that, under V(.), our model does not display indeterminacy. Therefore, the non-separability of public consumption and leisure in the utility function plays a key role in generating expectations-driven fluctuations in this economy. Under the U(c,g,n) utility function, the constant-consumption, reduced form labor supply curve can be obtained using Eq. (15) to eliminate g from Eq. (8): vs
Er 1qx B c Cfq F tAkan1yaqx, 1yt D tAkan1ya G
(27)
having a slope, in the neighborhood of the steady state: B dv Es
C D
Fs
dn G
w
x
vss Ž1yaqx.q nss
rŽay1. 1qf
y
tAŽkss ynss.Žay1. css ykss
z
|
.
(28)
~
Proposition 8. The constant-consumption, reduced form labor supply curve wEq. (27)x has a negative slope if and only if x-w3(r), where w3(r)sŽ1ya.= w z r y1 . t dqb 1qf 1yt dŽ1ya.qb y ~ Proof. Easy to show from Eqs. (16), (17) and (28).
x
|
Corollary 9. A necessary condition for the constant-consumption, reduced form labor supply curve wEq. (27)x to have a negative slope is that r)w4s1q t dqb f )1. 1yt dŽ1ya.qb Proof. Only if this inequality holds we will have a non-trivial range of values for x in Proposition 4. The next two propositions shows that if public consumption enters together with private consumption as a composite good in the utility function then, contrary to
420
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
what happens to the Frisch labor supply curve, the reduced-form labor supply curve obtained holding constant consumption can be upward-sloping when equilibrium is indeterminate. If public consumption enters the utility function just through the interaction with leisure, then the same necessary and sufficient conditions for indeterminacy apply to the labor supply curve obtained holding constant consumption than to the Frisch labor supply curve. This is because, as pointed out in Bennett and Farmer (2000), when preferences are separable in consumption and leisure, both supply curves coincide. This latter case is an illustration of the fact that nonseparability between private consumption and leisure is not needed for indeterminacy to arise with a downward sloping labor supply curve. Proposition 10. If f)0, there is a non-trivial set in the parameter space for which we have indeterminacy of equilibria with an upward sloping reduced form, constantconsumption, labor supply curve wEq. (27)x. ≠w1(r) s1ya) Proof. The previously defined w1(r) function satisfies: ≠r 1ya ≠w3(r) s , for all f,t,d,b,a and w1(0)-w3(0), since t dqb ≠r 1qf 1yt d(1ya)qb w1(0)sy1 and w3(0)sy(1ya). Being w1 (.) and w3(.) linear functions of r, they ˜ have a single intersection at a given r)0 , with r)r˜ ´w3(r)-w1(r). Hence, ˜ indeterminacy pairs (r,x) with r)r and w3(r)-x-w1(r) produce an upward sloping reduced form, constant-consumption, labor supply curve, while for those with x-w3(r) that labor supply curve is downward sloping. If there is any indeterminacy pair (r,x) with r-r˜ , the labor supply curve will be downward sloping.
Proposition 11. If fs0, there is no equilibrium indeterminacy with an upward sloping labor supply curve obtained holding constant consumption. The competitive equilibrium is then indeterminate if and only if the constant-consumption labor supply curve wEq. (27)x is downward sloping, but steeper than the labor demand curve. Proof. In this case, the Frisch labor supply curve and the constant-consumption labor supply curve are the same wsee Eqs. (26) and (28) under fs0x, and Proposition 7 shows the result. Our final result shows that requiring non-separability between public and private consumption in the utility function (f)0) to obtain indeterminacy of equilibria under a positively sloped constant-consumption labor supply curve, is not restrictive from the point of view of the sign of the elasticity of the labor supply function, relative to public consumption. That is, in an indeterminate competitive equilibrium with upward sloped labor supply curve an increase in public consumption could shift the labor supply curve either to the left or to the right.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
421
1 )w4 then 1ya the elasticity of the constant consumption labor supply curve wEq. (27)x relative to public consumption is always positive. When the competitive equilibrium is 1 -w4, the elasticity of the constant consumption labor supply indeterminate and 1ya curve relative to public consumption is positive, if r)w4, being negative if 1 -r-w4. 1ya Proof. Differentiating in Eq. (8) with dvsdcs0, we get, in the neighborhood of the steady state: Proposition 12. If the competitive equilibrium is indeterminate and
´n,gs
dn gss rf 1 ry1 rf 1 ry1 sy q sy q , dg nss x css ygssqf x x 1yt dŽ1ya.qb x qf t dqb
where we have used Eq. (18) to reach the last equality. Hence, ´n,g,0 if and only 1 if r,w4. If )w4, under indeterminacy of equilibria we will have, from 1ya 1 1 )w4, and consequently, ´n,g)0. If -w4-r, we will Corollary 5: r) 1ya 1ya 1 -r-w4, we will have ´n,g-0. again have ´n,g)0, while if 1ya 5. A numerical example In this section, we provide a numerical illustration of our analytical results identifying, for a given parameterization, the (x, r)-regions where the competitive equilibrium is either determinate or indeterminate. Parameter values are: bs0.01, as0.35, fs0.9 (the latter in the range estimated by Ni, 1995), ts0.2, As1, ds 0.0252. Assuming quarterly data, these values are compatible with annual depreciation of 10% and a steady-state real interest rate of 4%. In Fig. 1a we also identify (x,r)-pairs for which the constant-consumption labor supply curve is positively sloped. Fig. 1b and c reproduce Fig. 1a for alternative parameter values. In Fig. 1b, the only difference is fs0.6. In Fig. 1c we use the same values as in Fig. 1a except for the tax rate, which is taken to be ts0.4. Note that the indeterminacy condition is the same in the three graphs in Fig. 1 since it is independent of f and t. Comparing the three figures, we see that the bigger the substitution between private and public consumption (f) and bigger the tax rate, the larger the set of (r,x)-combinations compatible with indeterminacy of equilibria and an upward sloping labor supply curve. Finally, Fig. 2 shows, among the (r,x)-combinations for which there is indeterminacy of equilibria, those for which the two eigenvalues are complex, implying an
422
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
Fig. 1. Paramater regions for indeterminancy of equilibria and for a positively sloped labor supply curve. Points below each line satisfy the corresponding conditions. The shaded region presents the (r,x)-pairs for which equilibrium is indeterminate and the reduced-form labor supply curve obtained holding constant consumption has the standard slope.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
423
Fig. 2. (r,x)-combinations for which competitive equilibrium is either determinate, indeterminate, or indeterminate with oscillatory dynamics.
endogenous mechanism for the propagation of business cycles. Large values of either x or r (x)0.31, r)3.92) preclude this possibility. Parameter values are the same as in Fig. 1a. 6. Conclusions For a class of utility functions characterized by non-separability between public consumption and leisure, we have shown that the competitive equilibrium can be indeterminate for plausible values of the intertemporal elasticity of substitution, under endogenous government expenditures (financed through a fixed income tax rate) and constant returns to scale in production. The region in the parameter space leading to indeterminacy is characterized by: i. it is compatible with the constant-consumption labor supply curve and the labor demand curve having the standard slope signs, provided that private and public consumption are substitutes. However, the Frisch labor supply curve is downward sloping whenever there is indeterminacy; ii. the intertemporal elasticity of substitution must be below the labor income share on output, which is usually calibrated in the literature to be approximately 0.35. Equivalently, the relative risk aversion parameter must be above the inverse of that share. This condition only excludes values for the risk aversion parameter below 1.54, indeterminacy being a possibility for values above that threshold; iii. the tax rate and hence, the size of the public sector, measured by the participation of public consumption on output, does not enter the condition characterizing
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
424
indeterminacy. This means that the government cannot do anything to stabilize this economy, contrary to the result reached by Schmitt–Grohe´ and Martin Uribe in a different environment under constant government expenditures. In our case, a bigger public sector only increases the likelihood that indeterminacy will arise with the standard slopes for the constant-consumption labor supply and demand curves. 7. Uncited enunciations Proposition 8, Proposition 10, Proposition 11, Proposition 12 and Corollary 9 Appendix A: Part 1: Deriving the elements of G: From Eqs. (14) and (15) we get: ˙ Ž1yt.AkanŽk,l.1yaydkycŽk,l.. ks
(29)
where c(k,l) is given by Eq. (22) and n(k,l) is given by Eq. (21), while from Eqs. (6), (10) and (11) we get: w ay1 ˙ lsyl nŽk,l.1yayŽdqb.z~. yŽ1yt.aAk x
(30)
|
The dynamics of system Eqs. (29) and (30) in (k,l) around steady-state can be characterized through the linear approximation:
CF C B˙ E
k
BG
11
G12EBky kss E
FC F
s
˙G Dl
DG21
(31)
G22GDlylssG
with: Bk
G11sŽ1yt.AaC
ss
Eay1
F
D nss G Bk
G12sŽ1yt.Ž1ya.AC
Bk
ydqŽ1yt.Ž1ya.AC
ss
Ea ≠n Z
F
D nss G
ss
Ea ≠n Z
F
D nss G
≠c Z Z y Z ≠k Zss ≠k Zss Z
Z
Z
Z
≠c Z Z y Z , ≠l Zss ≠l Zss Z
Z
Z
Z
w
x
ay2 1ya G21sylssŽ1yt.aA Žay1.kss nss qŽ1ya.kay1 nya ss ss y
(32)
(33) ≠n Z z Z , ≠k Zss Z Z
| ~
(34)
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
ay1 ya G22sylssŽ1yt.aAŽ1ya.kss nss
≠n Z Z , ≠l Zss Z
425
(35)
Z
≠x Z Z , xsn,c and zsk,l, denoting partial derivatives, evaluated at steady state. ≠z Zss From Eqs. (21) and (22), and using Eqs. (15), (16) and (20), the partial derivatives in Eqs. (32)–(35) are: with
Z Z
≠n Z 1 nss , Z s ≠l Zss D lss Z Z
z ≠c Z 1 w B 1qx E 1qx Fy fgss|, xC55C1yay Z s ≠l Zss lssD y D r G r ~ Z Z
≠n Z ar nss , Z s ≠k Zss D kss Z Z
≠c Z t Ždqb.Ž1qx. , Z syf ≠k Zss 1yt D Z Z
where Ds1qxyr(1ya)sxyw1 (r). B k Eay1 dqb ss Using Eq. (16) to eliminate C F s , together with these expresD nss G Ž1yt.Aa sions, yields: rz t Ždqb.Ž1qx. D nss G D~ 1yt D y Ždqb.Ž1qx. t Ždqb.Ž1qx. Ždqb.Ž1qx. B t E C1qf Fyd s ydqf s D D 1yt D D 1yt G Bk
G11sŽ1yt.AaC
ss
Eay1w
x1qŽ1ya. |ydqf
F
and using Eqs. (17) and (18) to eliminate Bk
G12sŽ1yt.Ž1ya.AC
ss
Ea
F
D nss G
css gss , , we have: kss kss
1 nss 1 w B 1qx E 1qx z Fyf y gss| xcssC1yay D lss Dlss y D r G r ~
s
kss 1 w 1ya 1 css 1qx gss z f | x Ždqb.q D q lss D y a r kss r kss ~
s
kss 1 w 1ya 1 dqbyad 1qx dqb t z q f x |, Ždqb.q D lss D y a r a r a 1yt ~
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
426
and, combining terms in dqb: B
Ždqb.Ž1qx.C1q D
ft E FyadD 1yt G
G12s
kss 1 lss D
G21s
lss 1qxyr lss Ždqb.Ž1ya. s Ž1yt.aAŽ1ya.Žkss ynss.ay1 Ž1qxyr., kss D kss D
G22sy
,
ar
Ždqb.Ž1ya. D
(36)
.
Part 2: Proof of Proposition 1: Denote FsŽdqb.f
t )0 and N1s(dq 1yt
b)(1ya))0, B
DetŽG.sG11G22yG12G21syC
ŽdqbqF.Ž1qx.
D
D
B
ŽdqbqF.Ž1qx.
yC D
arD
E
ydF
N1 G D E
y
d N1 F Ž1qxyr. rG D
B
E N 1qx N1 ŽdqbqF.Ž1qx. 1qx Fsy 1Ž . wydŽ1ya.qbqFz~, sy C yd DD ar r G arD x
|
so that Det(G))0 if and only if D-0, i.e. if and only if x-w1(r). Part 3: Proof of Proposition 2: First, notice that: xyw2Žr.sxy
1 w dyŽdqb.ayFydŽ1ya.rz~. Fqb y x
|
Then, TrŽG.sG11qG22s s
1w ŽdqbqF.Ž1qx.ydŽ1qxyrŽ1ya..yŽdqb.Ž1ya.z~ Dy x
|
1w FqŽbqF.xyd Dy x
xyw2Žr. 1 qdrŽ1ya.qŽdqb.az~s Žxyw2Žr..ŽFqb.sŽFqb. , D xyw1Žr. |
so that Tr(G)-0 if and only if
xyw2Žr. xyw1Žr.
- 0.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
427
Part 4. Proof of Proposition 6: Proof: For utility function V(c,g,n), the control variables (c,n), as functions of (k,l) are given by: w
x
ns
y
Ž1ya.Ž1yt. 1qx
z
|
1 xqa
Alka
,
~
w
x
csly1yryft
Ž1ya.Ž1yt. 1qx
y
z
1qx
1qx
|
A 1ya ka 1ya l
1ya xqa
,
~
which are the analogue of Eqs. (21) and (22), with partial derivatives: ≠n Z a nss , Z s ≠k Zss xqa kss Z Z
≠c Z a ly1yr ycss ss , Z syŽ1qx. ≠k Zss xqa kss Z Z
≠n Z 1 nss , Z s ≠l Zss xqa lss Z Z
Bk
and, again, C
ss
Eay1
F
D nss G
G11sbqŽ1ya. G22syŽ1ya.
s
dqb , so that Eqs. (32) and (35) become: (1yt)Aa
1 a ly1yr ycss ss , Ždqb.qŽ1qx. xqa xqa kss
1 Ždqb., xqa
a y1yr =Žlss ycss.. But, from Eq. (20), optixqa y1yr ycss)0. Hence, Tr(G))0 for any feasible set of mality implies: fgssslss parameter values, without the possibility of having indeterminacy. and, TrŽG.sG11qG22sbqŽ1qx.
References Aiyagari, S.R. 1995. The econometrics of indeterminacy: an applied study, a comment. CarnegieRochester Conference Series on Public Policy 43, 273–282. ´ Barinci, J.P., Cheron, A., 2001. Sunspots and the business cycle in a finance constrained economy. J. Econ. Theory 97, 30–49. Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. J. Econ. Theory 63, 19–41.
428
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
Benhabib, J., Farmer, R.E.A., 1996. Indeterminacy and sector specific externalities. J. Monit. Econ. 37, 421–443. Benhabib, J., Nishimura, K., 1998. Indeterminacy and sunspots with constant returns. J. Econ. Theory 81, 58–96. Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: transitional dynamics with multiple equilibria. J. Econ. Theory 63, 113–142. Bennett, R.L., Farmer, R.E.A., 2000. Indeterminacy with non-separable utility. J. Econ. Theory 93, 118–143. Browning, M.J., Deaton, A., Irish, M., 1985. A profitable approach to labor supply and commodity demands over the life-cycle. Econometrica 53, 503–543. Cazzavillan, G., 1996. Public spending, endogenous growth and endogenous fluctuations. J. Econ. Theory 71, 394–415. Farmer, R.E.A., 1997. Money in a real business cycle model. J. Money Credit Banking 29, 568–611. Farmer, R.E.A., Guo, J.T., 1994. Real business cycles and the animal spirits hypothesis. J. Econ. Theory 63, 42–73. Guo, J., Lansing, K.J., 1998. Indeterminacy and stabilization policy. J. Econ. Theory 82, 481–490. Ni, S., 1995. An empirical analysis on the substitutability between private consumption and government purchases. J. Monit. Econ. 36, 593–605. Peril, R., 1998. Indeterminacy, home production and the business cycle. J. Monit. Econ. 41, 105–125. ´ S., Uribe, M., 1997. Balanced-budget rules, distortionary taxes and aggregate inestability. Schmitt-Grohe, J. Polit. Econ. 105, 976–1000. Weder, M., 1998. Fickle consumers, durable goods and business cycles. J. Econ. Theory 81, 37–57. Wen, Y., 1998. Capacity utilization under increasing returns to scale. J. Econ. Theory 81, 7–36. Xie, D., 1994. Divergence in economic performance: transitional dynamics with multiple equilibria. J. Econ. Theory 63, 97–112.
Indeterminacy under non-separability of public consumption and leisure in the utility function ´ ´ Ruiz Esther Fernandez, Alfonso Novales*, Jesus Departamento de Economia Cuantitativa, Universidad Complutense de Madrid, Campus de Somosaguas, Madrid 28223, Spain Accepted 18 June 2003
Abstract In a one sector growth model with public consumption in the utility function, the competitive equilibrium can be indeterminate for plausible values of the elasticity of intertemporal substitution of consumption, under constant returns to scale and endogenous government expenditures. Non-separability between public consumption and leisure in the utility function is crucial for this result. 䊚 2003 Elsevier B.V. All rights reserved. JEL classifications: E32; E62; D90 Keywords: Indeterminacy; Multiple equilibria; Externalities; Non-separable utility; Public consumption
1. Introduction We characterize conditions on preferences under which competitive equilibrium can be indeterminate in a single sector growth model with public consumption in the utility function, constant returns to scale in production and endogenous public expenditures financed through a constant rate income tax. General equilibrium models that display indeterminacy have been the focus of attention in recent years. Indeterminacy implies that there will be multiple paths converging to a given steady state. Hence, indeterminacy guarantees existence of a continuum of sunspot stationary equilibria, i.e. stochastic rational expectations *Corresponding author. Fax: q34-91-394-26-13. E-mail address: [email protected] (A. Novales). 0264-9993/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 0 3 . 0 0 0 3 8 - 5
410
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
equilibria determined by perturbations unrelated to the uncertainty in economic fundamentals. The interest of sunspot equilibria is that they provide a theoretical justification for ‘animal spirits’ underlying economic instability. Characteristics that produce indeterminacy of equilibria in one- or multi-sector real business cycle models or in endogenous growth models have been widely studied. Initially, it was shown that when these models are extended to include either productivity externalities or some market imperfection, indeterminacy can arise if social returns to scale in production are sufficiently high, so that the labor demand curve has a slope which is not only positive, but also greater than that of the labor supply curve (see Benhabib and Farmer (1994) and Farmer and Guo (1994), for one-sector models). These models have been widely criticized because to produce multiple equilibria, they require larger returns to scale than observed in actual data (see Aiyagari (1995)). More recent work has described two situations in which increasing returns needed to produce indeterminacy are lower than initially thought, sometimes allowing for negatively sloped labor demand curves: (i) two sector economies with externalities in one sector (Benhabib and Farmer (1996), Peril (1998), Benhabib and Nishimura (1998) and Weder (1998), which introduces a durable consumption sector in addition to the investment and non-durable consumption sectors, or Barinci and ´ Cheron (2001) in an environment with financial constraints); (ii) one sector models with non-standard characteristics, such as non-separable preferences (Bennett and Farmer (2000)), or capacity utilization affecting production (Wen, 1998). These models are able to produce indeterminacy of equilibria with relatively low increasing returns to scale, but the elasticity of intertemporal substitution needed is often too large, relative to values considered in the real business cycle literature (Weder (1998) and Bennett and Farmer (2000), among others). This will generally lead to rather volatile consumption paths, against the observation in most actual economies. The same criticism applies to existing endogenous growth models producing indeterminacy (see Benhabib and Perli (1994) and Xie (1994)). We show that if preferences of the representative agent are non-separable in public consumption and leisure, there are conditions under which competitive equilibrium is indeterminate for standard values of the elasticity of intertemporal substitution and constant returns to scale in production. If, in addition, public and private consumption enter as a composite good in the utility function, indeterminacy is compatible with labor supply (defined as in Benhabib and Farmer (1994), i.e. holding constant consumption) and demand curves having the standard slopes. As for the most part of the literature on indeterminacy, our model is subject to an externality, public consumption entering into the utility function. This extension can be justified from the point of view that to design fiscal policy, it is important to bear in mind the effect that public consumption may have on private consumption and work-leisure decisions. In this respect, there is some empirical work that provides evidence on complementary or substitutability between private and public consumption (Ni (1995), among others). Schmitt-Grohe´ and Uribe (1997) also characterize conditions for indeterminacy of equilibria under constant returns to scale without including public consumption
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
411
in the utility function. These authors show that under a constant level of public expenditures and endogenous proportional income taxes that balance the budget every period, indeterminacy arises for a range of values of the tax rate and the elasticity of intertemporal substitution consistent with those in the real business cycle literature. However, indeterminacy in their economy would be precluded under fixed income tax rates and endogenous government expenditures. Cazzavillan (1996) studies indeterminacy in a simple one sector model with public expenditures entering into the utility and production functions. The presence of both externalities is crucial to get indeterminacy in his economy, which displays endogenous growth. Hence, the contribution of our work is to show that there can be indeterminacy in a simple neoclassical growth model with only a source of externality: public consumption entering in the utility function non-separably with leisure. Relative to fiscal policy, we assume that (i) public consumption is financed through distorting taxes (a proportional tax on labor and capital income); and (ii) the government budget balances every period. Our economy can display indeterminacy for fixed income tax rates, plausible values of the intertemporal elasticity of substitution of consumption, constant returns to scale and endogenous government expenditures. Furthermore, (a) the region of the parameter space producing indeterminacy is independent of the income tax rate, and (b) the elasticity of the constant-consumption labor supply function with respect to the level of public expenditures can have either sign. Finally, our indeterminacy condition can be interpreted in a similar manner to Bennett and Farmer (2000): the economy displays indeterminacy if the Frisch labor supply curve (holding constant marginal utility of consumption) crosses the labor demand curve with the ‘wrong slope’. The model is described in Section 2. In Section 3 we characterize the transitional dynamics of the model and the conditions for indeterminacy. In Section 4 we provide an economic interpretation of the condition that generates indeterminacy. We provide a numerical example in Section 5, and the paper closes with some remarks in Section 6. 2. The economy 2.1. The consumer’s problem There is a continuum of identical consumers, all with the same preferences defined on private and public consumption and leisure, according to the utility function:
UŽc,g,n.s
Žcqfg.1yr 1yr
yg1yrn1qx, r)0, r/1,
(1)
where c is private consumption, g public consumption and n is the supply of
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
412
labor.1 To simplify notation, time dependence of all variables is suppressed. The parameter fG0 determines the degree of substitution between private and public consumption (as in Ni (1995)), xG0 defines the elasticity of labor supply (as in Benhabib and Farmer (1994)) and r is the inverse of the elasticity of intertemporal substitution of consumption, defined in a broad sense (Cscqfg). The possibility that rc1 allows for public consumption to either be complement or substitute for leisure. The single period utility function in Eq. (1) is homogeneous of degree 1yr in c and g. In our model, this condition allows for the existence of a balanced growth path although, for simplicity, we do not consider growth in the economy.2 Each consumer receives income on labor and capital, which can be used to consume, save, and pay taxes at a constant rate tg(0,1) on the two sources of income: ˙ cqkqdks Ž1yt.Žvnqrk.,
(2)
where v is the real wage and r is the return on capital (k). Hence, the representative consumer faces the optimization problem: Max
|
`
eybtU(c,g,n) dt
(3)
0
subject to Eq. (2) and given k0, with b)0 being the time discount. First order conditions for this problem are:
Žcqfg.yrsl,
(4)
Ž1qx.g1yrnxslŽ1yt.v,
(5)
l˙ y sŽ1yt.ryŽdqb., l
(6)
lim eybtls0,
(7)
t™`
l being the multiplier associated to the consumer’s budget constraint. Combining Eqs. (4) and (5): vs
1qx Žcqfg.rg1yrnx, 1yt
(8)
1 Farmer (1997) considers a similar class of utility functions, which includes real balances instead of public consumption. 2 Farmer (1997) showed that, in the case of a monetary model with real balances, MyP, in the utility function, the condition that allows for the possibility of balanced growth is that U(c,MyP,n) be homogenous in c and MyP.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
413
so that the marginal rate of substitution between private consumption and leisure must be equal to the after tax real wage. Combining Eqs. (4) and (6) and using Cscqfg: ˙ 1w C s Ž1yt.ryŽdqb.z~, C ry x
|
(9)
so that consumption growth is every period proportional to the return on capital, net of taxes and depreciation. 2.2. Firms There is a continuum of identical firms operating in a competitive environment. For simplicity, we normalise their number to one. Aggregate production exhibits constant returns to scale: ysAkan1ya, agŽ0,1.
(10)
y being aggregate output and k, n the demand for the two production factors. Under perfectly competitive markets for the two factors, profit maximizing conditions are: y rsa , k
(11)
y vsŽ1ya. . n
(12)
2.3. Government The government chooses a tax rate t and balances its budget every period. Hence, the instantaneous government budget constraint is: gstŽvnqrk.,
(13)
where g represents government spending on goods and services that contribute to household’s utility. 2.4. Equilibrium Given t and k0, a competitive equilibrium is a set of trajectories {c,n,k,g,v,r} such that: (i) given the paths for {v,r}, {c,n,k} solve consumer’s problem wEqs. (2), (7)–(9)x, with Cscqfg; (ii) given {v,r}, {n,k} solve the firm’s problem
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
414
wEqs. (11) and (12)x; (iii) government budget constraint Eq. (13) holds every period; and (iv) markets clear, in particular, from Eqs. (2), (10) and (13): ˙ Akan1yascqŽkqdk .qg,
(14)
where gstAkan1ya.
(15)
3. Local dynamics Let us now discuss the local properties of the equilibrium dynamics of the system. A steady state is a vector (css kss, nss, lss, gss) satisfying the equations for competitive equilibrium such that if it is ever reached, the system will stay at that ˙ sssc˙ sssk˙ sssl˙ sss0.: point forever ŽC w
z
1
Ž1yt.Aa 1ya kss | , sx nss y dqb ~
(16)
css dŽ1ya.qb s , kss a
(17)
css dŽ1ya.qb 1yt s , gss dqb t
(18)
w
1
B
Eyrz 1qx
dŽ1ya.qb 1yt 1yt 1ya Cfq F nsssx 1qx D dqb t G y t lsssŽcssqfgss.yr,
| ~
,
(19) (20)
where Eq. (16) comes from Eqs. (6), (10) and (11); Eq. (17) refers to Eqs. (14)– (16); Eq. (18) is obtained from Eqs. (15)–(17) and (19) comes from Eq. (8) together with Eqs. (10), (12), (15), (16) and (18); and Eq. (20) is obtained from Eq. (4). All these equations are evaluated at steady-state. To characterize the local dynamics of the system, it is enough to analyze the transition of the state and co-state variables (k,l). That requires to write the control variables (c,n) as a function of (k,l). w
1 Ž1ya.Ž1yt.Ar arz 1qxyr Ž1ya. ns lk , tŽ1yr.Ž1qx. y ~
x
|
(21)
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428 wB
xC
csly1yryf
Ž1ya.Ž1yt. E
Ft
1qx
yD
xqa
1qx
aŽ1qx.
1ya
1ya
Ž1ya.
G
A
k
z
|
l
415
1ya 1qxyrŽ1ya.
(22)
,
~
where Eq. (21) comes from Eqs. (5), (10) and (12); and Eq. (22) is obtained from Eq. (4) together with Eqs. (15) and (21). The first order approximation to the dynamic system made up by Eqs. (6) and (14) is:
CF C F B˙ E
Bkyk
k
E
,
sG
˙G Dl
ss
DlylssG
with transition matrix G (see Appendix A):
C B
Gs
w
x
B
Ždqb.Ž1qx.C1qf
Ždqb.Ž1qx. B t E C1qf Fyd 1qxyrŽ1ya. D 1yt G
Bk
Bl
Ždqb.Ž1ya. y 1qxyrŽ1ya.
E
C
E
D
F
arŽ1qxyrŽ1ya..
D lss Gy
1qxyr C FŽ1ya.Ždqb. D G k 1qxyr Ž1ya. D ss ss
ss
F
zE
t E F 1yt G y
|
d r~
.
G
(23) Since one of the variables in the system is predetermined and the other is free, the system will have a unique, locally determined equilibrium when the steady state is a saddle point, which requires two characteristic roots of matrix G to be of opposite sign. Indeterminacy of equilibria arises only when the two roots of G are negative. This means that trace of G be negative, and its determinant be positive. For any given values of the vector (a,d,t,f,b), let w1(r), w2 (r) denote the linear functions:
w1Žr.sy1qŽ1ya.r, w
dyŽdqb.xaqf y
w2Žr.s
t z | 1yt ~
t bqŽdqb.f 1yt
y
dŽ1ya. t bqŽdqb.f 1yt
r.
Proposition 1. Det(G))0 if and only if x-w1(r). Proof. See Appendix A.
Proposition 2. Tr(G)-0 if and only if
xyw2Žr. xyw1Žr.
-0.
416
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
Proof. See Appendix A. Let r*,r** be the zeroes of w1(Ø) and w2(Ø). We have: 1 1 w dqb B t Ez Caqf F| , r**s x1y 1ya 1ya y d D 1yt G~ and r*)r** for all a,d,t,f,b. r*s
Proposition 3. Let (x0,r0) be a point in the (x,r)-plane with 0Fx0-w1(r0). x0yw2Žr0. Then -0. x0yw1Žr0. Proof. Since r* is the single root of w1(r)s0, which is an increasing linear function of r, 0Fx0-w1(r0)´r0)r*)r** and hence, w2(r0)-0, since r** is the single root of w2(r)s0, w2(Ø) being a decreasing linear function of r. This implies that x0yw2(r0))0, and, hence x0yw2Žr0. -0. x0yw1Žr0. Proposition 4. The competitive equilibrium is indeterminate in economy Eqs. (1)– (15), if and only if 0Fx0-w1(r0). Proof. Let us assume the competitive equilibrium is indeterminate. Then, Det(G)) 0 and Tr(G)-0. By Proposition 1, x0-w1(r0), while x0 is constrained to be nonnegative in Eq. (1). Conversely, let us assume that 0Fx0-w1(r0) Proposition 1, Proposition 2 and Proposition 3 then imply that Det(G))0 and Tr(G)-0 and hence, the competitive equilibrium is indeterminate.
Corollary 5. A necessary condition for indeterminacy of the competitive equilibrium 1 is r) 1ya Proof. This condition is needed for w1(r))0, and only then condition in Proposition 4 can hold. The previous results show that, in this economy: i. For values of a as usually calibrated in the real business cycle literature (approx. 0.35), indeterminacy can arise for values of the relative risk aversion parameter r)1.54, (i.e. intertemporal elasticity of substitution smaller than 0.65), a range of values consistent with existing empirical estimates. ii. fiscal policy parameters do not affect the conditions guaranteeing indeterminacy of equilibria, at a difference of Schmitt-Grohe´ and Uribe (1997). In their model, indeterminacy is possible in a range of intermediate steady-state tax rates. In particular, indeterminacy is not possible for tax rates in the decreasing part of
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
417
the Laffer curve and, as the authors point out, it is crucial for their results that labor income taxes be endogenous. In the economy, we consider, indeterminacy can arise for any value of the income tax rate and hence, for any size of the public sector, defined as the ratio of government expenditures to output. The goverment cannot do anything to stabilize the economy. Guo and Lansing (1998) show indeterminacy in a one-sector real business cycle under increasing returns to scale with a tax policy similar to ours, but without public consumption in the utility function. They consider alternative parameterizations allowing for regressive, proportional and progressive taxes, showing that progressive taxes may act as an economic stabilizer, precluding indeterminacy of equilibria. These same fiscal rules could be used in our framework to discuss whether they could be used to stabilize the economy, in the same sense of these authors. Non-separability of public expenditures and leisure in the utility function is crucial for the previous results, as the next proposition shows. Žcqfg.1yr 1qx yn , r)0, r/1, xG0, fG0, be an utility funcLet V(c,g,n)s 1yr tion similar to Eq. (1) except for the fact that it is separable in public consumption and leisure. Proposition 6. If preferences of the representative agent in the economy are represented by utility function V(c,g,n), there is no point in the parameter space for which equilibrium is indeterminate. Proof. See Appendix A.
4. Indeterminacy and the labor market Most work discussing indeterminacy of equilibria in one sector models shows that it is associated with upward sloping demand curves, downward sloping supply curves, or both. The main result of Bennett and Farmer (2000) is that the Benhabib– Farmer condition wBenhabib and Farmer (1994)x, that labor demand and supply curves cross with the ‘wrong slopes’ generalizes to the case when preferences are non-separables in private consumption and leisure. However, Bennett–Farmer show that it is the Frisch labor supply curve (i.e. holding constant the marginal utility of consumption3) rather than the constant–consumption labor supply curve, which should be considered in this analysis. In this section, we relate our conditions for indeterminacy to the slopes of labor demand and both labor supply curves. In our model, under the utility function U(c,g,n), the Frisch labor supply curve and the constant–consumption labor supply curve differ. The Frisch labor supply 3
The use of tins term follows Bennett and Farmer (2000) and Browning et al. (1985).
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
418
curve (from Eq. (5)) is: vs
1qx
Ž1yt.l
g1yrnx,
(24)
and using again Eq. (15) to eliminate g from Eq. (24), we get the reduced-form Frisch labor supply curve: vs
1qx ŽtAka.1yrnxqŽ1yr.Ž1ya., Ž1yt.l
(25)
having a slope in the neighborhood of the steady state: B dv Es
C D
F s
dn GF
vss w xqŽ1yr.Ž1ya.z~, nss y x
|
which will be negative, when r)
(26) 1 (which implies r)1) and (ry1)(1ya)) 1ya
x, being positive otherwise. From Eq. (12), the slope of the labor demand curve in the neighborhood of steady state is: B dv Ed
C D
v F sya ss , dn G nss
which is always negative. Proposition 7. The competitive equilibrium is indeterminate in our economy if and only if the reduced-form Frisch labor supply curve is downward-sloping, and steeper than the labor demand curve. Proof. From Proposition 4 and its corollary, if equilibrium is indeterminate we have 1 r) and x-y1qr(1ya). Since (ry1)(1ya))y1qr(1ya) for all a, r, 1ya the labor supply curve slopes downward. It will also be steeper than the labor demand curve, since the former condition also implies: (ry1)(1ya))yx)a. Conversely, if the latter condition holds, then x-y1qr(1ya) and equilibrium is indeterminate, by Proposition 4. Hence, the appropriate condition for indeterminacy of the competitive equilibrium to arise in the economy, is that the Frisch labor supply curve crosses the labor demand curve with the ‘wrong slope’, the former being steeper (i.e. the Benhabib– Farmer condition holds).
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
419
The intuition behind the existence of stationary sunspot equilibria are as follows. If the private agents expect future public consumption to be above average, then they will also expect higher future income taxes. Future hours worked for any given capital stock will then be expected to increase because of Eq. (25), raising the expected marginal product of capital. The increase in the expected return to capital leads to an increase in current labor supply and output. Since the income tax rate is fixed, current public consumption will also be higher. Thus, expectations of aboveaverage public consumption of next period lead to a higher current public consumption. Under the V(c,g,n) utility function, it is easy to see that the Frisch labor supply curve does not depend on the public consumption, and the previous argument breaks down. This is consistent with Proposition 6, which shows that, under V(.), our model does not display indeterminacy. Therefore, the non-separability of public consumption and leisure in the utility function plays a key role in generating expectations-driven fluctuations in this economy. Under the U(c,g,n) utility function, the constant-consumption, reduced form labor supply curve can be obtained using Eq. (15) to eliminate g from Eq. (8): vs
Er 1qx B c Cfq F tAkan1yaqx, 1yt D tAkan1ya G
(27)
having a slope, in the neighborhood of the steady state: B dv Es
C D
Fs
dn G
w
x
vss Ž1yaqx.q nss
rŽay1. 1qf
y
tAŽkss ynss.Žay1. css ykss
z
|
.
(28)
~
Proposition 8. The constant-consumption, reduced form labor supply curve wEq. (27)x has a negative slope if and only if x-w3(r), where w3(r)sŽ1ya.= w z r y1 . t dqb 1qf 1yt dŽ1ya.qb y ~ Proof. Easy to show from Eqs. (16), (17) and (28).
x
|
Corollary 9. A necessary condition for the constant-consumption, reduced form labor supply curve wEq. (27)x to have a negative slope is that r)w4s1q t dqb f )1. 1yt dŽ1ya.qb Proof. Only if this inequality holds we will have a non-trivial range of values for x in Proposition 4. The next two propositions shows that if public consumption enters together with private consumption as a composite good in the utility function then, contrary to
420
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
what happens to the Frisch labor supply curve, the reduced-form labor supply curve obtained holding constant consumption can be upward-sloping when equilibrium is indeterminate. If public consumption enters the utility function just through the interaction with leisure, then the same necessary and sufficient conditions for indeterminacy apply to the labor supply curve obtained holding constant consumption than to the Frisch labor supply curve. This is because, as pointed out in Bennett and Farmer (2000), when preferences are separable in consumption and leisure, both supply curves coincide. This latter case is an illustration of the fact that nonseparability between private consumption and leisure is not needed for indeterminacy to arise with a downward sloping labor supply curve. Proposition 10. If f)0, there is a non-trivial set in the parameter space for which we have indeterminacy of equilibria with an upward sloping reduced form, constantconsumption, labor supply curve wEq. (27)x. ≠w1(r) s1ya) Proof. The previously defined w1(r) function satisfies: ≠r 1ya ≠w3(r) s , for all f,t,d,b,a and w1(0)-w3(0), since t dqb ≠r 1qf 1yt d(1ya)qb w1(0)sy1 and w3(0)sy(1ya). Being w1 (.) and w3(.) linear functions of r, they ˜ have a single intersection at a given r)0 , with r)r˜ ´w3(r)-w1(r). Hence, ˜ indeterminacy pairs (r,x) with r)r and w3(r)-x-w1(r) produce an upward sloping reduced form, constant-consumption, labor supply curve, while for those with x-w3(r) that labor supply curve is downward sloping. If there is any indeterminacy pair (r,x) with r-r˜ , the labor supply curve will be downward sloping.
Proposition 11. If fs0, there is no equilibrium indeterminacy with an upward sloping labor supply curve obtained holding constant consumption. The competitive equilibrium is then indeterminate if and only if the constant-consumption labor supply curve wEq. (27)x is downward sloping, but steeper than the labor demand curve. Proof. In this case, the Frisch labor supply curve and the constant-consumption labor supply curve are the same wsee Eqs. (26) and (28) under fs0x, and Proposition 7 shows the result. Our final result shows that requiring non-separability between public and private consumption in the utility function (f)0) to obtain indeterminacy of equilibria under a positively sloped constant-consumption labor supply curve, is not restrictive from the point of view of the sign of the elasticity of the labor supply function, relative to public consumption. That is, in an indeterminate competitive equilibrium with upward sloped labor supply curve an increase in public consumption could shift the labor supply curve either to the left or to the right.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
421
1 )w4 then 1ya the elasticity of the constant consumption labor supply curve wEq. (27)x relative to public consumption is always positive. When the competitive equilibrium is 1 -w4, the elasticity of the constant consumption labor supply indeterminate and 1ya curve relative to public consumption is positive, if r)w4, being negative if 1 -r-w4. 1ya Proof. Differentiating in Eq. (8) with dvsdcs0, we get, in the neighborhood of the steady state: Proposition 12. If the competitive equilibrium is indeterminate and
´n,gs
dn gss rf 1 ry1 rf 1 ry1 sy q sy q , dg nss x css ygssqf x x 1yt dŽ1ya.qb x qf t dqb
where we have used Eq. (18) to reach the last equality. Hence, ´n,g,0 if and only 1 if r,w4. If )w4, under indeterminacy of equilibria we will have, from 1ya 1 1 )w4, and consequently, ´n,g)0. If -w4-r, we will Corollary 5: r) 1ya 1ya 1 -r-w4, we will have ´n,g-0. again have ´n,g)0, while if 1ya 5. A numerical example In this section, we provide a numerical illustration of our analytical results identifying, for a given parameterization, the (x, r)-regions where the competitive equilibrium is either determinate or indeterminate. Parameter values are: bs0.01, as0.35, fs0.9 (the latter in the range estimated by Ni, 1995), ts0.2, As1, ds 0.0252. Assuming quarterly data, these values are compatible with annual depreciation of 10% and a steady-state real interest rate of 4%. In Fig. 1a we also identify (x,r)-pairs for which the constant-consumption labor supply curve is positively sloped. Fig. 1b and c reproduce Fig. 1a for alternative parameter values. In Fig. 1b, the only difference is fs0.6. In Fig. 1c we use the same values as in Fig. 1a except for the tax rate, which is taken to be ts0.4. Note that the indeterminacy condition is the same in the three graphs in Fig. 1 since it is independent of f and t. Comparing the three figures, we see that the bigger the substitution between private and public consumption (f) and bigger the tax rate, the larger the set of (r,x)-combinations compatible with indeterminacy of equilibria and an upward sloping labor supply curve. Finally, Fig. 2 shows, among the (r,x)-combinations for which there is indeterminacy of equilibria, those for which the two eigenvalues are complex, implying an
422
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
Fig. 1. Paramater regions for indeterminancy of equilibria and for a positively sloped labor supply curve. Points below each line satisfy the corresponding conditions. The shaded region presents the (r,x)-pairs for which equilibrium is indeterminate and the reduced-form labor supply curve obtained holding constant consumption has the standard slope.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
423
Fig. 2. (r,x)-combinations for which competitive equilibrium is either determinate, indeterminate, or indeterminate with oscillatory dynamics.
endogenous mechanism for the propagation of business cycles. Large values of either x or r (x)0.31, r)3.92) preclude this possibility. Parameter values are the same as in Fig. 1a. 6. Conclusions For a class of utility functions characterized by non-separability between public consumption and leisure, we have shown that the competitive equilibrium can be indeterminate for plausible values of the intertemporal elasticity of substitution, under endogenous government expenditures (financed through a fixed income tax rate) and constant returns to scale in production. The region in the parameter space leading to indeterminacy is characterized by: i. it is compatible with the constant-consumption labor supply curve and the labor demand curve having the standard slope signs, provided that private and public consumption are substitutes. However, the Frisch labor supply curve is downward sloping whenever there is indeterminacy; ii. the intertemporal elasticity of substitution must be below the labor income share on output, which is usually calibrated in the literature to be approximately 0.35. Equivalently, the relative risk aversion parameter must be above the inverse of that share. This condition only excludes values for the risk aversion parameter below 1.54, indeterminacy being a possibility for values above that threshold; iii. the tax rate and hence, the size of the public sector, measured by the participation of public consumption on output, does not enter the condition characterizing
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
424
indeterminacy. This means that the government cannot do anything to stabilize this economy, contrary to the result reached by Schmitt–Grohe´ and Martin Uribe in a different environment under constant government expenditures. In our case, a bigger public sector only increases the likelihood that indeterminacy will arise with the standard slopes for the constant-consumption labor supply and demand curves. 7. Uncited enunciations Proposition 8, Proposition 10, Proposition 11, Proposition 12 and Corollary 9 Appendix A: Part 1: Deriving the elements of G: From Eqs. (14) and (15) we get: ˙ Ž1yt.AkanŽk,l.1yaydkycŽk,l.. ks
(29)
where c(k,l) is given by Eq. (22) and n(k,l) is given by Eq. (21), while from Eqs. (6), (10) and (11) we get: w ay1 ˙ lsyl nŽk,l.1yayŽdqb.z~. yŽ1yt.aAk x
(30)
|
The dynamics of system Eqs. (29) and (30) in (k,l) around steady-state can be characterized through the linear approximation:
CF C B˙ E
k
BG
11
G12EBky kss E
FC F
s
˙G Dl
DG21
(31)
G22GDlylssG
with: Bk
G11sŽ1yt.AaC
ss
Eay1
F
D nss G Bk
G12sŽ1yt.Ž1ya.AC
Bk
ydqŽ1yt.Ž1ya.AC
ss
Ea ≠n Z
F
D nss G
ss
Ea ≠n Z
F
D nss G
≠c Z Z y Z ≠k Zss ≠k Zss Z
Z
Z
Z
≠c Z Z y Z , ≠l Zss ≠l Zss Z
Z
Z
Z
w
x
ay2 1ya G21sylssŽ1yt.aA Žay1.kss nss qŽ1ya.kay1 nya ss ss y
(32)
(33) ≠n Z z Z , ≠k Zss Z Z
| ~
(34)
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
ay1 ya G22sylssŽ1yt.aAŽ1ya.kss nss
≠n Z Z , ≠l Zss Z
425
(35)
Z
≠x Z Z , xsn,c and zsk,l, denoting partial derivatives, evaluated at steady state. ≠z Zss From Eqs. (21) and (22), and using Eqs. (15), (16) and (20), the partial derivatives in Eqs. (32)–(35) are: with
Z Z
≠n Z 1 nss , Z s ≠l Zss D lss Z Z
z ≠c Z 1 w B 1qx E 1qx Fy fgss|, xC55C1yay Z s ≠l Zss lssD y D r G r ~ Z Z
≠n Z ar nss , Z s ≠k Zss D kss Z Z
≠c Z t Ždqb.Ž1qx. , Z syf ≠k Zss 1yt D Z Z
where Ds1qxyr(1ya)sxyw1 (r). B k Eay1 dqb ss Using Eq. (16) to eliminate C F s , together with these expresD nss G Ž1yt.Aa sions, yields: rz t Ždqb.Ž1qx. D nss G D~ 1yt D y Ždqb.Ž1qx. t Ždqb.Ž1qx. Ždqb.Ž1qx. B t E C1qf Fyd s ydqf s D D 1yt D D 1yt G Bk
G11sŽ1yt.AaC
ss
Eay1w
x1qŽ1ya. |ydqf
F
and using Eqs. (17) and (18) to eliminate Bk
G12sŽ1yt.Ž1ya.AC
ss
Ea
F
D nss G
css gss , , we have: kss kss
1 nss 1 w B 1qx E 1qx z Fyf y gss| xcssC1yay D lss Dlss y D r G r ~
s
kss 1 w 1ya 1 css 1qx gss z f | x Ždqb.q D q lss D y a r kss r kss ~
s
kss 1 w 1ya 1 dqbyad 1qx dqb t z q f x |, Ždqb.q D lss D y a r a r a 1yt ~
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
426
and, combining terms in dqb: B
Ždqb.Ž1qx.C1q D
ft E FyadD 1yt G
G12s
kss 1 lss D
G21s
lss 1qxyr lss Ždqb.Ž1ya. s Ž1yt.aAŽ1ya.Žkss ynss.ay1 Ž1qxyr., kss D kss D
G22sy
,
ar
Ždqb.Ž1ya. D
(36)
.
Part 2: Proof of Proposition 1: Denote FsŽdqb.f
t )0 and N1s(dq 1yt
b)(1ya))0, B
DetŽG.sG11G22yG12G21syC
ŽdqbqF.Ž1qx.
D
D
B
ŽdqbqF.Ž1qx.
yC D
arD
E
ydF
N1 G D E
y
d N1 F Ž1qxyr. rG D
B
E N 1qx N1 ŽdqbqF.Ž1qx. 1qx Fsy 1Ž . wydŽ1ya.qbqFz~, sy C yd DD ar r G arD x
|
so that Det(G))0 if and only if D-0, i.e. if and only if x-w1(r). Part 3: Proof of Proposition 2: First, notice that: xyw2Žr.sxy
1 w dyŽdqb.ayFydŽ1ya.rz~. Fqb y x
|
Then, TrŽG.sG11qG22s s
1w ŽdqbqF.Ž1qx.ydŽ1qxyrŽ1ya..yŽdqb.Ž1ya.z~ Dy x
|
1w FqŽbqF.xyd Dy x
xyw2Žr. 1 qdrŽ1ya.qŽdqb.az~s Žxyw2Žr..ŽFqb.sŽFqb. , D xyw1Žr. |
so that Tr(G)-0 if and only if
xyw2Žr. xyw1Žr.
- 0.
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
427
Part 4. Proof of Proposition 6: Proof: For utility function V(c,g,n), the control variables (c,n), as functions of (k,l) are given by: w
x
ns
y
Ž1ya.Ž1yt. 1qx
z
|
1 xqa
Alka
,
~
w
x
csly1yryft
Ž1ya.Ž1yt. 1qx
y
z
1qx
1qx
|
A 1ya ka 1ya l
1ya xqa
,
~
which are the analogue of Eqs. (21) and (22), with partial derivatives: ≠n Z a nss , Z s ≠k Zss xqa kss Z Z
≠c Z a ly1yr ycss ss , Z syŽ1qx. ≠k Zss xqa kss Z Z
≠n Z 1 nss , Z s ≠l Zss xqa lss Z Z
Bk
and, again, C
ss
Eay1
F
D nss G
G11sbqŽ1ya. G22syŽ1ya.
s
dqb , so that Eqs. (32) and (35) become: (1yt)Aa
1 a ly1yr ycss ss , Ždqb.qŽ1qx. xqa xqa kss
1 Ždqb., xqa
a y1yr =Žlss ycss.. But, from Eq. (20), optixqa y1yr ycss)0. Hence, Tr(G))0 for any feasible set of mality implies: fgssslss parameter values, without the possibility of having indeterminacy. and, TrŽG.sG11qG22sbqŽ1qx.
References Aiyagari, S.R. 1995. The econometrics of indeterminacy: an applied study, a comment. CarnegieRochester Conference Series on Public Policy 43, 273–282. ´ Barinci, J.P., Cheron, A., 2001. Sunspots and the business cycle in a finance constrained economy. J. Econ. Theory 97, 30–49. Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. J. Econ. Theory 63, 19–41.
428
´ E. Fernandez et al. / Economic Modelling 21 (2004) 409–428
Benhabib, J., Farmer, R.E.A., 1996. Indeterminacy and sector specific externalities. J. Monit. Econ. 37, 421–443. Benhabib, J., Nishimura, K., 1998. Indeterminacy and sunspots with constant returns. J. Econ. Theory 81, 58–96. Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: transitional dynamics with multiple equilibria. J. Econ. Theory 63, 113–142. Bennett, R.L., Farmer, R.E.A., 2000. Indeterminacy with non-separable utility. J. Econ. Theory 93, 118–143. Browning, M.J., Deaton, A., Irish, M., 1985. A profitable approach to labor supply and commodity demands over the life-cycle. Econometrica 53, 503–543. Cazzavillan, G., 1996. Public spending, endogenous growth and endogenous fluctuations. J. Econ. Theory 71, 394–415. Farmer, R.E.A., 1997. Money in a real business cycle model. J. Money Credit Banking 29, 568–611. Farmer, R.E.A., Guo, J.T., 1994. Real business cycles and the animal spirits hypothesis. J. Econ. Theory 63, 42–73. Guo, J., Lansing, K.J., 1998. Indeterminacy and stabilization policy. J. Econ. Theory 82, 481–490. Ni, S., 1995. An empirical analysis on the substitutability between private consumption and government purchases. J. Monit. Econ. 36, 593–605. Peril, R., 1998. Indeterminacy, home production and the business cycle. J. Monit. Econ. 41, 105–125. ´ S., Uribe, M., 1997. Balanced-budget rules, distortionary taxes and aggregate inestability. Schmitt-Grohe, J. Polit. Econ. 105, 976–1000. Weder, M., 1998. Fickle consumers, durable goods and business cycles. J. Econ. Theory 81, 37–57. Wen, Y., 1998. Capacity utilization under increasing returns to scale. J. Econ. Theory 81, 7–36. Xie, D., 1994. Divergence in economic performance: transitional dynamics with multiple equilibria. J. Econ. Theory 63, 97–112.