Carine Nourry

Thomas Seegmuller

Alain Venditti

Aggregate instability under balanced-budget consumption taxes: a re-examination

Keywords: Indeterminacy, endogenous business cycles, consumption taxes, balanced-budget rule, infinite-horizon model Classification Numbers: C62, E32, H20

We re-examine the destabilizing role of balanced-budget fiscal policy rules based on consumption taxation. Using a one-sector model with infinitely-lived households, and assuming that preferences are of the Greenwood-Hercovitz-Huffman [8] (GHH) type, we show that non-linear consumption taxation may destabilize the economy, promoting expectation-driven fluctuations, if the tax rate is countercyclical. We also exhibit a Laffer curve, which explains the multiplicity of steady states when the tax rate is counter-cyclical. All these results are mainly driven by the absence of income effect. Finally, a numerical illustration shows that consumption taxation may be a source of instability for most OECD countries.

Introduction

Balanced-budget rules recommendations to governments is a recurrent subject of controversies among economists and politicians. In the recent years, the financial crisis has generated a strong revival of interest of this debate through its consequences in terms of government debt sustainability. Some countries, characterized by too large debt and public deficits, are constrained to borrow on the financial markets against excessively high interest rates and this generates an increasing debt burden. In Europe, this has been experienced by several countries, starting with Greece, but US faces the same kind of difficulties. These concerns related to the solvability of public debt also enforce governments to reduce their deficits and implement public policies characterized by a balanced-budget.

From a theoretical point of view, several arguments have been considered in favor or against balanced-budget fiscal policy rules. In their seminal contribution, Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF] focus on the fact that balanced-budget rules may induce belief driven aggregate instability. Considering an economy with infinitely-lived households, their basic model introduces a proportional tax on labor income. Since public expenditures are assumed to be constant, the tax rate is counter-cyclical, which is shown to be a source of endogenous fluctuations. 1 The economic intuition relies on self-fulfilling expectations. If households expect a higher tax rate at the next period, they will reduce their labor supply, which has a negative effect on capital returns. Therefore, they currently reduce investment. Needing less income to finance capital accumulation, they also supply less labor. Since the labor tax rate is counter-cyclical, it becomes larger, explaining that expectations are self-fulfilling.

Contributing to this debate, Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] argues that consumption taxation should be preferred to income taxation. As in Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF], she assumes a balanced-budget rule with a constant level of public spendings, but entirely financed by consumption taxation, and she shows that saddle-path stability always prevails. 2 Therefore, she suggests that a 1 See also Gokan [7] and Pintus [START_REF] Pintus | Aggregate instability in the fixed-cost approach to public spending[END_REF] for similar analysis of finance constrained economies. 2 Consumption taxation is shown however to have destabilization effects by Lloyd-Braga 1 government running policies with a stabilization target should increase the share of public spendings financed by consumption taxation.

Although Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF] and Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] show that the (de-)stabilizing impacts of labor income and consumption taxations are opposite, both tax rates introduce a similar distortion, which affects the price of leisure in terms of the consumption good. An increase in either the labor income or the consumption tax rate decreases the relative price of leisure, reducing both labor supply and consumption through a substitution effect. As mentioned previously, this lowers labor supply and generates expectationdriven fluctuations and aggregate instability. In the case of consumption taxation, the decrease of consumption is also compensated by an increase of the labor supply through an income effect. When this latter effect is sufficiently large with respect to the former, any instability due to self-fulfilling expectations is ruled out. This explains the clear-cut conclusions obtained by Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] in the case of additively-separable preferences. Nevertheless, this economic intuition also suggests that the final impact of a tax on consumption crucially depends on the relative size of the substitution and income effects, and thus on the specification of the utility function.

The aim of this paper is to re-examine the (de-)stabilizing role of consumption taxation considering the extreme case of the Greenwood-Hercovitz-Huffman [START_REF] Greenwood | Investment, capacity utilization and the real business cycle[END_REF] (GHH) utility function, characterized by the absence of any income effect. In addition, we assume a general one-sector technology to take into account any degree of input substitution. Finally, we focus on a balanced-budget rule that generalizes the one introduced in Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF]. We consider indeed public spendings that can be pro-cyclical, constant or counter-cyclical. Note that when public spendings are counter-cyclical, constant or weakly pro-cyclical, the consumption tax rate is counter-cyclical, whereas when public spendings are sufficiently pro-cyclical, the consumption tax rate becomes pro-cyclical.

Our main results are the following: First we prove the existence of a Laffer curve and thus of multiple steady states if the tax rate is counter-cyclical. Second, we show that the existence of local indeterminacy also requires a counter-cyclical consumption tax rate. More precisely, under GHH preferet al. [START_REF] Lloyd-Braga | Tax rate variability and public spending as sources of indeterminacy[END_REF] in a finance constrained economy with heterogeneous households. ences, expectation-driven fluctuations arise for a non-empty range of tax rates when public spendings are constant, the case considered in Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF]. This conclusion is even reinforced by counter-cyclical public spendings: the minimum level of the tax rate compatible with local indeterminacy becomes lower.

A numerical illustration allows finally to confirm all these results. Using a standard parameterization of the fundamentals, we show that aggregate instability and endogenous sunspot fluctuations are compatible with tax rates observed in most OECD countries, depending on the degree of countercyclicality of the public spendings. This paper then proves that in the absence of any income effect, a balanced-budget rule financed by consumption taxation may be destabilizing. As a result, if the government's interventions have a stabilization target, any policy recommendation leading to an increase of the share of public spendings financed by consumption taxation may not be worthwhile. The precise impact of a larger tax rate on consumption has to be evaluated depending on the size of income effects.

The rest of this paper is organized as follows. We present the model in the next section. In Section 3, we analyze the multiplicity of steady states in connection with the existence of a Laffer curve. In Section 4, we give conditions for the existence of aggregate instability and endogenous sunspot fluctuations under GHH preferences. Section 5 focuses on a numerical illustration and in Section 6, we provide economic intuitions for our results. Some concluding remarks are provided in Section 7, and all the proofs are in a final Appendix.

The model

The production structure

Consider a perfectly competitive economy in which the final output is produced using capital K and labor L. The production function of a representative firm is thus AF (K, L), with F (K, L) homogeneous of degree one and A > 0 a scaling parameter. Denoting, for any L = 0, x = K/L the capital stock per labor unit, we define the production function in intensive form as Af (x). Assumption 1. f (x) is defined over R + , C r over R ++ for r large enough, increasing (f ′ (x) > 0) and concave (f ′′ (x) < 0). Moreover, there exist M > 0 and N > 0 such that f ′ (0) > M and f ′ (+∞) < N.

The rental rate of capital r(t) and the wage rate w(t) then satisfy:

r(t) = Af ′ (x(t)) (1)
w(t) = A[f (x(t)) -x(t)f ′ (x(t))] (2)
We can also compute the share of capital in total income:

s(x) = xf ′ (x) f (x) ∈ (0, 1) (3)
and the elasticity of capital-labor substitution:

σ(x) = -(1-s(x))f ′ (x) xf ′′ (x) > 0 (4)

Households' behavior

We consider an economy populated by a large number of identical infinitelylived agents. We assume without loss of generality that the total population is constant and normalized to one. At each period a representative agent supplies elastically an amount of labor l ∈ [0, ℓ], with ℓ > 1 his endowment of labor. He then derives utility from consumption c and leisure L = ℓ -l according to a function U(c, L/B), where B > 0 is a scaling parameter,3 which satisfies:

Assumption 2. U(c, L/B) is defined over R + × [0, ℓ], C r over R ++ × (0, ℓ)
for r large enough, increasing with respect to each argument and concave. Moreover,

U 12 U 1 L B -U 22 U 2 L B = 1, lim X→0 XU 2 (c, X)/U 1 (c, X) = 0 and lim X→+∞ XU 2 (c, X)/U 1 (c, X) = +∞, or lim X→0 XU 2 (c, X)/U 1 (c, X) = +∞ and lim X→+∞ XU 2 (c, X)/U 1 (c, X) = 0.
We also introduce a standard normality assumption between consumption and leisure which ensures that the demands for these two goods are increasing functions of the agent's total income Assumption 3. Consumption c and leisure L are normal goods.

Actually, within these general properties for the utility function, we mainly consider in this paper the Greenwood-Hercovitz-Huffman [START_REF] Greenwood | Investment, capacity utilization and the real business cycle[END_REF] (GHH) formulation such that

U(c, L/B) = u(c + G(L/B)) (5)
with u(.) and G(.) some increasing and concave functions. Such a specification then satisfies Assumption 3 and implies that the marginal rate of substitution between consumption and leisure depends on the latter only as

U 2 (c,L/B) BU 1 (c,L/B) = G ′ (L/B
)/B As this expression does not depend on consumption, there is no income effect associated with the agent's labor supply decision. Assumption 2 imposes the following properties:

Assumption 4. ǫ G LL ≡ -G ′ (X)/(G ′′ (X)X) = 1, lim X→0 G ′ (X)X = 0 and lim X→+∞ G ′ (X)X = +∞, or lim X→0 G ′ (X)X = +∞ and lim X→+∞ G ′ (X)X = 0. 4
Remark 1 : Three other specifications of preferences are also quite popular in the literature:

-the additively separable formulation U(c, L/B) = u(c) + v(L/B), with u(c) and v(L/B) some positive increasing concave functions, 5-the King-Plosser-Rebelo [START_REF] King | Production, growth and business cycles[END_REF] (KPR) formulation

U(c, L/B) = [cv(L/B)] 1-θ 1-θ (6)
with v(L/B) a positive increasing function and θ ≥ 0, -the linearly homogeneous formulation which is characterized by the share of consumption within total utility α(c, L/B) ∈ (0, 1) defined as follows:

α(c, L/B) = U 1 (c,L/B)c U (c,L/B) (7)
As we are interested in finding conditions for local indeterminacy, i.e. the existence of a continuum of converging equilibrium paths, under reasonable (standard) parameterizations, we do not consider these formulations. Indeed, under Assumptions 1-3, it can be shown that for the additively separable and KPR specifications, local indeterminacy is always ruled out. This conclusion generalizes the main result of Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] to any production function and any utility function, be it additively separable or KPR. Moreover, with the linear homogeneous specification, local indeterminacy is ruled out under constant government spendings, and requires extreme conditions with counter-cyclical government spendings. 6Since the population is constant and of unit size, we have L = l. The intertemporal maximization program of the representative agent is given as follows:

max c(t),l(t),K(t) +∞ t=0 e -ρt U (c(t), (ℓ -l(t))/B) s.t. h(c(t))c(t) + K(t) + δK(t) = r(t)K(t) + w(t)l(t) K(0) > 0 given (8)
where ρ > 0 is the discount factor, δ ∈ (0, 1) the depreciation rate of capital and K(0) the initial capital stock. In the budget constraint, h(c) denotes one plus the tax rate on consumption. This function is assumed to have the following properties:

Assumption 5. h (c) : [0, +∞) → [1, +∞) is continuous, and C 1 on (0, +∞).
Assumption 5 provides quite large degrees of freedom for the formulation of government income Ω(c) = h(c)c -c. This allows us to consider in the same framework pro-cyclical (Ω(c)/c and h(c) increasing), or counter-cyclical (Ω(c)/c and h(c) decreasing) consumption tax rules. Let us introduce the following elasticity:

ζ(c) = h ′ (c)c h(c) (9)
The consumption tax rate is then pro-cyclical (counter-cyclical) if ζ is positive (negative). We assume in the following that the representative agent considers as given the tax rate on consumption. Let us introduce the Hamiltonian in current value:

H = U(c(t), (ℓ -l(t))/B) + λ(t) r(t)K(t) + w(t)l(t) -h(c(t))c(t) -δK(t)
with λ(t) the shadow price of capital K(t). Considering the prices ( 1)-( 2) and h(c) as given, we derive the following first order conditions

U 1 (c(t), (ℓ -l(t))/B) = λ(t)h(c(t)) (10)
(1/B)U 2 (c(t), (ℓ -l(t))/B) = λ(t)w(t) (11) λ(t) = -λ(t) [r(t) -ρ -δ] (12)
Any solution needs also to satisfy the transversality condition

lim t→+∞ e -ρt U 1 (c(t), ℓ -l(t)) h(c(t)) K(t) = 0 (13)

Government

Taxes on consumption are used to finance an endogenous level of public spending G(t). Given the level of government income Ω (c(t)) derived from the tax function, the government expenditures G(t) are endogenously determined by the following balanced-budget rule:

G(t) = Ω (c(t)) = h(c(t))c(t) -c(t) (14)
We further assume that the public expenditures G(t) neither affect the consumers' preferences nor the production function.

In the following we will consider a sufficiently general formulation of public expenditures G(t) that encompasses the case of Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF], in which the government faces an exogenous stream of constant expenditure G(t) = G that is financed by levying proportional taxes on consumption. Our formulation allows indeed for counter or pro-cyclical government spendings. We choose consumption c as a proxy of the measure of the business cycle and we assume that public spending is a function of c, namely G(t) ≡ G(c(t)). Using our notations we get

G(c(t)) = Ω(c(t)) = h(c(t))c(t) -c(t) = τ c (c(t))c(t) (15)
This expression endogenously determines the level of the consumption tax rate. Let us then denote:

η ≡ G ′ (c)c/G(c)
We will say that public expenditures are counter(pro)-cyclical when η < 0 (η > 0). The Giannitsarou's case of constant government spending is obtained when η = 0. Using the balanced-budget rule it follows that:

ζ = τc(η-1) 1+τc (16)
The tax rate τ (c) = G(c)/c is therefore counter-cyclical for η < 1, pro-cyclical for η > 1 and constant for η = 1.

Intertemporal equilibrium

Under Assumptions 2 and 3, substituting equation ( 2) in [START_REF] Kapour | Elasticity of intertemporal substitution in consumption: an empirical evidence from a natural experiment[END_REF] and solving (10)-( 11) with respect to c(t) and l(t) gives consumption demand and labor supply functions c(K(t), λ(t)) and l(K(t), λ(t)). Using (1)-( 2), we get equilibrium values for the rental rate of capital r(t) and the wage rate w(t):

r(t) = Af ′ (x(t)) ≡ r(K(t), λ(t)) w(t) = A[f (x(t)) -x(t)f ′ (x(t))] ≡ w(K(t), λ(t)) (17)
with x(t) = K(t)/l(K(t), λ(t)). From the budget constraint in the program ( 8) and ( 12), we finally derive the following system of differential equations in K and λ:

K(t) = r(K(t), λ(t))K(t) + w(K(t), λ(t))l(K(t), λ(t)) -δK(t) -h(c(K(t), λ(t)))c(K(t), λ(t)) λ(t) = -λ(t) [r(K(t), λ(t)) -ρ -δ] (18)
An intertemporal equilibrium is then a path {K(t), λ(t)} t≥0 , with K(0) > 0, that satisfies equations [START_REF] Mendoza | On the ineffectiveness of policy in altering long-run growth: Harberger's superneutrality conjecture[END_REF] and the transversality condition [START_REF] King | Production, growth and business cycles[END_REF].

3 Steady state analysis

Multiplicity of steady states and Laffer curve

A steady state is a 4-tuple (K * , l * , x * , c * ) such that x * = K * /l * , which is solution of:

δ + ρ = Af ′ (x) (19)
h(c)c = l [Af (x) -δx] (20) U 2 (c, (ℓ -l)/B) BU 1 (c, (ℓ -l)/B) = A[f (x) -xf ′ (x)] h(c) (21)
Obviously, when M > δ + ρ > N in Assumption 1, there is a unique x = x * satisfying [START_REF] Mendoza | Effective tax rates in macroeconomics: Cross-country estimates of tax rates on factor incomes and consumption[END_REF]. However, given x = x * , nothing guarantees the existence nor the uniqueness of c, l and k.

Lemma 1. Let Assumptions 1-3 and 5 hold with M > δ + ρ > N and ζ = -1 for all l ∈ (0, ℓ). Then there exists at most one steady state if the consumption tax rate is pro-cyclical, i.e. ζ 0. Multiplicity requires ζ < 0, at least for some values of l ∈ (0, ℓ).

Proof : See Appendix 8.1.

In the case of the balanced-budget rule as given by ( 15) and the expression (16), we conclude that uniqueness is ensured as soon as η ≥ 1 and multiplicity requires η < 1.

The multiplicity of steady states can also be related to the existence of a Laffer curve. Using the balanced-budget rule as given by ( 15) and considering a given x = x * , a steady state can be defined by a solution (τ c , c, l) satisfying:

G(c) = Ω(c) = τ c c (22)
(1 + τ c )c = [Af (x * ) -δx * ] l (23)
(1 + τ c ) U 2 (c, (ℓ -l)/B) BU 1 (c, (ℓ -l)/B) = A[f (x * ) -x * f ′ (x * )] (24)
Substituting the labor supply l derived from ( 23) into [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF], we can define consumption as c = c(τ c ). Therefore, government spendings and incomes can be expressed as functions of τ c . We can then define the following elasticities

ǫ G (τ c ) ≡ dG dτc τc G , ǫ Ω (τ c ) ≡ dΩ dτc τc Ω (25)
The existence of a Laffer curve is obtained if ǫ Ω (τ c ) is a non-monotonic function of τ c and the multiplicity of steady states can be analyzed by comparing ǫ G (τ c ) and ǫ Ω (τ c ) when τ c ∈ [0, +∞). With GHH preferences, from the elasticity of the G(.) function ǫ G LL defined in Assumption 4, we need to introduce ǫ G ll = -ǫ G LL L/l which also gives the elasticity of the labor supply with respect to the wage rate as

ǫ ℓw = -ǫ G ll . It follows that ǫ G (τ c ) = ητc(ǫ G ll -1) 1+τc , ǫ Ω (τ c ) = 1+τcǫ G ll 1+τc (26)
We then derive the existence of a Laffer curve in the case of a constant elasticity of the labor supply:

7 Proposition 1. Let U(c, L/B) = u(c + G(L/B)) with G(L/B) = -(ℓ - L) 1+1/ǫ G ll /(1 + 1/ǫ G ll ), and Assumptions 1, 2, 4 hold with M > δ + ρ > N. Then ǫ Ω (τ c ) 0 if and only if τ c τ * c ≡ -1/ǫ G ll . Proof : See Appendix 8.2.
The Laffer curve is illustrated by Ω(c(τ c )) on Figure 1. The number of steady states is determined by the intersections of the curves G(c(τ c )) and Ω(c(τ c )). As shown in Lemma 1, when η ≥ 1, i.e. ζ ≥ 0, we have ǫ G (τ c ) < ǫ Ω (τ c ) for all τ c ≥ 0 and there exists a unique steady state. On the contrary, if government spendings are too counter-cyclical with η ≪ 0 and the initial value G(c(0)) is too large, there is no steady state.

In the case of constant government spendings with G(t) = G, the existence of two steady states is generic as soon as G is not too large. Actually, Figure 1 shows that there exist η < 0 and η ∈ (0, 1) such that the same property holds when variable government spendings are considered with η ∈ (η, η) and

ǫ G ll < -η/(1 -η). 8 G, Ω G(c(0)) τ c τ * c Ω(c(τ c )) η ≪ 0 η < 0 η = 0 η ∈ (0, 1) η ≥ 1
δ + ρ = Af ′ (1) (27) U 2 (c(1, 1), (ℓ -1)/B) BU 1 (c(1, 1), (ℓ -1)/B) = A[f (1) -f ′ (1)] h(c(1, 1)) (28)
Using this, we can show: In the rest of the paper, we evaluate all the shares and elasticities previously defined at the NSS. From Remark 2 : Using a continuity argument we derive from Proposition 2 that there exists an intertemporal equilibrium for any initial capital stock K 0 in the neighborhood of K * .

Aggregate instability with consumption taxes

Let us introduce the following elasticities:

ǫ cc = -U 1 (c,L) U 11 (c,L)c , ǫ Lc = -U 2 (c,L) U 21 (c,L)c , ǫ cL = -U 1 (c,L) U 12 (c,L)L , ǫ LL = -U 2 (c,L) U 22 (c,L)L ,
As it is more convenient to study our model in terms of elasticities with respect to labor, we consider:

ǫ lc = ǫ Lc , ǫ cl = -ǫ cL ℓ-l l , ǫ ll = -ǫ LL ℓ-l l < 0 (29)
The linearization of the dynamical system (18) around the NSS gives the characteristic polynomial P(λ) = λ 2 -λT +D with T and D which obviously depend on all the previous elasticities evaluated at the NSS. 9 Our aim is to discuss the local indeterminacy properties of equilibria, i.e. the existence of a continuum of equilibrium paths starting from the same initial capital stock and converging to the NSS. Our model consists in one predetermined variable, the capital stock K, and one forward variable, the shadow price λ of capital. Any solution from (18) that converges to the NSS satisfies the transversality condition and is an equilibrium. Therefore, given K(0), if there is more than one initial price λ(0) solution of equations ( 18), the equilibrium path from K(0) will not be unique. In particular, if both roots of the characteristic polynomial have negative real parts, there will be a continuum of converging paths and thus a continuum of equilibria. Definition 1. If D > 0 and T < 0 the NSS is locally indeterminate. 9 The precise expression of the characteristic polynomial is given in Appendix 8.4.

We provide immediately a general result. Lemma 2. Under Assumptions 1-3 and 5, the NSS is locally indeterminate only if the consumption tax rate is counter-cyclical, i.e. ζ < 0.

We then derive from Lemmas 1 and 2 that if ζ ≥ 0, there exists a unique steady state which is locally determinate.

In the case of a GHH utility function, let us consider the elasticity of the function G(L/B), denoted ǫ G LL , as defined in Assumption 4. We easily get from (29) that

ǫ cc = ǫ lc , 1 ǫ ll = 1 ǫ cl + 1 ǫ G ll = -(1-s)(δ+ρ)
δ(1-s)+ρ

1 ǫcc + 1 ǫ G ll , ǫ G ll = -ǫ G LL ℓ- l l < 0 (30)
We derive the following results:

Proposition 3. Let U(c, L) = u(c + G(L/B
) and Assumptions 1, 3 and 4 hold. When ζ = τ c (η -1)/(1 + τ c ) with η < 1 and ǫ G ll < -η/(1 -η), there exist σ > 1, ρ > 0, ǭcc > 0, ǫ cc > 0 and 1 > τc > τ c > 0 such that the NSS is locally indeterminate if one of the following sets of conditions are satisfied:

i) if σ ≥ σ, η < [δ(1 -s) + ρ]/(ρσ), ǫ cc ≥ ǫ cc and τ c ∈ (τ c , τc ), ii) if σ ∈ (0, σ), ǫ cc ≥ max{ǫ cc , ǭcc } and τ c ∈ (τ c , τc ), iii) if σ ∈ (0, σ), ρ ∈ (0, ρ), ǫ cc ∈ (ǫ cc , ǭcc ) and τ c ∈ (τ c , τc ).
Proof : See Appendix 8.5.

There is a range of tax rates such that indeterminacy occurs even when the balanced budget is characterized by constant public expenditures, i.e. η = 0. This is in contrast to the result obtained in Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] with additivelyseparable preferences. Note also that local indeterminacy is compatible with both pro-cyclical (η ∈ (0, 1)) and counter-cyclical (η < 0) government spendings. It is worth pointing out that the upper bound τc is lower than τ * c as given in Proposition 1, i.e. sunspot fluctuations arise in the increasing part of the Laffer curve, when government spendings are counter-cyclical (η ≤ 0).

Constant/variable government spendings

We provide now a numerical exercise to illustrate some of our results. On the basis of yearly data, we consider the benchmark parameterization (s, δ, ρ) = (0.3, 0.1, 0.01). Concerning the other structural parameters, the numerical illustration will be based upon the following empirical evidences: i) While Cobb-Douglas technologies are widely used in growth theory, recent papers have questioned the empirical relevance of this specification and find that capital and labor have an elasticity of substitution significantly different than unity. However, empirical evidences for both gross substitutability (elasticity above one) and gross complementarity (elasticity below one) of capital and labor are obtained in the literature. For instance, Duffy and Papageorgiou [START_REF] Duffy | A cross-country empirical investigation of the aggregate production function specification[END_REF] report robust estimates that are contained in [1.24, 3.24] and Krusell et al. [START_REF] Krusell | Capital-skill complementarity and inequality: a macroeconomic analysis[END_REF] find an elasticity of substitution between unskilled labor and equipment of 1.67. On the contrary, Chirinko [START_REF] Chirinko | The CES production function in the theory and empirics of economic growth[END_REF], Klump et al. [START_REF] Klump | Factor substitution and factor-augmenting technical progress in the United States: a normalized supply-side system approach[END_REF] and León-Ledesma et al. [START_REF] León-Ledesma | Identifying the elasticity of substitution with biased technical change[END_REF] provide robust estimates in the range [0.4, 0.6].

In the following, in order to cover both configurations, we assume that σ = 2 and σ = 0.6.

ii) The elasticity of intertemporal substitution in consumption is usually assumed to be low, i.e. less than one. However, recent contributions provide divergent views. Mulligan [START_REF] Mulligan | Capital interest and aggregate intertemporal substitution[END_REF], Vissing-Jorgensen [START_REF] Vissing-Jorgensen | Limited asset market participation and the elasticity of intertemporal substitution[END_REF] and Vissing-Jorgensen and Attanasio [START_REF] Vissing-Jorgensen | Stock-market participation, intertemporal substitution and risk aversion[END_REF] repeatedly obtained estimates of this elasticity which are significantly larger than one. More recently, Gruber [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] and Kapoor and Ravi [START_REF] Kapour | Elasticity of intertemporal substitution in consumption: an empirical evidence from a natural experiment[END_REF] provide robust estimates in the range (2, 3). 10 We consider in the following ǫ cc = 2.2 and ǫ cc = 2.8.

iii) Concerning the elasticity of the labor supply with respect to the wage rate, Rogerson and Wallenius [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF] have shown that the elasticities at the macroeconomic level are virtually unrelated to the micro elasticities, and are much larger than expected. The reason for this discrepancy is attributed to the role of the participation decision of women and the extent of early retirement. According to Rogerson and Wallenius [START_REF] Rogerson | Micro and macro elasticities in a life cycle model with taxes[END_REF] the macroeconomic elasticity of the labor supply with respect to the wage is in the range of (2.25, 3.0). We finally assume that ǫ ℓw = -ǫ G ll = 2.5. Assuming first σ = 2 and ǫ cc = 2.2, Figure 5 shows the region of indeterminacy corresponding to Proposition 3, case iii), when τ c is between τ c (dash line) and τc (solid line). We can see how the indeterminacy area is modified when the elasticity η varies from -1 to 0, thus from strongly counter-cyclical to constant public spendings. Considering consumption tax rates provided by Mendoza et al. [START_REF] Mendoza | On the ineffectiveness of policy in altering long-run growth: Harberger's superneutrality conjecture[END_REF][START_REF] Mendoza | Effective tax rates in macroeconomics: Cross-country estimates of tax rates on factor incomes and consumption[END_REF], 11 and more recently by Volkerink and De Haan [START_REF] Volkerink | Tax ratios: a critical survey[END_REF], depending on the degree of counter-cyclicality of public spendings, local indeterminacy arises for almost all OECD countries. We get indeed: η = 0 and τ ∈ (0.338, 0.4) Denmark, Norway η = -0.3 and τ ∈ (0.241, 0.282) Ireland, Finland η = -0.36 and τ ∈ (0.228, 0.266) Iceland, Luxembourg, Sweden η = -0.49 and τ ∈ (0.204, 0.237) Austria, France η = -0.7 and τ ∈ (0.175, 0.202) New Zealand, Portugal η = -0.8 and τ ∈ (0.163, 0.188) Belgium, Germany, Greece, Netherlands, UK η = -1 and τ ∈ (0.144, 0.166) Italy

Note that Spain also belongs to the indeterminacy area when η = -1.06.

On the contrary, Australia, Canada, Japan, Switzerland and the US are characterized by much lower consumption tax rates and are strongly out of the indeterminacy region, at least for reasonable values of the elasticity η.

11 Updated estimates up to 1996 are available online from the authors.

Assuming now σ = 0.6 and ǫ cc = 2.8, Figure 3 again shows the region of indeterminacy corresponding to Proposition 3, case iii), when τ c is between τ c (dash line) and τc (solid line). Although we now consider a complementarity between capital and labor, local indeterminacy still arises for the same OECD countries as in the previous case. We get indeed: η = 0 and τ ∈ (0.348, 0.4) Denmark, Norway η = -0.3 and τ ∈ (0.248, 0.282) Ireland, Finland η = -0.36 and τ ∈ (0.234, 0.266) Luxembourg, Sweden η = -0.49 and τ ∈ (0.209, 0.237) Austria, France, Iceland η = -0.7 and τ ∈ (0.179, 0.202) New Zealand, Portugal η = -0.8 and τ ∈ (0.167, 0.188) Belgium, Germany, Greece, Netherlands, UK η = -1 and τ ∈ (0.148, 0.166) Italy

This numerical exercise clearly shows that the existence of indeterminacy is a robust property that occurs for a wide range of values for the elasticity of capital-labor substitution. This point is worth to mention as such a robustness is usually not satisfied. Indeed, in a Benhabib and Farmer [START_REF] Benhabib | Indeterminacy and increasing returns[END_REF] type model with productive externalities, Guo and Lansing [START_REF] Guo | Capital-labor substitution and equilibrium indeterminacy[END_REF] prove that local indeterminacy becomes much more difficult if not impossible when capital and labor are gross complement. 12 Similarly, in a Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF] type model with labor taxes, Ghilardi and Rossi [START_REF] Ghilardi | Aggregate stability and balancedbudget rules[END_REF] show that balancedbudget rules deliver determinacy for a broad range of OECD countries when the elasticity of capital-labor substitution is in the range [0.4, 0.6].

Let us finally complete this exercise by discussing the value of η. Using annual data over the interval 1960 -1998 for 22 OECD countries, Lane [START_REF] Lane | The cyclical behaviour of fiscal policy: evidence from the OECD[END_REF] provides some empirical estimates of the elasticity of government expenditure with respect to output growth. Since consumption is almost perfectly correlated with output, we then have a measure of η. Lane shows that in most OECD countries, namely Austria, Belgium, Finland, France, Germany, Italy, Netherlands, Sweden and UK, government spending is counter-cyclical, i.e. η < 0. Moreover, his estimates perfectly match the values of η used in the two previous tables for Finland, France, Sweden and UK. We have then proved that with GHH preferences, aggregate instability based of expectation-driven fluctuations can be generated by balanced-budget consumption taxes under empirically plausible parameters' configurations.

Economic interpretations

To establish the intuition for our results, let us explain why the expectations of an increase of the consumption tax rate may be self-fulfilling. To simplify, as in Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF] and Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF], we restrict our attention to a balanced-budget rule with constant spendings. In order to clearly emphasize the economic mechanism behind the occurrence of expectation-driven fluctuations, let us come back to the intratemporal choice between consumption and leisure. When both labor income and consumption are taxed, at rate τ l and τ c respectively, we have:13

U 2 (c,ℓ-l) U 1 (c,ℓ-l) = 1-τ l
1+τc w This equation shows that both tax rates affect the choice between consumption and leisure in a similar way. Indeed, an increase of τ l or τ c reduces the price of leisure in terms of the consumption good, which decreases the labor supply. This is the main channel that explains expectation-driven fluctuations in Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF] where only labor taxation is considered. To be more specific, assume that households expect a higher labor tax rate at the next period. They will reduce their future labor supply. Therefore, the return on capital will be lower, reducing current investment. To sustain a lower investment, households need to work less today. Since taxation is counter-cyclical, this implies a higher tax rate, explaining that expectations are self-fulfilling.

Since we argue that consumption taxation can a priori have the same effect on labor supply than labor taxation, let us focus on the case considered in this paper where the consumption tax is the only distortion. The (inverse) labor supply is given by:

w = (1 + τ c ) U 2 (c,ℓ-l) U 1 (c,ℓ-l)
(31) On the one hand, given τ c and c, this equation describes a positive relationship between w and l (under Assumption 3), as drawn in Figure 4. On the other hand, an increase in either τ c or c shifts the labor supply to the left, reducing labor at equilibrium. If households expect a higher future consumption tax rate, they will reduce their future labor supply, as above. This corresponds to a substitution effect. The labor supply will move from l s 0 to l s 1 in Figure 4. However, households also expect to decrease their future consumption. This induces an income effect, which goes in the opposite direction and increases labor at equilibrium. Indeed, as shown in Figure 4, this shifts the labor supply to the right from l s 1 to l s 2 . However, if this second effect is dominated by the first one, a higher expected consumption tax rate implies a lower labor supply. This decrease of next period labor reduces the real interest rate, which has a negative effect on current investment. Needing less income today, households work less. This induces a lower consumption, and thus an increase of the current consumption tax rate in order to match the balanced-budget rule. Therefore, expectations are self-fulfilling.

These substitution and income effects, that play in opposite directions, allow to explain why indeterminacy may occur under some particular preferences, while this is not possible with others. Starting with the GHH case, equation (31) rewrites:

w = (1 + τ c )G ′ (ℓ -l)
The absence of income effect in this formulation explains why indeterminacy appears easily. Consider as in Giannitsarou [START_REF] Gruber | A Tax-based estimates of the elasticity of intertemporal substitution[END_REF] the separable case, with a log-linear utility of consumption, i.e. U(c, ℓ -l) = lnc + v(ℓ -l). Equation (31) becomes:

w = (1 + τ c )cv ′ (ℓ -l) = (c + G)v ′ (ℓ -l)
where the second equality is obtained using the balanced-budget rule. When G is constant, a modification of the tax rate does not have any impact on the labor supply. This explains that expectation-driven fluctuations cannot occur in this case. Actually, such a property still holds with general additivelyseparable preferences.

When the utility is of the King-Plosser-Rebelo [START_REF] King | Production, growth and business cycles[END_REF] type, equation ( 31) is given by: w

= (1 + τ c )c v ′ (ℓ-l) v(ℓ-l) = (c + G) v ′ (ℓ-l) v(ℓ-l)
Again, the labor supply does not depend on the tax rate and expectationdriven fluctuations are also ruled out. Finally, when preferences are linear homogeneous, we can show that indeterminacy may occur, but this requires sufficiently counter-cyclical public spendings and a strong departure from the Cobb-Douglas specification. Indeed, for constant public expenditures and U(c, ℓ -l) = c α (ℓ -l) 1-α , one obtains:

w = (1 + τ c )c 1-α α 1 ℓ-l = (c + G) 1-α α 1 ℓ-l
and expectation-driven fluctuations are again ruled out.

Concluding comments

This paper re-examines the impact of balanced budget fiscal policy rules on the aggregate instability that may arise from self-fulfilling expectations.

The recent debate has focussed indeed on two main contributions showing that while distortionary taxes on labor income could have expectation-driven destabilizing effects (Schmitt-Grohé and Uribe [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF]), when the government finances instead its constant expenditures via taxes on consumption, the equilibrium is always unique and determinate (Giannitsarou [10]). Based on that, consumption taxes appear to be more appropriate than income taxes if the government is willing to avoid instability. We show in this paper that such a conclusion has to be taken with caution as it heavily relies on the specification of the fundamentals, in particular the utility function. We prove indeed that GHH preferences, which are characterized by the absence of income effects, generate expectation-driven instability under counter-cyclical consumption taxes and plausible restrictions on the main parameters, namely, the elasticity of capital-labor substitution, the elasticity of intertemporal substitution in consumption and the elasticity of the labor supply. Moreover, numerical illustrations, based on empirically relevant parameterizations of the model, show that consumption taxation can be a source of instability for most OECD countries.

Appendix

Proof of Lemma 1

Let us consider equations ( 19)- [START_REF] Pintus | Aggregate instability in the fixed-cost approach to public spending[END_REF]. Equation ( 20

H(l) ≡ h(c(x * , l)) U 2 (c(x * ,l),(ℓ-l)/B) BU 1 (c(x * ,l),(ℓ-l)/B) = A[f (x * ) -x * f ′ (x * )] (33)
Multiplicity of equilibria may occur as soon as H(l) is non-monotonic. Differentiating equation (33) with respect to l, we get:

ǫ H (l) ≡ H ′ (l)l H(l) = 1 1+ζ 1 ǫcc -1 ǫ lc + 1 ǫ cl -1 ǫ ll + ζ 1 + 1 ǫ cl -1 ǫ ll (34)
Under the normality Assumption 3 we conclude immediately that if ζ ≥ 0 H(l) is monotonic and uniqueness holds.

Proof of Proposition 1

Let us consider equations ( 22)- [START_REF] Schmitt-Grohé | Balanced-budget rules, distortionary taxes, and aggregate instability[END_REF]. Equation ( 23) rewrites:

l = (1+τc)c Af (x * )-δx * ≡ l(τ c , c) (35)
Substituting l(τ c , c) into ( 24), we get:

(1 + τ c ) U 2 (c,(ℓ-l(τc,c))/B) BU 1 (c,(ℓ-l(τc,c))/B) = A[f (x * ) -x * f ′ (x * )] (36)
which defines c as a function of τ c . Differentiating this equation gives:

dc dτc τc c = -τc 1+τc 1+ 1 ǫ cl -1 ǫ ll 1 ǫcc -1 ǫ lc + 1 ǫ cl -1 ǫ ll < 0 (37)
Using [START_REF] Pintus | Sunspots in real business-cycle models with complementary inputs[END_REF], we conclude that:

dΩ dτc τc Ω = 1 ǫcc -1 ǫ lc + 1 ǫ cl -1 ǫ ll -τc 1+τc " 1+ 1 ǫ cl -1 ǫ ll " 1 ǫcc -1 ǫ lc + 1 ǫ cl -1 ǫ ll ≡ ǫ Ω (τ c ) (38)
Consider now the GHH utility function. We have ǫ cc = ǫ lc and 1/ǫ cl -1/ǫ ll = -1/ǫ G ll . We conclude that:

ǫ Ω (τ c ) = 1+τcǫ G ll 1+τc (39)
with ǫ Ω (0) = 1 and ǫ Ω (+∞) = ǫ G ll < 0. Therefore, ǫ Ω (τ c ) 0 if and only if τ c τ * c ≡ -1/ǫ G ll .

Proof of Proposition 2

To establish the existence of the normalized steady state (x * , l * ) = (1, 1), we have to prove the existence and uniqueness of solutions A * and B * solving ( 27) and (28). We first obviously conclude that there is a unique value A = (δ + ρ)/f ′ (1) ≡ A * solving [START_REF] Volkerink | Tax ratios: a critical survey[END_REF]. Considering A * , we show now that there exists a unique B * solving

Γ(B) = (δ+ρ)[1-s(1)] h(c(1,1)) with Γ(B) = U 2 (c(1,1),(ℓ-1)/B)

Proof of Lemma 2

Considering all these elasticities evaluated at the NSS, the linearization of the dynamical system (18) around the NSS gives:

Proposition 8.1. Under Assumptions 1-2, the characteristic polynomial is

P(λ) = λ 2 -λT + D with D = δ+ρ ∆ (1-s)[δ(1-s)+ρ] sσ 1 ǫcc -1 ǫ lc -1 ǫ ll + 1 ǫ cl -ζ 1 ǫ ll -1 -1 ǫ cl T = ρ + (δ+ρ)(1-s)ζ ∆σǫ lc and ∆ = 1 ǫcc + ζ 1 ǫ ll -s σ -1 ǫ cl ǫ lc
Proof : Using (29) and the first order conditions ( 10) and ( 11), we get ǫ cl = -ǫ lc (ch(c)/wl). Using the expression of w at the NSS given in [START_REF] Lloyd-Braga | Tax rate variability and public spending as sources of indeterminacy[END_REF] together with (3) and ( 19) we find wl = K(1 -s)(δ + ρ)/s. Similarly we derive from ( 20)

h(c)c + δK = (δ + ρ)K/s and thus h(c)c = [δ(1 -s) + ρ]K/s (40)
Then at the NSS we get

ǫ cl = -δ(1-s)+ρ (1-s)(δ+ρ) ǫ lc (41)
Consider again the first order conditions ( 10) and [START_REF] Kapour | Elasticity of intertemporal substitution in consumption: an empirical evidence from a natural experiment[END_REF]. Under Assumptions 2 and 3, solving with respect to c(t) and l(t) gives consumption demand and labor supply functions c(K(t), λ(t)) and l(K(t), λ(t)). Using (29), the implicit function Theorem allows to get the partial derivatives of these functions evaluated at the NSS

dc dK = c K∆ s σǫ cl , dc dλ = -c λ∆ 1 ǫ ll -s σ -1 ǫ cl dl dK = -l K∆ s σ 1 ǫcc + ζ , dl dλ = -l λ∆ 1 ǫcc + ζ -1 ǫ lc with ∆ = 1 ǫcc + ζ 1 ǫ ll -s σ -1 ǫ cl ǫ lc
From these results and ( 17) we also derive at the NSS

dr dK = -r(1-s) Kσ 1 + s ∆σ 1 ǫcc + ζ , dr dλ = -r(1-s) λ∆σ 1 ǫcc + ζ -1 ǫ lc dw dK = ws Kσ 1 + s ∆σ 1 ǫcc + ζ , dw dλ = ws λ∆σ 1 ǫcc + ζ -1 ǫ lc
Consider then the system of differential equations in K and λ:

K(t) = r(K(t), λ(t))K(t) + w(K(t), λ(t))l(K(t), λ(t)) -δK(t) -h(c(K(t), λ(t)))c(K(t), λ(t)) λ(t) = -λ(t) [r(K(t), λ(t)) -ρ -δ]
Linearization around the NSS using (40), (41) and the above results gives

d K dK = ρ -(δ+ρ)(1-s) σ 1 ǫcc + ζ -1+ζ ǫ lc d K dλ = -K(δ+ρ)(1-s) λ∆s 1 ǫcc + ζ -1 ǫ lc -(1 + ζ) δ(1-s)+ρ (δ+ρ)(1-s) 1 ǫ ll -s σ -1 ǫ cl d λ dK = λ(δ+ρ)(1-s) Kσ 1 + s ∆σ 1 ǫcc + ζ d λ dλ = (δ+ρ)(1-s) ∆σ 1 ǫcc + ζ -1 ǫ lc
The expression of the characteristic polynomial follows after tedious computations and straightforward simplifications.

We may now prove Lemma 2. Note first that concavity of the utility function implies Let us now consider the expression of T in (44). We get T < 0 if and only if

(1 + τ c ) 1 ǫ G ll -s σ < τ c (1 -η) ǫ cc

«

≡ ǭcc so that the rhs of ( 46) is still negative. We derive from (47) that T < 0 in two distinct cases: 1-if ǫ cc ≥ (1 + ǭcc )/(1 -η) and τ c < τ 2 c , 2-if η > -1/ǭ cc and ǫ cc ∈ (ǭ cc , 1 + ǭcc ) without any restriction on τ c . c) Assume finally that σ ∈ (0, σ) and ǫ cc ≤ ǭcc . Then the rhs of ( 46) is positive and T < 0 without any restriction on τ c . But considering condition (45), we need to check that ǫ cc < ǭcc , i.e.

ρsǫ G2 ll + ǫ G ll [(δ + ρ)(1 -s) -ρ(σ + s)] -4ρ 2 σs ≡ h(ρ) < 0 Direct inspection of this inequality shows that there exists ρ ∈ (0, +∞] such that when ρ ∈ (0, ρ), ǫ cc < ǭcc . Considering all the above computations, when ǫ G ll < -η/(1 -η), local indeterminacy occurs in three different cases: i) if σ ≥ σ, η < [δ(1 -s) + ρ]/(ρσ), ǫ cc ≥ ǫ cc and τ c ∈ (τ c , τc ), with τc = min{τ 1 c , τ 2 c }, ii) if σ ∈ (0, σ), and -either ǫ cc ≥ max{ǭ cc , (1 + ǭcc )/(1 -η)} and τ c ∈ (τ c , τc ), with τc = min{τ 1 c , τ 2 c }, -or η > -1/ǭ cc , ǫ cc ∈ (ǭ cc , (1 + ǭcc )/(1 -η)) and τ c ∈ (τ c , τc ), with τc = τ 1 c , iii) if σ ∈ (0, σ), ρ ∈ (0, ρ), ǫ cc ∈ (ǫ cc , ǭcc ) and τ c ∈ (τ c , τc ), with τc = τ 1 c . The result is proved.

Figure 1 :

 1 Figure 1: Laffer curve and multiple steady states

Proposition 2 .

 2 Let Assumptions 1-3 and 5 hold and ζ(c * ) = -1. Then there exist A * and B * which are the unique solutions of (27) and (28) respectively such that when A = A * and B = B * , (x * , l * ) = (1, 1) is a steady state. Proof : See Appendix 8.3.

  (3), (4) and c * = c(1, 1), we consider indeed s(1) = s, σ(1) = σ and ζ(c * ) = ζ.

Figure 2 :

 2 Figure 2: Indeterminacy, consumption taxes and public spendings with σ > 1

Figure 3 :

 3 Figure 3: Indeterminacy, consumption taxes and public spendings with σ < 1

Figure 4 :

 4 Figure 4: The role of labor market

  ) implicitly gives c = c(x * , l), where implying that c can be defined as a function of l for all ζ = -1. Substituting x = x * and c = c(x * , l) into (21), one obtains:

BU 1 (

 1 c(1,1),(ℓ-1)/B) Under Assumption 2, either lim B→0 Γ(B) = 0 and lim B→+∞ Γ(B) = +∞ or lim B→0 Γ(B) = +∞ and lim B→+∞ Γ(B) = 0. Since Assumption 2 also ensures Γ ′ (B)B/Γ(B) = 0, this proves existence and uniqueness of B * .

1 ǫccǫ ll - 1 ǫ 1 Dσ

 111 cl ǫ lc ≤ 0 (42) while Assumption 3 implies1 ǫcc -1 ǫ lc ≥ 0 and 1 ǫ cl -1 ǫ ll ≥ 0(43)If ζ ≥ 0, we conclude that ∆ < 0 and thus, from (42) and (43), that D < 0.8.5 Proof of Proposition 3Using (30), we derive from Proposition 8.Since ζ < 0, we derive that T < 0 requires ∆ > 0, i.e.Let us first consider the expression of D in (44). When ∆ > 0, we get D > 0 if and only ifζ > 1 ǫ G ll -1 ≡ ζ 1 We then need to show that ζ 1 < ζ which is satisfied if and only if ǫ cc > that ζ = τ c (η -1)/(1 + τ c ) with η < 1. We derive that indeterminacy requires ζ 2 < τ c (η -1)/(1+τ c ) < ζ. This is obtained if ǫ Gll < -η/(1-η), ǫ cc > ǫ cc and τ c ∈ (τ c , τ 1 c )

  rhs of (46) is negative. Moreover (46) can be written as followsc [ǫ cc (1 -η) -1] T < 0 if ǫ cc ≥ 1/(1 -η) and τ c < τ 2 s)(δ+ρ) ρ+(1-s)δ -(1-s)(ρ+δ) ρσ Note that τ 2 c > τ c when η < [δ(1 -s) + ρ]/(ρσ) < 1, and that ǫ cc > 1/(1 -η) when ǫ G ll < -η/(1 -η). b) Assume now that σ ∈ (0, σ) and ǫ cc > (ρ+δ)(1-s) δ(1-s)+ρ [δ(1-s)+ρ(1-σ)]

The constant B is used to prove the existence of a normalized steady state which remains invariant with respect to preference parameters such that the elasticity of intertemporal substitution in consumption or the wage elasticity of the labor supply.

If G(X) = X 1-γ /(1 -γ) with γ ≥ 0 the inverse of the elasticity of the function, the first part of Assumption 4 is satisfied when γ ∈ [0, 1) while the second part holds if γ > 1.

Giannitsarou [10] mainly uses such an additively separable specification with u(c) = lnc, v(L/B) = -[ℓ -L)/B] 1+χ /(1 + χ) and χ ≥ 0.

The proofs of these claims can be provided upon request.

In the case of a general function G(L/B) the elasticity ǫ G ll depends on τ c and the Laffer curve may have an unconventional shape with multiple local maxima.

This inequality ensures ǫ G (+∞) > ǫ Ω (+∞) and thus a second crossing of the curves G(c(τ c )) and Ω(c(τ c )). Note that the inequality is obviously satisfied if government spendings are counter-cyclical with η ≤ 0.

Barro [1] uses the lower bound of this interval to evaluate the welfare costs of rare disasters in a representative-consumer model.

See also Pintus[START_REF] Pintus | Sunspots in real business-cycle models with complementary inputs[END_REF].

To simplify the presentation, we omit the scaling parameter B in this section.

* This work was supported by French National Research Agency Grant (ANR-08-BLAN-0245-01). We thank


Conrad Boton

Sylvain Kubicki

Gilles Halin

Understanding pre-construction simulation activities to adapt visualization in 4D CAD collaborative tools

Pre-construction phase is very important in construction project life cycle. It enables minimizing risks, waste and overheads, maximizing productivity and reliability during construction phase, and to design more accurate and feasible schedules. The collaborative use of 4D CAD models and simulations appears to be an innovative solution to support many of simulation activities during this phase. But the visualization adaptation to business requirements of actors remains a challenging issue. This paper presents the pre-construction collaborative simulation activities and describes a method aiming at adapting visualization in 4D collaborative tools. A formula for ranking visualization modes and related metamodels are also proposed.

Introduction: importance of visualization in pre-construction collaborative practices

Increasing productivity and efficiency is an important issue in the Architecture, Engineering and Construction (AEC) sector. This sector is mainly characterized by fragmentation, heterogeneous teams with low lifetimes and many uncertainties. 4D CAD modeling and simulation is one of the greatest innovations in recent years. It consists in linking a 3D model of the building with the works planning in order to simulate the construction development over time. 4D CAD techniques can fill several needs from design to project management through constructability analysis and tasks planning [START_REF] Tommelein | Acknowledging Variability and Uncertainty in Product and Process Development," 4D CAD and Visualization in Construction[END_REF]. Literature shows that several applications have been proposed to improve the 4D CAD use ( [START_REF] Chau | Four-Dimensional Visualization of Construction Scheduling and Site Utilization[END_REF]; [START_REF] Lu | Construction planning methodology integrating operations simulation and four dimensional computer aided design (4D-CAD)[END_REF]; [START_REF] Seok | Improvement of Link Process in 4D CAD Viewer by Using Interface Board for Construction Project Management[END_REF]). In addition, studies have shown the real impact of the use of 4D CAD in construction projects ( [START_REF] Staub-French | 3D and 4D modeling for design and construction coordination: issues and lessons learned[END_REF]; [START_REF] Dawood | Measuring the effectiveness of 4D planning as a valuable communication tool[END_REF]).

More recently, [START_REF] Mahalingam | An evaluation of the applicability of 4D CAD on construction projects[END_REF] showed that the collaborative use of 4D CAD is particularly useful during the pre-construction phase for comparing the constructability of working methods, for visually identifying conflicts and clashes (overlaps), and as visual tool for contractors, subcontractors and suppliers to discuss and to plan project progress. So the advantage of the 4D collaborative use has been largely addressed. Moreover, some studies have been conducted both in the scientific world as well as in the industrial world to improve it ( [START_REF] Zhou | An interactive approach to collaborative 4d construction planning[END_REF]; [START_REF] Kang | Empirical study on the merit of web-based 4D visualization in collaborative construction planning and scheduling[END_REF]).

But an important issue that remains in today's collaborative use of 4D CAD in construction projects is related to the adaptation of visualization to the actors' business needs. As collaboration is particular from one project to another, the business requirements regarding visualization also evolves.

Collaborative practices [START_REF] Zignale | A model-based method for the design of services in collaborative business environments[END_REF] can help to describe them and to understand most common pre-construction activities. Moreover, in the AEC field several visualization modes can represent the same concept and actors choose one or the other of these modes according to their specific needs related to the task they have to perform. For example, tasks planning may be represented by a Gantt chart or by a PERT network and building elements can be depicted with a 3D model or a 2D plan. The classical view (3D + Gantt) usually proposed to all practitioners in most of the current 4D tools seems therefore not suiting the needs of every actors and situations of use. So, adapting the visualization to individual business needs could significantly improve the collaboration.

We rely on Model-Driven Engineering (MDE) to propose a method aiming at designing business views. The aim is not to automatically generate user views but to guide the groupware designers in selecting the appropriate visualization modes and in composing them. To support the method, we propose a business view metamodel to firstly describe and then compare view models. A metamodel to describe the user's visualization tasks is also defined. The goal is to design adapted views according to actor's business practices in collaborative 4D simulation, so it is important to understand the simulation activities related to preconstruction phase. Relying both on literature review and on a preliminary survey with AEC professionals, we propose in this paper to understand these simulation activities, to find how they can be described and how this description can be used in our view adaptation method.

Understanding site preparation collaborative simulation activities 2.1 Pre-construction phase and site preparation activities

A facility development life cycle process involves a number of phases and activities that complement each other. According to [START_REF] Fischer | Circle integration[END_REF], these phases include "pre-construction, construction, operation and maintenance, and commissioning." More recently, [START_REF] Ibrahim | Emerging technology to model dynamic knowledge creation and flow among construction industry stakeholders during the critical feasibility-entitlements phase[END_REF] divided this process into five main phases: feasibility, entitlements, building permit, construction, and property management phases. We consider that pre-construction includes feasibility, entitlements and building permit and starts "when a parcel of land becomes available for consideration" [START_REF] Ibrahim | Emerging technology to model dynamic knowledge creation and flow among construction industry stakeholders during the critical feasibility-entitlements phase[END_REF]. It "encompasses a broad set of characteristics of a facility, including client requirements, specifications of functional systems, dimensions, given and calculated loads, computed stresses, bill of materials, construction plans, schedules, and cost estimates" [START_REF] Fischer | Circle integration[END_REF].

So, one important goal in the pre-construction phase is the site preparation. This involves taking steps to ensure that the construction phase takes place under the best conditions. According to [START_REF] Neufert | Les éléments des projets de construction[END_REF], site preparation is based on studies and specific plans (drawings, installation plans, etc.), administrative preparation (site opening notice, demolition permit, registries opening, etc.), material preparation (temporary roads, site board, etc.) and organizational management setup (contractors' mailing list, site rules, meetings and activities planning, etc.). During this phase it is necessary to anticipate problems, clashes and other defects both linked to the construction process as well as the designed building itself. Then, construction processes simulation is necessary. With the evolution of technology, more and more aspects can be simulated in order to make a reliable construction planning.

[14] identifies 3 groups of planning processes: core processes, facilitating processes and controlling processing. Core processes, that have clear dependencies, concern scope planning and definition, activity definition and sequencing, activity duration estimating, schedule development, resource planning, cost estimating and budgeting, and project plan development. Facilitating processes are quality planning, organizational planning, staff acquisition, communication planning, risk identification and quantification, risk response development, procurement planning and solicitation planning. Controlling processes enable to measure the performance of the executing process and to send feedback to the planning process. They are overall change control, scope change control, schedule control, cost control, quality control and performance reporting.

Collaborative simulation activities

AEC projects teams are comprised of a network of interdisciplinary players, with varied roles, responsibilities, goals, and objectives. "They are created to produce complicated project solutions through the sharing of highly specialized knowledge" [START_REF] Chinowsky | Managing interdisciplinary project teams through the Web[END_REF]. "Most actors only take part for a limited time. Therefore, the organization is continuously changing during the process" [START_REF] Josephson | The causes and costs of defects in construction A study of seven building projects[END_REF]. During the site preparation, it is essential to early involve these stakeholders in order to take into account their viewpoints, to make relevant the simulation and to design a planning that takes into account the needs of all. This is especially true because "effective collaboration in simulation modeling is possibly one of the key factors in the success of simulation project" [START_REF] Taylor | Netmeeting: a tool for collaborative simulation modeling[END_REF].

Collaborative simulations during site preparation concern activities sequencing, estimating activities duration, scheduling resources and costs, identifying conflicts and clashes (overlaps), managing reservations and comparing the constructability of construction methods. It also serves as a monitoring tool during the construction phase.

These collaborative activities in a project can mostly be described as processes that show the actors' activities. An example of process that show collaboration among actor's in order to build the site planning and to manage reservations is presented in [START_REF] Boton | Collaborative construction planning: towards 4D visualizations adapted to practitioners requirements[END_REF]. But if "some of the most important roles in a building project organization are played by the client, designer, contractor, sub-contractor, material supplier, machine supplier, and the user" [START_REF] Johansson | GeoAnalytics visual inquiry and filtering tools in parallel coordinates plots[END_REF], as said, each construction project sets up a specific collaborative context in which the different roles of actors are not always the same. Then processes are not repeatable and are often specific to the particular situations of each construction project [START_REF] Kubicki | Assistance to building construction coordination -towards a multi-view cooperative platform[END_REF]. It could be therefore appropriate to describe them as more common and widespread collaborative practices [START_REF] Zignale | A model-based method for the design of services in collaborative business environments[END_REF].

Based on literature review and a preliminary survey with some industry professionals, a certain number of pre-construction collaborative practices can be identified, including site scheduling, cost budgeting, clashes and conflict management, safety and risk management, reservation management, procurements planning, communication planning. Moreover quality aspects are also simulated and different kinds of defects are anticipated.

Describing pre-construction collaborative practices

Relying on the literature review and according to our specific needs regarding collaboration situations description, collaborative practices can be described as sets of sub-practices (collaborative or individual) with a group of involved actors. Sub-practices are characterized by an information need and outputs. The actors have a business profile (architect, engineer, etc.) and play a role in the collaborative context. If the business profile of an actor is fixed as it is linked to his every day's occupation, the role concerns the practices assigned to him in the project's collaborative context and can be different from a project to another.

For instance, "collaborative site scheduling" can be considered as a collaborative practice. So, in a given collaborative context, one can see in table 1 the sub-practices, the responsible actors, their information needs and related outputs. We note that the outputs of a given sub-practice is often the information needed by another sub-practice.

Collaborative practice

Sub-practices Responsible actors Information need

Collaborative site scheduling

Building

Collaborative use of 4D CAD and limits

Collaborative construction simulations are important in pre-construction phase, including for site preparation. Recent advances in the Computer-Support Collaborative Work (CSCW) technologies make it possible to develop a shared virtual workspace to support close teamwork and project integration between construction organizations and professionals physically located at dispersed places [START_REF] Sun | Integration technologies to support organisational changes in the construction industry[END_REF]. But some specific construction simulation practices require some special tools. 4D CAD appears to be an innovative simulation solution. Many research efforts have been made to advance the 4D CAD technology from simple 4D animation of construction progression to interactive 4D simulation of alternative construction processes ( [START_REF] Liao | Development and Application of 4D Construction Simulation Tools[END_REF]; [START_REF] Mckinney | Interactive 4D-CAD[END_REF]). Moreover, research works showed that the collaborative use of 4D CAD is particularly useful during the preconstruction phase ( [START_REF] Mahalingam | An evaluation of the applicability of 4D CAD on construction projects[END_REF]) and efforts have been made to improve the collaborative 4D planning ( [START_REF] Sriprasert | Multi-constraint information management and visualisation for collaborative planning and control in construction[END_REF]; [START_REF] Zhou | An interactive approach to collaborative 4d construction planning[END_REF]; [START_REF] Kang | Empirical study on the merit of web-based 4D visualization in collaborative construction planning and scheduling[END_REF]). An important aspect of the construction industry is the flexibility that characterizes the management of projects [START_REF] Kubicki | Assistance to building construction coordination -towards a multi-view cooperative platform[END_REF]. Indeed, each stakeholder of a cooperative project's context has his own interests, his methods and specific practices. In this context generic solutions are difficult to adopt successfully. It is then important to adapt solutions to the needs of actors. One of the key points in adapting CSCW tools is to design visualizations matching the business requirements of actors. Considering previous 4D developments, the usual standard view (i.e. Gantt planning associated with 3D model) currently proposed to different actors in most of 4D CAD tools does not take into account this adaptation. Specific requirements related to practices performed by particular roles (planning of interventions and reservations request for a contractor; simulation and integration of works' interfaces for an engineer, etc…) are not considered.

In the next section, we propose a method for designing user views in 4D collaborative tools. This method takes into account the actors' business needs and enables to compare visualization modes, in order to choose the most appropriate ones and to compose multiple views adapted to the business needs. These views are called "business views".

Towards a method to design business views

The method

To adapt the visualization to actor's business needs in 4D collaborative tools, we propose a 4-steps method (Fig. 1).

• The first step identifies the business needs of actors. This consists in formalizing the collaborative practices in order to identify the subpractices performed by the different involved actors. Knowing these sub-practices helps to better define the business needs. Business needs can be described as the set of information and operations that actors will need in order to perform their business activities related to a collaborative practice. • In the framework of the collaborative practices, actors use groupware to perform their business activities. During these activities, we can identify specific usages that we have to understand and describe. The purpose of the second step is to determine the visualization needs for each actor. Visualization needs are interactions and visualization tasks that a user will need to perform in front of a computer-supported tool. Visualization tasks are "the analytic and exploratory tasks that a user might need or want to perform on the data" [START_REF] Valiati | A taxonomy of tasks for guiding the evaluation of multidimensional visualizations[END_REF]. At this stage we use the taxonomy proposed by [START_REF] Valiati | A taxonomy of tasks for guiding the evaluation of multidimensional visualizations[END_REF] to describe such visualization tasks with a single and accurate formalism. • Once visualization needs are known, it is then possible to compare the possible visualization modes in order to choose the most appropriate ones in relation to the expressed needs. In the third step the visualization modes are described and compared. Visualization modes are the techniques used to represent information. According to [START_REF] Boton | Adaptation of user views to business requirements: towards adaptive views models[END_REF] they can be described through the structure of the technique, the graphical attributes and the supported interaction principles. The most appropriate visualization mode is the one that matches better to the user's visualization needs. To this end, we can use the formula presented below to rank visualization modes for a given business need. • When appropriate modes are selected for each actor, it is possible to select those that may be composed together to form multiple coordinated views. To this end, it is important to study the coordination mechanisms and the interaction principles associated with the visualization modes (fourth step). Note that this step is not well defined for now and will be an important issue in our future works. At the end, adapted business views composed for each actor, will be validated to ensure that the propositions respond effectively to the expressed business needs.

Comparing adapted views (step 3): business views metamodel

Model Driven Engineering (MDE) approach recommends the use of metamodels to define domain languages, so each model has to conform to its metamodel. In order to model visualization modes and to choose the most adapted ones, we propose to adapt and use the business view metamodel described in [START_REF] Boton | Adaptation of user views to business requirements: towards adaptive views models[END_REF]. This metamodel, based on a literature review in the fields of Human-Computer Interaction and Information Visualization, shows that a model of business view has to describe content, using a technique with some interaction principles (Fig. 2).

User view

Describing visualization tasks (step 2): visualization tasks metamodel

The taxonomy proposed by [START_REF] Valiati | A taxonomy of tasks for guiding the evaluation of multidimensional visualizations[END_REF] leads us to define a user visualization tasks metamodel as shown on Fig. 3. Such a metamodel allows describing abstracted form of user's visualization tasks according to a same formalism. A visualization task could be a simple visualization task or group of visualization tasks. Simple visualization tasks are Compare (e.g. dimensions, items, data, values, etc), Identify (e.g. clusters, correlations, categories, properties, etc), Visualize (e.g. data, domain attributes, n dimensions, etc.), Determine (e.g. mean, median, variance, amplitude, differences, statistics, etc.), Locate (e.g. items, clusters, distances, graphical primitives, etc.), Configure (e.g. normalizations, classifications, zoom, dimensions orders, etc) and Infer (e.g. hypothesis, rules, trends, probabilities, cause/effect, etc.).

As an example we describe the business task of an engineer having to verify validity of contractors' reservations with a 4D tool. In

Formula for visualization modes ranking

We are working on a scoring system that would rank visualization modes according to usage needs. By assessing the criteria (technique, content, interaction principle, visualization tasks, etc.) an adaptation score (As) should be assigned to each visualization mode. This score is calculated for each actor and each sub-practice with the formula below. So, the relevance of a criterion is related to the information needs and to the visualization tasks. Note that As is the average of the Nc i and n is the number of criteria while m is the number of proprieties for a criterion i. The score (Nc i ) of a criterion i is then the average of its properties relevance (P j ) scores according to a visualization requirement. The visualization requirement is both an information need and a need for visualization tasks. The properties relevance scores (P j ) are -1, 0 or 1 depending on whether the property j is clearly unsuited, poorly adapted or well suited to the sub-practice. Each Nc i may vary between -1 and 1.

Case study

We consider a case study related to site preparation collaborative context and apply our method to a specific collaborative practice: "Collaborative site scheduling". We assume that in specific situation some given actors are responsible for each sub-practice shown on table 2.

Step 1: Identifying business needs

The first stage is to describe the specific collaborative practice, identifying sub-practices, responsible actors and information needs. For the example of collaborative site scheduling practice, Table 1 could be the result of this step. So, we know which sub-practices will be performed and the responsible actors. The information needs are also identified. But this is not sufficient enough to choose appropriate visualization modes. We need to add the visualization tasks to be sure to take into account the actor's visualization needs.

Step 2: Identifying visualization tasks

After understanding business practice context, we will identify in this section the visualization tasks the actor will have to perform. For this, we will divide the sub-practices into elementary usages that will call for specific visualization tasks. The visualization tasks model will help us in this description (table 2).

After this step, we know exactly which kind of information actors need to visualize and their visualization tasks. For each information need, many visualization modes may be possible. We need to compare them in order to choose the best adapted ones according to actor's visualization tasks.

Step 3: Choosing adapted visualization modes

In this step, we will use the business view metamodel to describe the possible visualization modes for every expressed visualization need. In instance, for the sub-practice "Activities sequencing", actor need to visualize the dates, the activities durations and a building representation. The building representation could be a 2D plan or a 3D representation (Fig. 4). In this case, we need to describe these two visualization modes in order to compare them. According to the metamodel, we can have the description of views shown in table 3. Emphasizes parts

Numericity

Non numeric Non numeric

Dynamism

Static Static

Possible interactions

Interactive zoom, Dynamic projection, Interactive deformation, Link&Brush Interactive zoom, Interactive filtering, Link&Brush Table 3 shows a comparison between two visualization modes that could be used to represent building elements. For example, 3D representation is easy to understand and more attractive than 2D plan. More interactions are possible with 3D representations than with 2D plan. At the contrary, 2D representation is widely known and used compared to 3D.

Using the adaptation score formula enables to establish that it's rather to choose a 3D representation for this specific sub-practice (table 4). Same work for each other sub-practices will lead to know which visualization modes are appropriate. So, for each actor, we can know all modes he needs to achieve his usage. In order to compose multiple views and to coordinate them, we need to make sure that the selected visualization modes are compatible with other ones. That is the subject of the next step.

Step 4: Composing adapted business multi-visualization

After selecting visualization modes, we need to compose into multiple coordinated visualizations according to the need of each actor. This will include defining visualizations coordination mechanisms and how to associate and to manage adapted interaction principles. As said, this step is not well defined yet. So we will devote to this work in the future.

Conclusion and perspectives

In this paper, we presented the pre-construction phase, site preparation activities and related simulation needs. We focused particularly on collaborative simulation activities and showed how to describe collaborative simulation practices in the framework of a visualization adaptation method. 4D CAD appears to be an innovative simulation tool for construction projects, particularly in pre-construction phase. Its collaborative use is very interesting according to the AEC industry characteristics. But the adaptation of the visualization to actors' business needs in such 4D collaborative tools remains a challenge.

A method and related models were presented to design business views in collaborative 4D tools. A case study showed how to use the method to compose the most adapted views for actors' business requirements.

Future work will strengthen the method, operationalize the ranking formula and deepen the fourth step related to multiple views coordination. We will also develop a tool to support the method and validate both the method and the tool.

Fig. 1 :

 1 Fig. 1: Method to compose business view

Fig. 2 :

 2 Fig. 2: Business view metamodel

Fig. 3 :

 3 Fig. 3: User's visualization tasks metamodel

Table 1 :

 1 An example of description of collaborative site scheduling practice

	elements listing	Architect	Pre-list of building elements, building representation
	Activities definition	Supervisor	WBS, pre-list of construction activities
	Activities duration estimation Sub-contractors	Activities description
	Activities sequencing	Contractor	Dates, activities durations, building representation
	Schedule development	Supervisor	Activities planning

  determine any variance (i.e. Determine if there is any variance between simulation sequences and dates list); infer trends (i.e. infer if reservations are valid or no).

							terms of
	visualization, he has to:
	visualize data (i.e. visualize the elements of the building
	representation);
	locate position (i.e. locate the positions of the reservations);
	identify dependencies (i.e. identify the dependencies among tasks
	and reservations);
	compare values (i.e. compare the values of simulation sequences
	with the dates list);
				Visualization task
			Simple visualization task	ordered	visualization tasks group
	Compare	Identify	Visualize	Determine	Locate	Configure	Infer
	+focus: EtypeCompare	+focus: EtypeIdentify	+focus: EtypeVisualize	+focus: EtypeDetermine	+focus: EtypeLocate	+focus: EtypeConfigure	+focus: EtypeVisualize
	<<enumeration>>	<<enumeration>>	<<enumeration>>	<<enumeration>>	<<enumeration>>	<<enumeration>>	<<enumeration>>
	EtypeCompare	EtypeIdentify	EtypeVisualize	EtypeDetermine	EtypeLocate	EtypeConfigure	EtypeInfer
	+Dimensions	+Clusters	+x dimensions	+Mean	+Items	+Classification	+Hypotheses
	+Items	+Corelations	+x items	+Median	+Data	+Filtering	+Rules
	+Data	+Categories	+Data	+Variance	+Values	+Zoom	+Probabilities
	+Values	+Properties	+Domain parameters	+Standard deviation	+Clusters	+Dimension order	+Trends
	+Clusters	+Patterns	+Attribute information	+Amplitude	+Properties	+Derive attributes	+Cause/Effect
	+Properties	+Characteristics	+Metadata	+Percentile	+Position/Locations	+Graphical primitives
	+Proportions	+Thresholds		+Sum	+Distances	+Normalization
	+Positions / Location	+Similarities		+Proportions	+Graphical primitives
	+Distances	+Dependencies		+Differences
	+Graphical primitives	+Independencies		+Correlation coefficients
		+Uncertainty		+Probabilities
		+Variation		+Other statistics

Table 3 :

 3 Visualization modes description

	Criteria	Proprieties	3D rep.	2D plan
	Technique	Structure	0	-1
		Graphical elements	1	0
		Retinal attributes	1	0
		Business use	0	1
		Nc1	0,5	0
	Content	Data Format	1	1
		Mental perception	0	-1
		Data nature	0	-1
		Nc2	0,33	-0,33
	Interaction principles	Interaction level	0	-1
		Interaction type	0	-1
		Nc3	0	-1
	Visualization tasks	Visualisation tasks	0	-1
		Nc4	0	-1
		As	0,21	-0,58

Table 4 :

 4 Example of visualization modes adaptation score

Acknowledgment

This article is supported by the National Research Fund, Luxembourg.