Jean-Pascal Bénassy
email: benassy@pse.ens.fr

Pse

Cepremap

Competitiveness, market power and price stickiness: A paradox and a resolution

Keywords: Sticky prices, calvo prices, cost of changing prices

Are prices less sticky when markets are more competitive? Our intuition would naturally lead us to give an affirmative answer to that question. But we first show that DSGE models with staggered price or wage contracts have actually the opposite and paradoxical property, namely that price stickiness is an increasing function of competitiveness. To eliminate this paradox, we next study a model where monopolistic competitors choose prices optimally subject to a cost of changing prices as in Rotemberg (1982a,b). For a given cost function, we find the more intuitive result that more competitiveness leads to more flexible prices.

Introduction

Is there a connection between competitiveness, market power and the "stickiness" of prices1 ? This is an old question and, as early as 1935, Gardiner Means noted that, in the face of the general slump, prices were falling less in sectors with high market power.

Recently price or wage rigidities have been successfully introduced into dynamic stochastic general equilibrium (DSGE) models 2 . The corresponding models succeed in reproducing the persistent response of output and inflation found in the data. Modelling nominal rigidities has notably taken the form of staggered wage or price contracts à la [START_REF] Taylor | Staggered wage setting in a macro model[END_REF][START_REF] Taylor | Aggregate dynamics and staggered contracts[END_REF] or [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF].

It is possible to answer in such a framework our question on competitiveness, since these models include a parameter which explicitly depicts the degree of competitiveness, more specifically the elasticity of substitution between goods. But the answer to our question is highly paradoxical since, as we shall see below, in models with staggered price contracts, one finds that more competitiveness leads to more sticky prices.

The intuition is easy to grasp if one goes to the extreme case of perfectly substitutable goods, corresponding to a competitive market. Consider to simplify a Taylor model with two periods contracts. In any period t half of the agents renew their contract. Because of perfect substitutability these agents will have to align their price on the lowest price set in t-1 (which still holds, since the contracts last 2 periods). And for the same reason so will agents in period t+1. By that time all prices will be the same and, continuing the same reasoning, all future prices will be equal to that price also, so that prices are totally sticky irrespective of the shocks to which the economy may be subjected. So full competitiveness leads to complete stickiness of prices in that model.

Clearly this result is quite counterintuitive and our intuition would rather lead us to believe that more competitive markets should lead to more flexible, less sticky, prices.

So in a second step we want to construct a model where more competitiveness leads to more flexible prices. For that we shall use a different model of price stickiness proposed by Rotemberg (1982a,b), where prices are sticky because there is a cost of changing prices, measured by an explicit convex cost function. We shall show that, for a given cost function, the "flexibility" of prices increases with competitiveness.

The model 2.1 The agents

Households have an intertemporal utility function:

U = X β t • LogC t + σLog µ M t P t ¶ -ξ L ν t ν ¸ν > 1 (1)
and are submitted in each period to a budget constraint:

P t C t + M t = W t L t + Π t + µ t M t-1 (2)
where µ t is a multiplicative money shock à la [START_REF] Lucas | Expectations and the neutrality of money[END_REF] and Π t is distributed profits.

Output is produced with intermediate goods indexed by j ∈ [0, 1] by competitive firms. They all have the same constant returns to scale production function:

Y t = µZ 1 0 Y θ jt dj ¶ 1/θ 0 < θ ≤ 1 (3)
where Y t is the level of output and Y jt the amount of intermediate good j used in production. The elasticity of substitution between intermediate goods is equal to 1/ (1θ), so the parameter θ is a good index of competitiveness. These output firms competitively maximize profits:

P t µZ 1 0 Y θ jt dj ¶ 1/θ - Z 1 0 P jt Y jt (4)
Intermediate goods themselves are produced by monopolistically competitive firms indexed by j ∈ [0, 1]. Firm j has a production function:

Y jt = Z t L α jt 0 < α ≤ 1 (5)
where Z t is a common productivity shock.

Price rigidities

We shall consider in this article two alternative models of price rigidities:

-In the first model we shall consider, prices are set according to [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] contracts. The probability of a price contract to continue unchanged is γ. Conversely every contract can break with probability 1γ. If a contract breaks in firm j at time t, this firm sets a new price X jt , based on all information available up to time t.

-In the second model we shall consider, as in Rotemberg (1982a,b), a firm producing intermediate product j incurs, in addition to production costs, a cost of changing prices, denoted as χ jt in real terms, where:

χ jt = λ 2 Y t µ P jt P jt-1 -1 ¶ 2 (6)

Common equilibrium conditions

We shall thus study two models, and begin by deriving in this section the first order and equilibrium conditions that are common to these two models.

Households

Households maximize the discounted utility (1) subject to the budget constraints (2). The Lagrangean is:

X t β t • LogC t + σLog µ M t P t ¶ -ξ L ν t ν + ζ t (W t L t + µ t M t-1 -P t C t -M t ) (7)
and the first order conditions in C t , L t and M t :

ζ t P t = 1 C t ( 8
)
ζ t W t = ξL ν-1 t ( 9
)
ζ t = σ M t + βE t ¡ ζ t+1 µ t+1 ¢ (10)
Combining ( 8) and ( 9) we find:

W t P t C t = ξL ν-1 t (11)
Now combining ( 8) and ( 10) and using µ t+1 = M t+1 /M t , we obtain:

M t P t C t = σ + βE t µ M t+1 P t+1 C t+1 ¶ (12)
which solves as:

M t P t C t = σ 1 -β (13)

Output firms

Output producing firms competitively maximize profits (eq. 4) subject to the production function (eq. 3). This yields the demand for intermediate good j:

Y jt = Y t µ P jt P t ¶ -1/(1-θ) (14)
The elasticity of these demand curves goes from 1 to infinity (in absolute value) when θ goes from zero to one, so again θ appears as a natural index of competitiveness.

Since the output firms are competitive, the aggregate price is equal to the usual CES index:

P t = •Z 1 0 P -θ/(1-θ) jt ¸-(1-θ)/θ (15)
4 Price dynamics under Calvo contracts

We shall now show that under Calvo contracts persistence is, quite counterintuitively, positively related to competitiveness.

Optimal price setting

We shall first derive optimal price setting by intermediate firms under Calvo price contracts. Consider a firm j which has to decide in period t its "new price" X jt . If that contract is still in effect in period s ≥ t (with probability γ s-t ), the corresponding real profit in period s will be:

Π js P s = X jt Y js -W s L js P s = X jt Y js P s - W s P s µ Y js Z s ¶ 1/α (16)
Firm j maximizes expected discounted real profits, multiplied by the marginal utility of consumption. Since utility (1) is logarithmic in consumption, this marginal utility is equal to 1/C s = 1/Y s . So firm j maximizes the expected value of the following criterion, where the term γ s-t represents the probability that the price contract X jt signed in t is still in effect at time s:

X s≥t β s-t γ s-t Π js P s Y s = X s≥t β s-t γ s-t " X jt Y js P s Y s - W s P s Y s µ Y js Z s ¶ 1/α # (17)
Now inserting ( 5) and ( 14) (with P js = X jt , since we are dealing with the contract signed in t) into [START_REF] Rotemberg | Monopolistic price adjustment and agregate output[END_REF], we obtain the maximand:

E t X s≥t β s-t γ s-t " µ X jt P s ¶ -θ/(1-θ) - W s P s Y s µ Y s Z s ¶ 1/α µ X jt P s ¶ -1/α(1-θ) # (18)
Let us differentiate with respect to X jt . We obtain the first-order condition:

X (1-αθ)/α(1-θ) jt E t X s≥t β s-t γ s-t P θ/(1-θ) s = 1 αθ E t X s≥t β s-t γ s-t W s P s Y s µ Y s Z s ¶ 1/α P 1/α(1-θ) s ( 19
)
We see that X jt is actually independent of j. All firms changing their price choose the same price, given by ( 19), which we shall denote as X t . We finally use C s = Y s and replace in [START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the euro area[END_REF] W s by the value given by formula [START_REF] Huang | Staggered price-setting, staggered wage-setting, and business cycle persistence[END_REF] to obtain the final value of X t :

X (1-αθ)/α(1-θ) t = ξ αθ E t P s≥t β s-t γ s-t (Y s /Z s ) ν/α P 1/α(1-θ) s E t P s≥t β s-t γ s-t P θ/(1-θ) s (20)

Price dynamics

Let us start with newly set prices and loglinearize eq. ( 20). We obtain, omitting constant terms3 :

1 -αθ α (1 -θ) x t + (1 -βγ) X s≥t β s-t γ s-t E t θ 1 -θ p s = (1 -βγ) X s≥t β s-t γ s-t E t • ν α (y s -z s ) + 1 α (1 -θ) p s ¸(21)
We have from (13) y t = c t = m tp t , so that ( 21) is rewritten:

x t = (1 -βγ) X s≥t β s-t γ s-t E t [(1 -δ) (m s -z s ) + δp s ] (22)
with:

δ = 1 -αθ -ν + θν 1 -αθ = 1 - ν (1 -θ) 1 -αθ (23)
Note that δ = 1 when θ = 1, i.e. when we are in the competitive case. If θ < 1, then δ < 1. Differentiating ( 23) we find:

∂δ ∂θ = ν (1 -α) (1 -αθ) 2 > 0 (24)
Let us now turn to aggregate prices, which are given by eq. ( 15) above. In view of the "demographics" of price contracts, a fraction (1γ) γ i of price contracts comes from period ti, i = 0, 1, ....∞, so that ( 15) is rewritten:

P t = " (1 -γ) ∞ X i=0 γ i X -θ/(1-θ) t-i # -(1-θ)/θ (25)
which yields after loglinearization:

p t = (1 -γ) ∞ X i=0 γ i x t-i = 1 -γ 1 -γL x t (26)

Resolution

The dynamic system consists of Eqs. ( 22) and ( 26) above. Let us forward [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF] one period and take the expectation as of period t:

E t x t+1 = (1 -βγ) X s≥t+1 β s-t-1 γ s-t-1 E t [(1 -δ) (m s -z s ) + δp s ] (27)
Combining with [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF] we obtain:

x t = βγE t x t+1 + (1 -βγ) [(1 -δ) (m t -z t ) + δp t ] (28)
Now ( 26) can be written:

p t -γp t-1 = (1 -γ) x t ( 29
)
Forwarding one period and taking the expectation as of t yields:

E t p t+1 -γp t = (1 -γ) E t x t+1 (30)
Let us insert into (28) the values of x t and E t x t+1 given by eqs. ( 29) and (30). We obtain:

(1 -δ) (1 -γ) (1 -βγ) (p t -m t + z t ) + γ (p t -p t-1 ) + βγ (p t -E t p t+1 ) = 0
(31) This equation is of the form:

a (p t -m t + z t ) + b (p t -p t-1 ) + c (p t -E t p t+1 ) = 0 (32)
with:

a = (1 -δ) (1 -γ) (1 -βγ) b = γ c = βγ (33)
This solves as (see the appendix):

p t = φp t-1 + ∞ X j=0 κ j E t (m t+j -z t+j ) (34)
where:

κ 0 = a a + b + c (1 -φ) κ j = c a + b + c (1 -φ) κ j-1 (35)
and the autoregressive root φ is solution of the characteristic equation:

Φ (φ) = cφ 2 -(a + b + c) φ + b = 0 (36)

Competitiveness and price stickiness

We shall study how the response of prices to monetary and technology shocks depends on competitiveness. Let us define the composite shock ω t :

ω t = m t -z t (37)
Now let us first consider as our "benchmark" the case where all prices are reset every period, i.e. where γ = 0. In that case (31) immediately yields:

p t = m t -z t = ω t (38)
We shall now consider the case γ 6 = 0, and see that the parameter φ in eq. (34) will appear as a natural measure of the dynamic price stickiness. To make things particularly clear, let us take the following simple process for ω t :

ω t -ω t-1 = ε t (39)
where the ε t are i.i.d. Then it is shown in the appendix that eqs. ( 34) and (35) simplify to (eq. 81):

p t = ω t - φε t 1 -φL (40)
We see that, following a shock, the discrepancy between the price and its benchmark market clearing value ω t is both higher on impact, and returns more slowly to zero when φ is high. So the parameter φ appears indeed as a natural parameter to characterize price stickiness.

We can also compute the expression of output. Using y t = m tp t , eq. (40) yields:

y t = z t + φε t 1 -φL (41)
We want to see finally how the index of price stickiness φ relates to the index of competitiveness θ. Combining (33) and (36) we obtain the characteristic equation giving φ:

Φ (φ) = βγφ 2 -[(1 -δ) (1 -γ) (1 -βγ) + γ + βγ] φ + γ = 0 (42)
We can compute:

Φ (0) = γ ≥ 0 Φ (1) = -(1 -δ) (1 -γ) (1 -βγ) ≤ 0 ( 43
)
So there is a root φ between 0 and 1. Let us now find out how this index of price stickiness φ relates to the index of competitiveness θ. Differentiate eq. ( 42). This yields:

∂φ ∂δ = φ (1 -γ) (1 -βγ) γ ¡ 1 -βφ 2 ¢ > 0 (44)
So finally, combining (24) and (44):

∂φ ∂θ = ∂φ ∂δ ∂δ ∂θ > 0 (45)
We therefore have the paradoxical relation between competitiveness and price stickiness.

Price dynamics under costs of changing prices

We shall now show that if price sluggishness is due to a convex cost of changing prices as in Rotemberg (1982a,b), then one finds the natural result that price stickiness is negatively related to competitiveness.

Optimal price setting

With the cost of changing prices χ jt the real profit of intermediate firm j in period t is:

Π jt P t = P jt Y jt -W t L jt P t -χ jt = P jt Y jt P t - W t P t µ Y jt Z t ¶ 1/α - λ 2 Y t µ P jt P jt-1 -1 ¶ 2 (46)
Firm j maximizes discounted real profits, multiplied by the marginal utility of consumption, equal to 1/C t = 1/Y t , so that firm j maximizes the expected value of the following criterion:

X t β t Π jt P t Y t = X t β t " P jt Y jt P t Y t - W t P t Y t µ Y jt Z t ¶ 1/α - λ 2 µ P jt P jt-1 -1 ¶ 2 # (47)
Insert into the discounted profits (eq. 47) the expression of Y jt (eq. 14). We obtain: [START_REF] Huang | Staggered price-setting, staggered wage-setting, and business cycle persistence[END_REF] and Y t = C t into (48). Keeping only the terms where P jt appears, we obtain the maximand: so that:

X t β t " µ P jt P t ¶ -θ/(1-θ) - W t P t Y t µ Y t Z t ¶ 1/α µ P jt P t ¶ -1/α(1-θ) - λ 2 µ P jt P jt-1 -1 ¶ 2 # (48) Now insert
0 ≤ φ ≤ 1 (63)
We want now to see finally how the index of price stickiness φ relates to the index of competitiveness θ. Considering first the extreme cases, we note that φ = 0 if θ = 1 and φ = 1 if θ = 0. Now let us compute ∂φ/∂θ. Differentiating (61) we find:

∂φ ∂θ = - νφ (1 -θ) [νθ + αλ (1 -θ) (1 + β -2βφ)] = - νφ 2 αλ (1 -θ) 2 ¡ 1 -βφ 2 ¢
(64) This is negative since φ ≤ 1, and we therefore have a negative relation between competitiveness and price stickiness, as our intuition would tell us, but unlike the model we considered in section 4.

Conclusions

We studied in this article simple dynamic models where prices are sticky either because of staggered price contracts, or because there is a convex cost of changing them. In these models competitiveness is measured by a parameter θ which is higher, the higher the substitutability between goods. Price stickiness is measured by the size of the autoregressive root φ in the dynamic price process.

We saw that in a model with staggered price contracts (section 4) price stickiness is, quite counterintuitively, positively related to the parameter θ representing competitiveness. Although this is clearly disturbing, this should not deter us from using DSGE models with sticky prices, since we saw that in an alternative model with convex costs of changing prices (section 5) price stickiness is, in accordance with intuition, a decreasing function of competitiveness 4 .

By more sticky prices we will mean prices that return more slowly to their market clearing value. In what follows prices will have an autoregressive root, denoted below as φ, and stickiness will be measured by the size of that root.

See for example[START_REF] Ascari | Optimising agents, staggered wages and persistence in the real effects of money shocks[END_REF],[START_REF] Ascari | Staggered wages and output dynamics under disinflation[END_REF],[START_REF] Bénassy | The Macroeconomics of Imperfect Competition and Nonclearing Markets: A Dynamic General Equilibrium Approach[END_REF]Bénassy ( , 2003a,b),b),[START_REF] Chari | Sticky price models of the business cycle: can the contract multiplier solve the persistence problem?[END_REF], Christiano, Eichenbaum and Evans (2001),[START_REF] Collard | Stochastic nominal wage contracts in a cash-in-advance model[END_REF],[START_REF] Hairault | Money, new-Keynesian macroeconomics and the business cycle[END_REF],[START_REF] Huang | Staggered price-setting, staggered wage-setting, and business cycle persistence[END_REF], Jeanne (1998),[START_REF] Kim | Constructing and estimating a realistic optimizing model of monetary policy[END_REF],[START_REF] Smets | An estimated dynamic stochastic general equilibrium model of the euro area[END_REF],[START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF],[START_REF] Yun | Nominal price rigidity, money supply endogeneity, and business cycles[END_REF], and several others.

Lowercase letters represent the logarithms of the corresponding uppercase letters.

We may note that convex costs of changing prices have already been used in DSGE models. See, for example,[START_REF] Hairault | Money, new-Keynesian macroeconomics and the business cycle[END_REF] and[START_REF] Kim | Constructing and estimating a realistic optimizing model of monetary policy[END_REF].

Let us differentiate with respect to P jt . We obtain the first-order condition:

Price dynamics

All firms j are actually in a symmetric situation, so that in equilibrium P jt = P t . Inserting this into (50) we obtain the dynamic equation for prices:

We first characterize the long run equilibrium where all prices are equal over time. Then (51) yields:

or:

Now let us go back to the dynamics. Loglinearizing (51) we find:

and since, from ( 13), y t = c t = m tp t , this becomes:

Resolution

Again we have an equation of the form:

with:

Eq. ( 56) is solved in the appendix, and the solution has been already described above (eqs. 34, 35 and 36).

Competitiveness and price stickiness

Let us first take as a benchmark the case where the cost of changing prices is zero, i.e. where λ = 0. In that case (55) immediately yields:

Let us now consider the case λ 6 = 0, and assume again the simple process for ω t :

Then the price is given by (appendix, eq. 81):

where φ is solution of the following characteristic equation, obtained by combining (36) and (57):

We can compute:

Appendix

We want to solve the dynamic equation:

with:

Let us define the composite shock:

We hypothesize a solution of the form:

From that we deduce:

Inserting these into the initial formula (65) we obtain:

Identifying to zero the term in p t-1 we find the characteristic equation giving φ:

We can compute:

so that:

Now identifying to zero the term in ω t in (70) yields:

Finally identifying to zero the term in E t ω t+j gives:

We want to check that η < 1. This will be the case if:

So we compute:

so we have indeed η < 1.

We want finally to compute the solution in the particular case where

In that case, using (74) and (75), equation (68) yields:

This, in view of (71), becomes:

which can also be rewritten:


Jean-Pascal Bénassy
email: benassy@pse.ens.fr

† Pse

Cepremap

The fiscal theory of the price level puzzle: A non Ricardian view

Keywords: Fiscal theory of the price level, fiscal policy, global determinacy, monetary policy

The fiscal theory of the price level says that the price level can be made determinate if the government uses fiscal policies such that government liabilities explode unless the price in the first period is at the "right" level. The policy implications are disturbing, as they call for rather adventurous fiscal policies. We show that these disturbing policy implications are specific to the "Ricardian" models that have been used to develop the theory. By moving to non Ricardian models we see that price determinacy is consistent with reasonable fiscal policies.

Introduction

In the recent years an intriguing theory of price determinacy in monetary economies has developed, the fiscal theory of the price level1 (FTPL). In a monetary economy, depending on fiscal and monetary policies, dynamic trajectories may have determinate prices, indeterminate prices, or no equilibrium. A wellknown result by Sargent and Wallace (1975) says that if the nominal interest rate is pegged, there is nominal indeterminacy [START_REF] Bénassy | Interest rate rules, price determinacy and the value of money in a non Ricardian world[END_REF] . Now considering more general interest rate rules reacting to inflation, there will be indeterminacy if the rule does not satisfy the wellknown "Taylor principle" [START_REF] Benhabib | The perils of Taylor rules[END_REF] .

What the FTPL says is that, even in such circumstances, adequate fiscal policies can restore determinacy. The fiscal policies that achieve determinacy are such that the government's intertemporal budget constraint is not balanced in all circumstances. In fact the intuition behind the result is that under such policies the government's real liabilities will evolve explosively in time unless one starts from a particular price level. This makes this initial price level the only feasible one. The problem with the FTPL is that the corresponding fiscal policies are rather risky, since the government does not plan to balance its budget in all situations, and this could lead in most circumstances to explosive real liabilities. These controversial policy implications have led to numerous contributions making explicitations or criticisms [START_REF] Benhabib | Avoiding liquidity traps[END_REF] .

What we want to show in this article is that the controversial policy implications are actually due to the particular "Ricardian" framework whithin which the results were derived, and we will show that moving to a "non-Ricardian" framework yields much less controversial results. By non-Ricardian models we mean models where, as in OLG models, new agents enter in time, so that in particular Ricardian equivalence fails (Barro, 1974). We shall see that in such a framework price determinacy is consistent with much more reasonable fiscal policies.

Before going to the analytics, we may give a brief intuition as to why one may dispense with the explosive FTPL policies in a non Ricardian economy. It was shown by Weil (1991) that, unlike in a Ricardian economy, financial assets represent real wealth to alive agents in a non Ricardian economy [START_REF] Blanchard | The solution of linear difference models under rational expectations[END_REF] . As a result dynamic equations (see for example equation 16 below) will include not only prices (which are non predetermined), but also financial assets (which are predetermined), and this link will help to make prices determined.

The model

We want to have a non Ricardian model that "nests" the traditional infinitely lived agent model, so we shall use a model due to Weil (1987, 1991), and assume that new "generations" of households are born each period, but nobody dies. Denote as N t the number of households alive at time t. So N t -N t-1 ≥ 0 households are born in period t. We will mainly work below with a constant rate of growth of the population n ≥ 0, so that

N t = (1 + n) t .
The Ricardian case is obtained by taking the limit case n = 0.

Consider a household born in period j. We denote by c jt , y jt and m jt his consumption, endowment and money holdings at time t ≥ j. This household maximizes the following utility function:

U jt = ∞ X s=t β s-t Log c js (1)
and is submitted in period t to a "cash in advance" constraint:

P t c jt ≤ m jt (2)
Household j begins period t with a financial wealth ω jt . First the bond market opens, and the household lends an amount b jt at the nominal interest rate i t . The rest is kept under the form of money m jt , so that:

ω jt = m jt + b jt (3)
Then the goods market opens, and the household sells his endowment y jt , pays taxes τ jt in real terms and consumes c jt , subject to the cash constraint (2). Consequently, the budget constraint for household j is:

ω jt+1 = (1 + i t ) ω jt -i t m jt + P t y jt -P t τ jt -P t c jt (4)
consumer, because of the government's intertemporal budget constraint, every dollar of financial assets is cancelled by future discounted taxes, so that these assets represent no net wealth. In the contrary in a non Ricardian economy agents not yet alive today will pay part of these taxes in the future, so that a fraction of financial assets is real wealth for currently alive agents.

Aggregate quantities are obtained by summing the various individual variables. There are N j -N j-1 agents in generation j, so for example aggregate assets Ω t and taxes T t are equal to:

Ω t = X j≤t (N j -N j-1 ) ω jt T t = X j≤t (N j -N j-1 ) τ jt (5)
Similar formulas apply to output Y t , consumption C t , money M t and bonds B t . We now must describe how endowments and taxes are distributed among households. We assume that all households have the same income and taxes, so:

y jt = y t = Y t /N t τ jt = τ t = T t /N t (6)
and that real income per head grows at the rate ζ:

y t+1 /y t = ζ Y t+1 /Y t = (1 + n) ζ (7)
Let us now consider government. Households' aggregate financial wealth Ω t has as a counterpart an identical amount Ω t of financial liabilities of the government. These are decomposed into money and bonds:

Ω t = M t + B t (8)
The evolution of these liabilities is described by the government's budget constraint:

Ω t+1 = (1 + i t ) Ω t -i t M t -P t T t (9)
Note that, to simplify the exposition and to concentrate on tax policy, we have assumed that governement spending is zero. If not, the results would be essentially the same, but the formulas would be more clumsy.

Monetary policy

In what follows we shall study two types of monetary policies. First, and since we want to concentrate on the effects of fiscal policy, we shall study in sections 4 and 5 a particularly simple monetary policy, interest rate pegging. To simplify the exposition, we shall assume that the pegged interest rate is constant in time, so that i t = i 0 . As we indicated above this rule is of particular interest because, in the usual Ricardian framework, it leads to nominal indeterminacy.

We shall then also study in section 6 more general policies where the nominal interest rate responds to inflation, i.e. denoting Π t = P t /P t-1 :

1 + i t = Φ (Π t ) Φ (Π t ) ≥ 1 Φ 0 (Π t ) ≥ 0 (10)

Fiscal policy

If the budget was balanced, taxes would be equal to interest payments on bonds i t B t , so that one would have P t T t = i t B t . We shall actually assume that the government can run a deficit or a surplus, so taxes have the form:

P t T t = i t B t -D (Ω t , P t Y t ) (11)
where the function D (Ω t , P t Y t ), which represents the fiscal deficit in nominal terms, is homogeneous of degree 1 in its two arguments [START_REF] Buiter | The fiscal theory of the price level: a critique[END_REF] .

Dynamics

Let us start with the dynamic equation for Ω t . Putting together equations ( 8), ( 9) and ( 11), we find:

Ω t+1 = Ω t + D (Ω t , P t Y t ) (12)
In view of the homogeneity property of the function D, it will be convenient to take as a working variable:

Z t = Ω t /P t Y t (13)
Then equation ( 12) can be rewritten:

Ω t+1 /Ω t = F (Z t ) (14)
where:

F (Z t ) = Ω t + D (Ω t , P t Y t ) Ω t (15)
Turning now to nominal income P t Y t , it is shown in the appendix that, assuming N t+1 /N t = 1 + n, its dynamics is given by:

P t+1 Y t+1 = β (1 + n) (1 + i t ) P t Y t -(1 -β) nΩ t+1 (16)
Equations ( 14) and ( 16) are the basic dynamic equations of our model. We should note that whereas in the traditional "Ricardian" case (n = 0) equation ( 16) is homogeneous in prices only, if n > 0 the equation is homogeneous in prices and financial wealth. Since financial wealth is predetermined, this gives a "nominal anchor" that will contribute, together with further conditions that we will see below, to make prices determined. Now, in order to contrast the results with what will follow, we shall examine some determinacy conditions in the traditional Ricardian case. 4 The Ricardian case and the fiscal theory of the price level

We shall begin our investigation with the traditional Ricardian version of the model [START_REF] Guillard | Politique monétaire et fiscale dans un monde non-Ricardien: une théorie fiscale de l'inflation[END_REF] . For that it is enough to take n = 0. We shall also assume a simple interest rate peg i t = i 0 , as this is a case where the fiscal theory of the price level is particularly relevant. Equation ( 16) is then rewritten as:

P t+1 Y t+1 = β (1 + i 0 ) P t Y t (17)
Combining ( 13), ( 14) and ( 17) we find the dynamics of Z t :

Z t+1 = Z t F (Z t ) β (1 + i 0 ) (18)
We shall denote as ξ the potential steady states (in Z t ) of this system. In view of (18), potential values of ξ are characterized by:

ξ = ξF (ξ) /β (1 + i 0 ) (19)

Determinacy and the FTPL

We are mostly interested in equilibria with nonzero financial assets. We shall assume that ( 19) admits at least one positive solution, and loglinearize (18) around it. Omitting constants we find 8 :

z t+1 = (1 + f ) z t ( 20
)
where f is the elasticity of the "fiscal" function F : [START_REF] Guillard | Politique monétaire et fiscale dans un monde non-Ricardien: une théorie fiscale de l'inflation[END_REF] Determinacy conditions in the Ricardian case were notably developed by Leeper (1991). [START_REF] Kocherlakota | Explaining the fiscal theory of the price level[END_REF] Lowercase letters correspond to the logarithms of the corresponding uppercase letters.

f = f (Z t ) = ∂Log [F (Z t )] /∂LogZ t (21)
Using the Blanchard Kahn (1980) criterion, the condition for local determinacy is thus [START_REF] Leeper | Equilibria under "active" and "passive" monetary and fiscal policies[END_REF] :

f > 0 (22)
This means that, if the ratio Z t of government liabilities to nominal income is already high, the government must increase the rate of growth of these liabilities. Such a strategy will lead to an explosive behavior of financial liabilities, which is actually the basis of the fiscal theory of the price level. It is also a rather adventurous fiscal policy, which contributed making the FTPL controversial.

An example

Let us consider the following tax function:

P t T t = i t B t + (1 -γ) Ω t + δP t Y t γ ≥ 0 δ ≥ 0 (23)
The term δP t Y t says that the government taxes a constant fraction δ of nominal income P t Y t . The term (1γ) Ω t says that the government may want to withdraw a fraction 1γ of its outstanding financial liabilities. If γ is greater than 1, this actually corresponds to an expansion of government liabilities.

Combining ( 13), ( 17) and (23), we obtain the dynamics of Z t :

Z t+1 = γZ t -δ β (1 + i 0 ) (24)
There will exist a determinate equilibrium only if:

γ > β (1 + i 0 ) (25)
We see that the government should engineer a minimal rate of expansion of its financial liabilities. Actually, if γ satisfies (25), then from (24) the ratio of financial liabilities to income will be explosive. This is obviously a very risky fiscal policy.

We may note that conditions (22) and (25) actually say the same thing. Computing indeed the elasticity f at the equilibrium we find:

f = γ -β (1 + i 0 ) β (1 + i 0 ) (26)
so that f > 0 (equation 22) yields γ > β (1 + i 0 ) (equation 25).

Fiscal policy and determinacy in the non-Ricardian case

We want to show now that, as soon as one moves to a non-Ricardian framework, adventurous fiscal policies like ( 22) or (25) are not necessary anymore for determinacy. We shall begin in this section with local determinacy, leaving global determinacy to the next section.

The dynamic equations

We shall now study the dynamics of the system in the non Ricardian case, i.e. n > 0. To compare it with the results of section 4 we continue to assume interest rate pegging i t = i 0 , so equation ( 14) still holds and ( 16) is written:

P t+1 Y t+1 = θP t Y t -κΩ t+1 (27)
with:

θ = β (1 + n) (1 + i 0 ) κ = (1 -β) n (28)
Dividing ( 14) by ( 27) we obtain the dynamics of Z t :

Z t+1 = Z t F (Z t ) θ -κZ t F (Z t ) (29)
From now on we shall assume that government liabilities can become neither negative nor infinite, so that: 0 ≤ F (Z t ) ≤ F max < ∞. The potential steady states ξ are given by: ξ = ξF (ξ) / [θ -κξF (ξ)]. There are two types of solutions, ξ 1 and ξ 2 , given by (when they exist):

F (ξ 1 ) = θ 1 + κξ 1 ξ 2 = 0 (30)

Determinacy

Now loglinearizing equation ( 29) around the positive solution ξ 1 we find:

z t+1 = θ F (ξ 1 ) (1 + f ) z t (31)
and the condition for local determinacy is:

θ (1 + f ) > F (ξ 1 ) (32)
We see that f > 0 (equation 22) is not necessary anymore for determinacy.

An example

Assume that the government engineers through fiscal policy a constant rate of growth γ of its liabilities, so that F (Z t ) = γ and f = 0. Then the determinacy condition (32) becomes:

γ < θ = β (1 + n) (1 + i 0 ) (33)
We see that the policy prescription is practically the inverse of that in condition (25)! Here a more rigorous fiscal policy (i.e. a low γ) is conducive to price determinacy, unlike in the FTPL where a very unrigorous fiscal policy (a high γ) was the key to determinacy (condition 25).

To interpret further (33), let us rewrite it as:

ζ (1 + n) (1 + i 0 ) γ > ζ β ( 34
)
Since inflation is equal to γ/ζ (1 + n), this says that the real rate of return of the financial assets (left hand side) is greater than the rate of return in the Ricardian model ζ/β. Such a condition was shown in Wallace (1980) and Bénassy (2002) to be a condition for the viability of a monetary equilibrium.

A global view

Let us continue with the example above, and consider now the issue of global determinacy. With F (Z t ) = γ, the dynamic equation (29) becomes:

Z t+1 = γZ t θ -κγZ t (35)
This is represented in figure 1. We see that the equilibrium ξ 1 is indeed locally determinate. But there is a second equilibrium ξ 2 = 0 which is indeterminate, and all trajectories initiating between ξ 1 and ξ 2 converge towards it. So it is important to move to the study of global determinacy, and we shall now show how adequate combinations of monetary and fiscal policies allow to obtain global determinacy.

Global determinacy

We shall now introduce interest rate rules that respond to inflation like (10) and show that we can achieve global determinacy under reasonable fiscal policies.

Let us recall that in the Ricardian framework there are two alternative conditions for price determinacy, corresponding to the Taylor principle and the FTPL, and expressed respectively as φ (Π t ) > 1 and f (Z t ) > 0. What we want to show is that in a non-Ricardian world it is possible to obtain global determinacy eventhough none of these two conditions is satisfied, i.e. if we have both:

φ (Π t ) < 1 and f (Z t ) ≤ 0 (36)
Policies like (36) would lead to indeterminacy in the Ricardian framework [START_REF] Mccallum | Indeterminacy, bubbles, and the fiscal theory of price level determination[END_REF] . Combining (10), ( 14) and ( 16), the dynamic equation for nominal income is written:

P t+1 Y t+1 = β (1 + n) Φ (Π t ) P t Y t -κΩ t+1 (37)
The dynamic system consists of equations ( 14) and (37). Because of the more general interest rate rule Φ (Π t ), it will not be possible to summarize the dynamics with one single variable as in (29), so we shall use two working variables [START_REF] Sala | The fiscal theory of the price level: identifying restrictions and empirical evidence[END_REF] , inflation Π t and the (predetermined) variable X t = Ω t /P t-1 Y t-1 . Then, calling ν = κ/ (1 + n) = n (1β) / (1 + n), the dynamic system ( 14), (37) is rewritten:

X t+1 = X t ζ (1 + n) Π t F • X t ζ (1 + n) Π t ¸(38) ζΠ t+1 = βΦ (Π t ) -νX t+1 (39)
" -. cc U'h c ..., -CD

A general interest rate rule

We shall first study here the case with a general interest rate rule Φ (Π t ), but, in order to make the exposition more transparent, continue with the fiscal example of section 5.3, i.e. F (Z t ) = γ, so that (38) is rewritten as:

X t+1 = γX t ζ (1 + n) Π t
(40)

Uniqueness of equilibrium

As a first step we shall look for conditions that yield a unique equilibrium. From (39) and (40) the potential steady states (X * , Π * ) are characterized by the two equations:

X * = γX * /ζ (1 + n) Π * (41) ζΠ * = βΦ (Π * ) -νX * (42)
This yields two types of potential steady states:

Π * = γ ζ (1 + n) X * = βΦ (Π * ) -ζΠ * ν ( 43
)
X * = 0 ζΠ * = βΦ (Π * ) (44)
The first solution, described by (43), always exists. So the only way to have a unique equilibrium is to suppress the solutions described by (44). It is easy to see that a sufficient condition for that is:

βΦ (Π t ) -ζΠ t > 0 ∀Π t (45)
We may note that (45) can be rewritten as:

Φ (Π t ) /Π t > ζ/β (46)
In words, the interest rate rule should generate a real rate of interest that is higher than the real rate that would prevail in the corresponding Ricardian economy. We should note the conceptual similarity of (46) with equation (34), as both express that the real return on financial assets is sufficiently high to induce agents to hold them in long run equilibrium. " cc c: .., CD N L

A general fiscal rule

The above global determinacy result has been obtained with the particular fiscal rule F (Z t ) = γ. We want to show now that global determinacy can actually be obtained with quite more general fiscal rules F (Z t ). In order to keep calculations simple, we shall use for monetary policy the simple linear interest rate rule seen in the preceding section, i.e.:

Φ (Π t ) = AΠ t + B A ≥ ζ/β (53)
As we indicated at the beginning of section 6, we are interested in "reasonable", non explosive fiscal rules such that f (Z t ) ≤ 0. We shall actually now prove that a sufficient condition for global determinacy is, together with (53):

-1 < f (Z t ) ≤ 0 (54)
Let us prove indeed that (53) and ( 54) are sufficient for global determinacy. The equations of the curves X t+1 = X t and Π t+1 = Π t are respectively:

ζ (1 + n) Π t = F • X t ζ (1 + n) Π t ¸(55) (βA -ζ) Π t + B = νX t ζ (1 + n) Π t F • X t ζ (1 + n) Π t ¸(56)
Assumption (54) implies that, in the (Π t , X t ) plane, the locus X t+1 = X t is downward sloping, whereas assumptions (53) and (54) imply that the locus Π t+1 = Π t is upward sloping. The dynamics is represented in figure 3, where it appears that we have saddle path dynamics and global determinacy.

Conclusion

We have seen that in the Ricardian case price determinacy is obtained if the fiscal authority expands government liabilities sufficiently for these liabilities to become explosive (see for example conditions 22 and 25), which is a central mechanism behind the fiscal theory of the price level. This controversial prescription is not necessary anymore in a non-Ricardian framework. We found indeed that in such a case an explosive expansion of government liabilities is not required for local or global price determinacy, r 1 -and that price determinacy can be associated to reasonable fiscal prescriptions (see for example condition 33). Finally we saw that global determinacy could be achieved with a combination of monetary and fiscal policies where monetary policy does not satisfy the "Taylor principle" and fiscal policies are nevertheless reasonable.

We saw the basic intuitions along the way and we can summarize them briefly: (1) The non Ricardian framework creates a wealth effect through which prices are linked to financial wealth, a predetermined variable (see notably equation 16). (2) In addition to n > 0, the determinacy conditions (equations 34 or 46) simply say that the combination of monetary and fiscal policies must make financial assets attractive enough (in rate of return) to be actually held by households. These conditions have nothing to do with the "explosive" FTPL policies like (22) or (25).

So it appears that the controversial policy prescriptions associated with the FTPL are linked with the Ricardian character of the economies in which they were derived. They disappear when one moves to a (more realistic) non-Ricardian framework.

Figure 1 :

 1 Figure 1: Local determinacy, global indeterminacy

Figure 3 :

 3 Figure 3: Φ (Π t ) = AΠ t + B and a general fiscal rule F (Z t )

See notably[START_REF] Leeper | Equilibria under "active" and "passive" monetary and fiscal policies[END_REF],[START_REF] Sims | A simple model for the determination of the price level and the interaction of monetary and fiscal policy[END_REF],[START_REF] Woodford | Monetary policy and price level determinacy in a cash-in-advance economy[END_REF][START_REF] Woodford | Price level determinacy without control of a monetary aggregate[END_REF].

This means that, if a price sequence is an equilibrium one, any price sequence multiple of this one is also an equilibrium sequence.

The "Taylor principle" says that the nominal interest rate should respond to inflation with an elasticity greater than one. As a consequence the real interest rate will respond positively to inflation[START_REF] Taylor | Discretion versus policy rules in practice[END_REF][START_REF] Taylor | Monetary policy and the long boom[END_REF].

See, for example,[START_REF] Buiter | The fiscal theory of the price level: a critique[END_REF],[START_REF] Kocherlakota | Explaining the fiscal theory of the price level[END_REF],[START_REF] Mccallum | Indeterminacy, bubbles, and the fiscal theory of price level determination[END_REF]. An empirical evaluation of the theory can be found in[START_REF] Sala | The fiscal theory of the price level: identifying restrictions and empirical evidence[END_REF].

In a nutshell, the eonomic intuition is the following: if there is a single infinitely lived

Note that we express taxes as a function of P t Y t and Ω t . Some authors use instead as arguments P t Y t and B t . Since Ω t = M t + B t = P t Y t + B t , it is easy to go from one formulation to the other, and all subsequent formulas can be rewritten with B t as an argument, but the results are sometimes more clumsy.

The reader can actually check that this condition does not only hold for an interest rate peg, but with a general interest rate rule Φ (Π t ) as well, as long as it does not satisfy the Taylor principle.

Conditions for global determinacy in traditional Ricardian economies have been notably studied in Benhabib, Schmitt-Grohe and Uribe(2001, 2002).

[START_REF] Sala | The fiscal theory of the price level: identifying restrictions and empirical evidence[END_REF] This representation is borrowed from[START_REF] Guillard | Politique monétaire et fiscale dans un monde non-Ricardien: une théorie fiscale de l'inflation[END_REF].

Global determinacy

The equations of the curves X t+1 = X t and Π t+1 = Π t are respectively:

We may further note that:

Figure 2 depicts the two curves Π t+1 = Π t and X t+1 = X t , as well as the dynamics of the economy given by ( 49) and (50), in the case corresponding to condition (46). It appears clearly that the dynamics around the unique equilibrium is a saddle path, and there is global determinacy.

There remains now to check that there exist functions Φ (Π t ) such that the Taylor principle does not hold, and nevertheless condition (45) is satisfied.

An example: linear interest rate rules

We shall consider here simple linear interest rate rules:

They have the property that their elasticity is always below 1 since:

so they do not satisfy the Taylor principle. Now condition (45) will be satisfied if A > ζ/β, and then global determinacy is ensured.

Appendix

In this appendix we shall derive the fundamental dynamic equation ( 16). Consider the household's budget equation (4), and assume that i t is strictly positive. The household will thus satisfy the "cash in advance" equation exactly, so that m jt = P t c jt and the budget constraint is written:

Let us define the following discount factors:

Maximizing the utility function ( 1) subject to the sequence of budget constraints (57) from time t to infinity yields household j's consumption function:

Summing this across the N t agents alive in period t, and using the equilibrium condition C t = Y t we obtain the equilibrium equation:

Let us divide both sides by N t , subtract from it the corresponding equation for t + 1, and then divide by R t+1 . We obtain:

Divide the government's budget equation ( 9) by N t and insert it into (61). This yields:

Now multiply (62) by N t+1 , and assume N t+1 /N t = 1 + n. We obtain equation (16).


Jean-Pascal Bénassy
email: benassy@pse.ens.fr

† Pse

Cepremap

Liquidity effects in non Ricardian economies

Keywords:

It has often been found difficult to generate a liquidity effect (i.e. a negative effect of monetary injections on the nominal interest rate) in the traditional "Ricardian" stochastic dynamic model with a single infinitely lived household. We show that moving to a non-Ricardian environment where new agents enter the economy in each period allows to generate such a liquidity effect.

Introduction

The purpose of this article is to investigate a new mechanism through which liquidity effects can be introduced into dynamic stochastic general equilibrium (DSGE) models.

What we call here "liquidity effects" is the negative response of the nominal interest rate to monetary injections. That liquidity effect was already present in the famous IS-LM model, and it appears to be found in the data (see, for example, Christiano, Eichenbaum and Evans, 1997). As it turns out, this liquidity effect has been found difficult to obtain in standard monetary DSGE models. The reason is an "inflationary expectations effect" which tends to actually raise the nominal interest rate in response to a monetary injection.

We now briefly outline the mechanism behind this inflationary expectations effect 1 . It is found in the data that money increases are positively correlated in time. So when an unexpected money injection occurs, this creates the expectation of further money increases in the future, which will itself create the expectation of future inflation. Now from Fisher's equation the nominal interest rate is the sum of expected inflation and the real interest rate, so that, ceteris paribus, this will tend to raise the nominal interest rate.

Inspite of this "inflationary expectations effect" one can find in the literature a few models and mechanisms which introduce a liquidity effect in DSGE models. Two prominent ones are:

-Models of limited participation [START_REF] Lucas | Liquidity and interest rates[END_REF][START_REF] Christiano | Liquidity effects and the monetary transmission mechanism[END_REF][START_REF] Fuerst | Liquidity, loanable funds and real activity[END_REF], where households cannot adapt immediately their financial portfolios when a monetary policy shock occurs.

-Models of sticky prices (Jeanne, 1994, Christiano, Eichenbaum and[START_REF] Christiano | Sticky price and limited participation models of money: a comparison[END_REF], where prices are preset in advance. Actually the liquidity effect occurs if the intertemporal elasticity of substitution in consumption is sufficiently low.

What we want to do here is to explore a third avenue, which we will call "non-Ricardian". By non-Ricardian we mean models, like the overlapping generations model of [START_REF] Samuelson | An exact consumption-loan model of interest with or without the social contrivance of money[END_REF], where new agents are born every period, and where "Ricardian equivalence" [START_REF] Barro | Are government bonds net wealth?[END_REF] does not hold, by opposition to the traditional "Ricardian" model where the same agent or dynasty lives from the beginning to the end of time.

We shall argue that the non-Ricardian character of these economies is conducive to a liquidity effect. In a nutshell, the channel is the following: the existence of future, yet unborn, generations creates, as we shall see below, a "wealth effect" akin to those studied by [START_REF] Pigou | The classical stationary state[END_REF] and [START_REF] Patinkin | Money, Interest and Prices[END_REF]. This wealth effect itself produces a liquidity effect. We shall now proceed to demonstrate this formally.

The plan of the article is the following: section 2 presents the non-Ricardian model. Section 3 shows how a wealth effect arises in it. Section 4 derives the dynamic equations. Section 5 shows how the non-Ricardian economy displays a liquidity effect. Section 6 gives a generalization. Section 7 proves that the liquidity effect can be quite persistent. Section 8 concludes and the appendix gives an intuitive derivation of the results in an IS-LM model.

A non-Ricardian model

In order to have a non-Ricardian stucture that "nests" the usual Ricardian model, we shall use a model adapted from that of [START_REF] Weil | Permanent budget deficits and inflation[END_REF][START_REF] Weil | Is money net wealth?[END_REF]. Households never die, but new "generations" are born each period. Call N t the number of households alive at time t. Since households do not die, N t+1 ≥ N t . We will actually mainly work below with the case where the population grows at the constant rate n ≥ 0, so that N t = (1 + n) t .

Households

Consider a household j (i.e. a household born in period j). We denote by c jt and m jt his consumption and money holdings at time t ≥ j. This household receives in period t ≥ j an endowment y jt and maximizes the expected value of the following utility function:

U jt = ∞ X s=t β s-t Log c js (1)
Household j is submitted in period t to a "cash in advance" constraint:

P t c jt ≤ m jt (2)
Household j enters period t with a financial wealth ω jt . He then receives from the government a lump sum monetary transfer τ jt (if negative this will be called taxes), so that his available wealth is ω jt + τ jt . Then there are two successive "subperiods" where the bonds and goods markets open sequentially.

In the first subperiod the bonds market opens. All agents can exchange freely money against bonds at the nominal interest rate i t . So at this stage household j divides his available wealth ω jt + τ jt between bonds b jt and money m jt :

m jt + b jt = ω jt + τ jt (3)
Then in the second subperiod the goods market opens. Household j sells his endowment y jt and consumes c jt , subject to the cash constraint [START_REF] Bénassy | Interest rate rules and price determinacy in a non-Ricardian world[END_REF]. Consequently his budget constraint is:

ω jt+1 = (1 + i t ) b jt + m jt + P t y jt -P t c jt = (1 + i t ) (ω jt + τ jt ) -i t m jt + P t y jt -P t c jt (4)

Aggregation

Aggregate quantities are obtained by summing the various individual variables. There are N j -N j-1 agents born in period j, so for example:

T t = X j≤t (N j -N j-1 ) τ jt Ω t = X j≤t (N j -N j-1 ) ω jt (5)
Similar formulas apply to aggregate money M t , bonds B t , consumption C t and output Y t .

Endowments and transfers

Now we describe how endowments and transfers are distributed among households. In the main body of the article we shall assume that all households have the same income and transfers, so that:

y jt = y t = Y t N t τ jt = τ t = T t N t (6)
In section 6 we explore the more general assumption that households' resources diminish at a rate ξ < 1. We shall see that this still reinforces our results.

Government

Now an other important part of the model is the government. The households' aggregate financial wealth Ω t has as a counterpart an identical amount Ω t of financial liabilities of the government. The evolution of these liabilities is described by the government's budget constraint:

Ω t+1 = (1 + i t ) B t + M t = (1 + i t ) (Ω t + T t ) -i t M t (7)

Monetary policy

If we aggregate equation ( 3) across all generations we obtain:

M t + B t = Ω t + T t ( 8
)
Monetary policy is an "open market" policy whereby all agents, including government, freely exchange bonds against money on the bonds market. As in all studies on the "liquidity effect" we shall assume that the government uses the quantity of money M t as the policy variable, and that consequently the nominal interest rate i t is endogenously determined through the equilibrium on the bonds market. A positive shock on money M t corresponds to a purchase of bonds by the government.

Following the literature we shall make the assumption that M t is a stochastic process. As an example, it is often assumed in the literature that money increases are autocorrelated in time:

Log µ M t M t-1 ¶ = ε t 1 -ρL 0 ≤ ρ < 1 ( 9
)
where L is the lag operator, and ε t is i.i.d.

As indicated above, we shall say there is a liquidity effect if a positive shock on M t leads to a decrease in i t .

3 The wealth effect2 As we indicated in the introduction an important part of the story is the "wealth effect" through which financial wealth influences consumption and the dynamic equations.

One may of course wonder why part of financial assets, now and in the future, represents actual purchasing power in the non-Ricardian model, whereas it does not in the Ricardian model. The reason is simple: some of the future taxes that are the counterpart of current nominal wealth will not be paid by the currently alive agents, but by future, yet unborn, generations, so that this part of Ω t represents actual purchasing power. We shall now make this intuition more formal. As the intertemporal government budget constraint will play an important role in the reasoning, we start with it.

The government intertemporal budget constraint

In what follows we shall repeatedly aggregate discounted values. It is convenient to compute in monetary terms, and we shall thus use the following discount factors:

R t = t-1 Y s=0 1 1 + i s R 0 = 1 (10)
Let us consider the government's budget constraint [START_REF] Lucas | Liquidity and interest rates[END_REF] in period s multiplied by R s+1 :

R s+1 Ω s+1 = R s Ω s + R s T s -(R s -R s+1 ) P s Y s (11)
Now let us define total taxes in period s, Γ s , as:

R s Γ s = R s µ i s M s 1 + i s -T s ¶ = (R s -R s+1 ) P s Y s -R s T s (12)
Total taxes consist of proper taxes -T s and the money economized by the state because of the cash in advance constraint i s M s / (1 + i s ), the "money tax". Using this definition, (11) can be rewritten:

R s Γ s = R s Ω s -R s+1 Ω s+1 (13)
Taking expectation of (13) as of time t, and summing from time t to infinity we get:

R t Ω t = E t ∞ X s=t R s Γ s (14)
We see that every single dollar of financial wealth is matched by discounted current and future taxes.

The consumption function

Let us now consider household j's budget equation [START_REF] Christiano | Sticky price and limited participation models of money: a comparison[END_REF]. We assume that i t is strictly positive, so that the cash in advance constraint is always satisfied exactly and m js = P s c js . Applying the discount factor R s+1 to this budget constraint, it becomes:

R s+1 ω js+1 = R s ω js + R s τ s + R s+1 P s y s -R s P s c js (15)
Household j maximizes the expected value of his utility (1) subject to the sequence of budget constraints (15). This yields the first order condition:

1 R s P s c js = βE s µ 1 R s+1 P s+1 c js+1 ¶ (16)
We can approximate (16) to the first order as:

E s (R s+1 P s+1 c js+1 ) = βR s P s c js (17)
and take the expectation as of time t of both sides:

E t (R s+1 P s+1 c js+1 ) = βE t (R s P s c js ) (18)
Now let us take the expectation of (15) as of time t, and aggregate all these constraints from time t to infinity. Assuming that R s ω js goes to zero as s goes to infinity (this is the usual transversality condition), we obtain the intertemporal budget constraint of the household (in expected terms):

E t ∞ X s=t R s P s c js = R t ω jt + E t ∞ X s=t (R s+1 P s y s + R s τ s ) (19)
Applying repeatedly formula (18) and inserting it into (19) we find the consumption function for agents born in period j:

R t P t c jt = (1 -β) " R t ω jt + E t ∞ X s=t (R s+1 P s y s + R s τ s ) # (20)
Summing over all N t households alive in t we obtain the aggregate consumption function:

R t P t C t = (1 -β) " R t Ω t + N t E t ∞ X s=t (R s+1 P s y s + R s τ s ) # (21)
Now using the definition of "total taxes" Γ s (equation 12), this can be rewritten:

R t P t C t = (1 -β) " N t E t ∞ X s=t R s P s y s + R t Ω t -N t E t ∞ X s=t R s Γ s N s # (22)
We see that generations alive in t will pay at time s > t only a fraction N t /N s of total taxes. From this the wealth effect will arise.

The wealth effect

Combining the two equations ( 14) and ( 22), we rewrite the consumption function as:

R t P t C t = (1 -β) " N t E t ∞ X s=t R s P s y s + E t ∞ X s=t N s -N t N s R s Γ s # (23)
Note that N s -N t is the number of agents alive in period s, but yet unborn at period t. So the wealth of agents currently alive consists of two things: (a) the discounted sum of their incomes (b) the part of taxes that will be paid by future generations in order to reimburse the current financial wealth. Now we may wonder what is the part of financial wealth that will be considered as "real wealth" by the currently alive generations. For that let us insert ( 13) into (22). We obtain:

R t P t C t = (1 -β) " N t E t ∞ X s=t R s P s y s + R t Ω t + N t E t ∞ X s=t R s+1 Ω s+1 -R s Ω s N s # (24)
Rearranging we find:

R t P t C t = (1 -β) N t E t " ∞ X s=t R s P s y s + ∞ X s=t R s+1 Ω s+1 µ 1 N s - 1 N s+1 ¶ # (25)
Consider the case where, by an adequate fiscal policy, Ω t remains constant in time and equal to Ω 0 . In such a case the "supplementary wealth" beyond discounted incomes is, from (25), equal to: "

N t ∞ X s=t R s+1 µ 1 N s - 1 N s+1 ¶ # Ω 0 (26)
It is easy to see that the coefficient into brackets is between 0 and 1, and other things equal, larger when the rate of increase of the population is larger.

Dynamic equilibrium

In equilibrium we have C t = Y t , so equation (25) becomes, after dividing by N t :

R t P t y t = (1 -β) E t " ∞ X s=t R s P s y s + ∞ X s=t R s+1 Ω s+1 µ 1 N s - 1 N s+1 ¶ # (27)
Let us rewrite it for t + 1, take the expectation as of time t, and subtract from (27). We find:

E t (R t+1 P t+1 y t+1 ) = βR t P t y t -(1 -β) µ 1 N t - 1 N t+1 ¶ R t+1 Ω t+1 (28)
Multiplying by N t+1 /R t+1 , and taking N t+1 /N t = 1 + n we obtain:

E t (P t+1 Y t+1 ) = β (1 + n) (1 + i t ) P t Y t -(1 -β) nΩ t+1 (29)
This will be our central dynamic equation. Note that the "non-Ricardian" character appears through the last term, which shows that, unlike in the Ricardian model where n = 0, financial wealth does matter. Now since the model is non Ricardian, the dynamics will depend on the actual fiscal transfer policy. To simplify the dynamics below, and since our focus is on monetary policy, we shall choose the simplest fiscal policy and assume that the government balances its budget period by period. Fiscal transfers will thus exactly compensate interest payments on bonds:

T t = -i t B t (30)
Combining ( 7), ( 8) and (30) we find that under the balanced budget policy (30) total financial wealth will remain constant:

Ω t = Ω 0 for all t (31)
The dynamic equation (29) then becomes:

E t (P t+1 Y t+1 ) = β (1 + n) (1 + i t ) P t Y t -(1 -β) nΩ 0 (32)
and since M t = P t C t = P t Y t , this can be rewritten:

E t M t+1 = β (1 + n) (1 + i t ) M t -(1 -β) nΩ 0 (33) θ > 1 (38)
Now combining (34), ( 36) and (37), we obtain:

1 + i t 1 + i 0 = 1 θ E t µ M t+1 M t ¶ + µ 1 - 1 θ ¶ M 0 M t (39)
We see that formula (39) gives a balanced view between the new non-Ricardian liquidity effect and the traditional inflationary expectations effect. We can note that the higher θ (and thus notably the higher n), the stronger the liquidity effect will be.

The real interest rate

The real interest rate r t is defined as:

1 + r t = (1 + i t ) P t P t+1 (40)
Now let us assume that endowments per head are constant in time, so that Y t+1 /Y t = 1 + n. Combining this with M t = P t Y t , and equations (36), (37) and (39) we obtain:

1 1 + r t = β M t+1 /M t E t (M t+1 /M t ) + (θ -1) (M 0 /M t ) (41)
and:

E t µ 1 1 + r t ¶ = β E t (M t+1 /M t ) E t (M t+1 /M t ) + (θ -1) (M 0 /M t ) (42)
In view of assumption (35) we see that the real interest rate will react negatively to a positive money shock, in the sense that:

∂ ∂M t • E t µ 1 1 + r t ¶¸> 0 (43)

A generalization

We shall now consider the more general case where households' resources diminish at the rate ξ < 1. The income and transfer of an agent of generation j are ξ t-j times those of an agent of generation t > j. To be more precise:

y jt = ξ t-j y t τ jt = ξ t-j τ t (44) y = y 0 (61)
where π e is the expected rate of inflation, ω = Log Ω, p = LogP and:

a > 0 b > 0 c > 0 d > 0 e > 0 (62)
The third equation expresses market clearing (which we assumed throughout the article). The IS equation says that output is equal to demand, which itself depends negatively on the real interest rate iπ e , and positively on real wealth ωp. Note that the presence of this "real wealth" term, i.e. b > 0, is specific of the non Ricardian framework, as was shown rigorously in this article. Now we can solve for the nominal interest rate i, the real interest rate r = iπ e , and price level p. Omitting irrelevant constants this yields: We first see, differentiating the expression of i in (63), that:

∂i ∂m = 1 a + bd µ a ∂π e ∂m -b ¶ (64)
We recognize the two effects identified in the article. First the "inflationary expectations effect", which is positive if a positive money shock raises inflationary expectations (∂π e /∂m > 0). Secondly the negative "liquidity effect", itself due to the wealth effect (b > 0). Now the underlying mechanism for the liquidity effect is the following: an increase in money creates a price increase (63). This price increase decreases demand because of the real wealth effect (the second term in equation 59). To maintain total demand at the market clearing level, the first term in (59) must increase, i.e. the real rate of interest must decrease. This decrease in the real interest rate creates the liquidity effect.

  i = aπ ebm a + bd r = -b m + dπ e a + bd p = adπ e + am a + bd (63)

This effect appears more formally in section 5 below.

The existence of a wealth effect for financial assets in a non-Ricardian economy was first uncovered by[START_REF] Weil | Is money net wealth?[END_REF].

The appendix shows in an intuitive manner in an IS-LM framework how the liquidity effect arises from the wealth effect.

We may note that condition (38), which is similar to[START_REF] Wallace | The overlapping generations model of fiat money[END_REF] condition for the viability of money in an OLG structure, appears in relation to other issues concerning money and monetary policy. For example it is shown in[START_REF] Bénassy | Interest rate rules and price determinacy in a non-Ricardian world[END_REF] that (38) is a condition for price determinacy when the central bank uses interest rate rules à la[START_REF] Taylor | Discretion versus policy rules in practice[END_REF].

The proof is similar to that for ξ = 1 in sections 3 and 4. It is available upon request from the author.

See the appendix for a more detailed elaboration of this reasoning in a very simple model.

Liquidity effects

We shall now see that the non-Ricardian character of the economy, i.e. the fact that n > 0, will produce a liquidity effect 3 .

The nominal interest rate

We can actually solve explicitly equation (33) for the nominal interest rate:

We see that the first term, which is present even if n = 0, displays the "inflationary expectations effect": indeed, the nominal interest rate will rise if a positive monetary shock announces future money growth, i.e. if:

which is what is generally found empirically. In the example above (equation 9) this will occur if ρ > 0. We shall assume in what follows that the money process satisfies condition (35). Now the second term, which appears only if n > 0, i.e. if we are in a non-Ricardian framework, clearly introduces a liquidity effect, since an increase in money directly decreases the nominal interest rate. The higher n, the stronger this effect.

We can give an even simpler expression. Assume that money M t is stationary around the value M 0 . From (33), the corresponding stationary value of the interest rate, i 0 , is related to M 0 and Ω 0 by:

Let us define:

If we want to have a "wealth effect", net financial assets must be positive, i.e. Ω 0 > 0. As a consequence, from (36) the parameter θ must satisfy 4 : where y t and τ t are the income and transfer of a newborn agent in period t.

In such a case it can be shown 5 that the dynamic equation ( 33) is replaced by the more general one:

We can again solve directly for the interest rate:

in which we see that the second term does indeed produce a liquidity effect. Now from (45) the stationary value of money M 0 and the stationary interest rate i 0 are related by:

We can now give a more general definition of the parameter θ:

Combining ( 46), ( 47) and ( 48) we find that the interest rate is given by:

This is exactly the same expression as (39), but the expression of θ has been generalized from (37) to (48). We see that a lower value of ξ increases the value of θ and therefore enhances the non-Ricardian liquidity effect. In the extreme case where ξ = 0 (i.e. when agents have all their income in the first period of their life), θ is infinite and the liquidity effect totally dominates.

The persistence of the liquidity effect

We shall now develop an example showing that our liquidity effect can be quite persistent. Let us loglinearize equations (39) or (49), which yields:

where the Ricardian particular case is obtained by taking θ = 1. Let us consider the following stationary money process:

where ε t is i.i.d.. Then:

If µ + ρ > 1, then a positive monetary innovation ε t > 0 creates the expectation of a monetary increase next period, which is the assumption traditionally associated with the "inflationary expectations effect". So we shall assume µ + ρ > 1 so as to have this effect. Now combining (50), ( 51) and ( 52) we can compute the full effect of monetary shocks on the interest rate:

We first see that if µ + ρ > 1, the Ricardian version of the model (θ = 1) always delivers an increase in interest rates on impact in response to monetary injections. We thus obtain the traditional "inflationary expectations effect".

Let us now move to the non-Ricardian case θ > 1. Looking at formula (53), we see that the first period impact µ + ρθ is negative as soon as:

We shall see that this liquidity effect is persistent, and that condition (54) is actually sufficient for a monetary injection to have a negative effect on the interest rate in all subsequent periods. Formula (53) can indeed be rewritten as:

This can be written as a distributed lag of all past innovations in money ε t-j , j ≥ 0:

with:

We want to show now that condition (54) is a sufficient condition for ω j < 0 for all j. This is done simply by rewriting (57) as:

The second term is always negative or zero. The first term is negative if θ > µ+ρ. So condition (54) is sufficient for the non-Ricardian liquidity effect to dominate the usual inflationary expectations effect, not only on impact, but for all subsequent periods as well.

Conclusions

We developed in this article a new mechanism through which liquidity effects are introduced into dynamic monetary models.

The basic channel is the following: (a) in a non Ricardian economy, accumulated financial assets represent, at least partly, real wealth to the generations alive, simply because part of the future taxes that are a counterpart to this financial wealth will be paid by the next generations. This is not the case in the "Ricardian" framework since there is no such thing as the "next generations", and: (b) this wealth effect gives rise to a liquidity effect as follows: An increase in money raises prices, which decreases the real value of financial wealth. Because of the wealth effect this reduces aggregate demand. In order to maintain aggregate demand at the market clearing level the real interest rate goes down. This creates, ceteris paribus, the liquidity effect 6 .

Appendix

To guide our intuition as to why the wealth effect leads to a liquidity effect, let us consider a simple traditional IS-LM model augmented with a "wealth effect" à la Pigou or Patinkin. To make the exposition particularly simple we write this model in loglinear form:


Jean-Pascal Bénassy
email: benassy@pse.ens.fr

† Pse

Cepremap

ÉCOLE NATIONALE DES PONTS ET CHAUSSÉES -ÉCOLE NORMALE

Keywords: E5, E52, E58 Taylor principle, interest rate rules, Taylor rules, inflation, optimal monetary policy

The purpose of this article is to characterize optimal interest rate rules in the framework of a dynamic stochastic general equilibrium model, and notably to scrutinize the "Taylor principle", according to which the nominal interest rate should respond more than one for one to inflation. This model yields explicit solutions for the optimal rule. We find that the elasticity of response depends on numerous factors, such as the degree of price rigidity, the autocorrelation of the underlying shocks, or which measure of inflation is used. In general the optimal elasticity of the interest rate with respect to inflation needs not be greater than one.

Introduction

The purpose of this article is to characterize optimal interest rate rules in the framework of a dynamic stochastic general equilibrium model, and notably to examine the much debated subject of how nominal interest rates should react to inflation.

Indeed following Taylor's (1993) influential article, many authors have studied interest rate rules where central bank policy is a function of endogenous variables such as inflation and output. A particularly scrutinized issue has been the response to inflation 1 , and notably the "Taylor principle", according to which the nominal interest rate should respond more than one for one to inflation 2 . In a nutshell, if the interest rate rule is of the form (omitting constants) i t = γπ t , then the Taylor principle says that γ should be greater than 1.

The basic framework we shall use for this investigation is that of a dynamic monetary economy subject to stochastic shocks, like productivity shocks. We shall develop a simple model for which we will be able to compute explicit solutions for the optimal interest rate rules 3 . The optimal interest rate policy is obtained through maximization of households' utility, subject to the laws of motion of the economy. In such a framework, and under rational expectations, the natural expression of optimal interest rules would be a function linking the interest rate to all observable shocks.

But, as indicated above, our main interest will be in rules expressed as a function of inflation, and in examining the size of their elasticity of response. We can identify two channels through which inflation enters the response function, which we will call "intrinsic" and "surrogate".

(a) The "intrinsic" channel actually concerns not current inflation but expected inflation, for the following natural reason: expected inflation appears in the households's demand functions, and therefore in the dynamic 1 The original Taylor contribution introduced both inflation and output as arguments of the interest rate rule, but in subsequent writings the role of output has been overshadowed by that of inflation. 2 There are actually two different aspects of the Taylor principle in the literature. The first, which is the subject of this paper, is concerned with the optimal response of the interest rate to inflation. The second one says that it is necessary to respect the Taylor principle to obtain price determinacy. The validity of that claim in non-Ricardian frameworks has been analyzed in [START_REF] Bénassy | Interest rate rules and price determinacy in a non Ricardian world[END_REF]. 3 There are already in the literature numerous contributions deriving optimal rules from explicit maximization, for example Rotemberg andWoodford (1997, 1999), [START_REF] Svensson | Inflation forecast targeting: implementing and monitoring inflation targets[END_REF][START_REF] Svensson | Inflation targeting as a monetary policy rule[END_REF], Clarida, Gali and Gertler (1999), [START_REF] King | What should the monetary authority do when prices are sticky[END_REF], [START_REF] Erceg | Optimal monetary policy with staggered wage and price contracts[END_REF], [START_REF] Henderson | The choice of a monetary policy reaction function in a simple optimizing model[END_REF], [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF]. Due to the complexity of calculations many contributions use numerical simulations or quadratic approximations. equations of the economy. Consequently it will also generally appear in the optimal interest rate rules.

(b) The second way inflation appears in the interest rate rules is as a "surrogate" for the underlying shocks when these are omitted from the policy function. In that respect, the argument of the function can be either expected inflation or current inflation (it could even be some past inflation).

So, in order to investigate the optimal response of interest rates to inflation, we shall derive a number of optimal interest rate rules taking into account the two above channels.

In order to disentangle them, we shall first derive an optimal interest rate rule when both expected inflation and shocks appear independently as arguments.

We shall then derive, in the tradition of dynamic stochastic general equilibrium (DSGE) models, interest rate rules that respond to shocks only. This will show that the optimal response to shocks depends on the nature of the underlying nominal rigidities 4 .

Finally we shall investigate, for various types of rigidities, a number of interest rate rules where current or expected inflation act as surrogates for the underlying shocks. We shall see that the optimal degree of response to inflation depends on numerous factors, like which measure of inflation (current or expected) is used, the nature and degree of price rigidities, and the autocorrelation of shocks. But we will find that the corresponding elasticity can be smaller or greater than one, depending on the values of the relevant parameters, and thus not systematically greater than one.

The model 2.1 The agents

We shall consider a monetary overlapping generations model [START_REF] Samuelson | An exact consumption-loan model of interest with or without the social contrivance of money[END_REF] with production. The economy includes representative firms and households, and the government.

Households of generation t live for two periods, work N t and consume C 1t in period t, consume C 2t+1 in period t + 1. They maximize the expected value of the following two period utility:

U t = α C 1-σ 1t 1 -σ + C 1-σ 2t+1 1 -σ -N t σ ≥ 0 (1)
Households are submitted in each period of their life to a "cash in advance" constraint:

M 1t ≥ P t C 1t M 2t+1 ≥ P t+1 C 2t+1 (2)
The total quantity of money is M t = M 1t + M 2t . Since the young household starts his life without any asset, he has to borrow P t C 1t from the bank at the interest rate i t in order to satisfy the cash in advance constraint. Consequently the bank makes profits Λ t , equal to:

Λ t = i t P t C 1t (3)
To simplify calculations we assume that these profits Λ t are redistributed lump-sum to the young households 5 .

The representative firm in period t produces output Y t with labor N t via the production function:

Y t = Z t N t (4)
where Z t is a technological shock common to all firms. We assume that the firms belong to the young households, to which they distribute their profits, if any.

Government policy and the optimality criterion

The government has essentially one policy instrument, the nominal interest rate i t . In order to evaluate the optimality properties of potential interest rate policies, we shall use the criterion proposed by Samuelson for the overlapping generations model [START_REF] Samuelson | A turnpike refutation of the golden rule in a welfare-maximizing many-year plan[END_REF][START_REF] Samuelson | The two-part golden rule deduced as the asymptotic turnpike of catenary motions[END_REF][START_REF] Abel | Optimal monetary growth[END_REF]) and assume that in period t the government maximizes the function V t :

V t = E t ∞ X s=t-1 β s-t U s (5)
The sum starts at s = t -1 because the old household born in t -1 is still alive in t. Rearranging the terms in the infinite sum [START_REF] Bénassy | Output and inflation dynamics under price and wage staggering: analytical results[END_REF], we find that, up to a constant, the criterion V t can be rewritten under the more convenient form:

V t = E t ∞ X s=t β s-t ∆ s (6)
with:

∆ t = α C 1-σ 1t 1 -σ + 1 β C 1-σ 2t 1 -σ -N t (7)
3 Market equilibrium

For the policy evaluations that will follow, we need to characterize the market equilibrium.

Consider first the problem of the old households in period t. We denote by Ω t-1 the financial wealth that the old households have accumulated at the end of period t -1, and which they own at the beginning of period t. With a hundred percent cash in advance constraint (formula 2), their consumption C 2t is simply given by:

P t C 2t = Ω t-1 (8)
Now let us write the maximization program of the young household born in t. When young, the representative household receives wages W t N t , firms' profits Ψ t = P t Y t -W t N t and central bank profits Λ t . If he consumes C 1t in the first period of his life, he will have accumulated at the end of period t a financial wealth:

Ω t = (W t N t + Ψ t + Λ t ) -(1 + i t ) P t C 1t (9)
In view of ( 8), the part of the household's expected utility that pertains to his second period consumption can be written:

E t µ C 1-σ 2t+1 1 -σ ¶ = Ω 1-σ t 1 -σ E t µ 1 P t+1 ¶ 1-σ (10)
where the term E t (1/P t+1 ) 1-σ represents young households' expectations.

Note that, although we will assume rational expectations in all that follows, this term could possibly differ from its rational expectations value, for example if households were involved in an ongoing learning process 6 . So the expected utility maximization program of the young household boils down to choosing C 1t and N t so as to solve:

Maximize α C 1-σ 1t 1 -σ + Ω 1-σ t 1 -σ E t µ 1 P t+1 ¶ 1-σ -N t s.t. Ω t = (W t N t + Ψ t + Λ t ) -(1 + i t ) P t C 1t
The first order conditions with respect to C 1t and N t are:

αC -σ 1t = (1 + i t ) P t Ω -σ t E t µ 1 P t+1 ¶ 1-σ (11) 1 = W t Ω -σ t E t µ 1 P t+1 ¶ 1-σ (12)
We have two other equilibrium equations. The first says that total consumption must equal production:

C 1t + C 2t = Y t = Z t N t (13)
The second says that the price is equal to marginal cost:

P t = W t Z t (14)
We should finally note that, since central bank profits Λ t are redistributed to the consumers, Ω t will remain constant in time. Indeed, combining (3), ( 8), ( 9) and ( 13), we obtain:

Ω t = P t Y t -(1 + i t ) P t C 1t + Λ t = P t C 2t = Ω t-1 = Ω (15)
We can also combine ( 12) and ( 14) to find the dynamic equation giving Walrasian prices, denoted as P * t :

1 P * t = Z t Ω -σ E t µ 1 P * t+1 ¶ 1-σ (16)
A solution to this dynamic equation is derived in appendix 1. We note that Walrasian prices do not depend on the interest rate.

We will further make the traditional assumption that a higher expected inflation leads to a higher current equilibrium price, which leads to the following restriction on σ:

0 ≤ σ ≤ 1 (17)
4 Optimal interest policy: the Walrasian case

As a benchmark, we shall now compute the optimal interest rate policy in the case where all markets clear.

Proposition 1: Under Walrasian prices and wages the optimal interest rate rule is:

i t = 0 (18)
Proof : The central bank must choose the interest rate so as to maximize, in each period and for each value of the shocks:

∆ t = α C 1-σ 1t 1 -σ + 1 β C 1-σ 2t 1 -σ -N t (19)
or, using (13):

∆ t = α C 1-σ 1t 1 -σ + 1 β C 1-σ 2t 1 -σ - C 1t + C 2t Z t (20)
We first note that C 2t (equation 8) does not depend on the interest rate, since P * t does not. Combining [START_REF] Evans | Expectations and the stability problem for optimal monetary policies[END_REF], ( 12) and ( 14) we find that C 1t is given by:

C 1t = µ αZ t 1 + i t ¶ 1/σ (21)
Inserting this value of C 1t into (20) and maximizing ∆ t with respect to i t yields immediately [START_REF] Rotemberg | Interest rate rules in an estimated sticky price model[END_REF].

Q.E.D. We see that, in the Walrasian case, no matter what are the shocks or inflationary expectations, the interest rate should remain totally unresponsive. This is actually the famous "Friedman rule" [START_REF] Friedman | The optimum quantity of money[END_REF].

So we have here a first example where the interest rate should not respond at all to inflation, and therefore the Taylor principle does not hold.

Preset prices

We will now move to the study of economies with nominal rigidities. We shall begin assuming preset prices 7 , and make the assumption, traditional in the literature since [START_REF] Gray | Wage indexation: a macroeconomic approach[END_REF], that the preset price is equal to the expected value of the Walrasian price (in logarithms), i.e.:

p t = E t-1 p * t ( 22
)
where 8 p t = Log P t , p * t = LogP * t , and P * t , the Walrasian price in period t, is the solution of the dynamic equilibrium equation [START_REF] Mankiw | Sticky information versus sticky prices: a proposal to replace the new-Keynesian Phillips curve[END_REF].

We note that, since the Walrasian price P * t does not depend on the interest rate, the preset price will not depend on it either.

Equilibrium conditions

Since the price is preset, equation ( 14) does not hold anymore, but the other equilibrium conditions are still valid:

αC -σ 1t = (1 + i t ) P t Ω -σ t E t µ 1 P t+1 ¶ 1-σ (23) 1 = W t Ω -σ t E t µ 1 P t+1 ¶ 1-σ (24)
C 2t = Ω P t (25)

The optimal interest rule

We shall now derive an optimal interest rate rule having as arguments both the shocks and the households' inflationary expectations:

Proposition 2: Under preset prices the optimal interest rate rule is:

1 1 + i t = min " 1, Z t µ Ω P t ¶ -σ E t µ P t P t+1 ¶ 1-σ # (26)
Proof : We have to maximize for all values of the shocks the quantity:

∆ t = α C 1-σ 1t 1 -σ + 1 β C 1-σ 2t 1 -σ - C 1t + C 2t Z t (27)
Now since P t does not depend on i t , from (25) C 2t does not either, so that we are left to maximize in i t for all values of the shocks:

α C 1-σ 1t 1 -σ - C 1t Z t (28)
where, from formula [START_REF] Svensson | Inflation targeting as a monetary policy rule[END_REF], C 1t is given by:

C 1t = " 1 + i t α P t Ω -σ E t µ 1 P t+1 ¶ 1-σ # -1/σ (29)
Let us take C 1t as an intermediate maximization variable. Maximizing (28) we find that the optimal value for C 1t is:

C 1t = (αZ t ) 1/σ (30)
Equating the two values ( 29) and (30), and taking into account the fact that i t must be positive, we find (26).

Q.E.D. We see that inflationary expectations enter with an elasticity 1-σ, which is always smaller than 1 9 . The intuition is simple: since the optimal value of C 1t (equation 30) does not depend on expectations, the optimal interest rate will be chosen so as to exactly neutralize these expectations (formula 29).

The interest rate rule and shocks

We shall now express, in the tradition of DSGE models, the interest rate rule as a function of shocks only. Let us assume the following process for the technology shocks z t = LogZ t :

z t = u t 1 -ρL ( 31
)
where L is the lag operator and the stochastic variable u t is normal and i.i.d.

Proposition 3: Under technology shocks (31), the optimal interest rate rule is:

i t = - u t 1 -(1 -σ) ρ (32)
Proof : Formula (26) yields in logarithms, and ignoring irrelevant constants:

i t = -z t -p t + (1 -σ) E t p t+1 (33)
It is shown in appendix 1 that, with the technology process (31), the Walrasian price p * t is given by (up to a constant):

p * t = -µz t ( 34
)
with:

µ = 1 1 -(1 -σ) ρ (35)
Consequently:

p t = E t-1 p * t = -µE t-1 z t ( 36
)
Then formula (33) becomes:

i t = -z t + µE t-1 z t -µ (1 -σ) E t z t+1 = - u t 1 -ρL + µρu t-1 1 -ρL - µ (1 -σ) ρu t 1 -ρL (37)
which, after simplification, yields (32). Q.E.D. We see that, in this simple case, the optimal interest rate is a function of the innovation of the shock only. In section 7 we shall study a more complex pricing scheme where the optimal interest rate depends on all past values of the innovation.

Inflation as a surrogate for shocks

In proposition 2 (and the similar proposition 7 in appendix 3 on preset wages) we considered an interest rate rule with both expected inflation and shocks as arguments. Propositions 3 and 8 give rules with only shocks as arguments.

We shall now move to interest rate rules that are functions of inflation (current or expected) only, and we shall see that optimal interest rates will depend on inflation simply because inflation acts as a surrogate to shocks.

In order to make that "surrogate" aspect particularly clear, we shall take the particular case σ = 1, i.e. the case where the "intrinsic" coefficient of response to expected inflation is exactly zero (propositions 2 and 7).

Equilibrium

For σ = 1 a number of formulas actually simplify. Notably formula [START_REF] Mankiw | Sticky information versus sticky prices: a proposal to replace the new-Keynesian Phillips curve[END_REF] giving the equilibrium price becomes:

P * t = Ω Z t ( 38
)
so that:

p t = E t-1 p * t = ω -E t-1 z t ( 39
)
with ω = LogΩ. The values of the preset price equilibrium quantities are:

C 1t = αΩ (1 + i t ) P t ( 40
)
C 2t = Ω P t (41) N t = αΩ (1 + i t ) P t Z t + Ω P t Z t (42)

Utility evaluation

Setting σ = 1 in equation ( 7), we find that the "period t utility" ∆ t is now:

∆ t = αLogC 1t + 1 β LogC 2t -N t (43)
Inserting into (43) the values in (40), ( 41) and (42), we obtain the following value for ∆ t :

∆ t = αLog • αΩ (1 + i t ) P t ¸+ 1 β Log µ Ω P t ¶ - 1 P t Z t µ αΩ 1 + i t + Ω ¶ (44)
or, keeping only the terms containing the interest rate:

∆ t = -Log (1 + i t ) - 1 P t Z t Ω 1 + i t (45)
We shall find the optimal interest rate rule by maximizing the expected value of ∆ t (equation 45) with respect to the parameters of the rule, assuming that the technological process is still given by (31).

As indicated above, we shall consider two different rules, where the interest rate reacts to current or expected inflation.

Proposition 5: Under preset prices and the technology process (31), the optimal interest rate rule is characterized by:

γ = 1 + ρ 2ρ (53)
A = exp µ V 2 ¶ (54) V = 1 -ρ 2 (55)
Proof : Appendix 4.

We see that this time γ is always greater than 1, and the Taylor principle is always verified. To give an intuition about the value of the coefficient γ, we can compute the value of expected inflation as a function of shocks:

E t π t+1 = π t+1 = p t+1 -p t = -E t z t+1 + E t-1 z t = - ρ (1 -L) u t 1 -ρL = -ρu t + ρ (1 -ρ) u t-1 1 -ρL (56)
while we still have i * t = -u t . We see this time that i * t and E t π t+1 are correlated, so that E t π t+1 can play, at least partly, the role of a surrogate for u t .

To further interpret the coefficient γ in (53), let us separate two cases. First, in the case where ρ = 1, expected inflation and the innovation u t are statistically equivalent, and we have:

i * t = -u t = E t π t+1 (57)
so that quite naturally γ = 1. Now if ρ < 1, E t π t+1 contains two terms (equation 56). The first one, -ρu t , is collinear to u t . If there was only this term, the coefficient γ would be equal to 1/ρ. But, because of the second term ρ (1ρ) u t-1 / (1 -ρL), an increase in γ creates some inefficient "noise", so that in the end the actual coefficient γ is between 0 and 1/ρ (formula 53).

We can compute the value of the expected utility (formula 45) corresponding to this interest rate rule:

E (∆ t ) = - V 2 -1 (58)
We can also compute the expected utility corresponding to the rule found in proposition 3, and we find:

E (∆ t ) = -1 (59)
We see that, in view of (55), unless ρ = 1 the rule using expected inflation will never do as well as the rule using the shocks directly.

The interest rate rule under staggered prices

We shall now study the optimal rule when prices are staggered 10 . The underlying idea [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF]) is that in each period every price contract has a probability φ of being maintained, and a probability 1φ of being cancelled, in which case it is renegotiated on the basis of the latest information. The actual price is given, up to an unimportant constant, by the following weighted average:

p t = (1 -φ) ∞ X j=0 φ j E t-j p * t (60)
These contracts have an average duration of φ/ (1φ). We see that it is a quite flexible formulation, ranging from full flexibility (φ = 0) to full rigidity (φ = 1).

The optimal rule

We continue with the case σ = 1, so that:

p * t = -z t (61)
Combining ( 60) and (61), we find that p t is equal to:

p t = -(1 -φ) ∞ X j=0 φ j E t-j z t (62)
We shall now characterize the optimal interest rate rule in the case where the interest rate is a function of current inflation.

Proposition 6: Assume that the interest rate rule is function of current inflation:

1 + i t = A exp (γπ t ) = A exp [γ (p t -p t-1 )] (63)
then this rule is characterized by:

γ = φ (1 + ρ) 2 (1 -φ) (64)
Proof : Appendix 5.

We see this time that we have a more balanced view than in propositions 4 and 5, since, depending on the values of φ and ρ, the coefficient γ may be greater or smaller than 1, so that the Taylor principle holds sometimes, but not as a general rule.

An interpretation

As before we shall now compute both the optimal response of interest rates to shocks, and the value of inflation as a function of shocks:

The optimal response of interest rates is (cf. formula 26 with σ = 1):

i * t = -z t -p t (65)
Now:

z t = u t 1 -ρL p t = - (1 -φ) u t (1 -ρL) (1 -φρL) (66)
So the optimal response is:

i * t = - u t 1 -ρL + (1 -φ) u t (1 -ρL) (1 -φρL) = -φu t 1 -φρL (67)
We see that this time the interest rate should respond to a weighted average of past innovations in the technology. Let us now compute the value of inflation:

π t = (1 -L) p t = - (1 -φ) (1 -L) u t (1 -ρL) (1 -φρL) (68)
We see that, if ρ = 1, π t and i * t are collinear: i * t = φπ t / (1φ), so that π t is a perfect surrogate for i * t , and therefore:

γ = φ 1 -φ (69)
which is what formula (64) tells us for ρ = 1. Now if ρ < 1, one can again decompose π t as the sum of a term proportional to i * t , and one term orthogonal to it (noise). This computation is easy, but a bit clumsy, and is therefore omitted.

Conclusions

We constructed in this article a dynamic stochastic model for which it was possible to compute simple explicit solutions for optimal interest rate rules. We paid particular attention to the optimal reaction of nominal interest rates to inflation, and notably whether the elasticity of this reaction function should be greater than one (the Taylor principle).

We characterized this elasticity as a function of various underlying factors, and found that it could be smaller than one as well as greater, so that the Taylor principle does not hold in general.

We also saw on some examples that a rule using inflation as a surrogate for shocks usually does not perform as well as the rule using the original shocks. Therefore it would be more efficient to base interest rate policy directly on the information about the shocks, if it were available, of course.

In order to obtain explicit solutions the model presented here has been voluntarily streamlined. In particular some traditional transmission mechanisms are absent. Their inclusion would most likely necessitate the use of numerical simulations, but it is conjectured that this would not change the central message of this article.

where w t = LogW t , w * t = LogW * t , W * t = Z t P * t , and P * t is the solution of equation [START_REF] Mankiw | Sticky information versus sticky prices: a proposal to replace the new-Keynesian Phillips curve[END_REF]. We note that, since the Walrasian wage W * t does not depend on the interest rate, the preset wage does not depend on it either.

Equilibrium conditions

Since the wage is preset, equation [START_REF] Friedman | The optimum quantity of money[END_REF] does not hold anymore, but the other conditions are still valid:

αC -σ 1t = (1 + i t ) P t Ω -σ E t µ 1 P t+1 ¶ 1-σ (77)
C 2t = Ω P t ( 78
)
W t P t = Z t ( 79
)
The optimal interest rule Proposition 7: Under preset wages the optimal interest rate rule is:

1 1 + i t = min " 1, W t Ω -σ E t µ 1 P t+1 ¶ 1-σ # (80)
Proof : From formulas (77) and (79):

C 1t = " 1 + i t αZ t W t Ω -σ E t µ 1 P t+1 ¶ 1-σ # -1/σ (81)
We have to maximize in i t for all values of the shocks:

α C 1-σ 1t 1 -σ - C 1t Z t ( 82
)
where C 1t is given by (81). Taking C 1t as an intermediate maximization variable, we find:

C 1t = µ 1 αZ t ¶ -1/σ (83)
Equating the two values (81) and (83), and taking into account the fact that i t must be positive, we find (80).

Q.E.D.

A = exp µ V 2 ¶ (104)
so that the value of the expected utility is:

E (∆ t ) = - V 2 -1 (105)

We shall notably study Walrasian prices and wages, preset prices and preset wages.

This assumption simplifies notably the asset dynamics (see equation 15 below). It is conjectured that the main results would still hold under alternative redistribution schemes, although explicit solutions might not be available throughout.

One may find explicit analysis of such a learning process in a similar context in[START_REF] Bullard | Learning about monetary policy rules[END_REF] and[START_REF] Evans | Expectations and the stability problem for optimal monetary policies[END_REF].

Preset wages are studied in appendix 2.

So in all that follows lowercase letters denote the logarithm of the corresponding uppercase letters.

We should note that this result does not mean that the Taylor principle is not valid, since we consider only expected inflation through the "intrinsic" channel. Sections 6 and 7 study rules which include the "surrogate" channel as well, and where the Taylor principle sometimes holds.

We use a variant of the famous[START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] contract. That variant was developed in microfounded models inBénassy (2002aBénassy ( , 2003a,b),b), to which the reader is referred. Similar contracts have been proposed by[START_REF] Devereux | Predetermined prices and the persistent effects of money on output[END_REF] and[START_REF] Mankiw | Sticky information versus sticky prices: a proposal to replace the new-Keynesian Phillips curve[END_REF].

Current inflation

We shall first assume that the interest rate is a function of current inflation:

Proposition 4: Under preset prices and the technology process (31), the optimal interest rate rule is characterized by: γ = 0 (47)

Proof : Appendix 3.

We see that in this case the Taylor principle does not hold since γ = 0. We can now give a simple intuition as to why the coefficient of reaction to inflation γ is equal to zero. Let us denote as i * t the "optimal" interest rate derived in proposition 3. With σ = 1 we have:

On the other hand we can compute, using formula (36) with µ = 1, the expression of current inflation as a function of shocks:

We see that, because prices are preset, inflation is function of past innovations, whereas the optimal interest rate i * t is function of the current innovation u t only. So π t and u t are statistically independent, and therefore current inflation π t cannot play the role of a surrogate for u t . Moreover a nonzero γ would only create unwanted "noise", reducing utility. These explain why the optimal γ is equal to zero.

Expected inflation

We shall now assume that the interest rate is a function of expected inflation:

Appendix 1

Walrasian Price Dynamics

We shall compute the dynamics of the Walrasian price for the following process for z t :

where u t is normal and i.i.d. Let us recall the basic equation:

Loglinearizing it and omitting constants we obtain:

We make the usual decomposition:

Consequently we have, up to irrelevant constants:

Preset Wages

We shall investigate here a different nominal rigidity, preset wages, and assume that the preset wage is equal to the expected value of the Walrasian wage (in logarithms), i.e.:

Interest rate rule and shocks

Let us now express the interest rate rule as a function of shocks only.

Proposition 8: Under preset wages the optimal interest rate rule is:

Proof : Formula (80) yields in logarithms:

Now:

Combining (85), ( 86) and (88), we obtain:

and, simplifying, formula (84). Q.E.D. We see that under preset wages the elasticity of reaction to the technology shock (formula 84) is substantially different from that under preset prices (formula 32).

Appendix 3 Proof of Proposition 4

Finding the optimal rule consists in finding the optimal values for the two parameters γ and A. So, combining (45) and (46), one will maximize with respect to these two parameters the following quantity:

Given relation (39) this becomes:

Because z t is normal, this simplifies to:

where V is the variance of the last term into brackets in expression (91). Now we have:

The term in brackets in (91) is equal to:

and its variance V is therefore equal to:

We see that minimum variance is reached for γ = 0. The corresponding variance is V = 1.

Appendix 4 Proof of Proposition 5

Combining (45) and (52), we will maximize with respect to γ and A:

(96) Given relation (39) this becomes:

Again this simplifies as formula (92):

where V is now the variance of the last term into brackets in (97). We have:

The last term in brackets in ( 97) is equal to:

so its variance V is equal to:

Minimization with respect to γ yields:

Inserting (102) into (101) we obtain the resulting variance:

Maximizing expected utility (98) with respect to the constant term A yields: Appendix 5

Proof of Proposition 6

From (31) and (62) the price is equal to:

We want to minimize:

Now:

so the expression in ( 107) is equal to:

We want to minimize the variance of the quantity in brackets. The term of order j in the brackets in (109) is equal to:

The variance of the term of order j is, omitting the ρ in the denominator:

Summing over all values of j from zero to infinity we find the total variance:

Differentiating with respect to γ yields the first order condition:

Let us factor the coefficient γ:

Now:

Combining the last four equations we obtain:

which simplifies as: