Daria Onori
email: daria.onori@univ-orleans.fr

Optimal growth and debt dynamics under GDP-based collaterals Croissance optimale, dynamique de la dette et contraintes d'endettement basées sur le PIB

Keywords: open economy, two-stage growth, external debt, GDP-based collaterals, imperfect nancial markets, multi-stage optimal control. JEL Classication: CO2, C61, F34, F43, H63, O16, O4 économie ouverte, dette extérieure, marchés nanciers imparfaits, contraintes d'endettement baséees

This paper analyzes the consequences of external debt collaterals on the optimal growth path of a country. To this end we develop a small open economy model of endogenous growth where public spending can be nanced by borrowing on imperfect international nancial markets, where the country's borrowing capacity is limited. In contrast to the existing literature, which assumes that debt is constrained by the stock of capital, we investigate the consequences and policy implications of GDP-based collaterals. First, we show that the economy may converge in a nite time to the regime with binding collateral constraint. Second, in such regime the steady state public expenditures-to-GDP ratio is greater than that of the existing literature's models. Finally, if the economy is not suciently developed, in nancial and economic terms, the country will stay in autarky forever.

Introduction

The rapid development of nancial globalization started in the mid-1980s allowed developing countries to open to capital inows and many of them accumulated high levels of external debt. This exposed them to credit risk and solvency problems, so as international nancial markets reacted by limiting external borrowing.

This paper is a contribution to the growth literature in open economy and studies the eects of external debt collaterals on the optimal growth dynamics of a country. We extend Barro (1990) and Turnovsky's (1996) models of productive public expenditures. Notably, in Barro, public spending is enhanced by the government only by levying income taxes, while in Turnovsky the country can also borrow without limits from the rest of the world.

In our model, the country's external debt is constrained by GDP which, being the tax base, also sets an upper bound to the governments scal capacity. In this context, nancial openness means that the country is allowed to borrow on international nancial markets. The literature on growth, collateral constraints and nancial markets is very limited and assumes that a country's debt is constrained by physical capital.

1 Our main assumption on the borrowing constraint is that the country's collateral is GDP, rather than (private or public) physical capital. This assumption is as appropriate as that of physical capital since public capital usually takes the form of public patrimony, portions of territory, monuments, infrastructures, etc., which are of dicult appropriability for foreign creditors. Similarly to what is assumed and found in Djankov et al. (2008), we suppose that the presence of enforcement costs implies that creditors can seize just a fraction of country's collaterals.

One can conjecture that enforcement costs will be lower in such a scenario (consider, for example, the pre-existing transfer mechanisms within the European Union).

We focus on the policy implications of a collateral constraint (based on GDP, a proxy of scal capacity) for the optimal growth path of a country which opens to international borrowing. In this perspective, an increase in the economy's growth rate corresponds to a slackening of the scal constraint, as the borrowing capacity of the country increases.

2

We also introduce some technical novelties. In order to verify if the debt stock actually reaches the constraint's upper bound once the country moves from nancial autarky to openness, and to endogenously determine the date at which it can happen, we solve the model by using the multi-stage optimal control technique. Here, the standard optimal control problem is constructed as a dynamic programming problem, which allows to nd the instant at which the economy changes regime. Optimal switching problems that can be found in the literature typically involves regimes which are characterized by dierent technological or ecological states and each regime is dened as a certain set of parameters values (see Tomiyama 1985 and Boucekkine et al. 2004). A more recent branch of research uses this technique to analyze the problem of irreversibility: when the state variable reaches [START_REF] Ai | A normative analysis of public capital[END_REF] See: e.g. Eaton and Gersovitz (1981); Cohen and Sachs (1986); Brock (2009) and Boucekkine and Pintus (2012). Turnovsky (1997) studied the economy's dynamics when limited borrowing is represented by an upward-sloping supply of debt, where the cost of borrowing increases with the level of foreign debt, relative to the stock of capital.

2 Domestic revenue is clearly non seizable. One should think that, in case of default, a government can seize whatever he wants in the country. Hence the collateral condition should be interpreted as a signal of solvency of the debtor that can be compared to the situation of an individual ling for a loan with a bank. a certain threshold, the economy switches to a dierent regime in which the economy's parameters are dierent (see Boucekkine et al. 2013a, Boucekkine et al. 2013b, Boucekkine et al. 2013c, Prieur et al. 2011). In this paper we focus on another type of optimal switching time problem. The fact that at the initial date the constraint is not binding and then, if switching is optimal, it becomes binding modies one of the state equation, thus dening two possible states of the economy.

We obtain a number of novel results and some broad policy implications. In the regime where the constraint is not binding (non binding regime) the optimal public expenditures-to-GDP ratio is equal to the income share of public expenditures as in Barro and Turnovsky's models. However, this phase may be transitory: under certain conditions, the economy will switch to the regime in which the constraint is binding (binding regime). Here, we show that the economy admits a balanced growth path (BGP hereafter), which is not immediately reached because of the presence of investment adjustment costs and we determine analytically the conditions for the BGP's stability. This state of the economy displays an important dierence with respect to the Barro-Turnovsky model: the public expenditures-to-GDP ratio is greater than the income share of public expenditures. This happens because a higher investment in public goods relaxes the borrowing constraint. The increased borrowing capacity allows to nance greater public expenditures, which in turns boosts growth. Finally, we determine the endogenous switching date. In the papers previously mentioned, the endogenous switching date can be found because all of the regimes admit closed form solutions and the date can consequently be determined by replacing these solutions in the matching conditions between the regimes. Our binding dynamic system is non linear and cannot be solved in closed form. Nevertheless, we are able to give analytical conditions for corner solutions and then simulate the inter-regime transition. First, we nd that if there are no adjustment costs of capital, the economy immediately jumps to the regime where the constraint is binding. In other words, the presence of investment adjustment costs makes the inter-regime transition slower. Second, there exist some threshold conditions that must be satised by the relevant parameters, for the economy to switch to the binding regime and to fully benet from nancial openness. This happens when the interest rate and the adjustment costs parameter are low and the productivity is high. If, on the opposite, the economy is not suciently developed, in nancial and economic terms, the country will stay in autarky forever. We also perform a comparative statics analysis of the eect of the relevant parameters on the switching date. This date turns out to be an increasing function of the degree of nancial openness, the international interest rate, the adjustment costs and it is a decreasing function of the productivity parameter.

3

The paper is structured as follow. In Section 2 we present and solve our model. Section

The model

In this section we rst describe the economy's framework under the assumption of unlimited borrowing capacity, as in Turnovsky's (1996)

Y = AG α K 1-α , A > 0, 0 < α < 1 (2.1)
where K is the rm's stock of physical capital, A is the technology parameter and G is the ow of productive government expenditures. α represents the income share of public services. For a given G the economy faces diminishing returns to scale in the stock of capital and constant returns to scale in G and K. We normalize population to one throughout the paper: labor is supplied inelastically and population growth rate is exogenous. Capital can be interpreted in a broad sense by including human capital.

We dene public expenditures as a ow variable. This assumption is usually adopted in the literature (see, e.g., Futagami et al. 2008).

5

The economy's resource constraint (current account) can be expressed as

Ḃ = C + I + G + h 2 I 2 K + rB -Y (2.2)
which states that the economy accumulates debt to nance its total expenditures on consumption, C, public expenditures, private capital, interest payments on debt B (r is the exogenous international interest) net of produced output. We assume here that the country is a net borrower on international nancial markets, hence B ≥ 0. Capital accumulation involves convex investment adjustment costs of the form Φ (I, K) = I 1 + h 2 I K , h > 0. This is the traditional Hayashi's (1982) formulation, where the adjustment costs is proportional to the rate of investment in terms of units of installed capital. Their presence is a necessary assumption of small open economy models for two reasons. First, in a world where the country is not constrained by internal savings and has access to international nancial markets, the investment rate would be innity in the absence of adjustment costs. Second, without adjustment costs, this model would display the strong knife-edge condition according to which the internal marginal productivity of capital would be constrained to be equal to the exogenous interest rate (see Turnovsky 2002).

The agent's intertemporal utility is given by U (C) = ´+∞ 0 ln Ce -ρt dt where 0 < ρ < 1

is the discount factor.

6 Finally we assume that capital does not depreciate for the sake of simplicity, 7 so as

K = I (2.
3) [START_REF] Aschauer | Is public expenditure productive?[END_REF] The results of this model will out turn to be useful to study the emergence of corner solutions in our case. See Section 4. [START_REF] Barro | Government spending in a simple model of endogenous growth[END_REF] The consideration of the public stock of capital adds further diculties to the solution of the model.

2.1

Unlimited borrowing capacity: Turnovsky's model

Here the economy has unconstrained access to international nancial markets. For coherency, we also assume that the country is a net borrower on world capital markets.

8

The planner has to choose consumption, public expenditures, investment, debt and capital in order to maximize the intertemporal utility subject to (2.1), (2.2) and (2.3). The optimal public expenditures-to-GDP ratio is equal to the income share of public goods α, as in Barro's model. The growth rate of capital is K/K = (q -1)/h, where q ≡ -λ 2 /λ 1 is the market value of capital in terms of the price of bonds (Tobin's q),9 and that of consumption is

Ψ = r -ρ (2.4)
The growth rate of capital is determined by q, whose dynamics is given by q/q + (q -1) 2 /(2hq) + (1 -α) /qA 1 1-α α α 1-α = r. By setting q = 0 we obtain two solutions

q 1 = (1 + rh) -(1 + rh) 2 -1 + 2h (1 -α) A 1 1-α α α 1-α (2.5) q 2 = (1 + rh) + (1 + rh) 2 -1 + 2h (1 -α) A 1 1-α α α 1-α
As it is shown in Turnovsky (1996), the smaller equilibrium value, q 1 , is an unstable root, while q 2 is stable. However any stable arm converging to q 2 violates the transversality condition, lim t→∞ λ 2 (t) K (t) = lim t→∞ -q (t) λ 1 (t) K (t) = 0, where K (t) = K (0) e ´t 0 q(s)-1 h ds and λ 1 (t) = λ 1 (0) e -rt . In fact the stable adjustment path is q (t) = q 2 + (q (0) -q 2 ) e µt , where µ < 0 is the stable eigenvalue . Hence ´t 0 q(s)-1 h ds = q 2 -1 h t + q(0)-q 2 µ (e µt -1). By substituting this expression into the transversality condition it is possible to check that it is violated.

In the case of the unstable root q 1 , there is no transitional dynamics and the transversality condition is satised for q = q 1 . Hence q is always at its steady state value q 1 . This implies that capital is always on its balanced growth path, with a growth rate given by

Λ = q 1 -1 h (2.6)
By substituting (2.4), (2.5) and (2.6) into the economy's current account, we can solve for the stock of debt:

B (t) = -C(0) ρ e (r-ρ)t -θK (0) e q 1 -1 h t + B (0) + C(0) ρ + θK (0) e rt where θ ≡ q 1 -1 h +A 1 1-α α 1 1-α + (q 1 -1) 2 2h -A 1 1-α α α 1-α (r-q 1 -1 h )
. In order to ensure intertemporal solvency, the transversality condition lim t→∞ λ 1 (t) B (t) = lim t→∞ λ 1 (0) e -rt B (t) = 0 must be satised and this hold if and only if

C (0) = -ρ (B (0) + θK (0))
If C (0) > 0, it must be B (0) + θK (0) < 0. Given the denition of θ and given that the transversality condition implies r > q 1 -1 h , a necessary condition is

q 1 -1 h +A 1 1-α α 1 1-α + (q 1 -1) 2 2h - A 1 1-α α α 1-α < 0.
The equilibrium stock of debt is hence given by

B (t) = (B (0) + θK (0)) e (r-ρ)t -θK (0) e q 1 -1 h t (2.7)
Turnovsky's model has some important implications. First, the small open economy assumption implies that the growth rate of domestic consumption can be dierent from the growth rate of domestic output (capital), which is in sharp contrast with the closed economy version of the model, where all the variables grow at the same rate. Second, the growth rate of the stock of debt varies over time and asymptotically converges to q 1 -1 h , r -ρ .

In particular if q 1 -1 h < r -ρ, lim t→+∞ B (t) = -∞. On the contrary, if q 1 -1 h > r -ρ, lim t→+∞ B (t) = +∞ and the country would accumulate a forever increasing debt. Finally, the fact that the economy can borrow from the rest of the world does not inuence the part of domestic output invested in public expenditures, which is equal to the income share of public goods, α, as in Barro's (1990) model.

2.2

The model with GDP-based collaterals

We assume now that the country's debt capacity is bounded by GDP. The external debt constraint is:

0 ≤ B ≤ φY, φ ∈ [0, 1] (2.8)
The case φ = 0 corresponds to autarky, 10 while φ > 0 describes international nancial integration. Since nancial markets are imperfect and the highest obtainable collaterals is the entire GDP, it must be φ ≤ 1 11 in order to take into account the presence of enforcement costs.

12 Nevertheless, this restriction rules out debt-GDP ratios greater than 1. It is possible to show that the analysis remains valid for a wide range of parameters.

13

The social planner must choose the level of consumption, public expenditures, investment, [START_REF] Boucekkine | On the timing and optimality of capital controls: public expenditures, debts dynamics and welfare[END_REF] In this case the constraint (2.8) is binding. [START_REF] Boucekkine | Optimal regime switching and threshold eects[END_REF] See Cohen and Sacs (1986) and Boucekkine and Pintus (2012). Djankov et al. (2008) estimates that the values of φ range from 35% in middle-income countries to 85% in OECD countries. [START_REF] Brock | Collateral Constraints and Macroeconomic Adjustment in an Open Economy[END_REF] Imposing a ow variable as a constraint on a stock variable is reasonable in an endogenous growth model because GDP tends to innity in the long run. [START_REF] Chatterjee | Unilateral capital transfers, public investment, and economic growth[END_REF] In particular, φ must be such that φ < 1/(r -ρ).

external debt and capital which solve the problem

max {C,G,I} U (C) = ´+∞ 0 ln Ce -ρt dt s. t. K = I Ḃ = C + I + G + h 2 I 2 K + rB -Y B ≤ φY, B ≥ 0, ∀t K (0) , B (0) given (2.9)
where Y is given by (2.1). The collateral constraint, 0 ≤ B ≤ φY , denes three economic regimes. First, the stock of debt B can be equal to zero (autarky case) or positive. In the latter case the constraint (2.8) can be non binding or binding. The identication of the optimal autarky cases is quite straightforward. Hence we will focus on the two other regimes.

In order to endogenously nd the date at which the constraint becomes binding, we solve the model by using the two-stage optimal control technique. Optimal switching time problems that can be found in the literature typically involve technological or ecological states of the world that are exogenously given. For example Tomiyama (1985) In this paper we tackle a dierent type of optimal regime switching problems. The economy starts from a situation in which the stock of debt is such that the collateral constraint is not binding and it may become binding at a certain date (this is proved in Proposition 6) which has to be optimally determined. This modies one of the state equations. For this reason the problem can be set as an optimal switching time problem.

14 The two possible states of the economy can be summarized as follows

       Ḃ = C + I + G + h 2 I 2 K + rB -AG α K 1-α if B < φY t ∈ [0, T ] Ḃ = C + I + 1 φ 1 α A -1 α B 1 α K α-1 α + rB + h 2 I 2 K -1 φ B if B = φY t ∈ (T, +∞) (2.10)
Given (2.10), a natural approach is to decompose the problem into two sub-problems, solve them and nally identify the optimal timing T . In our case since

V (K (0) , B (0)) = ˆT 0 ln Ce -ρt dt + ˆ+∞ T ln Ce -ρt dt
one can view the problem sequentially starting from the second regime, for t ∈ (T, +∞) and then solving for the rst regime. In the binding regime the problem becomes:

max {C,I} V 2 = ´+∞ T ln Ce -ρt dt s. t. K = I Ḃ = C + I + 1 φ 1 α A -1 α B 1 α K α-1 α + rB + h 2 I 2 K -1 φ B for a given T, K (T ) , B (T ) (2.11)
The corresponding hamiltonian is

H (2) = ln Ce -ρt + λ (2) 1 C + I + 1 φ 1 α A -1 α B 1 α K α-1 α + rB + h 2 I 2 K - 1 φ B + λ (2) 2 I (2.12)
where the upper-script [START_REF] Aizenman | Economic growth with constraints on tax revenues and public debt: implications for scal policy and cross-country dierences[END_REF] means that the constraint binds. λ

(2)

1 < 0 is the co-state variable associated to the stock of debt and λ

(2) 2 > 0 is the co-state variable associated to capital when the economy is in the second regime.

Then we solve

max {C,G,I} V 1 = ´T 0 ln Ce -ρt dt + V * 2 (T, B (T ) , K (T )) s. t. K = I Ḃ = C + I + G + h 2 I 2 K + rB -AG α K 1-α K (0) , B (0) given, T free (2.13)
and the corresponding hamiltonian is

H (1) = ln Ce -ρt + λ (1) 1 C + I + G + h 2 I 2 K + rB -AG α K 1-α + λ (1) 2 I
(2.14)

where the upper-script (1) means that the regime is non binding.

This recursive scheme works exactly as a dynamic programming problem: the Bellman principle applies to the two sub-intervals implied by the timing problem.

We will discuss the Pontryagin optimality condition in the subsequent sections.

15 Here we shall focus on the optimality conditions with respect to the timing variable T . They are summarized by the following theorem:

Theorem 1. Let 0 < T < +∞ the optimal timing. Then

λ (1) 1 (T ) = λ (2) 1 (T ) (2.15) λ (1) 2 (T ) = λ (2) 2 (T ) (2.16) H (1) (T ) = H (2) (T )
(2.17) [START_REF] Djankov | Debt enforcement around the world[END_REF] The problem is concave hence each the two sub-intervals problems is well behaved.

A detailed proof is contained in Appendix A. Some comments are in order here. First, the above conditions can be read as continuity or matching conditions at the junction point: condition (2.17) imposes the continuity of the hamiltonian at the optimal junction time and conditions (2.15) and (2.16) impose the continuity of the co-state variables. Here both the co-state variables are continuous because the state variables do not jump at the switching time: the state variables do not have to satisfy any pure state constraint and both can be freely chosen. Second, one can interpret the matching conditions (2.15)-(2.17) as rst order optimal timing conditions with respect to the switching date T . In fact condition (2.17) can be rewritten as H [START_REF] Ai | A normative analysis of public capital[END_REF] (T ) -H [START_REF] Aizenman | Economic growth with constraints on tax revenues and public debt: implications for scal policy and cross-country dierences[END_REF] (T ) = 0. The left hand side can be interpreted as the marginal gain from extending the rst regime at the expense of the regime associated to the next interval. The right hand side represents the marginal cost. Since there is no direct switching cost, this cost is zero. Hence this condition equalizes the marginal benet to the marginal cost of delaying the switching time.

Preliminary analysis of economic regimes

In this section we will solve the model by starting with the Pontryagin problem of the binding regime and then by proceeding backward to the non binding case.

The binding regime

For a given T , B (T ) and K (T ), the Pontryagin necessary conditions of problem (2.11) are

1 C e -ρt + λ 1 = 0 (2.18) λ 1 1 + h I K + λ 2 = 0 (2.19) λ 1 1 α 1 φ 1 α A -1 α B K 1-α α + r - 1 φ = -λ1 (2.20) -λ 1 h 2
I 2 K 2 + 1 -α α 1 φ 1 α A -1 α B K 1 α = -λ2 (2.21) lim t→∞ λ 1 B = 0 lim t→∞ λ 2 K = 0
By using the shadow value of debt λ 1 as the numeraire and dening q = -λ 2 λ 1 , we nd the growth rate of capital

I K = q-1 h from
q q = 1 α 1 φ 1 α A -1 α B K 1-α α + r - 1 φ - 1 q 1 -α α 1 φ 1 α A -1 α B K 1 α - (q -1) 2 2hq
We nd it convenient to rewrite the arbitrage equation as a function of G Y in order to compare the results with Barro (1990) and Turnovsky's (1996) models. From the constraint we know that the ratio B K is linked to the ratio G K in the following way:

B K = φA G K α .
Moreover, by using the production function we know that:

G K = A 1 1-α G Y 1 1-α . Hence the arbitrage equation becomes q q = 1 α 1 φ G Y + r - 1 φ - 1 q 1 -α α A 1 1-α G Y 1 1-α - (q -1)
Ċ C = r - 1 φ + 1 α 1 φ G Y -ρ (2.23)
We nd it convenient to rewrite the dynamical system made up of the aggregate resource constraint, (2.22), and of (2.23) by dening b

≡ B K and c ≡ C K , so that        q q = 1 α 1 φ G Y + r -1 φ -1 q 1-α α A 1 1-α G Y 1 1-α -(q-1) 2 2hq ċ c = r -1 φ + 1 α 1 φ G Y -ρ -q-1 h ḃ b = 1 b c + q-1 h + A 1 1-α G Y 1 1-α + (q-1) 2 2h + rb -A 1 1-α G Y α 1-α -q-1 h (2.24) 2.3.2
The non binding regime

For a given T , B (T ) and K (T ), the rst order necessary conditions of problem (2.13) are [START_REF] Eaton | Debt with potential repudiation: theoretical and empirical analysis[END_REF] 1 C e -ρt + λ 1 = 0

(2.25)

λ 1 1 + h I K + λ 2 = 0 (2.26) G K = (Aα) 1 1-α ⇐⇒ G Y = α (2.27) λ 1 r = -λ1 (2.28) -λ 1 h 2
I 2 K 2 + A (1 -α) G K α = -λ2 (2.29)
where q = -λ 2 λ 1 and

I K = q-1 h
. The public expenditures-to-GDP ratio is equal to α. By using (2.29), (2.28) and the denition of q we get the arbitrage equation

q q = r - (q -1) 2 2hq -(1 -α) A 1 1-α α α 1-α 1 q (2.30)
Combining equation (2.25) with equation (2.28) we obtain the growth rate of consumption Ċ (t) [START_REF] Eaton | Debt with potential repudiation: theoretical and empirical analysis[END_REF] This regime is the same as Turnovsky's (1996) model, with exception of the transversality conditions: in our case the boundary conditions are given by the continuity conditions of Theorem 1.

C (t) = r -ρ (2.31)
Note that, in order to have a positive consumption growth rate, it must be r > ρ.

By dening c = C K , b = B K
we can write the reduced system as

     q q = r -(q-1) 2 2hq -(1 -α) A 1 1-α α α 1-α 1 q ċ c = (r -ρ) -q-1 h ḃ b = 1 b c + q-1 h + A 1 1-α α 1 1-α + (q-1) 2 2h + rb -A 1 1-α α α 1-α -q-1 h (2.32) 3
Balanced growth path analysis

In this section we study the balanced growth path of the economy (BGP hereafter) in the case where the economy ends up in the binding regime. The condition for optimal switching will be given in Section 4. Along the BGP, all the endogenous variables grow at the same, positive and constant growth rate, given by equation (2.23), and the endogenous variables (c, G/Y, q) of system (2.24) grow at a zero rate. This result can be veried by considering that when the collateral constraint binds, Y and B grow at the same rate. From the production function it can be seen that capital grows at the same rate of output. Finally, the terms of equation (2.10) are all proportional to K or B, 17 so that also C grows at the same rate.

By setting ċ = 0 in (2.24), we obtain q = h r -

1 φ + 1 α 1 φ G Y -ρ + 1.
By plugging it into the q = 0 equation and rearranging terms we end up with the equation determining the equilibrium (G/Y ) ratio

-1-α α A 1 1-α φ G Y 1 1-α + h 2 1 φ 1 α 2 G Y 2 + [h (rφ -1) + φ] 1 φ 1 α G Y + 1 φ h 2 (rφ -1) 2 + φ (rφ -1) -h 2 (ρφ) 2 = 0 (3.1)
Note that by setting φ = 0 we obtain Barro (1990) and Turnovsky's (1996) optimal public expenditures-to-GDP ratio: (G/Y ) = α. In the following proposition we give the conditions for the existence of a solution for the empirically interesting case α ≤ 1 2 .

18

Proposition 2. The dynamical system (2.24) admits one steady state.

The proof is given in Appendix B, where we also show that it satises the transversality condition.

In order to obtain analytical solutions for the ratio (G/Y ) and to study analytically the local stability of the steady state, we focus on the α = 1 2 case.

19 This case is perfectly equivalent to the case α < 1/2, hence the conditions for the existence of a BGP do not change.

17 Equation (2.10) can be rewritten as [START_REF] Futagami | debt poly rules, productive government spending, and multiple growth paths[END_REF] This range of values is consistent with the empirical estimations existing in the literature: Erberts's (1986) estimation for the income share of public expenditures is around 0.43, Aschauer (1989) nds that it is 0.39, Ai and Cassou (1995) nd estimations which range between 0.15 and 0.26, nally Chatterjee et al. (2003) assume a value of 0.2. [START_REF] Hayashi | Tobin's marginal q, average q: a neoclassical interpretation[END_REF] We analyze this case separately since the closed form solutions of the steady state variables are necessary to compute the Jacobian matrix for the stability analysis. Otherwise it is not possible to determine the sign of the trace and the determinant.

Ḃ B B = C + K K K + 1 φ 1 α A -1 α B K 1 α K + rB + h 2 I 2 K 2 K -1 φ B
Equation (3.1) becomes 2h φ -A 2 φ G Y 2 + 2 [h (rφ -1) + φ] 1 φ G Y + 1 φ h 2 (rφ -1) 2 + φ (rφ -1) -h 2 (ρφ) 2 = 0 (3.2)
The unique admissible solution is

G Y * = -[h (rφ -1) + φ] 1 φ + 1 2 (2 + 2h 2 ρ 2 + A 2 (h -2(1 + hr)φ + (r(2 + hr) -hρ 2 )φ 2 )) 2h φ -A 2 φ (3.3)
The next step is to study the stability of the BGP. System (2.24) allows transitional dynamics. This is due to the presence of investment adjustment costs: the economy does not reach immediately its steady state because the adjustment costs do not allow capital to immediately jump to its BGP. Proposition 3 describes the local stability properties of G Y * :

Proposition 3. If A 2 > - 2 h-2(1+rh)φ+(r(2+hr)-hρ 2 )φ 2
, the balanced growth path solution is saddle-point stable.

The proof is given in Appendix C.

Numerical analysis shows that the BGP is stable under the α ≤ 1/2 case, for standard calibrations of the parameters.

To conclude this section we will compare the optimal (G/Y ) of our model with that of Barro (1990) and Turnovsky's (1996; 1997) model. Proposition 4. The optimal public expenditures-to-GDP ratio of the model with GDP-based collaterals is greater than Barro (1990) and Turnovsky's (1996) solution.

In order to prove this result, consider the rst order condition with respect to public goods of the maximization problem (2.9), solved by attaching a Lagrange multiplier η ≥ 0 to the inequality constraint (2.8). 20

∂Y ∂G - η λ 1 ∂Y ∂G = 1 (3.4) where η = 1 φ 1 -1 α G Y λ 1 .
Being η > 0 when the constraint binds, and λ 1 < 0, equation (3.4) implies that in the binding regime (G/Y ) must be greater than α, i. e. greater than Barro (1990) and Turnovsky's (1996; 1997) value. Equation (3.4) states that the social planner will choose the optimal (G/Y ) by equating the marginal benet of increasing G to the marginal cost, which, by the economy's resource constraint, is equal to one. In the literature's models the corresponding expression is ∂Y ∂G = 1. Hence we can conclude that in the model with a GDP-based collateral constraint the overall marginal benet is equal to the marginal benet of increasing G on production augmented by the term -η λ 1 ∂Y ∂G , which represents the marginal benet of relaxing the collateral constraint, since a greater public expenditures increases the country's borrowing capacity. This is the reason why the public expenditures-GDP ratio is greater than that of the literature's models. [START_REF] Kose | Thresholds in the process of nancial integration[END_REF] This allows to carry out an economic interpretation.

3.1

Comparative statics

The aim of this section is to run a comparative statics analysis. The analytical results are included in the following proposition:

Proposition 5. The optimal public expenditures-to-GDP ratio is (i) decreasing in the world interest rate r;

(ii) increasing in the technology parameter A.

(iii) The long run growth rate is increasing in the TFP parameter.

Proof. The proof is contained in Appendix D.

An increase in the international interest rate makes borrowing more costly and given that public expenditures are nanced by debt, the optimal (G/Y ) ratio decreases. Results (ii) and (iii) stem from the double role played by the TFP in our model. On the one hand, A

rises the economy's productivity and this boosts growth. On the other hand, an increase in A, by making the economy richer, has the eect of increasing the collaterals, allowing the economy to borrow more from the rest of the world. The possibility to increase the nancing capacity enhances growth.

Unfortunately we cannot obtain analytical results for the main comparative statics, so we run numerical simulations. We x α ≤ 1/2, 21 h = 15 (Barro and Sala-i-Martin 2004, Chatterjee et al. 2003), r = 0.035 and ρ = 0.03 (these values allow to satisfy the conditions needed to move from the non binding to the binding regime that will be introduced later in Section 4). We set A = 1 in order to obtain a value for the capital-output ratio consistent with the real one, which is around 2.

22 We hence obtain the following results:

(iv) The optimal (G/Y ) and the economy's growth rate are increasing in φ. A higher φ implies higher growth because a greater nancial openness relaxes the borrowing constraint (growth enhancing eect of foreign borrowing).

(v) The growth rate is decreasing in the world interest rate r. Since (G/Y ) is decreasing in r, the same happens for the long run growth rate.

(vi) The optimal (G/Y ) and the economy's growth rate are decreasing functions of the installation cost parameter h. Higher capital adjustment costs reduce the (G/Y ) ratio and the growth rate because more resources are spent in installation.

The numerical analysis also shows that the growth rate of GDP is smaller than that of Turnovsky (1996). Recall that in his model the country can borrow as much as it wants because there is no collateral constraint and the growth rate of GDP coincides with the growth rate of capital, which can be dierent from the growth rate of consumption. This latter is set by the dierence between the exogenous international interest rate and the discount rate, and it is independent from any technological parameter. The growth rate in Turnovsky turns out to be greater than that of our model due to the absence of growthlimiting borrowing constraints. [START_REF] Prieur | Optimal emission-extraction policy in a world of scarcity and irreversibility[END_REF] We run the simulations for α = 0.2 (Chatterjee et al. 2003) and α = 1/2. [START_REF] Tomiyama | Two-stage optimal control problems and optimality conditions[END_REF] We repeated the exercise for other values of the parameter and the results are robust. The convergence to the binding regime

In this section we study the conditions for having the convergence to the binding regime.

We suppose that the economy is in the non binding regime and we assume that the system will be unconstrained forever, that is T = +∞. Hence the model turns out to be the same as Turnovsky's (1996). Here we extend his analysis in order to show when the corner solution T = +∞ is possible. In Section 2.1 we show that the dynamics of B (t), as given by equation (2.7), depends on the dierence between (r -ρ) and q 1 -1 h , whose sign is determined by q 1 , which is given by (2.5). These expressions represent the growth rates of consumption and capital (output), respectively, in Turnovsky's model. The growth rate of capital (and output) is positive if

q 1 > 1 , which happens when r < (1 -α) A 1 1-α α α 1-α .
We shall concentrate on this case. Proposition 6. Suppose that r -ρ < q 1 -1 h . Then there exists a date T < +∞ at which the economy hits the constraint.

In the case r -ρ < q 1 -1 h , [START_REF] Trimborn | Multidimensional Transitional Dynamics: A Simple Numerical Procedure[END_REF] B (t) is an increasing function of t and lim t→+∞ B (t) = +∞.

Hence in this case B (t) hits the constraint at a nite time.

Otherwise, Proposition 7. When r -ρ ≥ q 1 -1 h , lim t→+∞ B (t) = -∞: the economy will end up in autarky within a nite time.

The case r -ρ ≥ q 1 -1 h is more complex. When this condition is satised, lim t→+∞ B (t) = -∞, but B (t) can be either always decreasing or rst increasing and then decreasing, depending on the values of B(0) K(0)

. At t = 0, B (0) is equal to:

B (0) = θK (0) r -ρ - q 1 -1 h + (r -ρ) B (0) which is negative if B(0) K(0) < -θ 1 -(q 1 -1)/h r-ρ .
The other condition on B(0) K( 0) is given by the collateral constraint:

B(0) K(0) < φA 1 1-α α α 1-α . If φA 1 1-α α α 1-α < -θ 1 -(q 1 -1)/h r-ρ
, then B (t) is always decreasing and never hits the constraint.

If φA

1 1-α α α 1-α ≥ -θ 1 -(q 1 -1)/h r-ρ , then when B(0) K(0) < -θ 1 -(q 1 -1)/h r-ρ , B (t) is decreasing
and never hits the constraint. When B(0)

K(0) ≥ -θ 1 -(q 1 -1)/h r-ρ and B(0) K(0)
is close to the collateral constraint, B (t) is rst increasing then decreasing, hence it may hit the constraint twice. [START_REF] Trimborn | Multidimensional Transitional Dynamics: A Simple Numerical Procedure[END_REF] Since (r -ρ) is the growth rate of consumption in the non binding regime, we set r > ρ. It turns out that r -ρ < q1-1 h is possible only if q 1 > 1.

One might think that when r -ρ ≥ q 1 -1 h the binding regime starts at t = 0. However we have checked numerically that r -ρ ≥ q 1 -1 h is incompatible with the binding regime optimal from t = 0. Hence, the results of Proposition 6 translate into some threshold conditions that the parameters must satisfy for the economy to switch to the binding regime and hence to benet from nancial openness. Notably, this happens if the interest rate and the adjustment costs are small and the productivity is high. In the opposite case, an economy starting in the non binding regime will end up in autarky. A detailed numerical discussion is contained in Section 4.3.

4.1.2

The existence of the non binding regime: the role of investment adjustment costs

As previously shown, the assumption of investment adjustment costs in open economy models is necessary to rule out the unrealistic knife-edge condition on the marginal return of domestic capital. But this assumption has another important implications in our two-regimes model.

In the absence of capital adjustment costs, if the knife-edge condition is not satised, the regime with slack constraint does not exist, because the country would choose to invest or disinvest at an innite rate if the internal productivity is, respectively, greater or smaller than the interest rate (see Cohen and Sachs 1986). Since we assume that the country is a net debtor, the internal return must be greater than the world interest rate, so that the economy switches to the binding regime since the initial date. Hence the presence of adjustment costs of investment enriches the inter-regime transitional dynamics.

The inter-regime transitional dynamics

In this section we simulate the two-regimes model in order to nd the endogenous switching date. The existence of transitional dynamics in the binding regime does not allow us to nd analytical solutions for the date T . Our strategy is to apply the matching conditions of Theorem 1 in order to link two two-points boundary value problems, one for each regime.

Final boundary problems are commonly used for the simulation of the dynamics of growth models. We follow the method proposed by Trimborn et al. (2008a, 2008b).

We start by dening the matching conditions in our case. The conditions (2.15) and

(2.16) of Theorem 1 require the equality of the co-state variables between the two regimes.

Hence at T the following condition must hold:

q (1) (T ) = q (2) (T ) = q (T ) (4.1)
We replace (2.17), which is dicult to implement when a regime cannot be solved in closed form, with two other conditions, which emerge from our model. Given that at T the state variables can not jump, it must be b [START_REF] Ai | A normative analysis of public capital[END_REF] (T ) = b [START_REF] Aizenman | Economic growth with constraints on tax revenues and public debt: implications for scal policy and cross-country dierences[END_REF] (T ) = b (T ). This value can be obtained by the collateral constraint

B K = b = φA G K α = φA 1 1-α G Y α 1-α .
Given that b must be continuous at T , the same applies for the (G/Y ) ratio. In the non binding regime G Y = α, hence when b [START_REF] Ai | A normative analysis of public capital[END_REF] (t) hits the constraint it must be b [START_REF] Ai | A normative analysis of public capital[END_REF]

(T ) = b (2) (T ) = b (T ) = φA 1 1-α α α 1-α (4.2)
As a consequence the value of b (T ) is known.

The last condition stems from equation (2.15), the equality between the shadow values of external debt across regimes, and from the rst order conditions (2.18) and (2.25). These together implies that consumption is continuous at T , so that c [START_REF] Ai | A normative analysis of public capital[END_REF]

(T ) = c (2) (T ) = c (T ) (4.3) Equation (4.
3) also derives from the continuity of G Y at T and from the expression of the growth rate of consumption in the binding regime:

Ċ C = r -1 φ 1 -1 α G Y -ρ.
The adopted algorithm to simulate the two regimes separately is the relaxation algorithm proposed by Trimborn et al. (2008a, 2008b). Our numerical procedure starts backward with the simulation of the binding regimes and then of the non binding one. The goal of this exercise is to nd the switching date which satises the matching conditions (4.1), (

). The method employed to solve the inter-regime transitional dynamics is described in Appendix E.

We assume the same parameters' values of the baseline calibration of Section 4. We set α = 1/2 because in this case we know what conditions the parameters need to satisfy for the BGP to be stable (cf. Proposition The transition inter regime: the debt-to-capital ratio [START_REF] Trimborn | Multidimensional Transitional Dynamics: A Simple Numerical Procedure (Mathematica)[END_REF] The chosen value of initial conditions aects the switching date T : the smaller is b (0), the higher is T . These gures show that the transition occurs in a continuous way: the country starts in the non binding regime and moves to the binding one at T = 2.5. Namely, the endogenous variables do not jump: as stated in Section 2.2, the borrowing constraint is not a pure state constraint.

Note also that the non binding regime, described by system (2.32), does not admit a steady state. As shown in Section 2, the growth rates of the endogenous variables in level are dierent if the constraint does not bind.

During the transition the debt-to-capital ratio follows an increasing path, while the consumption-to-capital ratio and the Tobin's q are decreasing, until the economy attains its BGP. In order to interpret this result, we refer to the economy's current account, as given by equation (2.2). Throughout the transition, the stock of external debt is growing more than the stock of capital. The increasing international capital inows are spent in physical capital, installation costs, public goods and increasing interest payments. This diverts resources from consumption, reducing the consumption-to-capital ratio. The dynamics of the Tobin's q is consistent with that of capital. Once the switch to the binding regime occurs, the stock of debt is constrained, forcing the economy to reach an equilibrium where all the variables in level grow at the same rate. The economy's dynamics is consistent with Turnovsky's (1997) model, where nancial market imperfections are introduced by assuming that the cost of borrowing is an increasing function of debt, relative to capital. In particular, a small open economy with unlimited access to international nancial markets can sustain dierential growth rates of consumption ad output along the BGP, in sharp contrast to a closed economy where the economy's resources constraint all the variables to grow at the same rate. For a net debtor country, consumption grows less than capital (output). When a collateral constraint is added, the growth rates dier along the transition. However, once the steady state is reached, all the variables grow at the same rate.

Comparative statics

In this section we investigate the eects of some relevant parameters on the transition between the non binding and the binding regime. 25 The main results are summarized in the the ndings reported below.

(i) The optimal switching date T is increasing with respect to: nancial openness (φ), the international interest rate (r), the investment adjustment costs (h); it is decreasing with respect to the total factor productivity (A).

An increase of φ relaxes the collaterals, hence the initial value of the debt-to-capital ratio, b (0), will be farther away from the constraint and the economy will spend more time in the non binding regime. A rise of the international interest rate and of the adjustment costs makes the transition slower. The rst result is linked to the amount of interests to be paid: since debt cost is high, the country won't push the stock of debt until its upper bound immediately. Higher installation costs slow down the accumulation rate of external debt (see Section 4.1.2). Finally, for a greater productivity, the economy moves faster to the binding regime. When A becomes very large, the optimal switching date approaches zero. In our model the parameter A plays a double role. A greater TFP rises the economy's productivity making the country richer. This has in turns the eect of increasing the collaterals, allowing the country to borrow more from the rest of the world. Hence the economy will spend less time in the non binding regime, reaching faster the advantages of nancial globalization.

(ii) If the interest rate, the installation costs are too high and/or the total factor productivity is too low, the economy will stay forever in the non binding regime and the debt-to-capital ratio b (t) will tend to zero (autarky).

If the economy has to actually switch to the binding regime, the convergence conditions of Proposition 6 must be satised, that is: r -ρ < (q 1 -1) /h, where q 1 is given by (2.5). This translates into some threshold conditions on the relevant parameters that the economy has to satisfy in order to hit the constraint and to fully benet from nancial liberalization. [START_REF] Turnovsky | Fiscal Policy, Growth, and Macroeconomic Performance in a Small Open Economy[END_REF] The assumed parametrization is the baseline calibration of Section 4.2.

The numerical simulations show that when r and h are greater than their threshold and A is smaller than its threshold, the BGP value of (G/Y ) is smaller than or equal to α, thus implying that the binding regime is never optimal. Remember that the Kuhn-Tucker slackness condition (3.4) shows that the BGP value of (G/Y ) must be grater than α in the binding regime (see Proposition 4). The stock of debt will become zero at a certain point in time, and the economy will stay in autarky.

The above results can be linked to the recent empirical literature on nancial integration.

There is no consensus on the eects of nancial globalization on economic growth: the benets depend on the countries' situation. In particular, Kose et al. (2011) show that there exists a certain threshold of nancial and institutional development and economic features that an economy must attain in order to benet from nancial liberalization. Similarly, Aizenman et al. (2011) show that economy's GDP growth is lower in countries with high levels of past debt and weak institutions. This is consistent with our model's prediction: a country with a low level of productivity, with high adjustment costs of capital or facing a high interest rate will not benet from nancial liberalization, staying in autarky forever. In order to enjoy the advantages of international borrowing, the economy must reach a certain level of nancial and institutional development coupled with other favorable economic features. Moreover a high interest rate can be interpreted by the investors as an index of high risk and low institutions development, so that the country will be excluded from the main international capital ows. To conclude, for countries which are not developed enough it would be optimal to do not increase debt until its upper bound and rather stay in autarky.

Conclusions

This study analyzes the consequences of external debt collaterals on the optimal growth path of a country. We depart from the existing growth literature with collateral constraints which assumes that the collaterals are capital stock. We extended Barro (1990)- Turnovsky's (1996) small open economy model by assuming that public expenditures can be nanced by borrowing on imperfect international nancial markets, where external debt is bounded by GDP.

The collateral constraint denes three possible states of the economy: autarky, non binding regime and binding regime, where a balanced growth path exist. We determine the conditions that the economy must satisfy in order to switch from the non-binding to the binding regime, which also dene the corner solutions of the problem. First, in the absence of investment adjustment costs, the economy switches to the binding regime at the initial date.

Second, if installation costs are positive, for the economy to switch to the binding regime the TFP should be suciently high and/or the adjustment costs and the world interest rate should be low enough.

We then compare our results with those of the existing endogenous growth literature.

The optimal public expenditure-to-GDP ratio is greater than that of Barro-Turnovsky's model: the planner has incentive to nance public expenditures by debt because this relaxes the collateral constraint.

Reorganizing the terms

∂V 1 = ´T 0 ∂H (1)
∂B + λ(1)

1 ∂B + ∂H (1) ∂K + λ(1) 2 ∂K + ∂H (1) ∂C ∂C + ∂H (1) ∂G ∂G + ∂H (1) ∂I ∂I dt + H (1) (T ) + ∂V * 2 (T,B(T ),K(T )) ∂T ∂T + ∂V * 2 (T,B(T ),K(T )) ∂B(T ) -λ (1) 1 (T ) ∂B (T ) + ∂V * 2 (T,B(T ),K(T )) ∂K(T ) -λ (1)

(T ) ∂K (T )

A trajectory is (locally) optimal if any (local) departure from it decreases the value function:

∂V 1 ≤ 0 for any ∂B (t) and ∂K (t), t ∈ (0, T ), for any ∂C (t) and ∂G (t), t ∈ [0, T ], and for any ∂T , ∂B (T ) and ∂K (T ). This gives the following necessary conditions for an interior maximizer:

                             ∂H (1)
∂C = 0

∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂G = 0

∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂I = 0

∂H (1) ∂B = - λ(1) 1 ∂H (1) ∂K = - λ(1) 2 H (1) (T ) + ∂V * 2 (T,B(T ),K(T )) ∂T = 0 ∂V * 2 (T,B(T ),K(T )) ∂B(T ) -λ (1) 1
(T ) = 0 ∂V * 2 (T,B(T ),K(T )) ∂K(T ) -λ (1)

(T ) = 0

The rst ve equations are the standard Pontryagin conditions, the last three may be interpreted as the optimality conditions with respect to the switching time T and the free state values B (T ) and K (T ).

f G Y ≡ h 2 1 φ 1 α 2 G Y 2 +[h (rφ -1) + φ] 1 φ 1 α G Y + 1 φ h 2 (rφ -1) 2 + φ (rφ -1) -h 2 (ρφ) 2 and g G Y ≡ 1-α α A 1 1-α φ G Y 1 1-α . f G Y is a convex parabola, while g G Y is a convex increasing function such that g (0) = 0 and lim G Y →+∞ g G Y = +∞. The assumption α < 1 2 guarantees that lim G Y →+∞ f ( G Y ) g( G Y ) = 0.
2 1 φ 1 α 2 [h (rφ -1) + φτ ] 1 φ 1 α 1 φτ h 2 (rφ -1) 2 +φ (rφ -1) -h 2 (ρφ) 2 f G Y = 0 1 + + +
1. D ≥ 0 if h ≤ φ 1-rφ and E > 0 if h > 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ . Since φ 1-rφ < 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ the two conditions are incompatible. 2. D ≥ 0 if h ≤ φ 1-rφ , E ≤ 0 if h ≤ 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ
. Hence to have case 2 we must

require h ≤ φ 1-rφ . 3. D < 0 if h > φ 1-rφ , E ≤ 0 if h ≤ 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ . Hence φ 1-rφ < h ≤ 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ . 4. D < 0 if h > φ 1-rφ , E > 0 if h > 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ . This implies h > 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ
.

Cases 2 and 3 can be summarized by requiring h ≤

2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ .
Hence, the dynamical system admits

(a) One solution i h ≤ 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ ; (b) Two solutions i h > 2φ(1-rφ) (r 2 -ρ 2 )(φ) 2 +1-2rφ
.

Both the solutions satisfy the transversality condition:

lim t→+∞ -q * λ (2) 1 (t) K (t) =-q * λ (2) 1 (T ) exp [-ρt]
, where q * is the BGP value of the Tobin's q.

We can prove that case (b) implies just a false multiplicity.

A solution of equation (3.1) is a balanced growth path if the growth rate (2.23) is positive,

that is rφ -1 + 1 α G Y ≥ ρφ
This allows to write the following inequality

≤ h 2 1 φ 1 α 2 G Y 2 + h (rφ -1) 1 φ 1 α G Y + 1 φ h 2 (rφ -1) 2 -h 2 (ρφ) 2 + ρφ h 2 1 φ 1 α 2 G Y 2 + [h (rφ -1) + φ] 1 φ 1 α G Y + 1 φ h 2 (rφ -1) 2 + φ (rφ -1) -h 2 (ρφ) 2 = 1-α α A 1 1-α φ G Y 1

C

Proof of Proposition 3

The dynamical system is

           ċ = r -1 φ + 2 1 φ 2 A -2 b -ρ -q-1 h c q = r -1 φ q + 2 1 φ 2 A -2 bq -1 φ 2 A -2 b 2 -(q-1) 2 2h ḃ = c + q-1 h + 1 φ 2
A -2 b 2 + rb + (q-1) 2 2h -1 φ b -q-1 h b and the Jacobian matrix, J 1 , is given by [START_REF] Turnovsky | Equilibrium Growth in a Small Economy Facing an Imperfect World Capital Market[END_REF] Numerical analysis has shown that the result is valid for values of A commonly used in the simulations.

      r -1 φ + 2 1 φ 2 A -2 b 1 -ρ -q 1 -1 h -1 h c 1 2 1 φ 2 A -2 c 1 0 2 1 φ 2 A -2 b 1 + r -1 φ -q 1 -1 h 2 1 φ 2 A -2 q 1 -2 1 φ 2 A -2 b 1 1 1 h + q 1 -1 h -1 h b 1 2 1 φ 2 A -2 b 1 + r -1 φ -q 1 -1 h       where b 1 = φA 2
-[h (rφ -1) + φ] 1 φ + 1 2 (2 + 2h 2 ρ 2 + A 2 (h -2(1 + hr)φ + (r(2 + hr) -hρ 2 )φ 2 ))

2h φ -A 2 φ q 1 = h r - 1 φ + 2 1 φ 2 A -2 b 2 1 -ρ + 1 c 1 = q 1 -1 h b 1 + 1 φ b 1 - q 1 -1 h - 1 φ 2 A -2 b 2 1 -rb 1 - (q 1 -1) 2 2h det(J 1 ) = - 1 √ 2A 2 h(φ) 2 ρ 2 + 2h 2 ρ 2 + A 2 h -2 (1 + rh) φ + (r (2 + hr) -hρ 2 ) (φ) 2 -2hρ + √ 2 2 + 2h 2 ρ 2 + A 2 h -2 (1 + rh) φ + (r (2 + hr) -hρ 2 ) (φ) 2 tr(J 1 ) = 2ρ
The trace is positive.

The determinant is negative if A 2 > -2 h-2(1+rh)φ+(r(2+hr)-hρ 2 )φ 2 . The negative determinant implies that the product of the three eigenvalues is negative, implying that the number of negative eigenvalues is either one or three. However, the positive trace, being the sum of the three eigenvalues, rules out the latter case, thus implying one negative eigenvalue. Hence, the steady state is saddle-point stable.

D Proof of Proposition 4

In order to prove this proposition note that an increase in r moves the function g G Y of

E

The inter-regime transitional dynamics: methodology

In this section we illustrate the method which has been employed to simulate the interregime transitional dynamics. As we anticipated in Section 4.2, the aim of our numerical exercise is to link two two-points boundary problems (see Trimborn et al. 2008a, 2008b), in order to nd the switching date T .

The simulation proceeds as follows :

Step_1 Guess a nal date T

Step_2 Run the binding regime. Set:

1. The initial conditions. In our model they are given by the value of b (T ) given by condition (4.2).

2. The boundary conditions. The transversality conditions are replaced by the steady state values of the variables.

3. Run the simulations and write down the values of c (T ) and q (T ). 27

Step_3 Run the non binding regime. Set:

1. The initial conditions. In our case, they are represented by the initial value of the state variable, b (0).

2. The boundary conditions, that is the values of c (T ) and q (T ) found in Step_2, as well as the value of b (T ) given by condition (4.2).

3. Run the simulations. If the variables c (T ), q (T ) and b (T ) of the non binding regime converge to the nal conditions, as set in 2), the algorithm stops. Otherwise, it comes back to Step_1.

27 Condition (4.2) shows also that the values of the variables at T do not depend on the choice of T . This choice will only aect the duration of the transition to the steady state.

  model, 4 and subsequently discuss the case of GDP-based collateral constraints. The production function is the same as in Barro's (1990): government's purchases of goods and services enter the production function as pure public goods. The technology is Cobb-Douglas:

  and Boucekkine et al. (2004) index the successive regimes by a nite number of discrete parameter values: a newly available technological regime may exhibit a higher productivity parameter. A more recent branch of the literature applies the optimal switching problem to the analysis of irreversibilities (see Boucekkine et al. 2013a, Boucekkine et al. 2013b, Boucekkine et al. 2013c, Prieur et al. 2011): for example when the state variable (the stock of pollution) exceeds a certain threshold value, the natural decay rate goes down permanently. One can reformulate this as an optimal timing problem because the date at which the stock variable reaches the threshold becomes a control variables.

  3) and b (0) = 0.0145. 24 This parametrization satises the convergence conditions stated in Proposition 6. The gures below (Figure 4.1, Figure 4.2 and Figure 4.3) show the transition of the economy from the non binding to the binding regime. As indicated by the red dotted-line, the switching date is equal to 2.5 periods.

b

  Figure 4.1: The transition inter regime: the debt-to-capital ratio

Figure 4 . 2 :Figure 4 . 3 :

 4243 Figure 4.2: The transition inter regime: the consumption-to-capital ratio

  Combining these last three equations with equations (A.1), (A.2) and (A.3) obtained in the rst sub-problem we get the matching conditions stated in

  We study the intersections of f G Y with the positive G Y -axis because this helps in determining the number of intersections between g G Y and f G Y (the assumption α < 1 2 limits the number of possible intersections to two). The discriminant of the second degree equation dened by f G Y = 0 is positive, hence it can have either two negative solutions, one positive and one negative solution or two positive solutions:

  h

YFigure B. 1 :

 1 Figure B.1: Existence and Uniqueness of the BGP

Figure B. 1

 1 Figure B.1 up in the graph, so that (G/Y ) 2 decreases while an increase in A moves the f G Y curve of Figure B.1 up in the graph, so that (G/Y ) 2 increases. Given that (2.23) is increasing in (G/Y ), the growth rate increases as well.

analyzes the balanced growth path. Section

discusses the inter-regime transition and the determination of the optimal switching date. Finally, Section

concludes. The Appendix contains technical details.[START_REF] Aizenman | What is the risk of European sovereign debt defaults? Fiscal space, CDS spreads and market pricing of risk[END_REF] Aizenman et al. (2007) build a similar model to ours: countries are characterized by limited tax and debt capacity. However they do not tackle the problem of collaterals and imperfect nancial markets. Furthermore, the results and the technical analysis are dierent. First, the economy is closed. Second, they do not take into account any problem of endogenous switching.

We assume logarithmic utility throughout the paper in order to make the model tractable.

We will maintain this assumption throughout the paper, but the results do not change if the depreciation rate is taken positive.

[START_REF] Turnovsky | Fiscal Policy, Growth, and Macroeconomic Performance in a Small Open Economy[END_REF] assumes that the country may accumulate foreign bonds. This dierence does not aect the properties of the model.

λ 1 and λ 2 are the two co-state variables of the optimization problem attached to debt (negative) and capital (positive), respectively.

Of course the problem can be solved by using the traditional method of Lagrange, but this does not allow to determine the endogenous switching time in our case.

Acknowledgments I wish to thank E. Marchetti, C. Garcia-Peñalosa, M. Giannini, F. Mariani, R. Becker, P.

Villieu, M. Tidball, T. Trimborn, R. Dos Santos Ferreira, A. Venditti, and the participants in seminars and conferences at Université catholique de Louvain, GREQAM Marseilles, Université d'Orléans. Finally, I am grateful to the anonymous referees for their suggestions, which have signicantly improved the paper. The usual disclaimer applies.

Appendix

A

Proof of Theorem 1

In this section we prove Theorem 1. We make use of the calculus of variations techniques applied to the sequence of the two control sub-problems shown in the main text.

Second Regime. The problem is (2.11) and the associated hamiltonian is (2.12). We denote by the upper-script * the paths which satisfy the rst order conditions. The corresponding value function depends on T , B (T ) and K (T ):

since the problem is concave, the following envelope conditions apply

2 (T )

First regime. The corresponding control sub-problem is (2.13) and the corresponding hamiltonian is (2.14).

The value function can be rewritten in terms of the hamiltonian

Standard integrations by parts yield ˆT 0 λ

(1)

which allows to rewrite the value function as

The rst order variation of V 1 with respect to the state and the control variables' paths yields ∂V 1 = ´T 0 ∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂B ∂B + ∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂K ∂K + ∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂C ∂C + ∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂G ∂G + ∂H [START_REF] Ai | A normative analysis of public capital[END_REF] ∂I ∂I + λ(1)


Pierre Andreoletti
email: pierre.andreoletti@univ-orleans.fr

Pierre Gazé
email: pierre.gaze@univ-orleans.fr

Maxime Menuet
email: maxime.menuet@univ-orleans.fr

« Can a Platform Make Profit with Consumers' Mobility? A Two-Sided Monopoly Model with Random Endogenous Side-Switching »

Keywords: JEL Codes: L11, L13, L86 externalities, side-switching, two-sided markets externalités, mobilité inter-faces, marchés à deux faces

Can a platform make profit with consumers' mobility? A two-sided monopoly model with random endogenous side-switching ✩ Une plateforme peut-elle capter la mobilité des consommateurs en rente ?

Un modèle de marché à deux faces en monopole avec une mobilité inter-faces endogène et aléatoire.

Introduction

In two-sided markets (TSM), two sets of agents interact on a firm or an organization called a platform. Each group of agents generates indirect network externalities that positively (with some exceptions) affect the utility of the agents of the other group. A lot of activities are characterized by these specific cross externalities. For example, malls must attract buyers and sellers, newspapers sell its services to readers and advertisers, card payment systems must develop the merchants' acceptance in order to attract new cards holders etc.

An abundant literature has produced a fruitful analysis in several directions. [START_REF] Armstrong | Competition in two-sided markets[END_REF] For example, pricing within the platform (Hagiu 2006; Schmalensee 2002), competition between platforms (Armstrong 2006; Caillaud and Jullien 2003; Chakravorti and Roson 2006) and antitrust issues (Evans 2003; Rysman 2007; Evans and Schmalensee 2013) are important topics of TSM economics. We would like to contribute to the pricing analysis for a specific class of TSM characterised by agents' migration from one group to another one. We think that side-switching in TSM is a fruitful topic that deserves further theoretical investigation. Gazé and Vaubourg (2011) have identified this particular type of TSM. Indeed, there exist numerous examples of such TSM. In the auction website eBay, in all payment systems, in electronic marketplaces (like "Amazon Marketplace") people can achieve a transaction in one group then make another transaction in the other group. In a first time period, an agent can act as a buyer and in a second time period he can move to the other group and become e.g. a seller. These specific class of TSM raises new interesting economic questions. Do the side-switching behavior has consequences on the firm's strategy ? What impact has mobility on prices and profits?

One potential consequence of mobility is the raise of the platform's transactions number. If agents can move quickly and easily from a group to another group (e.g. with two mouse clicks), the platform can mechanically develop the market demand. For example, in the electronic market E-bay, a consumer can discover the platform from the buyer side and then have the idea and the possibility of selling second-hand objects on the sellers' side. Then, we conceive easily that mobility encourages a growth in market demand and potentially an increase of business for the firm. Actually, we are looking for more subtle mechanisms in order to have a better understanding of the "economics of mobility". We want to show that side-switching can affect prices and profit even without market growth.

In the case of a duopoly market Gazé and Vaubourg (op. cit.) demonstrate that the agent's mobility affects the equilibrium of the firms. They show that under a specific set of constraints, the firms' profit increase with the rate of the mobility of agents. The mechanism that generates profit is explained by changes in the set of prices that reflect a complex game of rewards and penalties. Firms obtain a profit from the heterogeneity of platform's consumers and from their mobility inside the TSM. Nevertheless, this analysis has a number of limitations. One of these limitations is the exogenous rate of changing face. In this paper we would like to remove this restriction; we will show that the platform considers the mobility rates as strategic variables.

We conceive a model to highlight the effects of mobility on prices and profit of the firm which produce TSM services. We suppose that the firm is a monopoly that faces two groups of agents who can switch to the other group when their migration enhances their satisfaction. So their probabilities of departure from a group to the other are totally endogenous to the pricing of the monopoly. The monopoly has a direct impact on the size of the two groups when it modifies its prices. We show that the firm encourages agents' mobility in order to raise its profit. Therefore, in our model, the mobility corresponds to the fluctuation of the sizes of the two sides, thus the side-switching allows the firm to adjust the size of both groups to an optimal level. Then, this framework suggests that the monopoly has the possibility of enhancing the quality of the platform by stimulating the side-switching. The rest of the paper is organized as follows. In Section 2, we describe the important assumptions of our model concerning consumers, firm and the timing of action. Equilibrium of period one and period two are calculated in Section 3. In Section 4, we discuss our theoretical results.

A platform that faces mobile consumers

The consumers

The firm sells its service to two groups of agents denoted a and b. We call "consumers" agents of both sides since they consume the output of the monopoly. So this denomination can be used to describe e.g. the buyers side as well as the merchants side. Conforming to a TSM framework, the participation of one group raises the value of participating for the other group. Consumers interact through the platform and their utility is positively affected by cross network externalities.

A consumer that joins face j, j ∈ {a, b}, yields at period i, i ∈ {1, 2}, obtains the following level of utility:

u a i := α a N b i -p a i , u b i := α b N a i -p b i . (1)
We denote α j as the cross network externality parameter and N j i as the number of face j's consumers at period i. The parameter α a (resp. α b ) measures the benefit a group-a (resp. group-b) agent enjoys from interacting with each group-b (resp. group-a) agent. Finally, p j i is the price charged to the consumers of group j during the period i. We assume that consumers must buy one platform's service during each period of time. The market demand is the same in period one and two but the respective size of groups can vary from one session to the other. In addition, the total number of consumers N > 0 is given by:

N := D a (u a 1 ) + D b (u b 1 ), (2)
where D j (•) are demand functions which will be defined in the Subsection 3.1. We do not focus our analysis on the total size of the market but on the demand repartition on the two faces. We assume that there is no market growth from period 1 to period 2. This hypothesis means that firm is not able to earn more profit by attracting new customers on the market during the period 2. This is not realistic but it is useful to analyse and isolate the role played by mobility. In formal word, we suppose that in period 2 utility is big enough so that consumers prefer join the platform instead of staying out 3 of the market (the market is fully served). Finally, participation's conditions are given by net consumer's expected utilities must be positive or null, i.e. E(u j i ) ≥ 0, i ∈ {1, 2}, j ∈ {a, b}, where E(•) is the expectation.

The platform

We focus our analysis on the most simple market structure since we assume that only one firm faces all the consumers. We choose the monopoly paradigm in order to highlight the effects of mobility on the economic link between the firm and the consumers. Obviously, in the monopoly market structure no competitive effects can alter the repercussions of side-switching on the firm's equilibrium. Our model allows us to consider the consumers probability of side-switching as an indirect strategic variable at the firm's disposal. We will show that through an appropriate set of prices, the firm can impact the repartition of the demand into the two groups and in fine make additional profits. For simplicity, we assume all production costs equal to zero. Finally, as usual, the constraint of the platform's budget is given by: p a i + p b i ≥ 0, i ∈ {1, 2}.

Timing of actions

We consider a two-stage game. Following Armstrong (2006), we suppose that the size of each face is directly determined by a function of the consumer's utilities. The platform set two prices in the first period that maximized its profit; one price for the face a and one price for the face b. Once the transactions are done, consumers and firm are engaged in the second period. Consumers move to the other face if they want, i.e. if their move increases their utility. In the same time, firm set the two second period prices. The two-session game is then finished. Note that pairs of profit-maximizing prices determine the relative size of each platform's face. In other words, the probability of switching from a face to the other is totally endogenous. In addition, consumers are homo economicus: they have no ex ante preference for being a member of one group instead of belonging to the other group. They choose the side they want to become a member by taking into account prices and externalities strength (i.e. the size of the other side and the level of the externality parameters).

Analysis and equilibrium

First period

In the first period, as Armstrong (2006), the number of consumers buying the service on each side, denoted N a 1 and N b 1 , and are given by the demand system:

N a 1 := D a (u a 1 ) = D a (α a N b 1 -p a 1 ), N b 1 := D b (u b 1 ) = D b (α b N a 1 -p b 1 ). ( 3
)
We assume that the functions D j (•), j ∈ {a, b}, are increasing and differentiable on a concave support. The demand on each side at monopoly prices is assumed to be positive.

Then, the plarform's program is to maximize the profit of first period Π

1 := p a 1 N a 1 +p b 1 N b 1 . Thus, the profit-maximizing prices (p a * 1 , p b * 1 ) satisfy: 2 p a * 1 = D a D ′ a -α b N b 1 and, p b * 1 = D b D ′ b -α a N a 1 , (4)
where D ′ j is the slope of the demand function at fixed participation of the other side. Replacing in (3) yields the optimal demands N a * 1 (resp group a, α a N a 1 ). The total number of consumers is then defined by

1 = D a (α a N b * 1 -p a * 1 ) and N b * 1 = D b (α b N a * 1 -p b * 1 ). Notice that p a * 1 (resp. p b * 1 ) is
N := N a * 1 + N b * 1 .
As usual, the first period allows to endogenous the numbers of the consumers in both sides. [START_REF] Caillaud | Chicken and egg: Competition among intermediation services providers[END_REF] In first period, the consumers enter or go out on a side of the market (a, or b) according to their level of utilities (D a (u a 1 ), or D b (u b 1 )). In second period, the market is closed (no entry and exit), but the consumers who entered in first period can now change side. For example, a consumer entered in first period on the side a, as buyer, can switch on the side b, as seller.

Second period

Our analysis of mobility is organized into three stages. First, we characterize equilibriums in absence of mobility, where consumers stay in their origin groups. Secondly, we determine equilibriums with stochastic mobility, where this mobility will be endogenous to expected consumers' utilities. Finally, we analyze differences between these two equilibriums and the consequences of the mobility for the firm.

Absence of mobility

In absence of mobility, the platform's program is to maximize the profit of second period

Π 2 := p a 1 N a 2 + p b 2 N b 2 .
By definition, the consumers stay in their first period group, so we can write: N j 2 = N j * 1 , j ∈ {a, b}, where N j * 1 are optimal demand functions defined in first period by equations ( 3) and (4). In addition, consumers' participation constraints are given by: u

a 2 = α a N b * 1 -p a 2 ≥ 0, and u b 2 = α b N a * 1 -p b 2 ≥ 0. Thus, the prices (p i * 2 , i = a, b) maximizing the profit are: p a * 2 = α a N b * 1 , and p b * 2 = α b N a *
1 , which are quite intuitive since the best the firm is to extract all the consumer surplus. Finally, the optimal profit on absence of mobility is simply

Π * 2 = tN a * 1 N b * 1 , (5)
where t := α a + α b measures the totality of external effect. Despite the simplicity of this results, the case without mobility allows to establish a brenchmark on the level of the profit. Our analysis is then to compare the profit in the case with stochastic mobility with the profit (5) without mobility.

Stochastic mobility

We defined the isidiosyncratic random variable, denoted by X n , which represent the membership group of the consumer n in period 2,

n ∈ {1, • • • , N }, where N = N a * 1 + N b * 1
is given by the first period. In other word, if the consumer n is a member of the site j in period 2, then X n = j, j ∈ {a, b}. In addition, we assume that these consumers make their decision independently, so that N a 2 (respectively N b 2 ) the number of agents in face a (resp. b) at the second period is a sum of N independent and identically distributed random variables given by:

N a 2 := N n=1 1 {Xn=a} , (6)
(resp. N b 2 := N -N a 2 )
, where 1 is the indicator function. [START_REF] Chakravorti | Platforms competition in two-sided markets: The case of payment networks[END_REF] We denote f a := P(X 1 = a) and f b := 1 -f a = P(X 1 = b), where f j represents the proportion of consumer in face j at period 2. Otherwise, there is an equivalent interpretation where f j represents the probability for a consumer to be in face j at the end of the second period. Thus, we determines the choice of belonging to each consumer probabilistically. The membership group of every consumers is not known in a certain way, but we know that they have a probability f a (resp. f b ) to belong in the side a (resp. b) at the end of the second period.

Obviously the mean of N a 2 , denoted E(N a 2 ), is given by: E(N a 2 ) = f a N , and similarly

E(N b 2 ) = N -f a N = f b N .
However, the mobility is a dynamic process, and assume that there is a price couple which ensures a stable equilibrium at the end of the second period. Condition i. guarantees an optimal equilibrium for consumers, and condition ii. ensures an optimal equilibrium for the platform. Thus, when i. and ii. are satisfied, the mobility stops, because neither the platform, nor consumers can hope for a better gain.

It is what we call stable equilibrium. The Definition 1 is constructive and implies the following Proposition. In a similar way we prove that u a 2 < u b 2 is impossible.

Proposition 1 implies that at equilibrium both faces bring the same utility to consumers. Indeed, for a given set of prices, external effects make converge the levels of utilities. We are now able to obtain explicit expressions for the proportions (f j , j = a, b).

Corollary 1. Let the prices (p a 2 , p b 2 ) assure one equilibrium at second period. Thus we have,

f a = p b 2 -p a 2 N t + α a t , and
f b = p a 2 -p b 2 N t + α b t , (7)
with the conditions 0 ≤ f a ≤ 1 and 0 ≤ f b ≤ 1.

Proof: See appendix.

We can see that (f j , j = a, b) are endogenous to prices and external effects. Also we notice that the proportion of agents migrating in face j increases with the utility of group j. [START_REF] Evans | The Antitrust Analysis of Multi-Sided Platform Businesses[END_REF] The intuition for this is that when u a 2 rises (with a p a 2 decrease or α a increase), ceteris paribus, the expected utility for the group a reduces, therefore, consumers are less incited to move in group a, thus f a decreases. In addition, the condition on the Corollary 1 easily implies:

0 ≤ f j ≤ 1, j ∈ {a, b} ⇔ -N α b ≤ p a 2 -p b 2 ≤ N α a . (8)
The computations of endogenous probabilities show that consumers go to the other face considering price's faces differential and the relative strength of network externalities. Not surprisingly, we see that a consumer is more likely to move to the cheapest face and also to the group that confers the highest externality.

In our model, the platform's profit at the second period is the random variable:

Π2 = p a 2 N a 2 + p b 2 N b 2
, where N j 2 are determined by (6). Moreover, the platform's program is to maximize the mean of Π2 ,6 denoted by Π 2 , which is given by

Π 2 (p a 2 , p b 2 ) := E[ Π2 (p a 2 , p b 2 )] = p a 2 N f a + p b 2 N f b . Namely, by (7), Π 2 (p a 2 , p b 2 ) = N (p a 2 α a + p b 2 α b ) t - (p a 2 -p b 2 ) 2 t . (9)
We now resume the constraints for the second period: the first is the condition (8), the second are consumers' participation constraints:

E(u j 2 ) ≥ 0, j ∈ {a, b} ⇔ p a 2 α b + p b 2 α a ≤ α a α b N.
And the constraint of the platform's budget is simply p 2 a + p 2 b ≥ 0. Thus, the space of prices at second period, denoted by C 2 , is

C 2 := {(p a 2 , p b 2 ) ∈ R 2 |p a 2 α b + p b 2 α a ≤ α a α b N ; -N α b ≤ p a 2 -p b 2 ≤ N α a ; p a 2 + p b 2 ≥ 0}. ( 10
)
As a consequence, the optimal prices (p a * 2 ,p b * 2 ) at equilibrium should satisfy

{p 2 * a , p 2 * b } := argmax (x,y)∈C2
Π 2 (x, y).

Theorem 1. (prices at equilibrium) The optimal prices (p a * 2 , p b * 2 ) are given by:

p a * 2 = α a N a * 1 + N b * 1 2 and, p b * 2 = α b N a * 1 + N b * 1 2 .
As a consequence the optimal proportion are given by f * a = f * b = 1 2 , and the mean of the profit at equilibrium by:

Π * 2 := E( Π * 2 ) = (N a * 1 + N b * 1 ) 2 t 4 . (11)
Proof: See appendix.

In the equilibrium, the price p a * 2 (resp. p a * 2 ) is equal to the mean of the consumers' number ((N a * 1 + N b * 1 )/2) adjusted by the external parameter α a (resp. α b ), which represent the benefit a group-a (resp. group-b) agent enjoys from interacting with each group-b (resp. group-a) agent. With this prices, each consumer will choose either side with equal probability (1/2), and the mean of the consumers' number in second period is the same in both sides:

E(N a * 2 ) = E(N b * 2 ) = N f * a = N/2.
We note that in the case without mobility (in the Subsection 3.2.1), if the number of consumers in each side was a strategic variable for the platform, the optimal profit Π * 2 = tN a * 2 N b * 2 would be maximized when N a * 2 = N b * 2 = N/2, which is the case at equilibrium with mobility. Thus, the case with mobility enables the firm to change the number of consumers to maximize the demand and the profit, but this case is risky for the platform. Therefore, we easily obtain the variance of Π * 2 : 7

Var( Π * 2 ) = N 4 (p a * 2 -p b * 2 ) 2 = N 3 16 (α a -α b ) 2 .
It is interesting to remark that mobility introduces uncertainty both for firm and consumers. The mobility gives birth to a specific kind of risk: the possibility that the group which gives utility (or profit) has a decreasing size in the future. For the monopoly, the variance of expected profit is increasing with the differential of externality parameters (α a and α b ). The larger is the difference between α a and α b the larger is the gap between expected prices p 2 * a and p 2 * b . For this reason, the firm could realize a smaller turnover if consumers go massively to the cheapest face. For consumers, the expected utility depends on the size of the other group. The higher is the consumers' mobility the higher is the variance of the size of the groups.

Thanks to Theorem 1 we can easily determine the impact of mobility on the platform's income. We call S the extra profit that the firm can earn from the consumer's mobility: [START_REF] Gazé | Electronic platforms and two-sided markets: A side-switching analysis[END_REF] Indeed the profit can be written:

Π2 = p b 2 N + (p a 2 -p b 2 )N a 2 . Thus, Var( Π2 ) = (p a 2 -p b 2 ) 2 Var(N a 2 ) = (p a 2 -p b 2 ) 2 fa(1 -fa).
Corollary 2.

S := E( Π * 2 -Π * 2 ) = t 4 (N a * 1 -N b * 1 ) 2
Proof: See appendix.

There are two noteworthy implications of this extra profit. First, we notice that the larger is the spread between the initial sizes of groups, the greater is the extra profit. The intuition is that in this model the mobility corresponds to the fluctuation of the sizes of the two sides by the platform. Indeed, in the first period, this sizes of the two groups are optimal according to the consumers' participations, i.e. the demand functions D a and D b given by (3). Therefore, the first period à la Armstrong (2006) determine the total level of demand:

N = N a * 1 + N b * 1 .
During the second period, as the market is closed, the optimal average sizes of two sides are determined to maximize this demand's repartition (EN a 2 = EN b 2 = N/2). Thus, the optimal profit with mobility is simply the maximum of the profit without mobility when the number of consumers in both sides are adjusted by the platform. In this way, the mobility which is actually the possibility for consumers to switch, allows the firm to set prices that maximize the level of demand, and determine the optimal average size of groups. Finally, the larger is the gap between the two populations of consumers in period one, the larger is the extra profit S, since the optimal size is the same for both groups. Second and not surprisingly implication, we notice that the profit is increasing with the sum of external effects t, with t := α a + α b .

Discussion

If mobility is not possible, standard TSM economics show that firm adjusts prices to the level of externality that a group of consumers generates. The central idea is that the price is adjusted by the external benefit that a group brings to the other group (the more a consumer brings value to the other group the smaller is the price). We can see this specific way of setting prices as "a reward pricing". If mobility is possible, modification in size group is a mechanism that raises profit in itself. The firm receives incitation to set prices that homogenizes the number of consumers in each group. At the end of this process (that is to say in period 2), there is no "reward pricing" anymore. Therefore, the mobility allows the firm to adjust the size of both groups to an optimal level that it yields a extra profit. But, we can wonder why the repartition of consumers in each group wouldn't be optimal during the first time period?

In several TSM, consumers discover the platform by one side without thinking about the benefit they can obtain by belonging to the other side. They progressively acquire information that lets them see the utility they can expect by switching to the other side. E.g a BlaBlaCar consumer can discover the site one day when he needs to travel. After experimenting rides as a passenger, he can become a driver who offers rides. Thanks to his experience, this consumer may maximize his satisfaction by belonging to the driver's side that is not the side that attracted him first to the site. These mechanisms can explain why consumers can choose the suboptimal side of the platform when they enter into the market. Actually, mobility gives the consumers an opportunity to enhance their satisfaction. This is the reason why it is profitable for the firm.

In addition, can we find in the industry some illustrations of these special relationships between the firm and their consumers? Do the firms really encourage mobility?

The booking site Airbnb is a good example of a TSM with mobility. Airbnb's simplicity has made it easy for people to find accommodation in private residences and to book them easily. Consumers can discover the site during a trip by the renter side and then, once back home, become an offeror of a room in his flat, i.e. make a side-switching. The same mechanism prevails for the electronic payment system Paypal. Paypal users can receive or sent funds (i.e. make a side-switching) very easily, with very low transaction costs and almost no constraints. That is true as well for the emblematic electronic platform Ebay which easily permits its consumers to become sellers through the Amazon Marketplace program. We think that common features of Ebay, PayPal and Airbnb (and other major electronic platforms) imply similarity in business management for these successful firms. Mobility is quality; it is the reason why these firms develop advertising programs, discount prices or didactic tutorials to give incentives to consumers to make some business into the other side of their favorite platform. We think that our model is consistent with these observations. Consumers' side-switching is a natural way to develop the business but it is also an attractive method of consumers' surplus extraction.

Conclusion

The possibility of changing from a face to the other appears to consumers like an improvement of product's quality. Since the platform is a monopoly, it can capture the increase of the consumer's utility and make additional profit compared to the first period of time. In this model, mobility is a valuable feature for consumers which enhance quality's product. Actually, this is a strong economic explanation for the profitability of the mobility. Finally our paper shows that, in a specific context, mobility is a strategical variable in the TSM industry.

According to our results, it would be interesting to study the TSM in a dynamic model. For example, we would define an intertemporal "cyclic" model with two phases. During phase one, we can imagine a firm that encourage the dissymmetry between the two sides. For example, the platform could attract many consumers to one side in the case of a new market. In the case of an existing market, the platform would encourage the whole population to move (or stay) to a certain side of its business. Then in the second phase, and according to our model's results, we could imagine a firm which takes advantage of the initial gap between the two groups. With such a dynamic model, conditions under which the alternation of phases is profitable for the firm should be obtained. The existence of one TSM implies that λ 3 = λ 4 = 0. Indeed, if it is not the case, the totality of the consumers may move to the same face, and the TSM disappear. So we now deal with the case λ 2 = 0 and then λ 1 = 0. First, if λ 2 = 0 and λ 1 > 0 the system (A1) becomes,

                         ∂ x L = -N t α a + 2 t (
       ∂ x L = -N t α a + 2 t (p a 2 -p b 2 ) + α b λ 1 = 0, ∂ y L = -N t α b -2 t (p a 2 -p b 2 ) + α a λ 1 = 0, p a 2 α b + p b 2 α a = α a α b N.
After resolution of this system, we find:

       p a 2 = N t α a , p b 2 = N t α b , λ 1 = N t > 0.
If λ 2 > 0 and λ 1 = 0, the system (A1) becomes,

       ∂ x L = -N t α a + 2 t (p a 2 -p b 2 ) -λ 2 = 0, ∂ y L = -N t α b -2 t (p a 2 -p b 2 ) -λ 2 = 0, p a 2 = -p b 2 .
Once the system is solved, we find λ 2 = -N 2 < 0 which is impossible because Lagrange multipliers are positive. So in this case, there are no points who verify KKT conditions. Finally, there is only one minimum p 0 := (N α a /t, N α b /t) complying the first order conditions and the constraint of TSM's existence. Taking (p a 2 , p b 2 ) = p 0 , we have,

f a = p b 2 -p a 2 N t - α a t = 1 2 = f b ,

  equal to the elasticity of the group's participation D a /D ′ a (resp. D b /D ′ b ), adjusted downward by the external benefit of group b, α b N b

Definition 1 .

 1 (Equilibrium at second period) The prices (p a 2 , p b2 ) define the equilibrium at second period if: i. No agent can improve his utility by changing group, ii. The mean profit of the platform is maximal.

Proposition 1 .b 2 )

 12 If prices (p a 2 , p b 2 ) assure equilibrium at second period, then u a 2 , defined in Definition 1, assuring the equilibrium at second period. If u a 2 > u b 2 , then agents migrate in face a. So, N a 2 increases and N b 2 decreases. Therefore by the external effects and since the prices are fixed, u a 2 decreases and u b 2 increases. According to Definition 1, this process is stopped when u a 2 = u b 2 .

p a 2 -p b 2 ) + α b λ 1 -

 221 λ 2 -λ 3 + λ 4 = 0, ∂ y L = -N t α b -2 t (p a 2 -p b 2 ) + α a λ 1 -λ 2 + λ 3 -λ 4 = 0, λ 1 (p a 2 α b + p b 2 α a -α a α b N ) = 0, λ 2 (p a 2 + p b 2 ) = 0, λ 3 (-N α b -p a 2 + p b 2 ) = 0, λ 4 (p a 2 -p b 2 -N α a ) = 0.(A1)

2 (p 0 ) = p a 2 f a N + p b 2 f b N = N 2 t 4 .

 24 Finally, it is obvious that O := (0, 0) ∈ C 2 , and Π 2 (O) = 0. So, there is a point (O), such as Π 2 (O) > -Π 2 (p 0 ) = -N 2 t/4, thus p 0 is a local minima of -Π 2 , i.e. a local maxima of Π 2 . As a consequence, we have Π * 2 = Π 2 (p 0 ).

For the proof seeCaillaud and Jullien (2012).

Another model could be to assume that the consumers' number in every side is an exogenous parameters, however this implies a loss of generality.

1 {Xn=a} = 1 if Xn = a, 0 else.

It is straightforward to show from equation (7) that: ∂fa/∂p b 2 ≥ 0, ∂fa/∂p a 2 ≤ 0, ∂fa/α b ≤ 0 and ∂fa/∂αa ≥ 0, reciprocally for f b .

Another method consists of maximizing the intertemporal profit Π = Π 1 + βΠ 2 , where β is the discount rate. But such a method does not distinguish simply the following two mechanisms: first the market entry (as our first period), and second the side-switching (as our second period). However, the maximization of the intertemporal profit, by setting prices for two periods is a possible extension of our model.

Appendix

Proof of Corollary 1 According to Proposition 1:

Proof of Theorem 1

The platform's program at the second period is given by the minimization problem (P ): min

where the mean profit Π 2 is the continuously differentiable function given by ( 9) and C 2 is given by (10). First note that C 2 is a compact space of R 2 , therefore (P ) admits at least one solution. Also the function Π 2 does not admit critical points: it does not exist (x, y) ∈ C 2 such that ∇Π 2 (x, y) = 0, where ∇ is the gradient. Consequently, the solution of the problem (P ) is on the boundary of the space C 2 . The Lagrangien L of (P ) is given by

where λ = {λ i ; i = 1 • • • 4} represent lagrange's multipliers. Then, Karush-Kuhn-Tucker (KKT) first order conditions are given by the following system, Proof of Corollary 2 By ( 5) and ( 11), we have

And by (2), N := N a * 1 + N b * 1 , thus (A2) becomes