Stochastic No-Ponzi-Game condition and government debt dynamics

Transcription

1 Stochastic No-Ponzi-Game condition and government debt dynamics Chen He Supervisor Prof.dr.Lex Meijdam Tilburg University Number of words: 6536 May 30, 2012 Abstract This paper concerns the optimal government debt policy in an overlappinggenerations model of a small open economy with stochastic government spending. A stochastic No-Ponzi-Game condition is provided. The government in one period is assumed to play a Stackelberg leader-follower game with the government in the next period. The Stackelberg Equilibria for both a nite time horizon game and in nite time horizon game are derived analytically. Finally, policy implications from the model are addressed. 1

2 1 Introduction The scal policy concerning government debt is becoming one of the key policy issues nowadays, which has triggered an intensive discussion both among academic scholars and within the political world. The most standard scal instrument for governments to nance their debt is tax. When a government determines how much tax it is going to levy from its citizens, at least two aspects are worthy to consider. First, the more tax that government is levying, the more funds it could use to nance its debt, as a consequence, the less likely the government will be in nancial distress. Second, the more tax that government is levying, the less disposable income individuals will have, and thus less consumption and less utility they will get. Inspired by this trade-o, this paper concerns the optimal government debt policy of a small open economy with stochastic government spending. Meijdam et al. addressed this issue in their publication The dynamics of government debt (1996). In their paper, they investigated the optimal debt policy by employing the overlapping-generations model (OLG) of a small open economy and explicitly derived the optimal solution for the time paths of both taxes and debt. Worthy to mention, their model is under a complete nonstochastic setting. That is, there is no uncertainty for the parameters of the model. This paper is an extension of the paper written by Meijdam et al. by adding uncertainty to the OLG model. To be more speci c, in this paper, government spending is stochastic and is assumed to have a binominal distribution. That is, in every period, government spending can have two values: high and low. Which value will actually be realized is unknown by the government ex-ante. The reason that we assume stochastic government spending is due to the uncertainty of economic circumstances. For example, consider an exogenous shock to this small open economy, as the one in the nancial sector in 2008 initially sparked by the subprime mortgage. Facing this exogenous shock, the government in a small open economy has no choice but to increase government spending in order to prevent a collapse of the banking sector. In this sense, government has no control over its spending. A second advantage of assuming a stochastic government spending is that this characterizes the so-called time inconsistency problem of government. In the real world, a government may run a risk of time inconsistency in the sense that it announces its government budget ex-ante, but for some reasons, (e.g. exogenous shock or political cycle) it overspends and thus creates a budget de cit ex-post. Assuming stochastic government spending explicitly takes this problem into account. When considering the optimal government policy, one crucial issue is government solvency. As discussed above, if a government levies a too low tax, it may not be able to repay its debt in the future and thus becomes insolvent. In order to avoid insolvency, government has to nance existing debt by issuing new debt. But can the government do this repeatedly forever? The famous No-Ponzi-Game (NPG) condition draws a bottom line on this. It says that gov- 2

3 ernment cannot completely repay the current debt by issuing new debt forever. In other words, this is no problem for government as long as the present debt is met by all future primary surpluses. Under non-stochastic setting, the NPG condition is su cient for government solvency. Bartolini (1994:9) also considers the Government Ponzi games and the sustainability of public de cits under uncertainty. The author nds that "the steady-state debt-to-income ration converges almost surelt to zero if and only if the interest rate on debt is smaller than the asymptotic average growth rate of the economy." Kuhle (2010:17) studies the structural di erences between implicit and explicit government debt under a stochastic OLG model. He shows that if governments could issue safe bonds and claims to an unfunded social security system to service a given initial obligation, there exists a set of Paretoe cient nancing policies. Bergman (1999:28) shows that the NPG condition is satis ed when government debt is integrated of any nite order but not when government debt is an explosive process. In this paper, we nd that under stochastic setting, the (stochastic) NPG condition is not su cient any more to ensure government solvency. As we shall see, even if government meets the stochastic NPG condition, the level of government debt could become very large and beyond any nite limit if the economy is dynamically e cient. Therefore, we need a second constraint, namely, a constant de cit ratio to keep the government solvent. This is the well-known Domar s rule, which says that the government debt will be constant if the budget de cit had been constant for a long time. In this paper, we also derive a stochastic Domar s rule which, together with stochastic NPG condition, constrains of scal policy space of the government. A government that determines the constant public de cit and debt ratio faces asymmetric information from capital market. That is, the government has no idea about what is the highest ratio permitted by the capital market. We argue that this asymmetric information will nally induce a social welfare loss borne by individuals. This paper employs an OLG model of a small open economy with stochastic government budget. The government is only concerned with current generation s utility. That is, there is no altruism for both government and individuals. However, the government explicitly takes future government decisions into account. That is, two governments in two consecutive periods play a Stackelberg leader-follower game, in which the government in the rst period moves rst, acting as the leader. The Set of Subgame Perfect Stackelberg Equilibria will be derived analytically. The rest of this paper is structured as follows. Section 2 lays out the model. Section 3 derives the solution of the model. Policy implications are discussed in section 4. Section 5 concludes. 3

4 2 The model In this section, we construct the model using the same setting as Meijdam et al. (1996:69). That is, we assume a small open economy where a xed interest rate is faced by both lenders and debtors. This, in turn, implies that the government borrowing behavior is not restricted by domestic capital market capacity. 2.1 The consumers Consider an overlapping generations economy in which two non-altruistic generations, young and old, are present in the same period. Each period a new generation of individuals enters the economy who live for two periods. The individuals are born with no assets and inelastically supply one unit of labor when young. The income that individuals earn in the rst period is normalized to 1, of which they save an amount s. Assume a consumption tax is levied by the government. The savings s, is used for consumption in the second period which is subjected to the exogenous interest rate r and the consumption tax. Individuals, therefore face budget constraints in both periods, which are: c y t = (1 t )(1 s t ) (1) c o t+1 = (1 t+1 )(1 + r)s t (2) where c o t+1 denotes the consumption in t+1 when individuals become old. For simplicity, assume individuals have a logarithmic utility function: U t = ln(c y t ) + ln(c o t+1) (3) where is the private discount factor re ecting the time preference. Since the consumption tax is stochastic, rational individuals optimize their expected utility function using s t as instrument, given the two budget constraints. Formally, let E t (.) denote the expectation operator, based on information up to t. individuals Maximize st E t (U t ) = E t [ln(c y t )] + E t [ln(c o t+1)] Subject to c y t = (1 t )(1 s t ) c o t+1 = (1 t+1 )(1 + r)s t 4

5 In particular, c o t+1 represents the consumption in period t+1. First notice that the maximum value of U t indeed exists. The proof of this statement is straight-forward: i) the function U t is a continuous real valued function and ii) the constraint set given by Eqs. (1) and (2) is a compact subset of R n ; According to the Theorem of Weierstrass, the function U t has a maximum on the compact subset. Solving this optimization problem using rst-order t (U t ) t(u t y (1 t ) t(u t ) o (1 + r)(1 t+1 ) = 0: (4) t+1 Substitute Eqs. (1) and (2) into Eq. (4), this yields s t = 1 + 8t 2 N ++: (5) Eq. (5) implies that the optimal saving rate chosen by individuals is independent of government behavior. Therefore, the introduction of stochasticity of government spending will not a ect individuals behavior. Individuals always choice their saving rate based on their private time preference at every time period. The possible theoretical explanation regarding this result is that the income e ect and the substitution e ect of a change in the tax rate exactly cancel out. 2.2 The government The government s budget constraint in period t consists of several elements. First, government in period t has to pay the debt (principal plus interest) per individual in t-1, b t 1 Second, government faces a stochastic government spending per individual in current period t. The government generates revenues from levying consumer taxes. Assume that government cannot nance its debt by seigniorage. This can be interpreted as the economy has an absolute independent central bank. In order to derive the stochastic No-Ponzi-Game condition, I rst introduce the stochastic setting. Let G t denote a random variable describing the government spending per individual in period t. For simplicity, assume G t has a binominal distribution, G t B(h p), which has only two values: high (h) and low (l). Let p be the probability that the government has a high government spending. The corresponding (discrete) probability density function (pdf) reads P (G t = h) = p; P (G t = l) = 1 p; h and l 2 R n (6) 5

6 Consistent with Meijdam et al. (1996:71), further assume that government spending per individual is smaller than 1, that is, government spending per individual at any period cannot be over GDP per capita at that time period. Let E t (.) denote the expectation operator based on information up to t, that is, before the realization of G t. Let n be the population growth rate. Then the government s expected budget identity in period t reads E t (b t ) = ( 1 + r 1 + n )b t 1 + E t (G t ) E t (T t ): (7) where T t denotes total tax revenue per individuals of generation t. Notice that in Eq. (7), b t 1 is non-stochastic, since in period t, the government debt in the previous time period t-1 has been realized. From Eqs. (2) and (3), government s expected total tax revenue per individuals of generation t is given by E t (T t ) = E t ( t )[(1 s t ) + ( 1 + r 1 + n )s t 1]: (8) Assume that the economy has no technological growth. Then the well-known No-Ponzi-Game (NPG) condition requires that the present value of outstanding debt converges to 0 asymptotically. In the current binominal stochastic setting, the corresponding stochastic NPG condition reads lim (1 + n t! r )t E t (b t ) = 0: (9) Aggregating Eq. (7) and applying Eq. (9), the stochastic NPG condition implies P b t = 1 E t (G t+i i=1 T t+i )( 1 + n 1 + r )i (10) The stochastic NPG condition is of our central interest. It requires the present value of government debt equals to the sum of future expected primary surpluses. Clearly it imposes a restriction on the government s policy space. The government has to choose its tax policy t in a way such that Eq. (10) has to be met. Whenever Eq. (10) fails to be met, the international capital market will doubt the solvency of the government and thus stop lending. In order to keep a robust solvency position, the government may levy a high consumption tax on individuals so that Eq. (10) is more likely to be met. However notice that 6

7 the government cannot levy a too high consumption tax t on individuals, for individuals are not happy and prevent the government from doing so. Assume the maximum tax rate max t 1. Eq. (10) has an essential di erence from its non-stochastic NPG condition in Meijdam et al. s work. In their paper, Meijdam et al show that total of future primary surplus is nite if the tax base grows at a rate smaller than 1+n 1+r (Meijdam et al. 1996:71). In the stochastic setting, as we shall see in section 3 and 4, if the government follows a "state-independent strategy", b t could become very large and beyond any nite limit (With a very small probability of course). Said di erently, b t is a divergent sequence without upper bound. The formal proof of this statement will be postponed in section 4. Escolano (2010:7) also con rms this unbounded governmnet debt trajectory. In his paper, Escolano explicitly argued that "the expected value of the debt ratio need not be bounded even if it is bounded in each of the possible trajectories". The fact that expected value of debt ratio is unbounded above is a very serious issue for the government, which implies that the stochastic NPG condition is not a su cient condition for government solvency. Even if the government meets the stochastic NPG condition in every period, there still exists a (small) probability such that the government debt is out of control. Dealing with this divergent government debt will be discussed in detail in section 4. The government at time t is trying to maximize the total welfare of current livings and does not take into account the future generations. Therefore, the government at time t Maximize t E t (W t ) = W [U t 1 ; (1 + n)e t (U t )]: (11) taking individuals optimal behavior, Eq. (5), into account. This implies that the government acts as a Stackelberg leader towards individuals. Furthermore, assume the future governments make their decision based on the decision of current government, which implies that the current government is also a Stackbelberg leader towards future governments. (Meijdam et al. 1996:72) 3 Solution of the model In this section, the solution of the model constructed in section 2 will be derived analytically. Broadly speaking, the government at any time period has two di erent types of tax rate strategy. First, the government could follow a "state-independent strategy". That is, the government will set tax rate based on the debt level in the last period, independent of (stochastic) government spending in the current period. Said di erently, government at time period t sets 7

8 t = t (b t 1 ) (12) At this moment, we just present t as some abstract function of b t 1. The exact form of this function will be derived later. Eq. (12) implies that the government in time period t will choose the optimal level of the tax rate in a way such that i) the social welfare presented by Eq. (11) is maximized and ii) the stochastic NPG condition ( Eq. (9)) holds, independent of government spending in the same period. Apart from this state-independent tax rate strategy, the government has another option, namely "state dependent strategy". As its name has already suggested, government will set its tax rate at time period t conditional on the stochastic government spending at the same period. In order to make this strategy possible, a crucial assumption needs to be made: At any period, the government set its tax rate after knowing its government spending at the same time period. One concern regarding this assumption is that given the government spending is stochastic, how could the government make its scal policy knowing its (stochastic) spending? To show the assumption is not too stylized, consider the following real world cases. For example, China has adjusted its scal policy after the banking crisis in the U.S in The same is also applied in the Eurozone. Governments could adjust its scal policy after knowing the random shock. According to this assumption, the government will set a high tax rate at time t if government spending during the same period is high and will set a low tax rate if corresponding government spending is low. Formally, government at time t sets l t = t (b t 1 ) if g t = g l h t (b t 1 ) if g t = g h (13) According to the probability density function in section 2 (Eq. (6)), in every time period t, there is a probability p that the government will set the high tax rate. In this state-dependent strategy, the government adjusts its tax policy every period conditional on its government spending at that period such that the stochastic NPG condition is ful lled. Given those two distinct strategies, one may wonder how to determine which strategy the government will actually choose. In the light of E cient Market Hypothesis (EMH), I document the following proposition: Proposition 1 In an e cient market, the government will only follow the state-independent strategy. Proof. Consider a government in an arbitrary period t and suppose there comes an exogenous shock such that g t = g h : Assume t (b t 1 ) < h t (b t 1 ): That is, the current tax rate is smaller than the high level tax rate. Given i) the individual utility function represented by Eq. (3), levying a high tax rate means reducing 8

9 individual utility; ii) this government tries to maximize its object function, Eq. (11); and iii) this government could choose between state-dependent strategy and state-independent strategy; It is in its best interest not to levy a high tax rate. If the capital market is e cient, that is, there is no asymmetric information, the state-dependent strategy will be ruled out because the market knows that the government will never levy a high tax even if the government claims to do so. Since we choose this government arbitrarily, this statement is true in any period. Proposition 1 excludes the state-dependent strategy. Our remaining task is therefore to derive an explicit expression of Eq. (12). The government in any time period t maximizes social welfare using tax as instrument. t =@U t 1 = 1 t =@U t =, the rst order condition of Eq. (11) then gives: 1 E t ( t+1) 1 E t ( t ) (1 + n)f 1 E t( t+1) [1 E t ( t )] t( t ( t ) g = 0 (14) Eq. (14) holds for t=1,...,+1. Notice the t ( t+1)=@e t ( t ) implies that the government in period t and the government in period t+1 play a Stackelberg leader-follower game. The government in period t, who moves rst, acts as the leader and the government in period t+1 makes its decision based on the incumbent government s decision. After period t+1, the government in period t+2 comes in and makes its decision based on the decision made by government in period t+1, so on and so forth. In short, this Stackelberg leader-follower game will be played an in nite number of times by an in nite number of di erent players (governments) with each player in each time period moves sequentially, best responds to the decision made in previous period. The optimal tax levied by each government in each period together constitutes the Stackelberg Equilibrium of this game. Notice also that Eq. (14) cannot hold with equality if =0, implying that there is no internal solution to the optimizing problem. In line with Meijdam et al (1996:74), can be naturally interpreted as the marginal political power of the young generation. t =@U t 1 = 1 is the marginal political power of the old generation) That is, if the young generation does not have any political power (=0), there is no internal solution and the government will levy the lowest tax rate given the constraint of stochastic NPG condition. This result meets intuition: there is no future for the elderly so that they prefer the tax rate is as low as possible. Standard dynamic programming techniques can only solve the nite horizon game. Therefore in order to derive the Stackelberg Equilibrium of this in nite periods game, we rst derive the Stackelberg Equilibrium of the nite horizon 9

10 game using dynamic programming. Then the Stackelberg Equilibrium to the in nite horizon game is derived by taking the limit of solution of the nite horizon game. Suppose the game will be played from period 1 to period T. In the last period T, the stochastic NPG condition implies that government in the last period has no choice but to repay all the government debt in order to remain solvency. That is, E T 1 (b T ) = 0 (15) The expected tax rate in the nal period, then, can be derived according to the government budget constraint (Eq. (7)): E T 1 ( T ) = (1 + ^r) [(1 + ^r)b T 1 + E T 1 (G T )] (16) where ^r=(r-n)/(1+n), which can be interpreted as the e ective interest rate. 1+^r, therefore equals to (1+r)/(1+n). Eq. (16) is the optimal expected tax rate in the nal period. The government in period T-1, who is the Stackelberg leader, explicitly takes Eq. (16) into account. From the government budget constraint given by Eq. (7) and the optimal expected tax rate given by Eq. (16), one could easily T 2 ( T ) = (1 + ^r) T 2 ( T 1 ) Before moving forward, further assume that the government in each period makes "equally good expectations" about the its government spending. That is, each government faces identical information and is assumed to have identical ability. No government in any period has excess information available to make "superior decision". This gives: E 1 (G 1 ) = E 1 (G 2 ) = ::: = E 1 (G T ) E(G) (18) Recall that the government cannot levy a tax rate higher than 1. This, in turn, implies that there exists a maximum level of expected government debt in each period, denoted by b max t, t=1,2,...,t: If in any period t, E t 1 (b t ) > b max t ;the e cient capital market will stop lending to that government because it knows that the government will be unable to repay in the future since the stochastic NPG condition cannot be met. Consequently, the government will go bankrupt. 10

11 To explicitly derive this maximum level of expected government debt, inserting T = 1 and Eq. (18) into Eq. (16): b max T 1 = 1+(1+^r) 1+ E(G) 1 + r (19) T Substituting Eq. (19) into government budget constraint Eq. (7), using 1 = 1 gets: b max T 1 = (1 + ^r)b max T 2 + E(G) 1 + (1 + ^r) 1 + (20) Solving b max T 2 using Eqs. (19) and (20) gives: b max 1 + (1 + ^r) T 2 = [ E(G)][ 1 + ^r + 1 (1 + ^r) 2 ] (21) An internal optimum is obtained if E T 3 (b T 2 ) b max T 2 :If E T 3(b T 2 ) = b max T 2 ;the boundary optimum will be obtained. That is, = max :Government in this case has no choice but to levy the maximum level of tax rate in order to meet the stochastic NPG condition. Furthermore, it is impossible to have E T 3 (b T 2 ) > b max T 2 : The reason for this follows from our EMH made earlier in this section. In an E cient capital market, the debt holders will stop lending to the government if they see the level of debt ratio is higher than a certain threshold. Rewriting the rst order condition given by Eq. (14), using Eqs. (17), (7) and the de nition of b max T 2 derived above gives the internal optimum solution: E int T 2( T 1 ) = 1 (1 + )(1 + ^r) 1 + (1 + ^r) 1 + [bmax T 2 E T 2 (b T 2 )] (22) where = [(1 + n) + ]=[(1 + n)]:given ET int 2 ( T 1)derived, the internal solution for ET int 3 ( T 2),ET int 4 ( T 3); :::; E1 int ( 2 ); E0 int ( 1 )can be derived sequentially. The vector V = (E int 0 ( 1 ); E int 1 ( 2 ); :::; E int T 2( T 1 ); E int T 1( T )) (23) constitutes one Stackelberg Equilibrium of this nite periods game. Other Stackelberg Equilibria could be obtained by one (or more) government(s) in one (or more) period(s) playing the boundary optimum strategies. In general, there 11

12 are nite numbers of Subgame Perfect Stackelberg Equilibriums to this nite periods game. To derive the solution of the in nite periods game, one could simply let the time index of nite game, T, go to in nite. From Eq. (21), let the time period T go to in nite: lim T!+1 bmax T 2 = b max E(G) 1 + (1 + ^r) + ^r (1 + )^r (24) The boundary optimum is the same as its nite counterpart. The government has no choice but levy the highest level of tax if the debt ratio reaches its upper bound. This is true even in an in nite-horizon game. Put the two cases together, the solution for the in nite-horizon game is: E 1;t ( 1;t) = E1;t 1 ( int 1;t) if E t 2 (b t 1 ) < b max max if E t 2 (b t 1 ) = b max (25) where E 1;t 1 ( int 1;t) = 1 (1 + )(1 + ^r) 1 + (1 + ^r) 1 + [bmax E t 2 (b t 1 )] (26) Compared to its nite counterpart, according to the Folk Theorem, there are in nite numbers of Subgame perfect Stackelberg Equilibriums to this in nite periods game. 4 Policy implications In this section, we pay attention to the policy implications of the stochastic model derived above. In order to do so, two crucial propositions will be made rst. After that, a set of policy conditions will be made for governments in order to keep them remain solvency. Consider again the stochastic NPG condition given by Eq. (9). Notice that Eq. (9) is always true if r < n: In other words, in a dynamically ine cient economy, the government does not need to worry about its debt, for it will converge to 0 in the long run, assuming government de cit is bounded. See, for instance, Blanchard and Weil (1992). This leads to the following proposition: 12

13 Proposition 2 In a dynamically ine cient economy, the stochastic NPG condition will always be held in the long run if government de cit is bounded. Proof. Consider the budget constraint Eq. (7), we prove this proposition by showing lim E t(b t ) is bounded above. t!+1 Let d t denote the public budget de cit in period t. Then E t (b t ) = (1 + ^r)b t 1 + E t (d t ) E t+1 (b t+1 ) = (1 + ^r) 2 b t 1 + (1 + ^r)d t + E t+1 (d t+1 ) E t+2 (b t+2 ) = (1 + ^r) 3 b t 1 + (1 + ^r) 2 d t + (1 + ^r)d t+1 + E t+2 (d t+2 ) Since we are in a dynamically ine cient economy where r < n; thus ^r = r n 1+r < 0 and 1 + ^r < 1: 1P 1P E t+1 (b t+1 ) (1 + ^r) +1 T b t 1 + lim (1 + ^r) ~ i d t+i lim (1 + ^r) ~ T T ~!1 i=1 i d max dmax ^r where d max = Max {d t, d t+1; d t+2 ; :::E +1 (d +1 )g T ~!1 i=1 According to proposition 2, there would be no need to worry about government debt if the economy is dynamically ine cient. Unfortunately, in real world, especially nowadays, countries are generally dynamically e cient, that is, real interest rate is larger than the population growth rate. This could happen, for instance, in a situation which central banks try to maintain a high real interest rate in order to reduce in ation. The relationship among output level, unemployment level, money supply level and (expected) in ation rate can be formally derived by Short-Run-Aggregate-Supply (SRAS) model, Phillips Curve model and Okun s law. I will not discuss them in detail here. See, for example, Mankiw (2007: ). Due to these reasons, it makes more sense to look at the stochastic NPG condition in a dynamically e cient economy. From the proof of proposition 2, it is immediately clear that the level of debt is not bounded from above, for (1 + ^r) > 1 if r>n. (following our de nition of ^r) Even worse, according to our stochastic overlapping generation model, we will further show that in such a dynamically e cient economy, the level of government debt could become very large and beyond any nite limit with a small probability, if the time horizon goes to in nity, which is the key proposition of this paper. Proposition 3 In a dynamically e cient economy, the level of government debt could become very large and beyond any nite limit if the time horizon goes to in nity. Proof. Consider again the budget constraint Eq. (7). We prove this proposition by showing lim E t(b t )! +1 under a dynamically e cient economy where t!+1 r > n. De ne the net government spending in the worst case g t = G h E t (T t ):From our previous discussion it is easy to see that g t > 0: Let g = Minfg 1 ; g 2 ; g 3 ; :::g Let 1 + ^r = ( 1+r 1+n ); since r > n; we have 1 + ^r > 1: 13

14 Substitute 1 + ^r and g into Eq. (7) and de ne b t = (1 + ^r)b t 1 + g yields b t = (1 + ^r)b t 1 + g < E t (b t ) since G t = G h, 8t. Now we show lim t! t!+1 +1 b t = (1 + ^r)b t 1 + g is a rst-order linear di erence equation. Solving this di erence equation: b t = (1+^r)b t 1 +g = (1+^r)[(1+^r)b t 2 +g]+g = (1+^r) 2 b t 2 +g[(1+^r)+1] =(1 + ^r) 3 b t 3 + g ((1 + ^r) 2 + (1 + ^r) + 1) =... =(1 + ^r) t b 0 + g ((1 + ^r) t 1 + (1 + ^r) t 2 + ::: + (1 + ^r) 2 + (1 + ^r) + 1) =(1 + ^r) t b 0 + g ( 1 (1+^r)t 1 (1+^r) ) Since(1 + ^r) > 1 and g ( 1 (1+^r)t 1 (1+^r) ) > 0 lim t = lim t!+1 t!+1 ^r)t b 0 + g ( 1 (1+^r)t 1 (1+^r) )]! +1 Therefore lim t(b t )! +1 t!+1 A key step of the proof is the realization of G h. The idea is that Debt will exceed every nite boundary if a su ciently large number of consecutive realization of G h takes place. This has a small, but strictly positive probability. Below, we formally show this small probability will realize (coverge to 1) at some point if t! 1. Proposition 4 In a dynamically e cient economy, the level of debt could exceed any nite level for sure if the time horizon goes to in nity. Proof. Let A t be the event "the level of debt reaches a certain level L in t th period. P (A t ) = and P (A 0 t) = 1. is small but strictly positive between 0 and 1. Therefore the probability that the level of debt does not reach a certain level L in the rst n periods is: Q P ( n A 0 i ) = Q n P (A 0 i ) = (1 )n i=1 i=1 Thus the probability that the level of debt reaches a certain level L in at least one of rst n periods is P P n = P ( n Q A i ) = 1 = n P (A 0 i ) = 1 (1 )n i=1 i=1 Let the time horizon goes to in nity, that is, Let n! 1, since 0 < < 1, thus 0 < 1 < 1, we have lim P n = lim 1 (1 n!1 n!1 )n = 1 0 = 1 Proposition 3 implies that If no further restrictions are imposed, under a dynamically e cient economy, the government debt will become very large in the long run and beyond any nite limit if the time horizon goes to in nity. Another implication from proposition 3 is that the NPG condition under a stochastic economy is not su cient for government solvency. Therefore, we need a second condition to prevent government from insolvency in the long run. 14

15 There exists the famous Domar s rule saying that the government debt will be constant if the budget de cit had been constant for a long time. We now derive the stochastic Domar s rule for our model described in section 2. The starting point is the government s expected budget identity, given by Eq. (7). Subtracting b t 1 from both sides gives E t (b t ) b t 1 = ( r n 1 + n )b t 1 + E t (G t ) E t (T t ) (27) Let E t (d t ) be the government s expected de cit at time t, which equals to interest payment on existing government debt and the expected net government spending in the current period. That is, E t (d t ) = rb t 1 + E t (G t ) E t (T t ) (28) Substitute Eq. (28) into Eq. (27) and rearrange terms gives E t (b t ) b t 1 = E t (d t ) ( 1 + r 1 + n )nb t 1 (29) Finally, dropping out the time index, we get the equilibrium level of government debt ratio: b = E(d) 1 + n n 1 + r (30) Thus we get the stochastic Domar s rule for our model which has a constant government debt level. Eq. (30) implies that, in order to prevent government debt level from continuously increasing, the government has to set their tax rate (thus primary surpluses) such that the expected public de cit ratio is constant. In other word, in a dynamically e cient economy where real interest rate is larger than population growth rate, primary surpluses have to be adapted to prevent unsustainable debt. The Domar s rule implies there exists a maximum level of government debt and expected per-period de cit level, which imposes a second restriction on government s tax space. In order to remain solvent, the government in any period has to adjust its tax rate so that i) the stochastic NPG condition holds and ii) E(d) is constant. Since in our model, government spending G is stochastic and can be either high or low. In order to meet Eq. (30), the government has to consider the worst scenario. That is, government has to meet Eq. (30) even if G = G h : Otherwise, E(d) will not be constant and consequently debt level will explode out. 15

16 Eq. (30) implies that there exists a one-to-one function relationship between government debt ratio and its (constant) expected public de cit ratio, given population growth rate and real interest rate. More precisely, assume population growth rate and real interest rate are positive, the lower the constant expected public de cit ratio is, the lower the equilibrium value of government debt will be, and vice versa. Therefore, an important and interesting question for government to concern is how to determine the constant level of expected public de cit ratio. In principle, any constant expected public de cit ratio corresponds to an equilibrium level of government debt ratio. The actually level of government expected public de cit ratio depends on the capital market. If the capital market believes that the government debt ratio will not explode out any more in the future. Then, high level of expected public de cit ratio and government debt ratio will not be a problem. In practice, di erent economies face di erent situations. In Eurozone, the upper bound of public de cit ratio set by the European Central Bank (ECB) is 3% of GDP. (Baldwin and Wyplosz, 2009: ) The idea can be explained by our simple binominal model in section 2 as follows: During "normal" times, the government de cit in one period is 0%. Thus d L = 0. During bad times, ECB allows government to have a maximum per-period de cit of 3%, that is, d H = 0:03:Then the expected de cit level, E(d) = 0:03p. Following the Domar s rule, the government debt level will be between 0 and 60%. In Japan, however, the government debt ratio as a percentage of GDP is 220% in (IMF 2011) One reason why Japan could have such a large government debt without (yet) sovereign default is that domestic capital market holds strong trust on Japanese national bonds. 95 percent of government debt is held domestically. (Cox and Hutchinson, 2011) The Japanese case implies that if the capital market has a high degree of trust on its political process, then the government can have a high debt ratio and as a consequence, have a larger tax space to choose its scal policy. In short, strong government accountability could remove part of constraint on government s strategy space. Another policy implication from our discussion is regarding the e ciency of government policy. Government in any period maximizes its social welfare function, given by Eq. (11), subject to i) stochastic NPG condition given by Eq. (9) and ii) stochastic Domar s rule given by Eq. (30). The presence of stochastic Domar s rule will force government to levy a tax rate higher than it would be without this constraint. This high tax rate, in turn, will result a decrease in total welfare of current livings. Therefore, our optimal tax solution derived in section 3 does not t any more in the presence of stochastic Domar s rule. Said di erently, governments, in essence, face a trade-o between total social welfare on one hand, and government solvency on the other hand. If government decreases its tax level, it increases the social welfare at an expense of deteriorate its budget position. In this sense, ceteris paribus, neither increasing nor decreasing tax will result in a Pareto-improvement. Is there a Pareto-improvement policy for government, namely, decreasing government tax without impairing its budget? The answer is that such policy is only a Paretoimprovement for the current generation, but not for future generations. To see this, recall Eq. (30), given any constant expected public de cit ratio, there is 16

17 a unique constant government debt ratio. That public de cit ratio in principle, can be any nite number as long as the capital market holds the belief that this de cit ratio and its corresponding debt ratio is constant and will not increase in the future. It is not di cult to think there is a maximum level of de cit ratio and debt ratio, above which the capital market will no longer trust government anymore and hence stop lending to the government. Assume that the maximum level of debt ratio is 200%. However, due to asymmetric information, government is too conservative and estimates this maximum level of debt ratio is 60% and hence overtaxes its citizens, resulting in an unnecessary welfare loss. In this case, the government could instead, decrease its tax level and consequently, increase government debt ratio to 200%. This is bene cial for the social welfare of the current generation. But such policy will hurt future generations because the future government will have to increase the tax so that the future generations will su er. In short, such policy is only Pareto-e cient for current generation, but if we are concerned with the intertemporal welfare, policies as such are not Pareto-improvement. The reason of implementing this sub-optimal policy is that in our hypothetical example, this government does not know what the maximum level of its debt ratio is permitted by the capital market. It could be the case that the actual maximum debt ratio permitted by the capital market is also 60% and if the government increases its debt level, the capital market will immediately lose trust on government bond and stop lending. In other word, the presence of asymmetric information produces a wedge between the actual maximum debt level permitted by the capital market and the estimated maximum debt level by the government itself. This wedge, in turn, contributes to social welfare loss borne by individuals. In the presence of asymmetric information, the government has no way to get rid of this social welfare loss. What the government can do is implementing a safe sub-optimal scal policy, taking into account the worst scenario, in order to remain solvency. In practice, consider the current Eurozone, it is unclear whether the 60% debt ratio and 3% public de cit ratio reach the limit of patience of the capital market or they are just too conservative. The same uncertainty holds also for Japan. We can only conclude ex-post which debt ratio is too high from revealed preference of capital market. For example, we can conclude that the current debt ratio in Greece exceeds the upper bound permitted by the capital market. There is no trust in Greek political process any more. In short, the main implication is that in the presence of asymmetric information, the governments are forced to tradeo part of their welfare to its solvency. 5 Conclusion This paper is concerned with the optimal government debt policy in a stochastic OLG model. Government spending is assumed to have a binominal distribution. Although this distribution is rather simple, it nevertheless characterizes 17

18 the essence of the problem faced by government under uncertainty. The government in our model is assumed to be only concerned with the utility of current livings, but explicitly takes decisions made by future governments into account. In this fashion, government in current period acts as a Stackelberg leader towards government in the next period. The sequence of all government policies constitutes a set of (Subgame Perfect) Stackelberg Equilibriums to this Stackelberg leader-follower game. We derive the set of (Subgame Perfect) Stackelberg Equilibriums for both nite horizon game and the in nite horizon game. One main nding of this paper is that the stochastic NPG condition is not su cient for government solvency. In a dynamically ine cient economy, the stochastic NPG condition will always hold in the long run. However, in a dynamically e cient economy, the level of government debt could become very large and beyond any nite limit if the time horizon goes to in nity. Due to this reason, we present a second constraint in order to keep government remaining solvent, namely, stochastic Domar s rule. However, Domar s rule, together with stochastic NPG condition, keeps government remaining solvent at an expense of reducing social welfare. We further argue that this loss of social welfare is inevitable in the presence of asymmetric information between government and capital market. The asymmetric information produces a wedge between the actual maximum debt level permitted by the capital market and the estimated maximum debt level by the government itself. Therefore, we conclude that government is forced to tradeo part of its social welfare to solvency risk. This analysis, however, is not the full picture of the government optimal tax policy problem. Some questions still remain. For example, we exclude the function of Central Bank in our analysis. In practice, a government may nance its debt through its central bank. By combining scal policies with monetary policies, government can extend its policy space. Second, in our model, we assume that the maximum tax rate is 100%. This assumption is of course unrealistic in practice. In the real world, consumption tax obviously cannot be 100% and there exists a minimum income standard for individuals. When disposable income falls below a certain threshold, individuals may complain to the government and force the government to reduce tax. In addition, a 100% tax rate is not optimal according to the well-known La er curve because when tax rate is 100%, the total government revenue is theoretically 0 due to the fact that individuals will change their behavior as a response to such a high tax rate. For example, in labor economics theory, individuals that face a too high tax rate will not join in the labor market because the post-tax wage they get is lower than their reservation wage. (Boeri and Ours 2008:4-7) This implies that the tax base is shrinking under high tax rate. The third concern of our model is that our assumption of binominal distributed government spending might be too simple. One could use more complicated distribution to capture the full picture. Furthermore, in reality, not only government spending, but also other factors (e.g. interest rate) can be stochastic. Finally, if government debt is in Dollar or Euro term, exchange rate and international capital ows also play a role. All those questions are interesting for future research. 18

19 References [1] Julio Escolano, 2010, A practical guide to public debt dynamics, scal sustainability, and cyclical adjustment of budgetary aggregates, International Monetary Fund, [2] Leonardo Bartolini, 1994, Government Ponzi games and the sustainability of public de cits under uncertainty, Ricerche Economiche 48, [3] Lex Meijdam, Martjin van de Ven and Harrie A.A. Verbon, 1996, The dynamics of government debt, European Journal of Political Economy Vol.12, [4] Michael Bergman, 1999, Testing government solvency and the No Ponzi Game condition, Applied Economics Letters, 2001, 8, [5] N.Gregory Mankiw, 2007, Macroeconomics, 6th edition, Worth Publishers. [6] Olivier Jean Blanchard and Philippe Weil, 1992, Dynamic e ciency, the riskless rate, and debt ponzi games under uncertainty, NBER Working Paper #3992, [7] Richard Baldwin and Charles Wyplosz, 2009, The Economics of European Integration, Third Edition, McGraw-Hill Higher Education Press. [8] Rob Cox and Martin Hutchinson, 2011, "Could Japan s debt lead to a crisis?", The New York Times, May ( [9] Tito Boeri and Jan van Ours, 2008, The Economics of Imperfect Labor Markets, Princeton University Press. [10] Wolfgang Kuhle, 2010, The optimum structure for government debt, MEA Discussion Paper No , [11] World Economic Outlook Database, International Monetary Fund, Sepeterber