Leveraged purchases of government debt and deflation


 Barrie Marshall
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1 Leveraged purchases of government debt and deflation R. Anton Braun Federal Reserve Bank of Atlanta Tomoyuki Nakajima Kyoto University October 5, 2011 Abstract We consider a model in which individuals have different beliefs on the amount of taxes the government collects in the future. We analyze how such uncertainty affects inflation for different asset market structures. We particularly focus on the environment where shortsales of government bonds are not allowed but individuals can borrow to purchase government bonds subject to the nodefault constraint. We show that, compared to the case with frictionless asset markets, such asset market structure generates significant deflationary pressure. We argue that our model helps understand why deflation persists in Japan in spite of the increasing possibility of fiscal crisis. The key factor is that Japanese banks use deposits (borrow from depositors) to purchase a large share of the government debt. Keywords: deflation; fiscal risk; leverage; borrowing constraint. JEL Classification numbers: E31, E62, H60. For helpful comments and suggestions, we thank Kosuke Aoki, Fumio Hayashi, Atsushi Kajii, Keiichiro Kobayashi, Masao Ogaki, Kozo Ueda, and seminar participants at the Bank of Japan, CIGS, Hitotsubashi University (ICS), Kyoto University, and the University of Tokyo. Part of this research was conducted while Nakajima was a visiting scholar at the Institute for Monetary and Economic Studies at the Bank of Japan, whose hospitality is gratefully acknowledged. Nakajima thanks the financial support from the JSPS. Federal Reserve Bank of Atlanta. Institute of Economic Research, Kyoto University, and the Canon Institute for Global Studies. 1
2 1 Introduction Recently, the government debt problem has become a serious issue in many developed countries. It is typically a politically difficult task for a government to increase taxes to repay its debt. Failing to raise tax revenues may result in high inflation to reduce the real amount of debt, unless the government chooses to default on its debt. Then how would such a possibility of high inflation in the future affect the inflation rate now? That is the question addressed in this paper. Our main finding is that the asset market structure plays a crucial role in determining how the evolution of the inflation rate is affected by a fiscal problem of the government. Our model is a variant of the fiscal theory of the price level (FTPL) studied, for instance, by Leeper (1991), Sims (1994), Woodford (1995), Cochrane (2001), and Bassetto (2002), among others. Following the convention of this literature, we suppose that the government sets a sequence of real amounts of taxes it collects as well as a sequence of the nominal interest rate. In equilibrium the price levels are determined so that the flow budget constraint of the government is satisfied in every period. For simplicity, we consider a finite horizon model in which the government collects taxes only in the last period. The amount of taxes collected in the last period may be high, T H, or low, T L. The price level in the last period will be higher when the taxes collected in that period are smaller. We let T L be so small that the equilibrium inflation rate will be sizable when a state with T L occurs. For convenience, we say that a fiscal crisis occurs in such an event. In addition, we assume that individuals have different beliefs over how likely a fiscal crisis (and high inflation) occurs in the last period. 1 We examine the dynamics of the inflation rate under two asset market structures. The first structure is the standard frictionless asset markets where a complete set of contingent claims are traded. In the second one, contingent claims are not traded and there is endogenous borrowing limit as in Geanakoplos (2003, 2010). If a fiscal crisis occurs, the inflation rate rises and, as a result, the expost real rate of return on government bonds falls. Thus, individuals who believe that the fiscal crisis occurs less likely are those who are relatively optimistic about the rate of return on government bonds. In the first asset market structure with complete markets, relatively optimistic individuals buy the Arrow security which pays off in a state without a crisis and sells the Arrow security which pays off in a state with a crisis. Relatively pessimistic individuals do the opposite transactions. Here, the markets are flexible enough that both optimistic and pessimistic individuals can freely bet on their beliefs. 1 Our modeling of heterogeneous beliefs follows Geanakoplos (2003, 2010). 2
3 With the second asset market structure, however, there is asymmetry between optimistic and pessimistic individuals in this respect. On the one hand, optimistic individuals borrow to purchase government bonds as much as possible. Indeed, such a transaction is similar to purchasing the Arrow security corresponding to a state without a crisis. On the other hand, since government bonds cannot be short sold by assumption, there is no transaction corresponding to the Arrow security associated with a state with a crisis. What pessimistic individuals can do is only lending to optimistic individuals. They cannot bet on their beliefs as much as optimistic individuals can. The two asset market structures imply very different dynamics of the inflation rate. The dynamics of the inflation rate with the frictionless asset markets has two notable features. First, the possibility of a fiscal crisis is immediately reflected in a high price level in the initial period. Second, along the path leading to the fiscal crisis, the inflation rate is distributed fairly evenly. In other words, there is a kind of inflation smoothing in the case with frictionless asset markets. The inflation dynamics with the second asset market structure is very different in these two respects. First, the initial price level does not react much to the possibility of a fiscal crisis in the future. Second, the inflation rate is distributed unevenly along the path leading to the crisis. That is, the inflation rate remains very low until the second to the last period, and then it jumps up suddenly in the last period if the crisis does occur. In this sense, our second market structure where optimistic agents can borrow to purchase government bonds generates significant deflationary pressure until the crisis occurs, but once it does occur, the inflation rate rises much more than the frictionless case. Intuition for this result is simple. As is discussed above, with frictionless asset markets, both optimistic and pessimistic individuals can bet on their beliefs, but with incomplete markets with limited borrowing, only optimistic individuals can do so. This implies that the real price of government bonds is higher, and thus the inflation rate is lower in the second case. Our model can be used to understand why deflation persists in Japan in spite of the increasing possibility of a fiscal crisis there. A major fraction of the Japanese government bonds are held by domestic banks, and the fraction held by households is very small. The Japanese banks use deposits, that is, borrow from depositors to purchase Japanese government bonds. Households in Japan, on the other hand, are glad to make deposits at banks but do not want to purchase Japanese government bonds by themselves. This fits well with the second market structure of our model. Then the persistent deflation observed in Japan is quite consistent with the implication of our model. As our model suggests, however, observing deflation now does not imply that the possibility of a fiscal crisis in the future is negligible. 3
4 2 Twoperiod model Our model is an exchange economy in which all agents consume the single consumption good in each period. For the sake of simplicity, we do not model money explicitly, as is common in the literature on the fiscal theory. The price level is determined as the relative price of the consumption good and the government bond. We begin with a twoperiod version of the model. Periods are indexed by t = 0, 1. There are two states of nature in period 1, U and D. As discussed below, they are distinguished by the amount of taxes collected by the government. We let s t denote the state of nature in period t, where s 0 = 0 and s 1 S {U, D}. Individuals: There is a continuum of agents indexed by h [0, 1]. They are identical except for their beliefs on the probability that state U occurs in period 1. Specifically, we assume that agent h believes that s 1 = U with probability h. All agents have linear preferences of the form: c 0 + γ h (s 1 )c(s 1 ) (1) s 1 S where γ h (s 1 ) denotes the subjective probability of agent h that state U occurs in period 1: { h, for γ h s1 = U, (s 1 ) = 1 h, for s 1 = D. Let y t denote the endowment of the consumption good that each agent receives in period t = 0, 1. Note that it does not depend on the state in period 1. Storage technology is available for all agents, which is risk free and whose gross rate of return is denoted by R > 1. Government: Let > 0 be the nominal amount of initial government debt, which is equally held by the agents in the economy. The government collects taxes and issues oneperiod bonds. There is no government consumption. Taxes are lump sum and imposed equally on all agents. Let T 0 and T (s 1 ) denote the real amount of taxes in period 0 and in state s 1 S of period 1, respectively. Let B 0 be the nominal amount of bonds issued by the government in period 0 and q 0 be the price of those bonds. The flow budget constraints of the government in period 0 and in state s 1 S of period 1 are given by 2 = T 0 + q 0 B 0, (2) B 0 = P (s 1 )T (s 1 ), s 1 S, (3) 2 As is often argued, the standard formulation of the fiscal theory has some ambiguity in exactly what constraints the government is subject to. To clarify it, Bassetto (2002) develops a game theoretic version of the fiscal theory. It is straightforward to rewrite our model in the same way as Bassetto (2002). 4
5 where and P (s 1 ) are the price levels in period 0 and in state s 1, respectively. The government sets {T 0, {T (s 1 )} s1 S} as fiscal policy, and q 0 (0, 1] as monetary policy (note that 1/q 0 is the oneperiod nominal interest rate). The amount of government bonds issued in period 0, B 0, is determined so as to satisfy demands from the private sector. We assume that the government collects taxes in the following fashion: T 0 = 0, (4) T (s 1 ) = { TH, if s 1 = U, T L, if s 1 = D, (5) where T H > T L > 0. In what follows we refer to state D in period 1 as the state with fiscal crisis, to emphasize the fact the amount of taxes collected by the government is very small. Also, we call (q 0, T H, T L ) (0, 1] R 2 ++ as a policy, which is given exogenously. 2.1 Complete markets We consider three different asset market structures and examine how the equilibrium inflation rates vary across them. Let us begin with the case without frictions. A complete set of contingent claims (Arrow securities) are traded in period 0. Let q(s 1 ) denote the price of the Arrow security that pays off one unit of account in period 1 if and only if s 1 occurs. Each agent h [0, 1] maximizes the utility (1) subject to the budget constraints: c 0 + k 0 + s 1 S q(s 1 ) b(s 1) + q 0 b 0 + y 0, (6) c(s 1 ) Rk 0 + b(s 1) P (s 1 ) + b 0 P (s 1 ) + y 1 T (s 1 ), s 1 S, (7) c 0, k 0, b 0, c(s 1 ) 0, s 1 S, (8) where k 0 is the amount goods stored in period 0, b 0 is the amount of government bonds purchased in period 0, and {b(s 1 )} s1 S period 0. are the amounts of Arrow securities purchased in Given an exogenously given policy (q 0, T H, T L ), a competitive equilibrium with complete asset markets consists of an allocation {c h 0, [ch (s 1 )] s1 S, k h 0, bh 0, [bh (s 1 )] s1 S} h [0,1], supply of government bonds B 0, and prices {, [P (s 1 )] s1 S, [q(s 1 )] s1 S} such that (i) for each agent h [0, 1], {c h 0, [ch (s 1 )] s1 S, k h 0, bh 0, [bh (s 1 )] s1 S} solves her utility maximization problem; (ii) the government flow budget constraints (2)(3) are satisfied, where taxes {T 0, [T (s 1 )] s1 S} are 5
6 given by (4)(5); and (iii) all markets clear: (c h 0 + k h 0 ) dh = y 0, (9) c h (s 1 ) dh = y 1 + R 1 0 k h 0 dh, s 1 S, (10) b h 0 dh = B 0, (11) b h (s 1 ) dh = 0, s 1 S. (12) Since a complete set of Arrow securities are traded, both the government bond and storage are redundant assets. Since the government bond pays off one unit of account irrespective of the state in period 1, the noarbitrage condition requires that q(s 1 ) = q 0. (13) s 1 S Similarly, since one unit of goods put into storage in period 0 yields R units of goods in each state of period 1, the noarbitrage condition requires that s 1 S as long as the storage technology is used in equilibrium. q(s 1 )P (s 1 ) R = 1, (14) It follows that the flow budget constraints (6)(7) are combined to obtain the lifetime budget constraint: Rc 0 + ( ) q(s 1 )P (s 1 ) Rc(s 1 ) R + y 0 + q(s 1 )P (s 1 ) R [ y 1 T (s 1 ) ]. (15) s 1 S s 1 S Each agent maximizes (1) subject to (15). Since R > 1, no one consumes in period 0: c h 0 = 0, for all h [0, 1]. (16) Hence 1 0 kh 0 dh = y 0 > 0 and thus (14) holds. Let us define h 0 as h 0 = q(u)p (U) R. (17) It follows from (14) that h 0 is between 0 and 1, and that 1 h 0 = q(d)p (D) R. Then we see that h 0 is the marginal agent in the sense that agents h > h 0 chooses c h (D) = 0 and agents h < h 0 chooses c h (U) = 0. 6
7 Next consider the government. It follows from (2) and (4) that It follows that the price level in period 1 becomes B 0 = q 0. (18) P (s 1 ) = 1, s 1 S. (19) T (s 1 ) q 0 Using the noarbitrage condition (13), the government s flow budget constraint (2)(3) are combined as = s 1 s q(s 1 ) P (s 1 )T (s 1 ) Using h 0 defined in (17), we can rewrite this equation as which determines. = h 0 T H R + (1 h 0) T L R, (20) Let us turn back to the agent s lifetime budget constraint (15). Using (20), its right hand side becomes Thus (15) can be rewritten as It follows that ( ) R + y 0 + q(s 1 )P (s 1 ) R [ y 1 T (s 1 ) ] = Ry 0 + y 1. s 1 S Rc 0 + h 0 c(u) + (1 h 0 )c(d) Ry 0 + y 1. Market clearing implies which leads to { 1 c h h (U) = 0 (Ry 0 + y 1 ), for h > h 0, 0, for h h 0, { 0, for h > h0, c h (D) = Ry 0 + y 1 = 1 1 h 0 (Ry 0 + y 1 ), for h h c h (U) dh = 1 h 0 h 0 (Ry 0 + y 1 ), (21) (22) h 0 = 1 2. Finally, the Arrow security prices q(s 1 ) are determined by solving q(s 1)P (s 1 ) R = 1 2 s 1 S. The next proposition summarizes the result. 7 for each
8 Proposition 1. In the two period model with complete asset markets, (i) the equilibrium consumption allocation {c h 0, [ch (s 1 )] s1 S} h [0,1] is given by (16), (21), and (22), where the marginal agent h 0 = 1 2 ; and (ii) the equilibrium price levels {, [P (s 1 )] s1 S} are given by (19) and (20). 2.2 No borrowing Now suppose that Arrow securities are no longer traded, so that government bonds and storage are the only asset available to agents. Here we assume that borrowing is not allowed. 3 Then the budget constraints for each individual become c 0 + k 0 + q 0 b 0 + y 0, (23) c(s 1 ) Rk 0 + b 0 P (s 1 ) + y 1 T (s 1 ), s 1 S, (24) c 0, k 0, b 0, c(s 1 ) 0, s 1 S. (25) Each agent maximize (1) subject to these constraints. A competitive equilibrium is defined similarly. Given a policy (q 0, T H, T L ), a competitive equilibrium without borrowing consists of an allocation {c h 0, [ch (s 1 )] s1 S, k h 0, bh 0 } h [0,1], supply of government bonds B 0, and prices {, [P (s 1 )] s1 S} such that (i) for each agent h [0, 1], {c h 0, [ch (s 1 )] s1 S, k0 h, bh 0 } solves her utility maximization problem; (ii) the government flow budget constraints (2)(3) are satisfied, where taxes {T 0, [T (s 1 )] s1 S} are given by (4)(5); and (iii) all markets clear, i.e., equations (9)(11) hold. Consider the utility maximization problem of agent h [0, 1]. Given the utility function (1) and the storage technology with rate of return R > 1, no agent consumes in period 0: c h 0 = 0 for all h [0, 1]. Between the two assets (government bonds and storage), each agent holds the one with the highest expected return. Let h 0 [0, 1] be the agent who is indifferent between government bonds and storage. That is, h 0 is the solution to { 1 q 0 P (U) h } P (D) (1 h 0) = R, (26) where the lefthand side is the real rate of return on government bonds, and the righthand side is that on storage. Agents h > h 0 have optimistic beliefs about the return on government bonds so that they hold only government bonds. To the contrary, agents h < h 0 are so 3 Note that there are two assets (storage and government bonds) and two states (U and D). Thus the asset markets are complete. However, because of the borrowing constraint, the competitive equilibrium in this case is different from the one considered in the previous subsection. 8
9 pessimistic about the return on government bonds that they do not hold government bonds at all. Thus, the solution to the utility maximization problem is summarized as follows: c h 0 = 0, h [0, 1], (27) k h 0 = b h 0 = { 0, h > h0, + y 0, h h 0, q 0 ( + y 0 ), h > h 0, 0, h h 0, ( ) 1 c h P (s (s 1 ) = 1 ) q 0 + y 0 + y 1 T (s 1 ), h > h 0, ( ) R + y 0 + y 1 T (s 1 ), h h 0. (28) (29) (30) As in the previous case, B 0 = /q 0, and hence the price level in state s 1, P (s 1 ), is given as (19). From (29) it follows that the market clearing condition for government bonds is ( ) q 0 B 0 = (1 h 0 ) + y 0, which implies that the price level in period 0,, is given by The next proposition summarizes the result. = h 0 1 h 0 y 0. (31) Proposition 2. In the two period model without borrowing, (i) the equilibrium consumption allocation and portfolio, {c h 0, kh 0, bh 0, [ch (s 1 )] s1 S} h [0,1], are given by (27)(30); (ii) the marginal buyer of government bonds, h 0, is given by (26); and (iii) the equilibrium price levels {, [P (s 1 )] s1 S} are given by (19) and (31). 2.3 Leveraged purchases of government bonds Now let us allow agents to borrow and lend to each other. Then optimistic agents, who expect that the rate of return on government bonds is greater than the borrowing rate, would like to borrow as much as possible to purchase government bonds. To obtain a loan, a borrower is required to put government bonds as collateral. How much can an agent borrow with one unit of government bonds as collateral? We use the theory developed by Geanakoplos (2003, 2010) to determine it. Then in our model the maximum amount that an agent can borrow is determined by the nodefault constraint, that is, the constraint that the amount of repayments does not exceed the value of the collateral in any state. 4 4 See Appendix for the details of this argument. 9
10 Since there is no default on loans, loans are riskfree. Thus the interest rate on loans is equal to R in equilibrium (as long as the storage technology is used). Consider an agent who borrows φ 0 and purchases government bonds b 0 in period 0. He/she must repay Rφ 0 in period 1. The nodefault constraint requires that Rφ 0 b 0 P (s 1 ), for all s 1 S. Thus the budget constraints for each individual becomes c 0 + k 0 + q 0 b 0 + y 0 + φ 0, (32) c(s 1 ) Rk 0 + b 0 P (s 1 ) + y 1 T (s 1 ) Rφ 0, s 1 S, (33) Rφ 0 b 0 P (s 1 ), s 1 S, (34) c 0, k 0, b 0, c(s 1 ) 0, s 1 S. (35) Given a policy (q 0, T H, T L ), a competitive equilibrium with borrowing limited by the nodefault constraint consists of an allocation {c h 0, [ch (s 1 )] s1 S, k h 0, bh 0, φh 0 } h [0,1], supply of government bonds B 0, and prices {, [P (s 1 )] s1 S} such that (i) for each agent h [0, 1], {c h 0, [ch (s 1 )] s1 S, k0 h, bh 0, φh 0 } solves her utility maximization problem; (ii) the government flow budget constraints (2)(3) are satisfied, where taxes {T 0, [T (s 1 )] s1 S} are given by (4)(5); and (iii) all markets clear, i.e., equations (9)(11) hold and 1 0 φh 0 dh = 0. Consider the utility maximization problem. As before, no one consumes in period 0: c h 0 = 0 for all h [0, 1]. Also, let h 0 be the solution to (26), that is, the agent who is indifferent between government bonds and storage. Then for agents with h > h 0, the expected rate of return on government bonds is greater than the riskfree rate R, and thus want to borrow as much as possible to purchase government bonds. Since P (U) < P (D) in equilibrium, borrowing constraint (34) is expressed as φ 0 b 0 RP (D). For each agent h, the expected rate of ( ) return on the portfolio (k 0, b 0, φ 0 ) = 0, b 0, equals Note that (26) can be rewritten as h b 0 RP (D) 1 P (U) 1 P (D) q 0 1 RP (D). 1 P (U) h 1 P (D) 0 q 0 1 RP (D) = R. (36) Thus, optimistic agents h > h 0 will indeed borrow as much as possible to purchase government bonds. On the other hand, pessimistic agents h < h 0 do not hold government bonds but are willing to lend to the optimistic agents at the interest rate R. 10
11 Thus, the solution to the utility maximization problem is summarized as follows: c h 0 = 0, h [0, 1], (37) ( ) 1 ( ) q0 b h P 0 = 0 1 RP (D) + y 0, h > h 0, (38) 0, h h 0, ( ) 1 ( ) k0 h φ h 1 q0 0 = RP (D) 1 RP (D) + y 0, h > h 0, (39) + y 0, h h 0, 1 P (s 1 ) 1 ( ) P (D) c h q 0 (s 1 ) = 1 + y 0 + y 1 T (s 1 ), h > h 0, ( RP (D) ) (40) R + y 0 + y 1 T (s 1 ), h h 0. Here, since storage and lending are perfect substitutes, only k0 h φh 0 is determined for agents with h h 0. Of course, agents with h > h 0 chooses k0 h = 0 and φh 0 = bh 0 RP (D). As before, B 0 and P (s 1 ), s 1 S, are determined by (18) and (19). Then it follows from (38) that the market clearing condition for government bonds is expressed as ( ) q0 1 1 ( ) = (1 h 0 ) + y 0. (41) q 0 RP (D) Given P (s 1 ), s 1 S, the initial price level and the marginal agent h 0 are determined as the solution to (36) and (41). The following proposition summarizes the result. Proposition 3. In the two period model with leveraged purchases of government bonds, (i) the equilibrium consumption allocation and portfolio, {c h 0, kh 0 φh 0, bh 0, [ch (s 1 )] s1 S} h [0,1], are given by (37)(40); (ii) the marginal buyer of government bonds, h 0, is given by (36); and (iii) the equilibrium price levels {, [P (s 1 )] s1 S} are given by (19) and (41). 2.4 Numerical example To illustrate how the asset market structure affects the inflation rate, let us consider a simple numerical example. We set the parameter values as = 1, y 0 = y 1 = 1, q 0 = 1, R = 1.02, T H = R 2, and T L = T H /2. Furthermore, we assume that in period t = 1 everyone believes that s 1 = U with probability one. That is, prior to period 0, no one thinks that fiscal crisis will occur. It follows that P 1 = 1. Table 1 shows the inflation rates at t = 0 and at s 1 = D for the three asset market structures considered in the previous subsections, which are referred to as complete markets, no borrowing, and leverage, respectively. The inflation rate in period 0, π 0, is highest in the model with complete markets and lowest in the model with leverage. To the contrary, the inflation rate when fiscal crisis does occur, π(d), is highest with leverage and lowest with 11
12 complete markets. In period 0, agents start to get worried that fiscal crisis may occur in the future, with different degrees of confidence. With complete markets, such a concern is immediately reflected in a rise in the inflation rate in period 0, π 0. In the model with leverage, this is much less so initially, which, in turn, makes the inflation rate during the fiscal crisis greater. There is simple intuition behind this result. In the model with complete markets, every agent can freely bet on her beliefs: optimistic agents buy Arrow security U and sell Arrow security D, and pessimistic agents do the opposite. On the other hand, in the model with leverage, such a bet is allowed only for optimistic agents who borrow to purchase government bonds. Indeed, borrowing to purchase government bonds with binding collateral constraint (34) is equivalent to buying Arrow security U. With this asset market structure, however, there is no corresponding bet available for pessimistic agents. The only thing they can do is to lend to optimistic agents. This asymmetry raises the price level and hence lowers the inflation rate π 0 compared to the case with complete markets. It is also reflected in the fact that the marginal agent h 0 is larger in the model with leverage than in the model with complete markets. 3 T period model In this section we shall see that the effect of the asset market structure on the dynamics of the inflation rate becomes more apparent in the model with a longer time horizon. Suppose that there are T +1 periods indexed by t = 0, 1,..., T. In each period t = 1,..., T, a shock s t S = {U, D} realizes. For t = 1,..., T, let s t = (s 1,..., s t ) S t denote the history of shocks, and also let s 0 0 and S 0 {0}. In each period t, regardless of the history s t, agent h believes that s t+1 = U with probability h and s t+1 = D with probability 1 h. Then her expected utility is expressed as T t=0 s t S t γ h (s t )c(s t ), (42) where γ h (s t ) denote the subjective probability that agent h assigns to history s t in period 0, that is, { hγ γ h (s t h (s t 1 ), if s t = U, ) = (1 h)γ h (s t 1 ), if s t = D, with γ h (s 0 ) 1. 12
13 In each period t, every agent is endowed with the same amount of the consumption good, y(s t ) for s t S t. For simplicity, we assume that y 0, for t = 0, y(s t ) = 0, for all s t with t = 1,..., T 1, y T, for all s T. In addition, at the beginning of period 0, each agent is endowed with the same amount of government debt > 0. As in the twoperiod model, the government specifies a statecontingent path of lumpsum taxes, {T (s t ) : s t S t, t = 0,..., T }, as fiscal policy, and a statecontingent path of the price of government bonds, {q(s t ) : s t S t, t = 0,..., T }, as monetary policy. To be specific, the fiscal policy takes the following form: 0, for all s t with t = 0,..., T 1, T (s t ) = T L, for s T = D T, T H, for all s T D T. As before, we say that fiscal crisis occurs when the government collects only T L of taxes in the last period. Under our assumption (43), it occurs only if s t = D for all t = 1,..., T. Given the fiscal policy (43), the government debt B(s t ) evolves in equilibrium as (43) = q 0 B 0, (44) B(s t 1 ) = q(s t )B(s t ), s t S t, t = 1,..., T 1, (45) B(s T 1 ) = P (s T )T (s T ), s T S T. (46) 3.1 Complete markets Let q(s t+1 s t ) denote the price of the Arrow security traded at s t that pays off one unit of account if and only if s t+1 occurs in the next period. With a complete set of Arrow securities, the flow budget constraints for each agent are given by c 0 + k 0 + s 1 S c(s t ) + k(s t ) + q(s 1 s 0 ) b(s 1 s 0 ) + q 0 b 0 s t+1 S b(s t s t 1 ) P (s t ) c(s T ) b(s T s T 1 ) P (s T ) q(s t+1 s t ) b(s t+1 s t ) P (s t ) + y 0, (47) + q(s t ) b(st ) P (s t ) + b(st 1 ) P (s t ) + Rk(st 1 ), t = 1,..., T 1, s t S t, (48) + b(st 1 ) P (s T ) + Rk(sT 1 ) + y T T (s T ), s T S T, (49) c(s t ), k(s t ), b(s t ) 0, s t S t, t = 0,..., T, (50) 13
14 where b(s t+1 s t ) denotes the quantities of the Arrow securities purchased at s t, and b(s t ) denotes the quantities of government bonds purchased at s t. Given a policy ({q(s t )}, T H, T L ), a competitive equilibrium is defined as in the twoperiod case. 3.2 Leveraged purchases of government bonds Now suppose that Arrow securities are no longer traded, and borrowing is limited by the nodefault condition. The flow budget constraints for agents are given by c 0 + k 0 + q 0 b 0 c(s t ) + k(s t ) + q(s t ) b(st ) P (s t ) + y 0 + φ 0, (51) b(st 1 ) P (s t ) + Rk(st 1 ) Rφ(s t 1 ) + φ(s t ), t = 1,..., T 1, s t S t, (52) c(s T ) b(st 1 ) P (s T ) + Rk(sT 1 ) + y T T (s T ) Rφ(s T 1 ), s T S T, (53) Rφ(s t ) b(s t ) P (s t, s t+1 ), t = 0,..., T 1, st S t, s t+1 S, (54) c(s t ), k(s t ), b(s t ) 0, s t S t, t = 0,..., T, (55) 3.3 Numerical example Now consider our numerical example. As before, let = 1, y 0 = y T = 1, q(s t ) = 1 for all s t and t, R = 1.02, T H = R T +1, and T L = T H /2. Also, assume that in period 1, everyone believes that Pr(s T = D T ) = 0, that is, everyone believes that the government collects taxes of amount T H in period T for sure. It follows that P 1 = 1. 4 Conclusion Appendix References [1] Bassetto, Marco A gametheoretic view of the fiscal theory of the price level. Econometrica, 70, [2] Cochrane, John H Long term debt and optimal policy in the fiscal theory of the price level. Econometrica, 69,
15 [3] Fostel, Ana, and John Geanakoplos Leverage cycles and the anxious economy. American Economic Review, 98, [4] Geanakoplos, John Liquidity, default, and crashes: Endogenous contracts in general equilibrium. In Advances in Economics and Econometrics: Theory and Applications, Eighth World Conference, Vol. 2, Econometric Society Monographs. [5] Geanakoplos, John The leverage cycle. In NBER Macroeconomics Annual 2009, ed. Daron Acemoglu, Kenneth Rogoff, and Michael Woodford, University of Chicago Press. [6] Leeper, Eric M Equilibria under active and passive monetary policies. Journal of Monetary Economics, 27, [7] Sims, Christopher A A simple model for study of the determination of the price level and the interaction of monetary and fiscal policy. Economic Theory, 4, [8] Woodford, Michael Price level determinacy without control of a monetary aggregate. CarnegieRochester Conference Series on Public Policy, 43,
16 Table 1: Inflation rates and marginal buyers in the twoperiod model π 0 π(d) h 0 (1) complete markets (2) no borrowing (3) leverage Table 2: Inflation rates and marginal buyers in the threeperiod model π 0 π(d) π(d 2 ) h 0 h(d) (1) complete markets (2) leverage
17 80 inflation rate leveraged complete
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