Mobile Infrastructure and Rural Business Enterprises: Evidence from SIM Registration Mandate in Niger

Business enterprises and non-agricultural startups in rural economies play crucial roles in ending the vicious cycle of poverty. The propagation of business enterprises are, however, subject to a high degree of institutional frictions and vacuums e.g., information; but mobile infrastructure which has the externality of flowing information can help overcome most of these vacuums through reduced fixed costs, lower cost of information or search, and outreach to a broader customer base. This paper studies the effects of mobile infrastructure ( "mobile use activity" ) on propagation of rural business enterprises in Niger. Instrumental variable estimates exploit the exogenous introduction of the 2013 national mandatory SIM registration reform which provides a quasi-experimental set-up for mobile use and activity. The mandate deactivated about one-third of all existing prepaid SIMs and led to a remarkable decline in mobile use activity. The results suggest that there is economically substantial effect of mobile infrastructure on propagation of business enterprises. Moving a household with mobile use activity to a no mobile use activity environment due to SIM deactivation results in roughly 33.1 percent points decline in the likelihood of engaging in non-agricultural business enterprises. Most of this effect come from three major sources: households' ownership of a business service or center; ownership of small income generating activities; and involvement in the transformation of agricultural products. There is suggestive evidence that the reform's impact is disproportionately larger for women. With the expansion of mandatory SIM registration reforms in developing countries, the findings have extended implications in these contexts.

The Fiscal Theory of the Price Level with a Bubble * Markus Brunnermeier † , Sebastian Merkel ‡ , Yuliy Sannikov § April 28, 2020 1 Introduction Different monetary theories emphasize different roles of money and different equilibrium equations to determine the price level. The Fiscal Theory of the Price Level (FTPL) stresses the role of money as a store of value and argues that the real value of all outstanding government debt, i.e., the nominal debt level divided by the price level, is given by the discounted stream of future primary government surpluses. Primary surpluses are the difference between government revenue and expenditures excluding interest payments. Absent government default, an increase in primary deficits leads to an increase in the price level, i.e., inflation, by devaluing outstanding debt.
Critics of the FTPL often point to Japan. Even though Japan has mostly run primary deficits since the 1960s (see Figure 1) and with no primary surpluses in sight, the price level has not risen much. Indeed, inflation levels are depressed even though the government and central banks leave no stone unturned to boost inflation closer to 2%.
In this paper, we revisit the key FTPL equation and argue that including the typically ignored bubble term allows us to reconcile the FTPL with Japan's experience. Indeed, we show that the transversality condition is often insufficient to rule out a bubble on the aggregate economy, refuting the usual justification to simply dismiss the bubble term. While the FTPL literature puts a lot of emphasis on distinguishing between monetary and fiscal dominance, the bubble term cannot be ignored under any policy regime. A bubble term emerges whenever the real rate paid on government debt is persistently below the growth rate of the economy, i.e., whenever r ≤ g. It is well known that this can be the case in overlapping generations models (Samuelson 1958), models of perpetual youth (Blanchard 1985), and incomplete market modelsà la Bewley (1980). In this paper we provide another simple example based on Brunnermeier and Sannikov (2016a,b) in which the r ≤ g outcome arises naturally and agents can invest in both physical capital and government bonds. Physical capital is subject to uninsurable idiosyncratic return risk. Hence, the expected return on capital exceeds the growth rate g since agents require a risk premium. Government bonds are the safe asset in the economy and allow agents to indirectly share part of their idiosyncratic risk. High idiosyncratic risk makes the government bond more attractive and depresses r below g.
By "printing" bonds at a faster rate, the government imposes an inflation tax that reduces the return on the bonds further. Since government bonds are a bubble, the government in a sense "mines a bubble" to generate seigniorage revenue. The resulting seigniorage revenue can be used to finance government expenditures without ever having to raise extra taxes.
For example, if the primary surpluses are always negative, then their discounted stream is also negative, and only the positive value of the bubble can ensure a positive price level. The size of the bubble, and hence the price level, is determined by wealth effects and goods market clearing. A larger bubble raises agents' wealth and hence their demand for output. To ensure goods market clearing, the bubble has to take on a certain size, which together with the FTPL equation determines the price level.
The price level is uniquely determined if the fiscal authority backs the bubble to rule out equilibria that lead to hyperinflation. Such fiscal backing is only required off-equilibrium.
Literature. Classic references for the FTPL are Leeper (1991), Sims (1994), and Woodford (1995). For more compehensive treatments see Leeper and Leith (2016) and a recent book draft by Cochrane (2019). All of these references do not discuss the existence of bubbles. There is an extensive literature on rational bubbles. Survey papers include Miao (2014) and Martin and Ventura (2018). More recently, Jiang, Lustig, Van Nieuwerburgh, and Xiaolan (2019) provide convincing empirical evidence that U.S. government debt has a bubble component.

The FTPL Equation with a Bubble
In this section, we derive the key equation of the fiscal theory of the price level in a generic partial equilibrium setting. We subsequently discuss when there may be a bubble term in this equation that has previously been ignored in the literature and briefly conclude what the general fiscal theory equation tells us about sources of seigniorage. In the following section, we elaborate more on these points in the context of a specific example in general equilibrium.

Revisiting the Derivation of the FTPL Equation
The derivation of the fiscal theory equation starts with the government flow budget constraint. In discrete time, this constraint is given by where B t is the nominal face value of outstanding government bonds, M t is the nominal quantity of money in circulation, P t is the price level, T t are (real) taxes, G t is (real) government spending, and i t , i m t are the nominal interest rates paid on bonds and money, respectively. i m t can be smaller than i t if money provides transaction services. If ξ t is a real SDF process that prices government bonds, then 1 = E t [ξ t+1 /ξ t · P t /P t+1 (1 + i t )]. Using this property, dividing the government budget constraint by P t and rearranging yields Iterating this forward until period T implies Up to this point, we have merely rearranged and iterated the government budget constraint and assumed that there is some SDF process ξ t that prices government bonds in equilibrium. To derive the fiscal theory equation, the literature now typically proceeds by invoking a private-sector transversality condition to eliminate a terminal value of government debt when passing to the limit T → ∞. In this paper, we focus on environments where the transversality condition does not eliminate the terminal value in the limit.
When taking the limit T → ∞, we therefore arrive at the more general equation 1 From now on, we switch to continuous time in order to make our formal arguments more elegant. The continuous-time version of the last equation is given by 2 This equation for the real value of government debt holds in any monetary model. While most conventional monetary models treat this equation as an intertemporal government budget constraint that holds on-and off-equilibrium, in the FTPL it is an equilibrium condition that determines the price level.

When Can a Bubble Exist?
Equation (1) differs from the standard fiscal theory only by the presence of an additional bubble term. When can this bubble term be nonzero? Well-known examples are bubbles in OLG (Samuelson 1958) and perpetual youth (Blanchard 1985) models. In Section 3 we present another example with incomplete idiosyncratic risk sharing. Here, we make some generic points that apply to all example models. For tractability, let us focus on environments with a stationary debt-to-GDP ratio and no risk. In this case, the real value of government debt B T +M T P T is Bt+Mt Pt e g(T −t) , where g is the growth rate of the economy, and ξ T /ξ t = e −r f (T −t) with r f denoting the real risk-free rate. By substituting these expressions into equation (1), we see that the bubble term does not vanish in the limit if r f ≤ g. More generally, the correct risk-adjusted discount rate compensating for the real risk inherent in B T +M T P T must be used in the comparison instead of the risk-free rate to determine whether a bubble is possible.
For any agent with recursive isoelastic utility (which includes CRRA utility) that is marginal in the market for government debt, the risk-free rate is (for environments with non-stochastic investment opportunities) where ρ > 0 is the agent's time preference rate, γ is the relative risk aversion coefficient, ψ is the EIS, µ c is the growth rate of agent-specific consumption, and σ c is a vector or relative risk exposures of agent-specific consumption to Brownian risk factors. 3 · denotes the standard Euclidian norm. This equation is linked to the growth rate of the economy through individual consumption growth µ c -e.g., in a representative agent economy with a balanced growth path µ c = g.
1 Mathematically, the sum of the three limits in the decomposition below may not be well-defined, even if the limit of the sum is. In this case, the right-hand side should be interpreted as the limit of the sum. For instance, this can happen if the bubble term is ∞, but the present value of surpluses is −∞. While this may seem a pathological case, it can make sense economically because the bubble and surpluses are not separately tradeable, but necessarily bundled together to one asset: government debt. As long as the value of this asset is well-defined and finite, infinite subcomponents do not imply arbitrage opportunities or infinite utility.
2 A formal derivation can be found in Online Appendix A.1. 3 Here, we assume that all risk takes the form of Brownian risk. The intuition derived from the argument is unaltered if more general sources of consumption risk are permitted.

Equation
(2) suggests three reasons for the growth rate to exceed the risk-free rate. First, a higher growth rate g may not always imply higher individual consumption growth µ c . This is the case, for example, in an OLG model with population growth. Second, if agents do not have strong desires to smooth consumption (large ψ), then the risk-free rate is very insensitive to changes in consumption growth and an increase in growth does not translate into an increase in the risk-free rate. Third, large individual risk exposure (large σ c 2 ) or risk aversion (large γ) may depress the risk-free rate through the last term in equation (2) and offset any positive effects of growth g on r f through ψ −1 µ c . In the example we provide below, the latter channel generates a bubble in government debt. However, the main insights we generate from this example are insensitive to this choice and would equally apply to other environments in which a bubble term in equation (1) is possible.
The possibility of r f ≤ g is not merely a theoretical curiosity. Historically, real interest rates on government bonds of advanced economies have mostly been below the growth rate. Even Abel, Mankiw, Summers, and Zeckhauser (1989), who are often cited as providing evidence against the existence of rational bubbles, report that the safe interest rate r f is smaller than g. With the more recent decline in r f , as stressed by Blanchard (2019), the evidence for r f < g has become more clear-cut.

Three Forms of Seigniorage
Equation (1) suggests three forms of seigniorage, which here we define simply as government spending that is not backed by offsetting future taxes. The first takes the form of a dilution of private claims to future primary surpluses through surprise devaluations of existing government debt or money. 4 Under rational expectations, this cannot be a regular source of revenue for governments. For the U.S., Hilscher, Raviv, and Reis (2014) assess the possibility of future surprise devaluations based on option-implied (risk-neutral) probabilities and conclude that this form of seigniorage is perceived to be a negligible source of revenue. The likelihood of a devaluation exceeding 5% of GDP is less than 1%.
A second form of seigniorage comes from exploiting the liquidity benefits (convenience yield) of "narrow" money (M in equation (1)). This form of seigniorage can only be extracted from the portion of government debt that takes the form of "narrow" money and provides liquidity services. It depends on the interest rate differential ∆i = i − i m between illiquid and liquid government debt. It is small if either that differential is small or if the stock of "narrow" money is only a small part of total government debt. 5 This form of seigniorage is not an important funding source for advanced economies. For example, in the U.S., Reis (2019) reports a flow revenue of approximately 0.36% of GDP and estimates a present value of ≈ 20% and, at most, 30% of GDP. Moreover, in the future the ∆i term is likely to decline, because central banks pay interest on reserves and as money becomes more digitalized, its velocity rises.
Besides these standard forms of seigniorage, equation (1) suggests a third form of seigniorage that has remained unexplored in prior work and is the focus of this paper. The government can "mine" the bubble by using its outstanding government debt to run an ever-expanding Ponzi scheme: letting the stock of government debt grow generates a steady revenue flow that does not have to be paid for by future taxes as long as a bubble term is present in equation (1). Unlike a surprise dilution through inflation, dilution of the bubble value is even feasible if it is fully expected by the private sector. This form of seigniorage is arguably larger than the officially measured seigniorage from growing narrow money M because all revenue from growing B + M is relevant for bubble mining.
3 A Simple Example with a Bubble: "I Theory without I" There are several model structures in which rational bubbles can exist and thus the bubble term in equation (1) does not necessarily disappear. We illustrate this in a simple example with incomplete idiosyncratic risk sharing, where a bubble and a productive asset (capital) can coexist, yet the model is sufficiently simple to have a closed-form solution. The model is a streamlined version of the "I theory of money" (Brunnermeier and Sannikov, 2016a) without banks and has been previously analyzed in Brunnermeier and Sannikov (2016b) and Di Tella (2019). Here, we add fiscal policy and reinterpret money in their model as bonds. For simplicity, we abstract from the presence of additional "narrow" money that yields transaction benefits. 6 After introducing the model, we discuss under which conditions the private-sector transversality condition is insufficient to rule out the bubble, how the standard fiscal theory argument for price level determination has to be adjusted to still guarantee a uniquely determined price level, and how the bubble can be "mined" for fiscal spending.

A Model with Idiosyncratic Return Risk
Environment. There is a continuum of households indexed by i ∈ [0, 1]. All households have identical logarithmic preferences with discount rate ρ. 7 Each agent operates one firm that produces an output flow ak i t dt, where k i t is the capital input chosen by the firm. Absent market transactions of capital, capital of firm i evolves according to where ι i t k i t dt are physical investment expenditures of firm i (in output goods), Φ is a concave function that captures adjustment costs in capital accumulation, δ is the depreciation rate, andZ i is an agent-specific Brownian motion that is i.i.d. across agents i.Z i introduces firm-specific idiosyncratic risk. To obtain simple closed-form expressions, we choose the functional form Φ (ι) = 1 φ log (1 + φι) with adjustment cost parameter φ ≥ 0 for the investment technology.
The key friction in the model is that agents are not able to share idiosyncratic risk. While they are allowed to trade physical capital and risk-free assets, they cannot write financial contracts contingent on individualZ i histories. As a consequence, all agents have to bear the idiosyncratic risk inherent in their physical capital holdings. 6 Other than adding an additional source of seigniorage, including transaction benefits into the analysis does not substantially alter our conclusions. For the FTPL with transaction benefits but no bubble, see, e.g., Sims (2019). 7 We assume log utility for maximum simplicity. At the expense of heavier notation and more involved algebra, the model can be solved in closed form with isoelastic recursive utility with EIS kept at 1 and arbitrary coefficient of relative risk aversion γ. The expressions for the model solution p, q, ι, ϑ remain identical under these generalized preferences, except thatσ has to be replaced everywhere with √ γσ.
Besides households, there is a government that funds government spending, imposes taxes on firms, and issues nominal government bonds. The government has an exogenous need for real spending gK t dt, where K t is the aggregate capital stock and g is a model parameter. The government imposes a proportional output tax (subsidy, if negative) τ t on firms. Outstanding nominal government debt has a face value of B t and pays nominal interest i t . B t follows a continuous process dB t = µ B t B t dt, where the growth rate µ B t is a policy choice of the government. In short, the government chooses the policy instruments τ t , i t , µ B t contingent on histories of prices taking g as given and subject to the nominal budget constraint 8 where P t denotes the price level as in Section 2. The model is closed by the aggregate resource constraint where C t := c i t di is aggregate consumption and ι t = ι i t k i t /K t di is the average investment rate.
Price Processes and Returns. Let q t be the market price of a single unit of physical capital. Then, q t K t is private capital wealth. Let further p t := Bt/Pt Kt be the ratio of the real value of government debt to total capital in the economy. 9 Then, the real value of the total stock of government bonds is p t K t and the real value of a single government bond is ptKt Bt . It is convenient to define the share of total wealth in the economy that is due to bond wealth, We postulate that q t and p t have a generic deterministic evolution dq t = µ q t q t dt, dp t = µ p t p t dt.
Whenever p t , q t = 0, the unknown drifts µ q t , µ p t are uniquely determined by the local behavior of q t and p t , respectively. In the following, we also use the notation µ ϑ t :=θ t ϑt = (1 − ϑ t ) (µ p t − µ q t ). Households can trade two assets in positive net supply (if p t = 0), bonds and capital. Assume that in equilibrium ι t = ι i t for all i (to be verified below) such that aggregate capital grows deterministically at rate Φ(ι t ) − δ. Then, the return on bonds is The return on agent i's capital is 8 At this point, we do not impose additional restrictions on government policy. In particular, policy can be characterized by either monetary dominance or fiscal dominance and the choice of policy regime is irrelevant for most of our discussion. We do make more restrictive assumptions on policy in Section 3.4 where we explain how one can adjust the fiscal theory arguments for price level determination based on fiscal dominance if government debt has a bubble component.
9 It is more convenient to work with this normalized version of the inverse price level 1/Pt, because the latter depends on the scale of the economy and the nominal quantity of outstanding bonds in equilibrium, whereas pt does not.
Using the government budget constraint (3) to substitute out τ t a yields Household Problem and Equilibrium. Denote by n i t the net worth of household i and let θ i t be the fraction of net worth invested into bonds. Then net worth evolves according to The household chooses consumption c i t , real investment ι i t , and the portfolio share θ i t to maximize utility V i 0 subject to (6). The HJB equation for this problem is (using the returns expressions from the previous paragraph) This is a standard consumption-portfolio-choice problem, so we conjecture a functional form V t n i = α t + 1 ρ log n i t for the value function, 10 where α t depends on (aggregate) investment opportunities, but not on individual net worth n i . Substituting this into the HJB and taking first-order conditions yields 11 Tobin's q

Merton portfolio
Using the functional form Φ (ι) = 1 φ log (1 + φι) and goods market clearing (4), the first two equations aggregated across agents imply 10 We relegate the technical but standard verification argument to Online Appendix A.2. 11 In particular, the first condition verifies ι i t = ιt, and the last condition already uses this fact to eliminate Φ ι i − Φ(ιt).
which determines the equilibrium uniquely up to the nominal wealth share ϑ t . Bond market clearing and the fact that all households choose the same θ i t imply θ i t = ϑ t and substituting this and goods market clearing into the first-order condition for θ i gives the additional condition (after solving for µ ϑ ) This is a backward equation for ϑ t that has been derived under the assumption that bonds have a positive value (ϑ t > 0). In particular, in these cases multiplying the equation by ϑ t represents an equivalence transformation. Furthermore, if ϑ t = 0, then by no arbitrage agents must expect alsoθ t = 0; otherwise, they could earn an infinite risk-free return from investing into bonds. Consequently, the ODĖ must hold along any equilibrium path, regardless of whether bonds have positive value or not.
Steady-State Equilibria. We now focus on government policies that holdμ B and τ constant over time and consider steady-state equilibria with constant p and q -and thus constant ϑ. All such equilibria must solve equation (4) withθ = 0. The right-hand side is a third-order polynomial, so there are three solutions to this equation. ϑ = 0 is always a valid equilibrium. In this equilibrium, government bonds have no value, p = 0, the price level is infinite, P = ∞, and the government does not raise primary surpluses, τ a = g. Substituting ϑ = 0 into the equations for ι, p, and q yields the nonmonetary steady state The other stationary solutions to (7) are 1 − ϑ = ± √ ρ+μ B σ . The "−" solution has always the property ϑ > 1 and would therefore imply either a negative capital price (if p > 0) or a negative value of government bonds. Both cases violate free disposal and therefore this solution cannot be a valid equilibrium. The remaining solution ϑ =σ − √ ρ+μ B σ corresponds to a valid equilibrium, ifσ ≥ ρ +μ B . In this case, there is a second monetary steady state In the following we focus exclusively on this monetary steady state. In particular, from now on we make the assumptionσ ≥ ρ +μ B . In Section 3.4 we show that an off-equilibrium modification to the fiscal policy rule can select this steady state as the unique equilibrium.

Transversality Condition and Existence of a Bubble
In our model, government debt can have value even in the absence of primary surpluses (μ B ≥ 0) because it has a bubble component. It provides a store of value that is free of idiosyncratic risk, which allows agents to self-insure against their risk exposure. In this subsection we discuss why the private-sector transversality condition may not rule out the existence of a bubble. For each individual agent, the transversality condition, is necessary for an optimal choice. We can write n i T = n b,i T + n k,i T , where n b,i T is bond wealth at time T and n k,i T is capital wealth at time T . Because n b,i , n k,i ≥ 0, the transversality condition for total wealth also implies individual transversality conditions for bond wealth and capital wealth, respectively. The former condition seems to suggest that there is a transversality condition on government bonds that should rule out a bubble. This argument overlooks that individual bond wealth n b,i T is optimally chosen to be risky, because agents constantly adjust their portfolio in response to idiosyncratic shocks. It is therefore discounted by the individual agent at a discount rate that takes into account the idiosyncratic risk. Specifically, n b,i T = ϑn i T and the SDF is = e −ρT ϑ and thus the individual transversality condition on bond wealth is trivially satisfied in the model. Yet, when valuing a marginal additional unit of bonds, the relevant discount rate from the perspective of all agents is the risk-free rate, because government bonds do not have idiosyncratic risk. Formally, the marginal time 0 valuation of any agent i of the entire time T bond wealth in the economy is and the latter expression does not converge to zero, if r f ≤ g. Nothing in the model implies that the risk-free rate must necessarily be above the growth rate. Indeed, the growth rate of the economy equals the growth rate of capital, g = Φ(ι) − δ, and the risk-free rate equals the return on bonds, by equation (5) (in steady state) Consequently, r f ≤ g, ifμ B ≥ 0. A nonnegative value ofμ B is consistent with the existence condition of a monetary equilibrium,σ ≥ ρ +μ B , as long as idiosyncratic risk is sufficiently large.

Mining the Bubble
In this section, we show how the government can mine a bubble, i.e., finance government expenditures without ever raising taxes for it. Primary surplus is defined as T t − G t = τ aK t − gK t =: sK t . Due to our assumptions on fiscal policy, it grows at the same rate as K. From the government budget constraint (3), s = −μ B p. Hence, in our model the fiscal theory equation (1) reduces to (this time more precisely with a limit of the sum): 12 Provided p > 0, equation (8) implies precisely three cases: 13 12 Since the model does not include "narrow" money, there is no ∆i term. 13 The apparent dichotomy, a positive bubble value and nonnegative surpluses or positive surpluses and no bubble, is due to the steady-state nature of our analysis. In a more general model, a positive present value of surpluses and a bubble can coexist.
1. s > 0,μ B < 0: then r f > g, P V S 0,∞ > 0 and a bubble cannot exist. This is the "conventional" situation considered in the literature.
2. s =μ B = 0: then r f = g, P V S 0,∞ = 0 and there is a finite positive bubble whose value exactly equals pK 0 and grows over time at the growth rate/risk-free rate.
3. s < 0,μ B > 0: then r f < g and thus the integral P V S 0,T converges to −∞ as T → ∞. Yet, p is still positive, which is only possible if there is an offsetting infinite positive bubble. These infinite values do not violate any no-arbitrage condition and are also not otherwise economically problematic, since the bubble cannot be traded separately from the claim on surpluses. Both are necessarily bundled in the form of government bonds. As long as Bt Pt = pK t is determined and finite in equilibrium, the model remains economically and mathematically sensible despite the infinite values in the decomposition of the value of government bonds. 14 In all three cases, the (possible) presence of a bubble grants the government some extra leeway. Clearly in case 3, the government can run a perpetual deficit, "mine the bubble" and never has to raise taxes to fully fund all government expenditures. In case 2, the existence of the bubble is beneficial, because the value of government debt is positive -allowing agents to share part of their idiosyncratic risk -despite the fact that the present value of primary surpluses is zero. Even in case 1, government debt is more sustainable since an unexpected drop of primary surpluses to zero results in a bubble instead of a total collapse of the value of debt.

Price Level Determination, Uniqueness, and Off-Equilibrium Policy
The key equation (1) of the fiscal theory of the price level without a bubble term can be solved for the price level as a function of the present value of primary surpluses and the outstanding quantity of nominal government debt. For a given real allocation and initial quantity of debt, this equation alone therefore pins down the price level. 15 In the fiscal theory with a bubble, this is no longer true because the size of the bubble is not determined by the present value identity itself. Instead, goods market clearing determines the price level. A larger real value of bonds, holding taxes constant, means bonds represent more net wealth for the private sector, which increases consumption demand through a wealth effect. The equilibrium price level is the price level at which consumption demand equals consumption supply. 16 The fiscal theory equation itself determines the size of the bubble as the residual value of government debt that is not explained by the present value of primary surpluses.
As a consequence, the presence of a bubble makes price level determination based on fiscal dominance more challenging because it eliminates the simple one-to-one relationship between the present value of primary surpluses and the price level. Even making primary surpluses completely exogenous may not be sufficient to determine the price level uniquely if government policy fails to pin down the value of the bubble. The same path of surpluses can be consistent with multiple paths for the bubble value and thus with multiple initial price levels. 17 The steady state equilibria derived above are consistent with a policy that fixesμ B at a constant level and adjusts taxes τ such that the government budget constraint (3) holds after any price history. 18 While simple, such a policy is clearly inadequate to determine the price level. This is evident from the existence of a nonmonetary equilibrium in which nominal government bonds are worthless (p = 0). However, government policy can easily be modified off-equilibrium to select the monetary steady state as the unique equilibrium. 19 To see this, recall that along any equilibrium path ODE (7) must hold. The converse is also true, provided ϑ is contained in [0, 1]: Lemma 1. An absolutely continuous function [0, ∞) → [0, 1], t → ϑ t corresponds to a model equilibrium, if and only if it satisfies equation (7).
Above, we have already provided the proof that equation (7) is necessary for an equilibrium. The proof of sufficiency requires (1) that equilibrium prices p and q consistent with market clearing can be expressed as a function of ϑ (see above) and (2) a number of technical arguments verifying that the resulting equilibrium satisfies all optimal choice conditions of households (see Online Appendix A.3).
With constantμ B , there is a continuum of solution paths for ϑ consistent with the requirement in Lemma 1, which can be indexed by the initial value ϑ 0 ∈ [0, ϑ * ], where ϑ * := √ ρ+μ B σ denotes the monetary steady-state level of ϑ. For any initial value but the right endpoint, ϑ asymptotically converges to 0. 20 Conversely, if for some reasons agents expected that the equilibrium value of ϑ could never fall below a positive threshold ϑ > 0, then all equilibria but the monetary steady state ϑ 0 = ϑ * could be ruled out.
These considerations suggest a simple off-equilibrium modification of the fiscal policy rule to achieve equilibrium uniqueness: fix an arbitrary threshold 0 < ϑ < ϑ * and, whenever ϑ falls below ϑ, switch from a constant debt growth rule (constantμ B ) to a positive surplus rule with a constant output tax rate τ > g/a for as long as ϑ ≤ ϑ. From the government budget constraint (3) it follows that under this modified fiscal policy ODE (7) becomeṡ It is easy to see that this modified ODE has a strictly negative left-hand side on the interval [0, ϑ * ) including the left endpoint, and therefore all solutions that start inside this interval turn negative in finite time. By Lemma 1, the only possible equilibrium path for ϑ is therefore the steady-state equilibrium ϑ = ϑ * . The above rule modifies fiscal policy only off-equilibrium, whereas along the equilibrium path of the remaining unique equilibrium the government is free to choose any debt growth rate net of interest paymentsμ B . This raises questions about the credibility and fiscal capacity to promise off-equilibrium surpluses. These issues are beyond the scope of the present paper.

Conclusion
This paper integrates the typically ignored bubble term in the FTPL equation, which is necessary to explain low inflation in countries with persistently negative primary surpluses. Brunnermeier, Merkel, and Sannikov (2020) argue that this model provides a tractable framework to discuss and evaluate Modern Monetary Theory and enrich the debate on sovereign debt sustainability analysis. The analysis can also be easily extended to include a transaction role of money. In this case "narrow money" has an interest rate advantage ∆i = i − i M , which constitutes another source of seigniorage besides the "bubble mining" emphasized in this paper.

A Online Appendix
A.1 Derivation of the Continuous-time Fiscal Theory Equation (Equation (1)) As in the discrete-time case, the derivation of the fiscal theory equation starts with the government flow budget constraint, which here is i t and i m t have the same meaning as in the main text and µ B t , µ M t are the growth rates of nominal bonds and money, respectively. 21 Multiplying the budget constraint by the nominal SDF ξ t /P t and rearranging yields Next, Ito's product rule implies Solving these last two equations for µ B t − i t ξt Pt B t dt and µ M t − i t ξt Pt M t dt, respectively, and substituting the results into equation (9) yields (after rearranging) or in integral form Up to this point, we have merely rearranged and integrated the government budget constraint. To derive the fiscal theory equation, the literature proceeds by using two equilibrium conditions. First, if the nominal SDF ξ/P prices the government bonds, then its expected rate of change must be the negative of the nominal interest rate. Then, the last stochastic integral on the right must be a martingale and disappears when taking conditional time-t expectations E t [·]. Second, a private-sector transversality condition is invoked to eliminate a terminal value of government debt when passing to the limit T → ∞. We perform the first operation, but do not want to restrict attention to environments where transversality can rule out a nonzero discounted terminal value. When taking the limit T → ∞, we therefore arrive at the more general equation (1).

A.2 Verification Argument for the Value Function
While one could adjust standard verification arguments for HJB equations, we present here a more direct verification proof for the functional form V t n i = α t + 1 ρ log n i that does not make reference to the HJB equation. In what follows, we drop all i superscripts as we always consider a single household's decision problem. Taking return processes dr B t , dr K t (ι t ) as given, the household chooses c, θ, ι to maximize subject to the net worth evolution (6). For any n > 0, denote by V 0 (n) := sup {ct,θt,ιt} E ∞ 0 e −ρt log c t dt the optimal objective at t = 0 22 , if n 0 = n. To make the dependence of n t on n 0 and the chosen policy {c t , θ t , ι t } explicit, we write furthermore n c,θ,ι By the previous auxiliary result, { n (2) n (1) c t , θ t , ι t } is then an admissible policy for n 0 = n (2) and thus where the last inequality follows from (10). In total, this argument implies the value function inequality for any arbitrary n (1) , n (2) , ε > 0. Because ε > 0 was arbitrary, the inequality must then also hold for ε = 0. Because n (2) , n (1) were arbitrary, the inequality must still hold, when the roles of n (1) and n (2) are interchanged. Taking both facts together implies the equality V 0 n (2) + 1 ρ log n (1) = V 0 n (1) + 1 ρ log n (2) .

A.3 Missing Steps in Proof of Lemma 1
It is to show that any solution ϑ : [0, ∞) → [0, 1] to (7) corresponds to a unique equilibrium of the model. For any such function, define ι, q, and p consistent with the expressions given in the main text, i.e., Because ϑ t ∈ [0, 1] at all times, q t , p t ≥ 0, so these expressions are consistent with free disposal of both money and capital. We now verify that ι t , q t , p t and θ t := ϑ t satisfy all household choice conditions and the aggregate resource constraint. One immediately verifies that ι t and q t satisfy households' optimal investment choice condition, q t = 1 Φ (ιt) = 1 + φι t . In addition, total wealth of all households is (p t + q t ) K t and because individual consumption demand c i t = ρn i t implies an aggregate consumption demand of C t = ρ (p t + q t ) K t , we obtain so this equilibrium candidate satisfies the aggregate resource constraint (4).
It is left to show that at the asset prices p and q, agents' capital portfolio share θ t = ϑ t is consistent with their optimal choice condition for θ t . We consider two cases: 1. If ϑ t > 0, then equation (7) (that ϑ satisfies by assumption) is equivalent to µ ϑ t = ρ +μ B t − (1 − ϑ t ) 2σ 2 and rearranging the latter equation and using θ t = ϑ t yields Next, by definition of ι t and q t (1 − ϑ t ) (1 + φ (a − g))
2. If ϑ t = 0, then p t = 0, hence money has no value and the return on money is not well-defined. Consequently, the household portfolio choice condition as stated in the main text is not directly applicable. Instead, households demand a finite quantity of money (which is consistent with equilibrium and θ t = 0), if and only ifṗ t ≤ 0, i.e., the value of money is expected to become positive in the infinitesimal future. Because p s ≥ 0 for all s, this condition reduces here toṗ t = 0 ⇔θ t = 0. We therefore have to show that ϑ t = 0 impliesθ t = 0. We do this in two steps.