Consider a simple production economy populated by a large number of identical infinitely lived individuals.
In each period \(t\), there are two goods: labor \(l_t\) and a consumption good, \(c_t\). The per-period utility function is \(U(c_t,l_t)\) and individuals discount the future at rate \(\beta\). A constant returns-to-scale technology is available to transform one unit of labor into one unit of output. The output can be used for private consumption or for government consumption.
The per-capita level of government consumption in each period, denoted \(g_t\), is exogenously specified. This government consumption is given by \(g_H\) in even periods and \(g_L = 0\) in odd periods. The initial stock of government debt is given and is zero. The government can default on inherited debt in any period. The government raises revenues by levying a proportional tax \(\tau_t\) on labor income.
Note: Suppose the endowment of potential labor each period is \(\bar{l}\). Let \(B_t\) denote the stock of government debt at the beginning of period \(t\). Let \(\bar{B}\equiv \sum_{t=0}^\infty \left(\prod_{s=0}^t {1\over R_s}\right) w_t \bar{l}\) denote a natural borrowing limit for the household. Let \(z_t\) denote whether the government defaults, with \[z_t = \begin{cases} 0, & \text{if default at period } t\\ 1, & \text{otherwise} \end{cases} \]
Given exogenous policy \(\left\{ g_t, z_t, B_{t+1}, \tau_t \right\}_{t=0}^\infty\), parameters \((\beta, \bar{l})\), and initial stock of debt \(\bar{B_0}=0\), a competitive equilibrium in this economy consists of the following sets:
Let \(U_{ct}\) be shorthand for \({\partial \over \partial c_t}U(c_t, l_t)\). Likewise for \(l_t\).
Assumming an interior solution where only the budget constraint binds in the household's problem, and letting \(\lambda_t\) be the Lagrange Multipliers for the Budget constraints, the first-order conditions are: \begin{align} {\color{name}c_{t}:} && {\beta^t U_{ct}} &= \lambda_t \\ {\color{name}l_{t}:} && {\beta^t U_{lt}} &= -\lambda_t (1-\tau_t) w_t \\ {\color{name}s_{t+1}:} && \lambda_t &= \lambda_{t+1}R_{t+1}z_{t+1} \\ {\color{name}l^f_{t}:} && w_t &= 1\\ \end{align} Combining the first order conditions, we get: \[\color{red}{U_{lt} \over U_{ct}}=-(1-\tau_t)\] \[\color{blue} {\lambda_t\over\lambda_{t+1}} = {U_{ct} \over \beta U_{ct+1}} = {(1-\tau_{t+1}) U_{lt} \over (1-\tau_t) \beta U_{lt+1} } = R_{t+1}z_{t+1}\] There is also the transversality condition: \[0 = \lim_{t\to\infty} \lambda_t s_{t+1} = \lim_{t\to\infty} {\beta^t U_{ct}} s_{t+1} \] These equations, along with the market clearing and budget constraints, characterize the equilibrium.
Rearranging the budget constraint and plugging in the above: \begin{align} R_t z_t s_t &= c_t - {\color{red}(1-\tau_t)} l_t + s_{t+1}\\ R_t z_t s_t &= c_t + {\color{red} {U_{lt} \over U_{ct}}} l_t + s_{t+1} \\ R_t z_t s_t U_{ct} &= c_t U_{ct} + l_t U_{lt} + s_{t+1} U_{ct} \mytag{*}\\ \end{align} Similarly, note that: \begin{align} {\color{blue} R_{t+1} z_{t+1} } s_{t+1} U_{ct+1} &= c_{t+1} U_{ct+1} + l_{t+1} U_{lt+1} + s_{t+2} U_{ct+1} \\ {\color{blue}{U_{ct} \over \beta U_{ct+1}}} s_{t+1} U_{ct+1} &= c_{t+1} U_{ct+1} + l_{t+1} U_{lt+1} + s_{t+2} U_{ct+1} \\ s_{t+1} U_{ct} &= \beta \left[c_{t+1} U_{ct+1} + l_{t+1} U_{lt+1} + s_{t+2} U_{ct+1}\right] \mytag{**}\\ \end{align} Repeatedly Plugging ** into *, we get that: \begin{align} R_0 z_0 s_0 U_{c0} &= c_1 U_{c1} + l_1 U_{l1} + s_{2} U_{c1}\\ &= c_1 U_{c1} + l_1 U_{l1} + \beta \left[c_{2} U_{c2} + l_{2} U_{l2} + s_{3} U_{c2}\right] \\ & \;\; \vdots \\ &= \sum_{t=0}^\infty\beta^t\left[c_{t} U_{ct} + l_{t} U_{lt}\right] + \lim_{t\to\infty} {\beta^t U_{ct}} s_{t+1} \\ \end{align} The left-hand side is equal to zero because \(s_0 = B_0 = 0\) is given by the problem, and the last term on the right-hand side is zero by the transversality condition. Thus we get the following: \[\boxed{0 = \sum_{t=0}^\infty\beta^t\left[c_{t} U_{ct} + l_{t} U_{lt}\right] } \mytag{Implementability Constraint}\]
Given exogenous spending requirements \(\left\{ g_t\right\}_{t=0}^\infty\), and parameters \((\beta, \bar{l})\), the Ramsey problem is to choose household allocations to solve: \[\max_{\left\{ c_{t}, l_t \right\}_t} \sum_{t=0}^\infty \beta^t U( c_{t},l_{t})\] such that \[0 = \sum_{t=0}^\infty\beta^t\left[c_{t} U_{ct} + l_{t} U_{lt}\right] \mytag{Implementability}\] and such that for all \(t\geq 0:\) \begin{align} c_{t}\geq 0, \; &\; l_t \in [0,\bar{l}] \mytag{Non-Negativity}\\ c_t + g_t &\leq l_t \mytag{Feasibility}\\ \end{align}
Let \(\pi_t \equiv (g_t, z_t, \tau_t, B_{t+1})\) be a policy at time \(t\). Let \(h_t \equiv (B_0, \pi_0, \pi_1, \cdots, \pi_t)\) be a history of government policies up to including that at time \(t\).
The government now has policy rules, which map histories onto next-period policies: \[\sigma^t \equiv (z^t:h_{t-1}\to \{0,1\},\;\;\; \tau^t:h_{t-1}\to \mathbb{R}_+, \;\;\; B^{t+1}:h_{t-1}\to \mathbb{R})\] And the household now has allocation rules: \[HH^t \equiv (c^t:h_{t}\to \mathbb{R}_+,\;\;\; s^{t+1}:h_{t}\to \mathbb{R},\;\;\; l^{t}:h_{t}\to [0,\bar{l}])\]
Additionally, the firm's problem in this environment will simply imply that \(w^t (h_t)=1\). So for simplicity, I will exclude the firm's problem and wages from the following definition.
Then given exogenous spending requirements \(\left\{ g_t \right\}_{t=0}^\infty\), parameters \((\beta, \bar{l})\), and initial stock of debt \(\bar{B_0}=0\), a sustainable equilibrium in this economy consists of:
Autarky is the worst sustainable equilibrium, if it is feasible.
By autarky, I mean that in the outcome induced by the sustainable equilibrium, \(B_t=s_t=0=z_t \;\forall t\), and so \(g_t\) is financed purely through taxes on labor income: \[\tau_t = \begin{cases} 0, & \text{in odd periods}\\ {g_H \over l_t}, & \text{in even periods} \end{cases} \] And \(c_t,l_t\) are such that \(c_t = (1-{g_t \over l_t})l_t = l_t-g_t\) and \[{U_{lt} \over U_{ct}}=-(1-\tau_t) = \begin{cases} -1, & \text{in odd periods}\\ -1 + {g_H \over l_t}, & \text{in even periods} \end{cases} \]
Note that autarky is sustainable. If the government has decided never to honor its debts in any possible history, then the household choosing to never lend money is a best response. And likewise, if the household never lends, then the government cannot borrow, and so may as well not honor its debts.
Autarky must be the worst sustainable equilibrium because at every possible history, deviating to autarky is feasible. The government could decide to start defaulting, and the household could decide to stop lending. Thus if there is any non-autarky sustainable equilibrium, the payoffs must be at least as good as autarky for it to satisfy the HH and Gov optimization problems in the definition of the sustainable equilibrium.
Going forward, let \(U^A_t\) denote the per-period utility to the household from the autarky allocation at time \(t\).
Consider the class of grim trigger strategies, where at time zero, the policy and allocation rules implement a competitive equilibrium, and if the government ever deviates from this plan, then the household stops lending. (And the government will always default if it deviated in the past.)
For any outcome capable of being induced by a sustainable equilibrium, this outcome can also be induced using a grim trigger strategy. The grim trigger also yields a sustainable equilibrium because, if the outcome is preferable to whatever happens in a deviation in the original sustainable equilibrium, then it will be preferable to deviation followed by autarky, and so the government will not deviate if this grim trigger is a credible threat. And secondly, this grim trigger is a credible threat because given that the government will start defaulting, it is a best response for the household to stop lending in any future history; while if the household stops lending, then it is a best response for the government to start always defaulting.
But what about the other direction? What outcomes can be induced by a grim trigger strategy? It is not enough for an outcome to be better than autarky. It must also be good enough that the utility loss of autarky is strong enough to overwhelm the benefits from deviation. Let \(U^D(h_{t})\) be the maximum possible utility that can be achieved in period \(t+1\) given history \(h_{t}\), and given that credit markets will subsequently close down:
\[U^D(h_t) = \max_{z, \tau, c, l}\] subject to \begin{align} c &\leq (1-\tau) l + R_{t+1}(h_{t+1}) z s_{t+1} (h_t)\\ g_{t+1} + R_{t+1}(h_{t+1}) z s_{t+1} (h_t) &= \tau l\\ {U_{lt}\over U_{ct}} &= -(1-\tau) \end{align}(The last condition comes from the fact that the government doesn't directly choose the deviation allocation.)
Then any outcome \(\left\{ c_t, s_{t+1}, l_t \right\}_t, h\) induced by a sustainable equilibrium satifies the constraint that at all points in time \(T\), the continuation of the planned outcome is better than the value of the best one shot deviation plus the value of autarky: \[\sum_{t=T}^\infty \beta^{t-T} U(c_t,l_t) \geq U^D(h_{T-1}) + \sum_{t=T}^\infty \beta^{t-T} U^A_t \mytag{Sustainability Constraint} \] Also note that this condition is sufficient to prevent the government from deviating from a plan which induces this outcome.
In order to find the best sustainable equilbrium, we modify the Ramsey problem to include this sustainability constraint.
Given exogenous spending requirements \(\left\{ g_t\right\}_{t=0}^\infty\), and parameters \((\beta, \bar{l})\), the allocations in the best sustainable equilibrium are found by solving: \[\max_{\left\{ c_{t}, l_t \right\}_t} \sum_{t=0}^\infty \beta^t U( c_{t},l_{t})\] such that \[0 = \sum_{t=0}^\infty\beta^t\left[c_{t} U_{ct} + l_{t} U_{lt}\right] \mytag{Implementability}\] and such that for all \(t\geq 0:\) \begin{align} c_{t}\geq 0, \; &\; l_t \in [0,\bar{l}] \mytag{Non-Negativity}\\ c_t + g_t &\leq l_t \mytag{Feasibility}\\ \sum_{T=t}^\infty \beta^{T-t} U(c_T,l_T) &\geq U^D(h_{t-1}) + \sum_{T=t}^\infty \beta^{T-t} U^A_T \mytag{Sustainability}\\ \end{align}
Suppose that \(g_H > \bar{l}\). Then government spending cannot possibly be financed by labor taxes alone, unless labor is taxed at a rate above 100%. And such a thing would make it impossible for the household's budget to be balanced. As such, being locked out of the bonds market would render the government's optimization problem impossible.
Assume that \[\lim_{c\to 0+} U(c,l) = -\infty\] Then as \(g_H\to\bar{l}\), \(c_t \to 0\) in even periods, and so \(U^A_t \to -\infty\) in even periods. Then because \(U^D\) is bounded, there is some sufficiently high \(g_H\) such that the Ramsey allocation satisfies the sustainability constraint.
Finally, note that the constraints in the Ramsey problem are looser than the constraints which define the best sustainable equilibrium's outcome. In other words, all sustainable outcomes are implementable and feasible. And the Ramsey outcomes are the best outcomes which are implementable and feasible, so if the Ramsey allocations are sustainable, they must be the best sustainable outcomes.