Debt Dynamics and the Government Budget Constraint

Start from the one-period government budget identity:

B_t = (1 + r_t)B_{t-1} + G_t - T_t

where B_t is end-of-period debt, r_t is the real interest rate, G_t is non-interest spending, and T_t is tax revenue.

Divide through by output Y_t:

\frac{B_t}{Y_t} = \frac{(1+r_t)B_{t-1}}{Y_t} + \frac{G_t-T_t}{Y_t}.

Define b_t \equiv B_t/Y_t, d_t \equiv (G_t-T_t)/Y_t, and let output grow at rate g_t, so Y_t=(1+g_t)Y_{t-1}. Then:

\frac{B_{t-1}}{Y_t} = \frac{B_{t-1}}{(1+g_t)Y_{t-1}} = \frac{b_{t-1}}{1+g_t}.

Substituting gives the standard debt-ratio law of motion:

\boxed{ b_t = \frac{1+r_t}{1+g_t}b_{t-1} + d_t }

If we instead define the primary surplus ratio as s_t \equiv (T_t-G_t)/Y_t, the same identity becomes:

\boxed{ b_t = \frac{1+r_t}{1+g_t}b_{t-1} - s_t }

Stabilizing the Debt Ratio

To keep the debt ratio constant at b_t=b_{t-1}=b^\ast, impose:

b^\ast = \frac{1+r}{1+g}b^\ast - s^\ast.

Solving for the stabilizing surplus gives:

\boxed{ s^\ast = \frac{r-g}{1+g}b^\ast }

• If r > g, debt snowballs unless the government runs a primary surplus.
• If g > r, growth helps dilute the debt burden.
• The larger the inherited debt ratio, the larger the required stabilizing surplus.

Strategic Debt Accumulation with Initial Debt

Inherited debt D_0 affects the level of resources available to the current government, but the strategic mechanism still comes from how today’s borrowing restricts tomorrow’s policymaker.

Write the two-period resource constraints as:

g_1 = \tau - (1+r)D_0 + D_1

g_2 = \tau - (1+r)D_1

where D_1 is the end-of-period-1 debt chosen by the current government.

Suppose the incumbent solves:

U = u(g_1) + \delta \mathbb{E}\bigl[v(g_2)\bigr].

Substitute the constraints into the objective:

U(D_1)=u\!\left(\tau-(1+r)D_0+D_1\right)+\delta\mathbb{E}\left[v\!\left(\tau-(1+r)D_1\right)\right].

Differentiate with respect to D_1:

\frac{dU}{dD_1} = u'(g_1) - \delta(1+r)\mathbb{E}[v'(g_2)].

The first-order condition is:

\boxed{ u'(g_1) = \delta(1+r)\mathbb{E}[v'(g_2)] }

What Changes When D_0 Rises?

• A higher D_0 lowers current resources one-for-one for any given D_1.
• But D_0 is inherited, so it enters the choice problem as a constant rather than a new marginal wedge.
• In the baseline unconstrained model, the strategic choice of additional debt is therefore unchanged by D_0.
• With borrowing limits, default risk, or strongly nonlinear marginal utility, high inherited debt can still reduce feasible new borrowing.

The practical lesson is that the strategic motive depends on how D_1 disciplines the next government, not on where debt started.

Sovereign Default as a Threshold Problem

Let next period’s fiscal capacity be a random variable T, and suppose promised gross debt service is RB. Default occurs when available resources are too low:

\text{default if } T < RB.

So repayment occurs iff

T \ge RB.

If fiscal capacity is uniformly distributed,

T \sim U[\mu-X,\mu+X], \qquad X > 0,

then default probability is:

\Pr(\text{default})= \begin{cases} 0, & RB \le \mu-X, \\ \dfrac{RB-(\mu-X)}{2X}, & \mu-X < RB < \mu+X, \\ 1, & RB \ge \mu+X. \end{cases}

Repayment probability is therefore:

\Pr(\text{repay})= \begin{cases} 1, & RB \le \mu-X, \\ \dfrac{\mu+X-RB}{2X}, & \mu-X < RB < \mu+X, \\ 0, & RB \ge \mu+X. \end{cases}

If competitive lenders require a gross safe return R_f, bond pricing satisfies:

\boxed{ q = \frac{\Pr(\text{repay})}{R_f} }

where q is the bond price. Lower repayment probability lowers q and therefore raises the promised yield that the sovereign must offer. If your lecture notation uses \pi for the borrowing level rather than B, the same threshold logic applies after relabeling the face value term.

Comparative Statics

• Higher \mu shifts fiscal capacity to the right and lowers default risk for any given promised repayment.
• Lower X compresses uncertainty and makes the default schedule steeper around the repayment threshold.
• Higher promised repayment RB raises default risk monotonically.

Sovereign debt crises are threshold events: once required repayment gets too close to what markets think the government can reliably raise, the bond price drops sharply and the interest rate jumps.