1. Public debt, Finite Horizons in NGM
Ec1430 ● Robert Barro ● Harvard University ● 2014

2. Budget Constraints
Government bonds, B, infinitesimal maturity, pay real interest rate r. Government b.c. is
T is lump-sum taxes. Household assets per person (when internal loans cancel) are a = k + b . Household b.c. is
 is per capita taxes.
To simplify algebra, assume r constant and n = 0 .

3. Budget Constraints / F.O.C.
“Lifetime” budget constraint (when TVC holds for households) is
F.O.C. same as before (if c, G separable effects on U):
Different if positive marginal tax rate on saving—for example, tax on asset income or time-varying consumption tax. Also effects on labor-leisure choice if taxes on labor or consumption. Assume lump-sum taxes here.

4. Dynamics Dynamic equation for is
Since F.O.C. for dc/dt same as before and path of is given, only way that shift between T and dB/dt can affect equilibrium is through household income effect. From lifetime b.c., this has to come through term
or, in aggregate,

5. Ricardian Equivalence
Can show from government b.c., for any B(0) and path of T(t), that
Path of G given. Therefore, no income effect from shift in B(0) or path of T(t) if chain-letter debt issue (Ponzi game by government) ruled out. Hence, shift in B(0) or path of T(t) (for given G path) leaves unchanged equilibrium path of etc.
Ricardian equivalence holds in NGM. (Equivalently, tax cut today is fully saved by households, so that national saving unchanged.)

6. Alternative approaches
To study finite-horizon effects, use Blanchard model (Ch. 3). Use for public debt and also for real money balances/inflation.
Alternative approach is overlapping-generations (OLG) structure, from Samuelson (1958) and Diamond (1965). See appendix to Ch. 3. In 2-period version, individuals work, receive wages, consume, and save in period 1; receive asset income and consume in period 2. May also pay taxes in period 1 and receive social security in period 2. Economy goes on forever (no end of world). Individuals have finite horizons (finite lifetimes) if no concern for future generations. Barro (1974) adds altruistic linkages across generations—may restore infinite horizon.

7. Blanchard – Weil
OLG useful for studying life-cycle, retirement, finite-horizon effects. But cumbersome for comparative statics. Blanchard (1985) captures finite-horizon effect in more tractable way. See extension in Weil (1989) and Ch. 3.
Blanchard extends NGM to have p per unit of time of dying as before. Individuals hold annuities—pay r + p if survive, 0 if die. Attractive asset because people have no desire to leave anything to descendants. For debts, r + p is payment on loan secured by life insurance. Annuity-life insurance companies hold claims on capital—which pay r—and break even.

8. Blanchard – Weil Individual j maximizes expected utility:
is probability of being alive at t if alive at 0. p constant—no dependence of mortality rate on age. (Can interpret “dying” as end of connected dynasty, rather than literal death.) No n because no concern with children. Blanchard has n = 0 anyway, but Weil adds n > 0—call this Blanchard-Weil model. Assume usual iso-elastic form for

9. Blanchard – Weil Budget constraint for individual survivor is
FOC for survivor is the usual:
New effects from aggregation of cj over persons of different ages. Assume people start life with zero assets (no bequests). Survivors have more assets as they age.

10. Blanchard – Weil
Assume wj = w, independent of age. Could introduce life cycle of wages (zero for children, then rising, then falling in old age). Assume, to simplify, θ = 1 (log utility), but could be dropped. Key to tractable aggregation is independence of p from age.
Recall in Ramsey model, with θ = 1, consumption is given by
In Blanchard-Weil, n does not appear (because no concern with children), but  becomes  + p (overall discount rate on future utils).

11. Blanchard – Weil
Aggregation of consumption function across ages (along with assumption that assets = 0 at age 0) leads to key equation:
Note: c is consumption per capita in overall economy—NOT consumption of representative person (who vary by age).
Note in equation that we assumed θ = 1. p + n is gross flow of new persons (n = 0 in Blanchard). With n > 0, could have p = 0 (infinite lives but new people born). However, remember that n refers to “disconnected” new persons—unloved children and immigrants.  + p is MPC out of assets (independent of r path because θ = 1).

12. Blanchard – Weil Model still has Second equation is different:
Phase diagram changes. Locus for now a curve that asymptotes to previous vertical line. See figure.

13. Blanchard – Weil
Phase Diagram in Blanchard Model:

14. Blanchard – Weil Steady-state lower than before:
Increase in p or n lowers (Note: n refers to population growth for disconnected people.) Finite-horizon effect is small—e.g. δ = 0.05, ρ = 0.02, x = 0.02, p = 0.02, n = 0.01 means that RHS is
One idea from OLG was equilibrium might have oversaving— Blanchard shows that finite horizon is NOT what might produce this result.

15. Blanchard – Weil
Life-cycle pattern—wages falling by age might do it. However, typical life cycle has w rising with age for some time, especially childhood.
Add public debt to Blanchard model. New persons have no government bonds but share in future taxes. In key dynamic equation, can think heuristically of being replaced by :
(Not fully accurate. should appear net of expected p.v. of taxes paid by each living person. This p.v. of taxes depends also on path of future budget deficits.)

16. Blanchard – Weil
Can use Blanchard phase diagram readily if constant. Rise in shifts locus for upward. Hence, falls. With finite horizons, people currently alive feel richer when more government bonds outstanding. Therefore, want to consume more. Equilibrium requires higher r* and, hence, lower
As with finite-horizon effect, impact on r* small. Depends on (More generally, public-debt effects in Blanchard model depend on current stock of bonds and on expected future budget deficits.)