L8: Fiscal Policy in a Two-Period & OLG Framework – Ricardian Equivalence, Social Security & Market Imperfections

Continue with the two–period (today $t$, tomorrow $t'$) micro-foundation used last week.
Prices (incl. the real interest rate $r$ ) are treated as exogenous for now → a partial-equilibrium setting. A full GE model, where $r$ becomes endogenous, comes later.
Household choice:
- Maximise intertemporal utility $U(c,c')$ s.t. individual intertemporal budget constraint (IBC):
  $c+\frac{c'}{1+r}\le y+\frac{y'}{1+r}.$
- Consumption-smoothing intuition retained: temporary income shocks mainly shift saving/borrowing, permanent income shocks shift consumption in both periods.
- Interest-rate changes decomposed into income & substitution effects.

Population size introduced: $N$ (today), $N'$ (next period). Later allow $N'\neq N$ via growth rate $n$ .
Taxes per person: $t$ (today), $t'$ (tomorrow). Total taxes: $T\equiv Nt$ , $T'\equiv N't'$ .
Government consumption/spending: $G,\;G'$ .
Government one-period debt (issued today, repaid next period): $B$ (positive = borrowing, negative = saving/surplus).
Period budget constraints:
- Today: $G+ B = T.$
- Tomorrow: $G' + (1+r)B = T'.$
Eliminate $B$ to produce Government IBC (GIBC):
$G+\frac{G'}{1+r}= \frac{T}{N}+\frac{T'}{N'(1+r)}=t+\frac{t'}{1+r}.$
Interpretation: PV of public spending must equal PV of per-capita taxes.

Credit market: household saving supply $S^p$ must equal government bond demand $B$ (with a closed economy & identical agents).
Implies familiar identity $Y=C+G$ (no investment nor external sector yet).

Policy experiment: cut taxes today (↓ $t$ ) financed by new debt B>0, repaid by raising $t'$ tomorrow so that $PV(t,t')$ unchanged.
Substitute GIBC into household IBC → present value wealth $WE$ unchanged → budget line unchanged.
Optimisation ⇒ ${c^<em>,c'^</em>}$ unchanged; interest rate $r$ unchanged; households merely save the windfall to buy the extra bonds.
Graphically: endowment shifts along an unchanged line; equilibrium stays at point $A$ .
Hence: timing of lump-sum taxes irrelevant under full rationality & perfect capital markets (classical Ricardian result).

Taxes are rarely lump-sum; most are distortionary (depend on labour, capital, trade, etc.).
Overlapping-Generations (OLG) structure: different cohorts coexist but have finite lifetimes.
Credit-market imperfections: borrowing rates > lending rates, collateral constraints, default risk.
Behavioural/psychological limits to foresight or self-control.

Below focus on (2) & (3).

Policy: tax the current young by $t$ , transfer $b$ to current old (so old face $t'=-b$ ).
Budget feasibility each period:
$N't = Nb \;\Rightarrow\; t = \frac{1}{1+n}\,b.$
Individual young budget line pivot:
- Vertical intercept shift: $-t$ today.
- Horizontal intercept shift (older self): $+b$ .
- Slope equals $-(1+n)$ (implicit return on the PAYG “asset”).
Welfare comparison hinges on return comparison:
• If (1+n) > (1+r) (i.e.
n>r): PAYG offers higher return than private saving → lifetime wealth rises → higher indifference curve → welfare-improving for all cohorts.
• If n < r: PAYG return inferior → PV wealth falls → lower indifference curve → welfare-reducing; burden rises because young cohort smaller relative to retirees → mirrors ageing crises in many countries.
Political economy: sustainability depends on future voters; declining birth rates threaten PAYG viability.
Real-world links: aged-pension systems, demographic subsidies, pro-fertility policies.

Government mandates each worker save a fixed amount (superannuation in Australia).
If agents were already optimally choosing savings, a binding mandate forces them inside their optimum (point moves to a lower indifference curve) → potential welfare loss.
Could be welfare-enhancing only if households suffer myopia / self-control problems (behavioural rationale).
Opens door for lobbying by financial-service providers to raise mandatory contribution rates for private gain.

Introduce kinked budget line:
- For c>y (borrowing): face interest rate $r<em>b > r</em>l$ (lending rate).
- Graph: single budget line replaced by two rays with a kink at endowment point.
Now a tax cut today shifts endowment rightwards; because borrowing is expensive, previously constrained borrowers may move onto a higher utility level immediately (consume more) rather than fully saving the rebate.
Thus timing of taxes matters when borrowing constraints bind.
Empirical relevance: mortgage vs. deposit rate spreads; low-income households face high $r_b$ ; evidence (e.g.
Andrew Leigh’s Australian studies) shows consumption responds to transfer timing.

PAYG intergenerational transfers create equity debates between older & younger voters.
Mandatory savings create vested interests (super funds, insurers) lobbying for larger pools.
Fiscal-deficit debates (e.g.
in Australia “too much deficit?”) partly reflect whether Ricardian neutrality is believed.

Household IBC: $c+\dfrac{c'}{1+r}= y+\dfrac{y'}{1+r}-\left(t+\dfrac{t'}{1+r}\right).$
Government IBC: $G+\dfrac{G'}{1+r}= t+\dfrac{t'}{1+r}$ (per capita).
PAYG tax/benefit link: $t=\dfrac{b}{1+n}.$
Lifetime wealth under PAYG:
$WE = y+\frac{y'}{1+r} + b\left[\frac{1}{1+r}-\frac{1}{1+n}\right].$
Welfare effect sign given by $\operatorname{sgn}\bigl[(n-r)\,b\bigr].$

Ricardian Equivalence is a helpful benchmark but rests on stringent conditions.
Once generational turnover, distortionary taxes, credit frictions or behavioural biases enter, fiscal timing matters.
PAYG works only when population (tax base) grows faster than market returns; otherwise it strains future cohorts.
Forced-saving schemes can fix under-saving but may over-reach and create rent-seeking.
Policymakers must weigh efficiency, distribution and political sustainability when designing fiscal tools.