Privatizing Social Security: Implicit Liabilities Become Explicit Debt

1. (a) From the young's budget, \(\tau^t_t=e^t_t-s_t-c^t_t\). From the old's, \(\tau^{t-1}_t=e^{t-1}_t+s_{t-1}(1+r_{t-1})-c^{t-1}_t\). From goods-market clearing, \(g_t=e^t_t+e^{t-1}_t-c^t_t-c^{t-1}_t\). Substituting all three into the government budget, consumption and endowments cancel, leaving \(B_t+s_t=(1+r_{t-1})(B_{t-1}+s_{t-1})\). Induction from \(B_0+s_0=0\) gives \(B_t+s_t=0\). Not used: asset-market clearing itself (it is implied), and household optimality (the identity holds for any plans satisfying the budgets).
2. (b) Each generation's present value of taxes is unchanged, \(\tilde\tau^t_t+\tilde\tau^t_{t+1}/(1+\bar r)=\theta-\theta/(1+\bar r)\), but nothing is paid or received when old. The young's tax falls from \(\theta\) to \(\theta\bar r/(1+\bar r)\), a cut equal to the PV of the abolished benefit. The initial old still receive \(\theta\).
3. (c) The lifetime budget is \(c^t_t+\frac{c^t_{t+1}}{1+\bar r}=e^t_t+\frac{e^t_{t+1}}{1+\bar r}-\big(\tau^t_t+\frac{\tau^t_{t+1}}{1+\bar r}\big)\). The tax PV is the same under both systems, so the budget set is the same and so is the optimal choice. The initial old face the same tax and consume all their wealth. Hence \(\tilde c^t_t=\alpha-\theta\) and \(\tilde c^t_{t+1}=1-\alpha+\theta\).
4. (d) At \(t=1\): \(\tilde B_1=\tilde\tau^0_1+\tilde\tau^1_1=-\theta+\theta\bar r/(1+\bar r)=-\theta/(1+\bar r)\). For \(t\ge2\): \(\tilde B_t=(1+\bar r)\tilde B_{t-1}+\theta\bar r/(1+\bar r)\). If \(\tilde B_{t-1}=-\theta/(1+\bar r)\), then \(\tilde B_t=-\theta+\theta\bar r/(1+\bar r)=-\theta/(1+\bar r)\). Errors in a circulating solution: it writes the government budget with \(\tilde\tau^{t-1}_t/(1+\bar r)\) and gets the sign of \(\tilde\tau^0_1\) wrong mid-derivation. The final answer is right.
5. (e) Subtract the two young-age budgets, using \(\tilde c=c\): \(\tilde s_t=s_t+(\tau^t_t-\tilde\tau^t_t)=0+\theta-\theta\bar r/(1+\bar r)=\theta/(1+\bar r)\). When old, this pays \((1+\bar r)\tilde s=\theta\), exactly replacing the benefit.
6. (f) \(\tilde B_t+\tilde s_t=0\) for every \(t\). Together with (a) and (c), this confirms that \(\tilde r=\bar r\) is an equilibrium of the privatized economy.
7. (g) At \(t=1\), the government pays \(\theta\) to the initial old but collects only \(\theta\bar r/(1+\bar r)\) from the young. It covers the gap by issuing debt \(\theta/(1+\bar r)\), which the young buy with their tax cut. At \(t\ge2\), each young generation pays \(\theta\bar r/(1+\bar r)=-\bar r\tilde B\), exactly the interest on the debt, and buys the outstanding bonds from the old. The principal is rolled over forever. The implicit PAYG liability has become explicit debt, while consumption, welfare and \(r\) are unchanged.

The tax system depends on the equilibrium price, so we look for a fixed point. Conjecture a price, compute the policy and the households' responses, and check that markets clear at the conjectured price. If they do, we have found an equilibrium. Suppose the government computed \(\tilde\tau\) with some \(r'\ne\bar r\). Then each generation's lifetime budget at \(\bar r\) would change by \(\theta\big(\frac{1}{1+r'}-\frac{1}{1+\bar r}\big)\). Consumption would change, \(\bar r\) would no longer clear the market, and the reform would redistribute across generations instead of being neutral.

(a) The after-tax endowments are \((\alpha-\theta,1-\alpha+\theta)\), so \(1+\bar r=\frac{(1-\beta)(1-\alpha+\theta)}{\beta(\alpha-\theta)}\). This is \(<1\) iff \((1-\beta)(1-\alpha+\theta)<\beta(\alpha-\theta)\). Expanding, that is \(1-\alpha+\theta<\beta[(1-\alpha+\theta)+(\alpha-\theta)]=\beta\). So the after-tax old endowment must be below the share \(\beta\) the old would choose at \(r=0\).
(b) \(\tilde\tau^t_t=\theta\bar r/(1+\bar r)<0\): the young receive a subsidy. The reform replaces a system whose PV of taxes is negative.
(c) \((1+\bar r)\tilde B+\tilde\tau^t_t=-\theta+\frac{\theta\bar r}{1+\bar r}=-\frac{\theta}{1+\bar r}=\tilde B\) ✓. With \(\bar r<0\), the debt's real value shrinks by \(-\bar r\,|\tilde B|>0\) each period without any action. That "negative interest" is the resource the government hands to the young.
(d) The PV of PAYG taxes is \(\theta\bar r/(1+\bar r)<0\), so each generation is a net recipient. PAYG's implicit return (0) beats the market return (\(\bar r<0\)), which is exactly when PAYG can be Pareto-improving.

(a) The initial old lose \(\theta\). Generations \(t\ge1\) now consume their endowment \((\alpha,1-\alpha)\) instead of \((\alpha-\theta,1-\alpha+\theta)\). With a positive interest rate in the no-PAYG economy, each young person's MRS at the endowment exceeds 1. Moving toward more young consumption along \(c_y+c_o=1\) raises utility, so they gain.
(b) No. The PAYG equilibrium is a CE of an exchange economy with after-tax endowments. Its \(\bar r\) is positive, because PAYG raises the old's endowment and hence \(\bar r\). So the value of the aggregate endowment is finite, and the 1WT says the PAYG equilibrium is Pareto optimal. Any change must hurt someone, here the initial old.
(c) Privatization keeps every generation's PV of taxes, including the initial old's, so it reallocates nothing. The cost of honouring the initial old's promise doesn't disappear; it becomes explicit debt that is rolled over forever. Abolition defaults on that implicit debt, which is why the initial old pay.

(a) Generation 1's lifetime budget changes by \(+\Delta-(1+r^*)\Delta/(1+r^*)=0\). It consumes as before and saves the extra \(\Delta\) in government bonds, so \(s_1=\Delta=-B_1\) and the asset market clears. This is the Ricardian experiment within a generation, and it is neutral.
(b) Asset clearing at date 1 forces \(s_1=\Delta\). So generation 1 consumes \(c^1_1=\alpha+\Delta-\Delta=\alpha\) when young and \(c^1_2=1-\alpha+(1+r_1)\Delta\) when old: it gains. Generation 2 pays \((1+r_1)\Delta\) when young and holds no bonds, so it consumes \((\alpha-(1+r_1)\Delta,\ 1-\alpha)\): it loses. The interest rate is not unchanged. Generation 1 must be willing to hold \(\Delta\) of bonds even though it is richer when old. So \(1+r_1=v_1/v_2\) evaluated at \((\alpha,1-\alpha+(1+r_1)\Delta)\), which exceeds \(1+r^*\) because old-age consumption is higher. The allocation changes: debt is not neutral when it moves PV across generations. This is the failure of Ricardian equivalence in OLG models, developed further in Lesson 20.

1. (a) No trade, so \(1+\bar r=\frac{(1-\beta)(1-\alpha)}{\beta\alpha}\). This is \(<1\) iff \(1-\alpha<\beta\), i.e. \(\alpha+\beta>1\).
2. (b) Constant \(B_t=-b\ne0\) with \(B_t=(1+r)B_{t-1}\) requires \(1+r=1\).
3. (c) Wealth is \(\alpha+(1-\alpha)/(1+r)\), and \(c_y=(1-\beta)\times\)wealth, so \(s(r)=\alpha-c_y=\beta\alpha-\frac{(1-\beta)(1-\alpha)}{1+r}\). Asset-market clearing \(s_t+B_t=0\) at \(r=0\) gives \(b=s(0)=\beta\alpha-(1-\beta)(1-\alpha)=\alpha+\beta-1\). This is positive iff \(\alpha+\beta>1\iff\bar r<0\).
4. (d) The young consume \(\alpha-b=1-\beta\), and the old consume \(1-\alpha+b=\beta\): the best symmetric allocation. Every generation \(t\ge1\) strictly prefers it to the endowment (unless \(b=0\)), and the initial old gain \(b>0\). So it Pareto-dominates the no-policy CE exactly when that CE has \(\bar r<0\).
5. (e) PAYG with \(\theta=b=\alpha+\beta-1\) gives \((\alpha-\theta,1-\alpha+\theta)=(1-\beta,\beta)\). The same job is done by a valued bubble (the zero-dividend tree of Exercise 12.6(f)) and by fiat money. In Blanchard–Yaari, the positive debt steady state \(\bar B\) exists iff the autarky rate \(\theta-\alpha\) (BY notation) is negative (Lesson 20). It is the same logic.