A The Social Planner’s Problem

Suppose that we have a social planner who maximizes

for some sequence of Pareto weights ρ_t ≥ 0, subject to the feasibility constraint that aggregate consumption and next period’s capital stock equal current output and any undepreciated capital

and given both k₀ > 0 and c_0,1 > 0.

The Lagrangian for the social planner’s problem is

which generates the first-order conditions

and

Combining these first-order conditions, we obtain the usual Euler equation

that applies in the absence of mortality risk. A steady state allocation, for which kt=k∗, ct,0=c0∗, and ct,1=c1∗ for all t ≥ 0, will be optimal with

if

for all t ≥ 0; i. e. if the allocation is dynamically efficient.^[9] Thus for any k^* that satisfies (34), we will have a stationary Pareto optimal allocation that satisfies

and

Comparing (35) to (8) and (36) to (10), we see that the steady state competitive equilibrium in the annuities regime will be a stationary Pareto optimal allocation if ka∗ satisfies (34). That is to say, given k0=ka∗ and c1,0=c1∗, there can be no feasible allocation {ct,0,ct+1,1}t=0∞ for which

for all t with strict inequality for some t. Otherwise, such an allocation would yield V > V^* when V^* maximizes (26) over the set of feasible allocations.^[10]

In contrast, if we compare (35) to (16), assuming Q < 1 these two equations will not be the same. Thus a competitive equilibrium in the bequest regime will not be Pareto optimal. Starting from the bequest-regime competitive equilibrium, there will be Pareto-improving transitions that go to steady states with higher steady state utility than the lifetime utility in the bequest regime. This does not, however, imply that it is possible to transition from the bequest-regime competitive equilibrium to the annuity-regime competitive equilibrium in a Pareto-improving fashion. There are infinitely many Pareto optimal steady state allocations. Only one of these corresponds to the annuity-regime competitive equilibrium. All of the other steady state allocations can only be achievable in a market setting if there are exogenous transfers. Households may do better in a competitive equilibrium with bequests than in a competitive equilibrium with perfect annuities if the bequest is a second-best mechanism that approximates a transfer which can yield a bigger lifetime utility if the presence of perfect annuity markets. For more on this result, see Bagchi and Feigenbaum (2019).