The Barro Becker Model

One of the principle economic models of fertility is that pioneered in the two papers Becker and Barro (1988) and Barro and Becker (1989). In this section, we show how our approach to e±cient fertility can be applied to that class of

5>From a formal point of view, this problem is similar to that studied in the Clubs literature: When is it true that a non-cooperative formulation gives rise to e±cient out-comes? (See Scotchmer (1997) for an example.) However, the mechanism at work here is quite di®erent. In club and other local public good environments, e±ciency is guaranteed by competition between the di®erent clubs for members. Here, since the dynasty is the analog of a club, no such competition across clubs is possible. Rather, here it is the natural allignment of incentives within a family that guarantees e±ciency within the group.

models. In that approach, at each date, t, the individuals alive make decisions about their own consumption, how many children to have and how large a bequest to leave each child. To make the model more tractable, B&B assume that fertility can take on any positive value, not just integers. Because of this, the analysis of the previous sections do not directly apply to the B&B model.

The modi¯cations necessary are straightforward however.⁶

We generalize the B&B framework here by allowing for more than one period-zero person. Each initial agent is the dynastic head of his own dynasty.

We allow dynasties to di®er in their initial capital stock k^¿₀, child-rearing costs µ^¿_t, discount factor ¯^¿, and skill a^¿_t. Let wt be the wage per skill unit at time t and rtreturn on capital. We also use a more general utility function. For most parts in this section, it is enough to focus on one dynasty. For these cases we drop the superscript ¿.

In the Barro and Becker model, it is assumed that each agent alive in period t, i^t, must choose his current consumption level, xt(i^t), his fertility level, ft(i^t)2 [0; ¹f] and a bequest level for each of his children, subject to his own budget constraint:

ptxt(i^t) + c(ft(i^t)) +

Z ft(i^t) 0

bi^t(it+1)dit+1· ptet(i^t) + bi^t¡1(it)

Note that this includes the bequest that he has received from his own parents, b_i^t¡1(i_t). It is assumed that his preferences are given recursively by:

U_t(i^t) = u(x_t(i^t)) + ¯g(f_t(i^t))

Z ft(i^t) 0

U_t+1(i^t+1)di_t+1:

Each agent in each dynasty views the entire sequence of prices as given when making this choice.

Since it is our goal to establish that an equilibrium is sequentially P-e±cient, when prices are determined by the interaction of multiple price-taking dynasties, we must ¯rst have a precise de¯nition of what the equilibrium is.

6Details on this are available in the appendix to this paper online. See Golosov, Jones and Tertilt (2003) at http://www.econ.umn.edu/~lej/lejresearch.html

To do this, we will model the formulation above as an in¯nite horizon game in which each period each child that is born must make decisions as given above. Then, the natural question is how does a time period t decision maker conjecture the future utility of his children? Of course, the answer is that they must correspond to the actual utility levels that these children receive if they optimally respond to the bequests that they receive from their parents, etc.

That is, it should be that the sequence of consumption/fertility/bequest plans should be a Subgame Perfect Equilibrium (SPE) of this in¯nite horizon game.

Of course, there are typically many SPE's of in¯nite horizon games involving di®erent `threats' of `punishments' on the o® equilibrium path. There is no easy way to select among these di®erent equilibria, but one selection criterion that is often used is that it not be too dependent on the assumption that time lasts forever. That is, it should be the limit of the equilibria of the ¯nite horizon truncations of the in¯nite horizon game. This is the criterion that we will use below. Formally, our de¯nition of equilibrium is:

De¯nition 9 An equilibrium are prices fp^tg and fx^¿t(i); f_t^¿(i)g^t;i for each dy-nasty ¿ such that

1. For each dynasty ¿ , fx^¿t(i); f_t^¿(i)g^t;i is the limit of the subgame perfect equilibrium outcome of the ¯nite dynasty game.

2. The allocation is feasible.

To prove that the BB equilibrium is sequentially P-e±cient, it needs to be shown that all dynasties are sequentiallyP-maximizing in the dynastic BB game. The details for this are available in Golosov, Jones and Tertilt (2003), where a T period truncation of the in¯nite horizon game is de¯ned. It is then shown that the subgame perfect equilibrium outcome of this game coincides with the dynastic head choosing his most preferred outcome, and moreover that this is unique. This result depends on the following four assumptions:

Assumption 5 u(¢) is continuous, strictly increasing, strictly concave and u(0) = 0.

Assumption 6 Assume c(f ) = µf .

Assumption 6 implies that f1c(f2) = c(f1f2) which will be used below.

Assumption 7 Assume that g(¢) satis¯es g(x)g(y) = g(xy).

Assumption 8 Assume that H(F; X)´ g(F)F u(X=F ) is strictly concave in (F; X).

Then, we have:

Proposition 6 Let the allocation z = fz^¿g¿, where z^¿ = (c^¿_t; f_t^¿; k^¿_t+1)_t, to-gether with prices fwt; r_tg be a BB equilibrium as de¯ned in De¯nition 9.

Then, under Assumptions 5-8, along the equilibrium path, all dynasties are dynastically sequentially P-maximizing given prices.

Given this result, we can then state the main result of this section:

Theorem 1 Let the allocation z = fz^¿g¿, where z^¿ = (c^¿_t; f_t^¿; k^¿_t+1)_t, together with prices fwt; r_tg be a BB equilibrium as de¯ned in De¯nition 9. Then, under Assumptions 5-8, z is sequentially P-e±cient.

Proof. This follows immediately from the ¯rst welfare theorem (for se-quentially P-maximizing dynasties) together with Proposition 6 above: z^¿ is sequentially P-maximizing for each dynasty ¿ by Proposition 6, it is also fea-sible, since it is an equilibrium, there are no externalities across dynasties by assumption, hence it must be sequentiallyP-e±cient. ¤ Note that an immediate corollary to Theorem 1 is that the equilibrium allocation is alsoP-e±cient. This follows from the fact that the set of sequen-tially P-e±cient allocations is a strict subset of the set of P-e±cient alloca-tions. That it is alsoA-e±cient follows from the fact that, for each dynasty, the equilibrium allocation is the unique maximizer of the utility function of the dynasty head at the equilibrium prices.

In the above we have assumed that it is feasible for parents to leave negative bequests to their children. One would like to know whether this is a necessary assumption. Assume that there is a non-negativity constraint on bequests, k_t^¿ ¸ 0. Note that if all dynasties were identical, then this constraint would never be binding in equilibrium, and hence the equilibrium allocation would still be (sequentially)P-e±cient. If dynasties are heterogeneous, but not too di®erent, then the same logic would still apply by continuity. However, if the heterogeneity is big, then prohibiting negative bequests can indeed lead to an ine±ciency in the B&B environment, as we will see in Section 5.3.