History and Notes

Advanced Concept

The Continuous Choice Problem with Backward Induction

Consider the choice problem presented in equation 13.1 for an individual who lives for three periods. Given consumption of c1 and c2 in the first two periods, the individual in the third period must solve

Given this solution to the third-period problem, the second-period problem is given by

This is exactly the two-period consumption problem presented in Chapter 11, and it is solved where the marginal utility of consumption in period 2 is equal to the marginal utility in period 3 weighted by the discount factor

Recall from Chapter 12 that this is also the condition for the solution in the last two periods when solving the standard (and not the recursive) optimization problem. Let c*2 c1 be the solution to equation 13.C, given the level of consumption in period 1. Then, in the first period, the decision maker must solve

This problem now behaves as the two-period problem, only now in the second period the decision maker receives uc2*c1 + δuw − c1 − c*2 c1. Now the condition for optimization is the first order condition given by

owing to application of the chain rule. Cancelling like terms in equation 13.E and combining with equation 13.C yields

Combining the optimization conditions in equations 13.F and 13.C, we find that the solution to the three-period recursive optimization problem must solve

subject to the budget constraint c3 ≤ w − c1 − c2, which under the condition that utility is increasing in consumption will yield the solution c*3 = w − c1 − c2. Next, we consider the second-period consumption decision. In this case, the decision maker will solve

which is solved where discounted marginal utilities are equal:

which is solved by some c*2 c1, similar to the second period of the three stage problem with exponential discounting. In the first stage, the decision maker now solves

with first-order conditions

Combining equation 13.N with equation 13.P, we can rewrite the relationship in terms of consumption in the first and second period:

− 1 < dc*2 dc1 < 0.

We can contrast this solution with the choice made if the decision maker could commit to a consumption path

If we compare this condition to equation 13.Q, the two consumption paths would be identical if dc*2 dc1 = − 1. But as we showed earlier, dc*2 dc1 > − 1. Moreover, if dc*2 dc1 = 0, then equation 13.Q would be identical to the condition in equation 13.C, representing the exponential discounting case with a discount factor of δ. But we also know that dc*2 dc1 < 0. In fact, the consumption path for the recursive optimization problem will always lie between the planned consumption path of the naïve quasi-hyperbolic discounter (which is identical to the optimal consumption path for a committed decision maker) and the consumption path of the rational exponential discounter with discount factor of δ.

Without the ability to commit, decision makers will choose consumption in the first period that is lower than consumption of the naïve quasi-hyperbolic discounter, because they know they will not be able to commit to the low level of consumption in the second period that would be necessary to preserve consumption possibilities in the third period. Moreover, decision makers will consume more in the first period than they would if they discounted all periods by δ because they still suffer from present-biased preferences—discounting all future consumption utility by β < δ. In the second period, the sophisticated decision maker will choose to consume more than the committed decision maker. This occurs because the decision maker first anticipates that owing to present biased preferences, the decision maker will value utility of consumption in the third period at only β that of consumption in the second period at the time the decision maker makes this decision. Secondly, the decision maker consumes more because more wealth is left (consumption was lower in period 1). The sophisticated decision maker still consumes more in the second period than would the rational decision maker with discount factor δ. Finally, in the third period, the sophisticated decision maker consumes less than the committed decision maker.

To see this, note that equation 13.Q and equation 13.N can be combined to find

First, consider what would happen if dc2*dc1 = 0. In this case, equation 13.X is similar to the condition for the committed consumer in equation 13.U, except that the sophisticated decision maker consumes less in the first period than the committed consumer does. This means that consumption in the third period would also be smaller given the utility functions are increasing and concave. However, the second term on the right side of equation 13.X is negative because dc*2 dc1 < 0. This means that the first term is actually greater than u′c1. Because of diminishing marginal utility of consumption, the only way u′w − c1 − c*2 c1 can be made larger is by further decreasing the argument—c3. Thus, the sophisticated consumer ends up consuming less in the third period.

The lack of ability to commit leads to a loss in utility to the decision maker (which is illustrated in Example 13.1). Because the decision maker must put off consumption today to make sure his period 2 self will not leave his period three self with too little, the decision maker in period one feels worse off. He would thus be willing to pay some amount to allow greater consumption now, without endangering his third-period consumption. Thus arises the desire for a commitment mechanism: Let me consume more now, but make sure I don’t indulge tomorrow. In longer-period problems, the commitment mechanism can allow indulgence today while ensuring that the consumer will consume at more-rea- sonable levels for long periods (potentially infinite) of time. Thus, the would-be dieter can eat with confidence knowing that tomorrow the diet will kick in whether he likes it or not.

Exactly how selfish would one need to be to satisfy the conditions of the first welfare theorem? A well-known parable in Christian scripture refers to a man who was beaten and robbed and left naked by the side of the road. Suppose that Homo economicus happens to be walking on that road carrying three extra changes of clothes, bandages, and plenty of money. The unfortunate man asks Homo economicus how much he would take in exchange for a coat.

Homo economicus replies, “This coat is worth $120 to me, I would not be willing to part with it for less.” The bleeding man tells him that he has just been robbed and beaten, but that when he is able to return to his house, he could give him $119 (all he has). “Sorry, that is just too little for this coat.” What about other clothing or bandages? “Well, I could sell you bandages for $5, and a set of clothes for $40. But that would be if you gave me the money now. If you cannot pay me until tomorrow, I would need to receive somewhat more. Moreover, if you only have $119 total, you sound like a bit of a credit risk to me. Do you have anything you can give me as collateral?” The bleeding man asks if Homo economicus could at least help the bleeding man up to his feet. “How much would you be willing to pay me to stand you upright? You don’t have any blood-transmitted diseases do you?”

Our abhorrence of anyone who would act in such a way reflects just how foreign Homo economicus is to our notions of common, acceptable, and reasonable behavior. Nonetheless, this assumption is critical to much of economic thought. Clearly the feelings of others, our regard for cultural norms, and our concern for those in dire circumstances matter in our decisions and thus should matter in economics. Though many who have studied such matters would not classify themselves as behavioral economists, the assumption of selfish actors is ubiquitous enough in economics that behavioral economists consider other-regarding preferences as part of their jurisdiction. Altruistic behavior might not be irrational, but it demonstrates a deviation from the most commonly made assumptions of economic models.

On the surface it might seem simple to delineate purely selfish behavior from behavior that is motivated by a desire to help others. It turns out, however, that it is difficult to determine which behaviors are actually altruistic and which are based on self-interest. In this chapter we examine the empirical evidence of altruism and some of the applications. Of course, altruism is not the only form in which others’ well-being can enter in to one’s decisions. We also explore important theories of how we regard others and base our own preferences on how we perceive our neighbors.

EXAMPLE 14.1 Spoiling the Child

A commonly cited example of altruistic behavior is that of parents’ willingness to sacrifice their own pleasure and consumption for the well-being of their child. It is estimated that including food and clothing costs, childcare, and other regular expenses, parents in the United States spend an average of $227,000 rearing their child. Beyond this, parents spend countless hours teaching, training, consoling, and nurturing children, all with the seeming goal of making them into independent and responsible adults capable of providing for their own happiness. Introspection might lead us to believe that this is truly altruistic behavior. Nonetheless, some point out that this behavior might all be self-interested.

Historically, many farmers had large families to provide labor for the farm. This provided plenty of incentive to keep a family well fed and in good health. One piece of

evidence for work being a motivation to have children comes from 19th-century Sweden, where orphans were auctioned off to would-be parents. Sofia Lundberg found that parents were willing to bid much more for orphans who were healthy and able to provide substantial labor. Moreover, the parents were also willing to bid more if more work was needed in their farm or other operation. More recently, children might not be desired for the work they provide around the house while they are children but for the support and care they will provide when parents become too old to care for themselves. Even a bequest to children upon death may potentially be viewed as a reward in exchange for good behavior. In fact it is difficult to separate any single act a parent performs for a child from a potential reward that the parent might receive in return, thus making it difficult to detect any pure altruism on the behalf of the parent.

If parents truly desire to provide the child utility without receiving anything in return, they might end up encouraging (or at least failing to discourage) bad behavior. Consider, for argument’s sake, parents with a child whom they do not care for except for what they will receive in return from that child in the form of elderly care. The parent might know that the child’s ability to provide elderly care for them will depend on the level of bequest they promise the child as a reward, as well as on the child’s eventual financial success. Thus, this parent decides to implement a schedule of penalties and rewards for grades, work, and other outcomes and habits that will prepare the child for financial success. These penalties and rewards might not be malicious, but they reflect the optimal incentive they can provide the child. Now, alternatively, consider altruistic parents who, in addition to caring about the level of elderly care they will receive, also care directly about how much fun, enjoyment, and utility their child obtains. In this case, the altruistic parent might not be willing to impose some of the same penalties because it would cause them some disutility. Thus, the child might not perform as well with respect to school, work, or other training because she does not face the same incentives. In this case, altruism could lead to an inefficient set of incentives for the child and also a reduction in the total resources available to the parent and child. Similar arguments suppose that charitable giving to those falling on hard luck could induce less effort on the part of those facing the hard luck.

Some evidence suggests that such inefficiency exists in some circumstances. Many firms (e.g., Walmart) are operated as part of a family organization. William Schulze, Michael Lubatkin, and Richard Dino theorize that if a family with altruistic preferences controls a firm, the same problem can arise in trying to provide the proper incentives for family workers to provide thoughtful, skilled, and timely labor and to prepare family members for the management task of taking over the business once the current CEO resigns, retires, or passes on. In fact the fraction of successful family businesses that manage the transition to the second generation is only about one third. Only about half of those that mange to pass the management of their family firm to the second generation are able to pass the business on to the third. In other words, people appear to be promoted to management positions because the current CEO cares about the family members rather than about their qualifications or ability to perform. Some have found, for example, that family firms are less effective at using strategic planning to obtain sales growth than firms not controlled by a family. This provides at least some weak evidence that family members may be held to a lower standard than others.