If we add time preference, why assume it is the same across dynasties and constant across time?

The problem of allowing heterogeneous discount rates for contemporaries was already anticipated by Frank
Ramsey (1928). He argued that if two dynasties have different discount rates and a loan market is in operation, then "equilibrium would be attained by a division of society into two classes, the thrift enjoying bliss and the improvident at the subsistence level" (Ramsey, 1928). This is confirmed by
R.A. Becker (1980). However, Robert Lucas and Nancy
Stokey (1984) demonstrate that if the utility function is *non*-additively separable, then this result is not necessarily true.

Ramsey (1928) also argued that if a *single* person is discounting his future utility, he *must* discount all of it at the *same* constant rate. Otherwise, what he planned for the future will be changed when that future get closer. This has become known as *dynamic inconsistency*.

To see why, first we need to convince ourselves that a single discount rate and
Koopmans's stationarity axiom are effectively the same thing. Recall that the stationarity axiom claims that preferences between two periods remain the same even if we shunt those periods forward. So, suppose that we have a baseline consumption path (1, 1, 1, .., 1) and one of the following two adjustments can happen: either we add X to consumption in time period t *or* we add amount Y to consumption to time period t + h. Suppose we have chosen adjustments X and Y in such a manner than the individual is indifferent between the two alternative paths, thus, assuming a constant discount rate, b
, then:

b

^{t}U(1 + X) + b^{t+h}U(1) = b^{t}U(1) + b^{t+h}U(1+Y)

Notice that if we divide through by b
^{t} and rearrange this, then::

U(1 + X) - U(1) = b

^{h}[U(1 + Y) - U(1)]

Notice that t disappears from this expression; only the absolute time difference, h, remains. In other words, we remain indifferent between the two adjustments X and Y as long as these two adjustment happen with a difference of h periods of each other. It doesn't matter whether X happens in period 3 and Y in period 3 + h or whether X happens in period 45 and Y in period 45 + h.

But now suppose that we have two different discount factors. Namely, let b
^{t} be the discount for period t and g
^{t+h} be the discount for period t+h. Then, once again, choose X and Y so that the agent is indifferent:

b

^{t}U(1 + X) + g^{t+h}U(1) = b^{t}U(1) + g^{t+h}U(1+Y)

But now, when this is rearranged:

U(1 + X) - U(1) = (g

^{t+h}/b^{t})[U(1 + Y) - U(1)]

Now, notice that t remains in the expression. Thus, our preferences over the adjustments are dependent not only on the absolute time difference, h, but also on the actual reference time t when the events happen. Stationarity is broken.

The absence of stationarity means that we can have *dynamic inconsistency*, i.e. plans that are made at one point in time, are contradicted by later behavior. The identification of this possibility is often credited to Robert
Strotz (1956). Its implications are teased out in Bezalel
Peleg and Menachem Yaari (1973).

Intuitively, recall that if the stationarity axiom is violated, then we can have it that we prefer one apple today to two apples tomorrow, but, at the same time, prefer two apples in 31 days to one apple in thirty days. Why this leads to inconsistency is obvious. If I make a consumption plan according to these preferences, I will plan to receive two apples in 31 days, but then, as time passes and that day approaches, I'll change my mind and choose to get the one apple one day earlier. My initial plans are inconsistent with my subsequent actions.

Interestingly, the old economists who came up with time preference allowed for the discount rate to change over time. For instance, we find William Stanley Jevons arguing that:

"An event which is to happen a year hence affects us on the average about as much one day as another; but an event of importance, which is to take place three days hence will probably affect us on each of the intervening days more acutely than the last." (Jevons, 1871: p.34-5)

Or, even more explicitly in Eugen von Böhm-Bawerk:

"I should like to call special attention, further, to the fact, that the undervaluation which resutls from these causes is not at allgraduated harmoniously, in the subjective valuation of the individuals, according to the length of the time that intervenes. I mean, it is not graduated in this way, for example, that the man who discounts a utility due in one year by 5%, must discount a utility due in two years by 10%, or one due in three months by 1ｼ%. On the contrary, the original subjective undervaluations are, in the highest degree, unequal and irregular. In particular, so far as the undervaluation is caused by defects of the will, there may be a strong difference between an enjoyment hich offers itself at the very moment and one which does not; while, on the other hand, there may be a very small difference, or no difference at all, between an enjoyment which is pretty far away, and one which is further away." (Böhm-Bawerk, 1889: p.257-8).

However, the relevance of this question to our context depends on our interpretation of the
social welfare function. If we maintain a "successive generations" interpretation, then it is not clear what this means because people do not live more than a period to begin with. If we argue on the basis of
"dynastic" utility, changing time preference makes some sense (cf.
Phelps and Pollack, 1968). Specifically, *within* a dynasty, different generations will have different time preference rates, e.g. the generation of time 0 discounts at rate b
, but the generation of time 1 discounts at rate g
, where b
¹
g
. This means that the "plans" that generation 0 sets for generation 1 (and all subsequent generations) are *not* followed by generation 1 when the time arrives, who go on to develop their own distinct plans instead. The ethical implications of this "time inconsistency" in a dynasty are, however, a bit harder to
distentangle.

These discussions have kicked off a series of experimental studies on time discounting in recent years. Although the results are mixed, they suggest that people often use non-constant "hyperbolic" discounting rather than constant "exponential" discounting. See Shefrin (1998) and Rubinstein (2000) for surveys and critical evaluations of this literature.

However, we should note that for *normative* purposes, changing time preferences are not necessarily a deep challenge. Ethically, there is no case for supporting hyperbolic discounting in the social planner's utility function and so, in that case, we can ignore whether people personally discount hyperbolically or not. However, if we derive the social welfare function via the "dynastic" utility argument, then hyporbolic discounting matters very much indeed, even if the ethical implications are unclear. Changing time preference matters most if we were to adopt the *positive*
"decentralization" thesis for our intertemporal social planner.

Finally, there is no need to assume that time preference is completely exogenous. Tjalling Koopmans, Peter Diamond and Richard Williamson (1964) and Hirofumi Uzawa (1968a, 1968b), for instance, have argued quite persuasively that the time preference factor should be dependent on the levels of consumption. This is what we obtain with non-additively separable utility. But the outcome of allowing this is unclear. Harl Ryder and Geoffrey Heal (1973) have incorporated changing time preferences into an optimal growth model and shown that the resulting dynamics can be quite complicated.

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