Chapter 21: The Discounted Utility Model
21.1: Introduction
This is an important chapter in that it introduces, and explores the implications of, an empirically
relevant utility function representing intertemporal preferences. In fact, it is not only empirically
relevant but it also has the important normative property that individuals with such preferences are
never dynamically inconsistent. What is meant by this is that the relative evaluation of consumption
in period t and in period s remains constant through time – for instance, if today an individual
regards £1000 in 2015 as equivalent to him or her as £1500 in 2017, they should think the same
way tomorrow and in 2003, and so on. This means that it represents the kind of intertemporal
preferences that people ‘ought’ to have – if they want to be consistent in their intertemporal
decision-making. These normative properties are not relevant in the two-period world that we have
been analysing so far, but become relevant in a more-than-two period world. We shall consider such
a world after we have described the model and explored its implications in a two-period world.
21.2: The Discounted Utility Model in a Two-Period World
This is a model of preferences over bundles of consumption (c1, c2) where c1 denotes consumption
in period 1 and c2 denotes consumption in period 2. It is simply given as follows:
U(c1, c2) = u(c1) + u(c2)/(1+ρ)
(21.1)
Do note carefully that there are two utility functions here: one is U, which is defined over the
consumption bundle (c1, c2) and which represents the utility of that bundle; the other is u, which is
defined over a single period’s consumption and which represents the utility gained from consuming
a particular amount in a particular period. The Discounted Utility Model states that the utility of a
consumption bundle (c1, c2) is given by the utility gained from consuming the amount c1 (in period
1) plus the utility gained from consuming the amount c2 (in period 2) divided by (1+ρ). In this
model, ρ is a parameter of the model and usually varies from individual to individual. For most
individuals the parameter ρ is positive, which means that (1+ ρ) is greater than 1, which means that
u(c2)/(1+ρ) is smaller than u(c2) and hence that greater weight is attached to a particular level of
consumption if it consumed in period 1 rather than in period 2.
Why is it called the Discounted Utility Model? Because it discounts the utility gained from
consumption in period 2 at the rate ρ. This parameter for a particular individual is called the
individual’s discount rate. It is called this for the following reason. Consider the present value of
the stream of income m1 in period 1 and m2 in period 2 when the market interest rate is r. The
present value is (see section 20.5) is
m1 + m2/(1+r)
As we have already noted in chapter 20, as viewed from period 1 the future income m2 is discounted
by the market at the rate r because it will not be received for a period. Now notice the similarity
between this expression and that for the Discounted Utility Model in (21.1). In the latter the
individual discounts the future utility because it will not be received for a period. He or she
discounts it at the rate ρ. We should emphasise that this discount factor is individual specific –
some individuals have a high value for ρ, some a low value. What it depends upon is how the
individual regards the utility of consumption in period 2 relative to the utility of consumption in
period 1. It may be the case that the individual regards them equally – in which case the parameter ρ
takes the value 0 – the individual in this case does not discount the future. However for most of us it
seems to be the case that we weight the present more highly than the future – in which case the
value of the parameter ρ is positive. Furthermore, the more the individual regards the present
relative to the future the higher is ρ - alternatively we can say the higher the individual regards the
present relative to the future, the more the individual discounts the future. We should also point out
that, as a parameter representing preferences, the value of ρ is independent of the value of the
market discount rate r.
The parameter ρ captures the individual’s preferences regarding the relative weighting of present
and future consumption. To complete the specification of the Discounted Utility Model we also
need to specify the function u. This will also be individual specific. Normally we expect it to be a
concave function – as consumption rises then so does the utility gained from that consumption, but
it rises less than proportionately. Or, in more casual terms, for every increase of 1 in consumption
there is an increase in utility but these increases get smaller as the amount consumed increases. In
what follows we will assume that the function u is the square root function (which is concave for
positive consumption1). An alternative is that the u function is the logarithmic function (which is
also concave for positive consumption). With this function we get slightly different demand
functions but the important property concerning the relationship of r with ρ (which we shall prove
shortly) is true for all concave u functions, including the logarithmic function. Of course, in
practice, the form of the function depends upon the individual’s preferences.
21.4: Indifference Curves with the Discounted Utility Model
We have now specified the Discounted Utility Model. In the next section we explore its
implications. But first, as we are going to use the framework for analysis developed in chapter 20,
we need to look at the properties of the indifference curves in (c1, c2) space implied by this model.
There follows a little bit of mathematics; if you do not like maths, look away until we discuss the
implications of the results.
An indifference curve in (c1, c2) space is given, as ever, by
U(c1, c2) = constant
If we substitute in the specification of the Discounted Utility Model from (21.1) we get the
following equation for an indifference curve in (c1, c2) space.
u(c1) + u(c2)/(1+ρ) = constant
From this we can find the slope of the indifference curve (a proof is provided in the Mathematical
Appendix to this chapter). The slope is given by:
slope of indifference curve = - (1+ρ) [du(c1)/dc1]/[du(c2)/dc2]
(21.2)
where du(c)/dc denotes the derivative of u(c) with respect to c – that is the rate at which utility is
increasing with consumption.
1
It is difficult to conceive of negative consumption.
The slope is negative so the indifference curves are downward sloping. Moreover if u is concave
then, as we move down and rightwards along an indifference curve, c1 is rising while c2 is falling,
and hence du(c1)/dc1 is falling while du(c2)/dc2 is rising, and so the magnitude of the slope is
falling. From this it follows that if u is concave then the indifference curves are convex. If however
u is linear then both du(c1)/dc1 and du(c2)/dc2 are constant and so the slope of the indifference
curves are constant – that is they are linear. If we continue this line of argument with a convex
function u, then we get the following result:
If u is concave, linear or convex then the indifference curves in (c1, c2) space are convex, linear or
concave.
As we have already argued the concave u function is the empirically more realistic – as people’s
consumption increases, their utility rises, but at a decreasing rate.
One final result is of particular importance. If, in (21.2) we put c1 = c2 we get that the slope of an
indifference curve is –(1+ρ). Calling the line c1 = c2 the ‘equal consumption line’ we get the
important result that:
Along the equal consumption line the slope of every indifference curve of an individual with
Discounted Utility Model preferences is equal to –(1+ρ).
You should remember this.
We illustrate in figure (21.1). In this figure I have used the square root utility function for u, and I
have put ρ = 0.2. (that is, the individual discounts the future at 20%). In figure 21.1 I draw some of
his or her indifference curves in (c1, c2) space. I have also inserted the equal consumption line. It
can be seen that the slope of every indifference curve along this line is –1.2.
For an individual with a higher discount rate than 20%, his or her indifference curves are
everywhere steeper along the equal consumption line.
21.5: The Implications in a Two-Period World
You may be able to guess some of the implications. We know that along the equal income line the
slope of the indifference curves are all –(1+ρ). It follows therefore that to the right of the equal
income line the magnitude of the slope is everywhere less than (1+ρ) and to the left of the equal
income line the magnitude of the slope is everywhere more than (1+ρ). Why is this important?
Because we know that the magnitude of the slope of the budget line is (1+r) and we know that at
the optimum point the magnitude of the slope of the budget line is equal to the magnitude of the
slope of the indifference curve at the optimum point. (Because the budget line must be tangential to
the indifference curve at the optimal point.) So at the optimal point the slope of the indifference
curve must be –(1+r).
There are three cases to consider. The simplest is when r = ρ. In this case the optimum point must
lie somewhere along the equal consumption line – for we know that the slopes of the indifference
curves there are –(1+ρ), and if this is equal to –(1+r) (because r = ρ) it follows that the optimum
point (where the slopes of the indifference curve and the budget line are equal) must be along the
equal consumption line. In this case the individual consumes the same in both periods – the reason
being that the market discounts the future at exactly the same rate as the individual.
When r > ρ we can argue that the optimal point must lie to the left of the equal consumption line –
so that the individual consumes more in the second period than the first. And when r < ρ we can
argue that the optimal point must lie to the right of the equal consumption line – so that the
individual consumes more in the first period than the second. What is crucial is whether the market
discounts the future more heavily than the individual.
These properties are confirmed in the following example, in which we keep fixed the initial
incomes of 40 (in period 1) and 40 (in period 2) and vary the rate of interest from zero (as in figure
21.3) upwards. We assume a value of ρ equal to 0.2 (20%) – that is, we use the preferences of
(21.1) above. You might like to verify that when the market rate of interest reaches 20% the optimal
point is on the equal consumption line.
The implied borrowing and lending are pictured in figure 21.4. (Recall that the rate of return is one
plus the rate of interest. The downward sloping line is the net demand for consumption in period 1
and the upward sloping line the net demand for consumption in period 2. Note that they cross the
axis at a rate of interest 20%.)
If we repeat this exercise for an individual with ρ = 0.4 we get figure 21.10. Note the similarities
and the differences between this and figure 21.4.
21.6: The Discounted Utility Model in a Many-Period World
Although a little outside the scope of this book, it is interesting to consider the extension of the
Discounted Utility Model to a many-period world. The extension is the natural one and follows the
extension of the discounting formula discussed in section 20.6.
Suppose we are in a world which lasts T periods, where T may be finite, infinite or random.
Suppose the individual has consumption c1 in period 1, c2 in period 2, …, ct in period t, …, and cT
in period T. We consider his or her preferences over consumption bundles over these T periods –
bundles which we denote by (c1, c2, …, ct , …, cT). The Discounted Utility Model states that
preferences over these bundles are given by
U(c1, c2, …, ct , …, cT) = u(c1) + u(c2)/(1+ρ) +…+ u(ct)/(1+ρ)t-1+…+u(cT)/(1+ρ)T-1
(21.3)
Once again the model is specified by a discount rate ρ and a utility function u. The model is the
natural extension of the two-period model. The utility of the bundle is the sum of the utilities of
consumption in each period discounted at the rate ρ. If ρ is positive the weight attached to future
consumption declines with the time from the present: period 1’s consumption is given a weight of
1; period 2’s consumption is weighted by 1/(1+ρ); …; period t’s consumption by 1/(1+ρ)t-1; … ;
and period T’s by 1/(1+ρ)T-1. You may find it constructive to compare (21.3) with the formula in
section 20.5 giving the present value of an income stream – there future incomes are discounted by
the market at the rate r because they are received in the future. In the Discounted Utility Model
future utilities are discounted by the individual at the rate ρ because they are experienced in the
future.
Whether this is a good description of actual intertemporal preferences is obviously an empirical
issue but it may be instructive to give the normative reason why such preferences might be
appropriate. In essence the idea is that an individual with such preferences is dynamically consistent
in the sense that they carry out plans that they formulate. In a certain world there is no reason why
one should ever want to change plans once formulated, so this property is an appealing one.
Let us discuss it in more detail. Suppose we start in period 1 and we know that the income stream is
going to be m1 in period 1, m2 in period 2, …, mt in period t, …, mT in period T. Then the individual
in period 1 formulates a plan for consumption (c1, c2, …, ct , …, cT) through time on the basis of
maximising the utility (21.3) subject to the income stream and the rate of interest (which we are
assuming is fixed and known). Accordingly in period 1 the individual consumes the planned c1. In
period 2 what happens? Does the individual implement the planned consumption c2? It depends.
Suppose the individual re-plans at this stage. At this stage he or she has a different objective
function – because period 1 has been and gone. Now only periods 2 through T remain. If we update
(21.3) to take account of this the objective function is now
U(c2, …, ct , …, cT) = u(c2) + u(c3)/(1+ρ) +…+ u(ct)/(1+ρ)t-2+…+u(cT)/(1+ρ)T-2
(21.4)
Does the maximisation of this, given the remaining stream of income, lead to the same optimal
values c2, …, ct , …, cT that were the optimal values in the original plan c1, c2, …, ct , …, cT
formulated at time period 1?
The answer to this question is ‘yes’ – though the reason for this answer may not be obvious. A
proof is provided in the Mathematical Appendix, which shows that we can express the utility from
period 1 onwards in the following form:
U(c1, c2, …, ct , …, cT) = u(c1) + U(c2, …, ct , …, cT)/(1+ρ)
(21.5)
So the choice of the best values of c2, …, ct , …, cT given the optimal value of c1 from the
maximisation of utility as viewed from the first period (with respect to the choice of c1, c2, …, cT)
leads to the same choice of c2, …, ct , …, cT as in the original choice of c1, c2, …, ct , …, cT in the
maximisation of utility as viewed from the first period. The individual would not want to change the
plans originally made.
In essence – and this should be clear from (21.5) - the reason is that if we compare any two periods
s and t the relative weight attached to the utility of consumption in period s is always the same value
(1+ρ)s-t irrespective of which period we are viewing it from. This means that the individual never
wishes to revise any plans that he or she has formulated. (Of course, in a certain world, there are no
reasons why one should ever wish to revise any plans.) If, instead, the individual used a varying
discount rate through time, then it is possible that the individual might want to revise plans made
earlier. In a sense this is because the individual, with a varying discount rate, is not just one
individual but several – and they have conflicting views what is best to do. This is exactly the kind
of person who resolves every morning not to drink in the evening – and then ends up doing so.
21.7: Summary
Most of this chapter has been concerned with the two-period Discounted Utility Model though the
final section contained an extension to the many-period model in which we provided a normative
justification for it. In the two-period case we showed the following.
The Discounted Utility model states that the intertemporal utility function is given by U(c1, c2) =
u(c1) + u(c2)/(1+ρ) where ρ is the individual's discount rate.
With the Discounted Utility Model the slope of all indifference curves along the equal-consumption
line are –(1+ρ)
Now we recalled that the perfect capital market budget constraint has a slope of –(1+r) where r is
the market interest rate. Combining these two results we showed that:
An individual with Discounted Utility preferences would consume the same in both periods if his or
her ρ equalled the market r…
... and that he or she will consume more in period 1 than in period 2 if r < ρ
.. and that he or she will consume more in period 2 than in period 1 if r > ρ
We then briefly extended the model to a many-period world and showed that one attractive
normative property of the model is that an agent with such preferences is never dynamically
inconsistent.
21.8: At what rate do you discount the future?
The Discounted Utility Model incorporates the notion that individuals do not value consumption in
all periods equally. In general, it would appear to be the case that individuals care more about the
present than the future, though the extent to which they do this differs from individual to individual.
Here you are invited to try and discover how the extent to which you do this.
The model in a T-period world takes the form of equation (21.3) and in a 2-period world, the form
of equation (21.1). This latter says that a two-period bundle of consumption (c1,c2) is valued by the
function
U(c1, c2) = u(c1) + u(c2)/(1+ρ)
where u(c) is the utility gained from consuming c in some period and ρ is the individual’s discount
rate. Both of these are specific to the individual. Here we will concentrate on discovering your value
of ρ, assuming, of course, that your preferences are in accordance with the Discounted Utility
Model2.
The method used is simple – we exploit the implications of the equation above, particularly those
results that we have already derived concerning the slope of indifference curves. If we can find two
points on the same indifference curve, then we can use these results to try to infer the value of ρ.
Specifically, let us take equation (21.2):
slope of indifference curve = - (1+ρ) [du(c1)/dc1]/[du(c2)/dc2]
If we do not know the individual’s utility function, then we start at a position where c1 and c2 are
equal. This enables us to derive the result stated in the text:
Along the equal consumption line the slope of every indifference curve of an individual with
Discounted Utility Model preferences is equal to –(1+ρ).
It follows that if we start at a point where c1 = c2 = c and we find a nearby point about which the
individual feels the same (that is, is on the same indifference curve) then we can infer the value of
ρ. To fix ideas suppose that the point (c - a, c + b) is such a point – that is, the individual feels
indifferent between the two-period consumption bundle (c, c) and the two-period consumption
bundle (c - a, c + b). Then the slope of the indifference curve is approximately equal to –b/a. This is
an estimate of –(1+ ρ). Hence we have that an estimate of ρ is b/a – 1 = (b – a)/a. If the individual
puts more weight on present than on future consumption, then b will be bigger than a (because to
compensate the individual for consuming a less today he or she will require more than a next
period), and hence the estimate of ρ is positive.
Obviously the value of ρ depends upon the length of the period we are considering. Let us assume
here that the period is of length one year. You can now try and implement the above ideas and
2
If they are not, then there is not a ρ to discover.
hence find your yearly discount rate. You have to do the following introspection. Suppose you start
from a position where you are consuming the same both this year and next year: for example, you
are consuming £5000 each year. Now suppose someone suggests that your consumption this year
will fall by say £100 (to £4900) but that you will be given some extra consumption next year to
compensate. You should ask yourself: what is the minimum compensation I would require? This is
quite a difficult introspection, but you should attempt it. Try and narrow it down. Would £1
compensation be enough? (Probably not.) Would £1000 compensation be enough? (Probably more
than enough.) Would £50 be enough? And so on.
The minimum compensation that you require gives an estimate of your discount rate ρ. Suppose
this minimum compensation is £120. Then we have that a = £100 and b = £120, so that your ρ is
0.2 ( = (120-100)/100). It should be clear that we can derive the following table of examples.
Minimum Compensation required Implied value
for a decrease in period 1 discount rate ρ
consumption of £100
£100
0.0
£110
0.1
£120
0.2
£130
0.3
£140
0.4
of
the
Note carefully that we are talking about changes in consumption and not about changes in money
income. If it were the latter and there was a perfect capital market in which you could freely borrow
and lend at the constant rate of interest r, then the answer to the question would have to be
£100(1+r). We would not learn anything about your discount rate – only about the rate of interest
in the perfect capital market!
Note the assumptions carefully: (1) we start along the equal consumption line (so that we do not
have to worry about your utility function, which is difficult to infer); (2) we consider a small
reduction in period 1 consumption (otherwise we are moving around the indifference curve and its
slope may change); (3) you tell us honestly the minimum amount of compensation in terms of
period 2 consumption you require. If you do all this, you can find your discount rate.
If you are interested, you could explore the implications for a many-period world. In the extension
to T periods, as given in equation (21.3), you will see that the Discounted Utility Model assumes
that the same discount rate is used throughout. Obviously this is a strong assumption, but one that
can be tested. You can try one such test yourself. Suppose you have found that the minimum
compensation you require (under the assumptions listed above) is £120. Then, starting from an
equal consumption point, you regard having £100 less today as being compensatable with an extra
£120 in one year’s time. Now do the same exercise, but now ask what is the minimum
compensation in two years’ time for having £100 less today. Denote again by b this minimum
compensation, but do remember that this will be consumed in two years’ time. Repeating the
argument that we used above it follows that –b/a is an estimate of –(1+ ρ)2. Thus to be consistent
with the Discounted Utility Model and with your previously derived estimate of ρ (which is 0.2), it
must be the case that the minimum compensation you require in two years’ time is £144. (So that
b/a = (1+ ρ)2.)
At first glance you may this odd – or, at least inconsistent with your introspection. Let us discover
why the Discounted Utility Model makes this prediction. We begin with your first introspection –
you needed £120 in one year’s time to compensate you for having £100 less today, that is, for each
£1 less today you needed £1.20 in compensation in one year’s time. If this story applies not only to
consumption deferred for one year from today, it should also apply to consumption deferred for one
year from next year. So, if you are to be compensated for having £100 less today, but will receive
the compensation in two years’ time, you can argue as follows: I would need £120 more in one
year’s time, and to defer each of these £120 for a further year, I will need a compensation of £1.20
for each of those £120 – that is a compensation of 1.2 times £120 = £144 in two year’s time. Was
your introspection consistent with this?
You may be interested to know that there have been many experimental tests of the Discounted
Model, and particularly its central assumption that the discount rate is constant3. The great strength
of experimental economics is its central tenet that participants should be given appropriate
incentives to behave in such a way that their behaviour reveals their preferences. In many areas it is
easy to give appropriate incentives – as we will see in the chapter on Game Theory – but in the area
of intertemporal choice it is more difficult. We have already noted some strong assumptions that
underlie the inferences we have made. These are difficult to enforce in the laboratory. The greatest
problem, however, is that correct incentives in intertemporal choice experiments necessary involve
the passage of time. It may be difficult to ensure that participants and experimenters are still around
after that passage of time.
21.9 Mathematical Appendix
We first derive the proposition concerning the slope of the indifference curves implied by the
Discounted Utility Model.
As stated in the text, an indifference curve in (c1, c2) space is given by
U(c1, c2) = constant
If we substitute in the specification of the Discounted Utility Model from (21.1) we get the
following equation for an indifference curve in (c1, c2) space.
u(c1) + u(c2)/(1+ρ) = constant
To find the slope of the indifference curve we differentiate this totally, thus getting
u'(c1) dc1 + u'(c2) dc2/(1+ρ) = 0
where u'(c) denotes the derivative of u(c) with respect to c. From this we get the slope of an
indifference curve
dc2/dc1 = - (1+ρ) u'(c1)/u'(c2)
This is equation (21.2) of the text.
We now derive equation (21.5).
Suppose c1*, c2*, …, ct* , …, cT* maximise
3
Many of these studies suggest that the discount rate is not constant.
U(c1, c2, …, ct , …, cT) = u(c1) + u(c2)/(1+ρ) +…+ u(ct)/(1+ρ)t-1+…+u(cT)/(1+ρ)T-1
given an income stream m1, m2, …, mt, …, mT,
and hence subject to the intertemporal budget constraint
m1 + m2/(1+r) + … + mT-1/(1+r)T-2 + mT/(1+r)T-1 = c1 + c2/(1+r) + … + cT-1/(1+r)T-2 + cT/(1+r)T-1
then it must be the case that c2*, …, ct* , …, cT* maximise
U(c2, …, ct , …, cT) = u(c2) + u(c3)/(1+ρ) +…+ u(ct)/(1+ρ)t-2+…+u(cT)/(1+ρ)T-2
subject to the constraint
m1 + m2/(1+r) + … + mT-1/(1+r)T-2 + mT/(1+r)T-1 = c1* + c2/(1+r) + … + cT-1/(1+r)T-2 + cT/(1+r)T-1
because
U(c1, c2, …, ct , …, cT) = u(c1) + U(c2, …, ct , …, cT)/(1+ρ).