Theory
In this section, we set out a model that supports our empirical findings – specifically the finding that in the early stages of working life, the average income of the comparison group may have either a positive or insignificant effect on reported happiness or lifesatisfaction.
The essential insight we wish to capture is that lifesatisfaction may depend on not just a comparison of a person’s own current income with the current income of their peers, but also on a comparison of how their life as a whole is going relative to their peers, thus on relative lifetime income. Of course, early in their working life people do not know for sure how their lives might pan out and, in particular, how not just their own lifetime income but that of the comparison group will evolve. So they use information about how their life has gone to date – specifically their current income and that of their peers – to draw inferences about how things might go in the future. In this context a high current income of the comparison group may signal that there has been a significant amount of promotion to date, and hence future promotion prospects and expectations of relative future lifetime income are good.
The aim of the model is to formalise this idea and show that there are indeed contexts in which, in the earlier part of working life, the current income of the comparison group may be positively associated with reported happiness.
The Model
The model is framed in a way that is consistent with the data on which the empirical analysis has been conducted. So it is assumed that individuals’ working lives are split into two periods.
We also assume that all individuals have a comparison/peer group with whom they compare how their lives are going. Accordingly, we consider a subpopulation of individuals who are identical in terms of some observable characteristics: age, educational attainment, location etc. This constitutes the comparison/peer group to which everyone within the subpopulation compares themselves.
Though identical in certain respects, individuals differ in some other characteristics that are unobservable but will manifest over the course of their lifetime in two different respects:

Individuals may turn out to be Hares or Tortoises. Hares show early promise and get promoted early (in period 1). Tortoises develop more slowly, and get promoted, if at all, later in life – in period 2. Individuals learn in period 1 whether or not they have been promoted and hence whether they are Hares or Tortoises. So in period 1, the current income of a Tortoise is {c}_{1}^{T}=b, where b >0 denotes basic income, while the current income of a Hare is {c}_{1}^{H}=b\left(1+\phi \right), where φ > 0 is the proportionate income supplement obtained through promotion in Period 1.

Individuals may turn out to be genuinely smart or basically dull. Smartness only manifests itself in period 2 and leads to smart people – Tortoises or Hares – being promoted (or further promoted) in Period 2. It is assumed that Smart Tortoises turn out to be equally smart as Smart Hares; therefore, in period 2, their current incomes are {c}_{2}^{ST}={c}_{2}^{SH}=b\left(1+\sigma +\phi \right), where σ > 0 represents a smartness factor – the extent to which promoted people get an extra income supplement to reflect the value of real smartness rather than the flashiness of a Hare. In Period 2, some of the Hares who were promoted in Period 1 will turn out not to actually have much substance and will be Dull Hares. Having already been promoted they tread water in terms of income and in period 2 get current income {c}_{2}^{DH}=b\left(1+\phi \right). Finally Dull Tortoises don’t get promoted in period 2 either and thus end up with current income {c}_{2}^{DT}=b.
For simplicity it is assumed that these two manifested characteristics – flashiness and smartness – are independently distributed in the population. Let p_{
H
}, 0 < p_{
H
} < 1 be the proportion of people who are Hares, and p_{
S
}, 0 < p_{
S
} < 1 be the proportion of people who are smart.
In period 1 the average current income of the group is
{\overline{c}}_{1}={p}_{H}{c}_{1}^{H}+\left(1{p}_{H}\right){c}_{1}^{T}=b\left(1+{p}_{H}\phi \right)\text{,}
while in period 2 it is
{\overline{c}}_{2}=b\left[1+{p}_{S}\left(\sigma +\phi \right)+\left(1{p}_{S}\right){p}_{H}\phi \right]={\overline{c}}_{1}+{p}_{S}b\left[\sigma +\phi \left(1{p}_{H}\right)\right]
It is assumed that the happiness experienced by each person in each period depends on

i.
A comparison of their current income with the average current income of their peers.

ii.
A comparison of their view of their lifetime income with the average lifetime income of their peers. In Period 1, lifetime income is not fully known, so individuals have to estimate both their own lifetime income and the average lifetime income of their peers.
It follows from the above assumptions that at the end of Period 1:

the expected lifetime income of a Hare is
{y}_{1}^{eH}=2{c}_{1}^{H}+{p}_{S}b\sigma

the expected lifetime income of a Tortoise is
{y}_{1}^{eT}=2{c}_{1}^{T}+{p}_{S}b\left(\sigma +\phi \right)

the expected average lifetime income of the peer group is
{\overline{y}}_{1}=2{\overline{c}}_{1}+{p}_{S}b\left[\sigma +\phi \left(1{p}_{H}\right)\right]\text{.}
Now suppose that although for individuals the probability of being smart is the same whether they are a Hare or a Tortoise, in the population as a whole, the proportion of smart people is related to the proportion of Hares by^{20}
It follows from this that at the end of Period 1:

the expected lifetime income of a Hare is
{y}_{1}^{eH}=2{c}_{1}^{H}+{p}_{H}b\sigma
(A2)

the expected lifetime income of a Tortoise is
{y}_{1}^{eT}=2{c}_{1}^{T}+{p}_{H}b\left(\sigma +\phi \right)
(A3)

the expected average lifetime income of the peer group is
{\overline{y}}_{1}=2{\overline{c}}_{1}+\left({p}_{H}b\right)\left(\sigma +\phi \right){\left({p}_{H}b\right)}^{2}\frac{\phi}{b}
(A4)
Information structure
The information structure of the model is as follows.

At the outset, and throughout their lives, individuals know: the values of φ and σ –the income premiums to flashiness and smartness respectively; the relationship between Period 1 and Period 2 incomes, conditional on being of various types; and the relationship between p_{
S
} and p_{
H
} as given by (A1).

However, initially they do not know the economic prospects for their cohort – whether they have skills that will turn out to be in high demand and lead to high opportunities for promotion. That is, initially they do not know the values of b and p_{
H
}.

However, in Period 1 they learn their own income and that of their peers, and so by comparing them, they know whether they have turned out be a Hare or a Tortoise. Formally, they learn: {c}_{1}^{j},\phantom{\rule{1em}{0ex}}j=H,T ; the average income of their peers, {\overline{c}}_{1}; their current income relative to that of their peers, {}^{c}r_{1}^{j}=\frac{{c}_{1}^{j}}{{\overline{c}}_{1}},\phantom{\rule{0.25em}{0ex}}j=H,T and hence their type H or T. Also from what they learn in Period 1, they can deduce the values of b and p_{
H
} and, hence, from (A1), the value of their future promotion prospects, p_{
S
}. Using this, they can use (A2), (A3) and (A4) to calculate their own expected lifetime income and the average of that of their peers.

In Period 2 everything is revealed. Individuals learn the value of their current income in Period 2 and the average current income of their peers. Comparing their current income in Period 2 to that earned in Period 1, they learn whether they are smart or dull. So they now fully know their type. They can now carry out a full comparison of how their life has gone relative to their peers in terms of both their relative current income and their relative lifetime income. Formally individuals learn their Period 2 income {c}_{2}^{jk},\phantom{\rule{1em}{0ex}}j=S,D;\phantom{\rule{1em}{0ex}}k=H,T and hence their type jk, j = S, D; k = H, T. They also learn the average Period 2 income of their peers {\overline{c}}_{2}.^{21} Individuals therefore know their full lifetime income {y}_{2}^{jk}={c}_{1}^{k}+{c}_{2}^{jk},\phantom{\rule{1em}{0ex}}j=S,D;\phantom{\rule{0.5em}{0ex}}k=H,T and the average lifetime income of their peers: {\overline{y}}_{2}={\overline{c}}_{1}+{\overline{c}}_{2}.
Implications
Having set out the assumptions of the model, we now derive the implications. The fundamental issue we want to investigate is how the average current income of the peer group in each of the two periods affects each individual’s reported happiness, taking as given their own income. In particular, we want to explore the possibility that although a higher level of peer income in Period 1 lowers relative current income, it might raise expected relative lifetime income since it sends a signal about higher promotion prospects in the future.
Unfolding Lives
Period 1
Hares
In Period 1, Hares learn their current income {c}_{1}^{H}=b\left(1+\phi \right) and the average income of their peers {\overline{c}}_{1}=b\left(1+{p}_{H}\phi \right). Hence they know their relative current Period 1 income
{}^{c}r_{1}^{H}=\frac{{c}_{1}^{H}}{{\overline{c}}_{1}}>1,
which is, of course, a strictly decreasing function of the average Period 1 income of their peers.
From this they calculate:
b=\frac{{c}_{1}^{H}}{1+\phi};\phantom{\rule{1em}{0ex}}b{p}_{H}=\frac{{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}}{\phi \left(1+\phi \right)}
(A5)
Substitute (A5) into (A2) and (A4) to get:
{y}_{1}^{eH}=\frac{2\phi \left(1+\phi \right){c}_{1}^{H}+\sigma \left[{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}\right]}{\phi \left(1+\phi \right)}
(A6)
{\overline{y}}_{1}^{H}=\frac{2\phi \left(1+\phi \right){\overline{c}}_{1}+\left(\sigma +\phi \right)\left[{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}\right]{\left[{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}\right]}^{2}\frac{1}{{c}_{1}^{H}}}{\phi \left(1+\phi \right)},
(A7)
where {\overline{y}}_{1}^{H} is the average lifetime income that Hares expect their peers to get on the basis of the information available to Hares in Period 1.
It is straightforward to show that
\frac{\partial {\overline{y}}_{1}^{H}}{\partial {\overline{c}}_{1}}=\frac{\left(\sigma +\phi \right)+2\left(1{p}_{H}\right)\phi}{\phi}>\frac{\sigma}{\phi}=\frac{\partial {y}_{1}^{eH}}{\partial {\overline{c}}_{1}}>0,
(A8)
so, other things being equal, the higher the current income of their peers, the higher the realised proportion of Hares in the population, and thus, from (A1), the greater the promotion prospects they face in Period 2. This raises Hares’ estimated value of their own lifetime income, but also that of their peers, and indeed the latter increases by more than the former.
Now from (A6) and (A7), in Period 1, Hares expect to end up with a relative lifetime income:
\begin{array}{l}{}^{y}r_{1}^{eH}=\frac{{y}_{1}^{eH}}{{\overline{y}}_{1}^{H}}\\ \phantom{\rule{3.3em}{0ex}}=\frac{2\phi \left(1+\phi \right){c}_{1}^{H}+\sigma \left[{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}\right]}{2\phi \left(1+\phi \right){\overline{c}}_{1}+\left(\phi +\sigma \right)\left[{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}\right]{\left[{\overline{c}}_{1}\left(1+\phi \right){c}_{1}^{H}\right]}^{2}\frac{1}{{c}_{1}^{H}}}\end{array}
(A9)
It is straightforward to show that
{}^{y}r_{1}^{eH}=\frac{2\left(1+\phi \right)+\sigma {p}_{H}}{\left[2\left(1+\phi \right)+\sigma {p}_{H}\right]\phi \left(1{p}_{H}\right)\phi {\left(1{p}_{H}\right)}^{2}}>1,
(A10)
and thus, as we know must be the case, the expected lifetime income of Hares is greater than the expected lifetime income of their peers.
By differentiating (A9) w.r.t {\overline{c}}_{1} we get:
\frac{\partial {}^{y}r_{1}^{eH}}{\partial {\overline{c}}_{1}}=\frac{\frac{\partial {y}_{1}^{eH}}{\partial {\overline{c}}_{1}}{}^{y}r_{1}^{eH}\frac{\partial {\overline{y}}_{1}^{H}}{\partial {\overline{c}}_{1}}}{{\overline{y}}_{1}^{H}},
(A11)
which from (A8) and (A10) is strictly negative. So the relative lifetime income expected by Hares in period 1 is a decreasing function of average current income of their peers, and so too is their happiness.
Tortoises
In Period 1, Tortoises learn their current income {c}_{1}^{T}=b and the average income of their peers, {\overline{c}}_{1}=b\left(1+{p}_{H}\phi \right). Hence they know their relative current Period 1 income
{}^{c}r_{1}^{T}=\frac{{c}_{1}^{T}}{{\overline{c}}_{1}}<1,
(A12)
which is, of course, a strictly decreasing function of the average Period 1 income of their peers.
From this information, Tortoises can also work out:
b={c}_{1}^{T};\phantom{\rule{2em}{0ex}}b{p}_{H}=\frac{{\overline{c}}_{1}{c}_{1}^{T}}{\phi}
(A13)
Substitute (A13) into (A3) and (A4) to get:
{y}_{1}^{eT}=\frac{2\phi {c}_{1}^{T}+\left(\sigma +\phi \right)\left({\overline{c}}_{1}{c}_{1}^{T}\right)}{\phi}
(A14)
{\overline{y}}_{1}^{T}=\frac{2\phi {\overline{c}}_{1}+\left(\sigma +\phi \right)\left({\overline{c}}_{1}{c}_{1}^{T}\right){\left({\overline{c}}_{1}{c}_{1}^{T}\right)}^{2}\frac{1}{{c}_{1}^{T}}}{\phi},
(A15)
where {\overline{y}}_{1}^{T} is the average lifetime income that Tortoises expect their peers to get on the basis of the information available to Tortoises in Period 1.
It is straightforward to show that
\frac{\partial {\overline{y}}_{1}^{T}}{\partial {\overline{c}}_{1}}=\frac{\left(\sigma +\phi \right)+2\left(1{p}_{H}\right)\phi}{\phi}>\frac{\sigma +\phi}{\phi}=\frac{\partial {y}_{1}^{eT}}{\partial {\overline{c}}_{1}}>0,
(A16)
so, just as with Hares, the higher the current income of their peers, the higher the realised proportion of Hares in the population, and thus so, from (A1), the greater the promotion prospects that Tortoises face in Period 2. This raises Tortoises’ estimated value of their own lifetime income, but also that of their peers, and indeed the latter increases by more than the former.
Now from (A14) and (A15), in Period 1, Tortoises expect to end up with a relative lifetime income:
{}^{y}r_{1}^{eT}=\frac{{y}_{1}^{eT}}{{\overline{y}}_{1}^{T}}=\frac{2\phi {c}_{1}^{T}+\left(\sigma +\phi \right)\left({\overline{c}}_{1}{c}_{1}^{T}\right)}{2\phi {\overline{c}}_{1}+\left(\sigma +\phi \right)\left({\overline{c}}_{1}{c}_{1}^{T}\right){\left({\overline{c}}_{1}{c}_{1}^{T}\right)}^{2}\frac{1}{{c}_{1}^{T}}}
(A17)
It is straightforward to show that
{}^{y}r_{1}^{eT}=\frac{2+\left(\sigma +\phi \right){p}_{H}}{\left[2+\left(\sigma +\kappa \right){p}_{H}\right]+\phi {p}_{H}\left(2{p}_{H}\right)}<1,
(A18)
and thus, as we know must be the case, the expected lifetime income of Tortoises is lower than the expected lifetime income of their peers.
By differentiating (A18) w.r.t {\overline{c}}_{1} we get:
\frac{\partial {}^{y}r_{1}^{eT}}{\partial {\overline{c}}_{1}}=\frac{\frac{\partial {y}_{1}^{eT}}{\partial {\overline{c}}_{1}}{}^{y}r_{1}^{eT}\frac{\partial {\overline{y}}_{1}^{T}}{\partial {\overline{c}}_{1}}}{{\overline{y}}_{1}^{T}}
(A19)
Consequently,
\frac{\partial {}^{y}r_{1}^{eT}}{\partial {\overline{c}}_{1}}\frac{>}{<}0\phantom{\rule{1em}{0ex}}\iff \phantom{\rule{1em}{0ex}}\frac{\frac{\partial {y}_{1}^{eT}}{\partial {\overline{c}}_{1}}}{\frac{\partial {\overline{y}}_{1}^{T}}{\partial {\overline{c}}_{1}}}\frac{>}{<}{}^{y}r_{1}^{eT}
(A20)
Substitute (A16) into (A20), and we get:
\frac{\partial {}^{y}r_{1}^{eT}}{\partial {\overline{c}}_{1}}\frac{>}{<}0\phantom{\rule{1em}{0ex}}\iff \phantom{\rule{1em}{0ex}}\frac{\sigma +\phi}{\left(\sigma +\phi \right)+2\left(1{p}_{H}\right)\phi}\frac{>}{<}\frac{2+\left(\sigma +\phi \right){p}_{H}}{2+\left(\sigma +\phi \right){p}_{H}+\phi {p}_{H}\left(2{p}_{H}\right)}
(A21)
It is clear that if p_{
H
} = 0, then \frac{\partial {}^{y}r_{1}^{eT}}{\partial {\overline{c}}_{1}}<0, whereas if p_{
H
} = 1, then \frac{\partial {}^{y}r_{1}^{eT}}{\partial {\overline{c}}_{1}}>0. So the conclusion is that if p_{
H
} is sufficiently large, then an increase in the average income earned by their peers in Period 1 raises the expected relative lifetime income of Tortoises and, hence, potentially their happiness.
Period Two
This is straightforward.
Each type of individual knows their current Period 2 income, {c}_{2}^{jk},\phantom{\rule{1em}{0ex}}j=S,D;\phantom{\rule{1em}{0ex}}k=H,T and the average Period 2 income of their peers, {\overline{c}}_{2}_{.} Consequently, they can work out their relative current income
{}^{c}r_{2}^{jk}=\frac{{c}_{2}^{jk}}{{\overline{c}}_{2}}\phantom{\rule{1em}{0ex}}j=S,D;\phantom{\rule{1em}{0ex}}k=H,T,
which is a strictly decreasing function of the average income of their peers.
Each individual also sees clearly their relative performance in terms of lifetime income
{}^{y}r_{2}^{jk}=\frac{{y}_{2}^{jk}}{{\overline{y}}_{2}}=\frac{{c}_{1}^{k}+{c}_{2}^{jk}}{{\overline{c}}_{1}+{\overline{c}}_{2}}\phantom{\rule{1em}{0ex}}j=S,D;\phantom{\rule{1em}{0ex}}k=H,T,
and this too is a strictly decreasing function of the average Period 2 income of their peers, {\overline{c}}_{2}_{.}
So, unambiguously, happiness of all individuals is a strictly decreasing function of the average Period 2 income of their peers, {\overline{c}}_{2}_{.}