Staff Working Paper 2003.2
Andrew Lilico
February 2003
STAFF
WORKING
PAPER
STAFF WORKING PAPER
When Might People Pay Too Much
For Their Housing?
FOREWORD
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Europe Economics Staff Working Papers are intended to provide a
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individuals within the firm.
The present paper has been prepared by Andrew Lilico. Andrew is a
consultant at Europe Economics. The paper sets out a formal economic
framework within which to consider conditions under which people might pay
“too much”, in some sense, for their housing.
It is hoped that this work will provide a relevant technical contribution to an
important area of current debate.
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This paper considers factors that might lead households to pay too much for their housing. It
provides a formal model of housing demand, shows theoretical grounds why the house price to
average earnings ratio might be fairly constant, and considers whether limited foresight might
drive cycles in house prices.
1
INTRODUCTION
In this paper we are interested in factors that may lead to households paying ''too much'' for their
housing. We shall consider both what might lead to ''too much'' being paid when agents are fully
rational, and also what happens in the oft-discussed (but rarely modelled) case where agents
look only finitely far into the future.
During 2002 house prices rose by over 25 per cent1 so that by the end of 2002 the house prices
to average earnings ratio had reached about 6 (compared with a long-term trend of about 3.75),
and it is widely debated whether a house price ''crash'' is about to occur (see Figure 1).2
Why might house prices cycle in this way, and what features of the housing market might lead to
booms and crashes in house prices?
In our model the major driver for house prices will be expectations of future wage growth. The
higher our expected future wages, the higher our lifetime wealth, and the more housing we can
afford (and want to afford).
The first headline result is that, given certain key assumptions, optimising behaviour will lead
house prices to be a roughly constant multiple of average wages (though perhaps with a
slight upwards trend over time). These key assumptions are:
•
Housing and other consumption are complements (e.g. you will enjoy your dinner more if
you eat it in a pleasant kitchen, and you won't be able to appreciate your kitchen at all if
you are starving.);
•
Inherited wealth is fairly small compared with lifetime wages;
•
The stock of housing is fairly constant;
•
The population of the country is roughly constant;
•
Average expectations of lifetime wages are fairly accurate.
1
2
25.3 per cent according to the Nationwide Quarterly Review, Winter 2002; 26.4 per cent according to the Halifax house price index see http://www.hbosplc.com/view/housepriceindex/nationalcommentary.asp
See, for example, Capital Economics Property Focus, December 12 2002
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1
Figure 1: The Relationship Between House Prices and Average Earnings
6
8%
6%
5
4.5
4%
4
3.5
2%
3
0%
2.5
2
-2%
1.5
Average ratio 1969-2002
02
01
20
00
20
99
20
98
19
97
19
96
19
95
19
94
19
93
19
92
19
91
19
90
19
89
19
88
Long-term average ratio
19
87
19
86
19
85
19
84
19
83
19
82
19
81
CML-based series
19
80
19
79
19
78
19
77
ODPM-based series
19
76
19
75
19
74
19
73
19
72
19
19
19
19
1969
71
-4%
70
1
GDP Growth (right-hand axis)
Source: ODPM, CML, National Statistics & Treasury
On the other hand, if, say, our expectations of wage growth3 are too optimistic, then we may pay
an amount for our housing that, with the benefit of hindsight, we shall consider too high. This is
one possible source for house price booms and crashes - some kind of cycle in people's future
wage expectations.
A related possibility is that people misperceive, in some way, the price they will actually pay for
their housing. Such misperceiving might arise if, for example, agents believed that real mortgage
interest rates would be lower than in fact they would be, for example if they thought that low
interest rates would persist but low inflation were a temporary phenomenon4. In recent years
many commentators have argued that low interest rates mean that houses are more ''affordable''
at high prices than in the past. This, they say, justifies higher house prices. The idea of this
argument appears to be that if, at 10 per cent mortgage interest rates service a mortgage
involved a £500 interest payment and a £100 loan repayment, then if mortgage interest rates fall
to 6 per cent (so the mortgage interest payment falls to £300) that justifies a rise in loan
repayments and hence in house prices. Perhaps house prices could rise significantly, raising loan
repayments to £300 delivering the same nominal monthly servicing cost?
3
4
or, equivalently (if we can regard the wage as being adjusted for the probability of job loss) expectations about prospects of avoiding
unemployment
Note, for example, Mervyn King's November 2002 speech in which he suggested that ''It may take longer for households to work
out the impact of low inflation on real interest rates than to realise that the rate of increase of prices of everyday purchases has
fallen. Learning takes time. One possible consequence of a slow adjustment to low inflation is that households may mistake too
much of the reduction in nominal interest rates for a permanent fall in the real rate. As a result, asset prices are bid up to levels that
prove unsustainable when learning finally occurs'' - http://www.bankofengland.co.uk/speeches/speech181.pdf
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2
GDP growth rate
House prices to average earnings ratio
5.5
However, this argument as stated is bogus. If low interest rates reflect low inflation (and hence
low wage rises) and house prices rise so as to keep the nominal monthly servicing cost the same,
the real burden of housing debt will rise, because the same nominal payments would be being
serviced by lower nominal wages, as illustrated in Figure 2.
It is interesting to note from Figure 2 that if mortgage interest rates fall from 10 per cent to 5 per
cent (which is very roughly the fall over the past five years) the house price which maintains the
same nominal monthly mortgage payment rises by approximately 50 per cent - an order of
magnitude rise very similar to the rise from house prices being about four times average earnings
in 1997 to six times average earnings today. Though hardly conclusive, this coincidence may
suggest that increased "affordability" is an important factor in higher house prices.
It is worth considering the ''affordability'' idea slightly further. As the chart in Figure 2 illustrates, for
the same house price (£100,000) the time profile of the burden of mortgage payments is different
when mortgage rates and inflation are higher than when they are lower. At higher mortgage rates
and inflation the burden of debt early in the term of the mortgage is higher but later in the term the
burden is lower. This may imply that the risk of default is higher in the early stages of a mortgage
under higher inflation and interest rates, and if this were the determining constraint which set what
mortgages people took out, we might then expect that under lower inflation and lower mortgage
rates higher mortgages could be taken out.
Of course, it would be possible to get around the differences in early burden of debt by having
mortgages that had lower payments near the start under higher inflation or higher payments near
the start under lower inflation. And indeed there were some attempts to establish products along
these lines during the period of higher inflation. However, these were not very successful. That
may have been because people did not believe that high inflation would really persist into the
future, or there could be other difficulties with such mortgages, but it seems most likely that
actually the cash-flow implications of having a higher burden early in the mortgage were not a
critical constraint on mortgages in the past. Mortgage lenders use other types of constraint to
reduce the risk of default, such as only lending 90 per cent or so of the value of a house. In fact,
under higher inflation it would appear that this gives more comfort to lenders than under lower
inflation, because if inflation is lower the risk of fluctuations in house prices leading to negative
equity is greater. Since they would have more comfort from this source under higher inflation it
seems unlikely that they would require higher loan-to-income ratios.
Thus low inflation suggests a cash-flow impact that might appear to argue for higher loan-toincome ratios and a collateral risk effect that might appear to argue for lower loan-to-value ratios.
As the UK Financial Services authority pointed out in 20015, there is a net implication for
mortgage loans of these two effects from lower inflation: they imply that the term of loans should
be shortened through an increase in the rate of capital payments equal to the reduction in interest
rates, but without any change in either loan-to-value or loan-to-income ratios. (For example, in
Figure 2 this would imply that in the case of inflation of 2 per cent and mortgage rates 5 per cent
5
''Low inflation, implications for the FSA'', Ed Harley and Stephen Davies, 2001
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3
the loan term should be reduced to about 12.5 years, giving the same monthly payment over a
shorter period.) Low inflation does not require lower loan-to-value ratios, provided that loans are
repaid faster. But if loans are repaid faster, then the cash-flow benefits of low inflation are offset.
However, the authors also point out that the repayment profile of mortgage borrowing never fully
adapted to high inflation (e.g. repayment terms did not increase as much as analysis would have
suggested), possibly suggesting that relatively high inflation was never expected to be a longterm phenomenon. Hence there may be little reason to expect mortgage terms to change with
low inflation.6
Another possibility suggested by Figure 2 is that agents may have problems if they do not look far
enough ahead. Under lower inflation the burden of debt early in the mortgage is lower, and so if
people only take account of that period that may lead them to mis-perceive their debt burden. In
the current housing market it has been commonly suggested that people are being ''shortsighted'' in their decisions over whether to pay high house prices, and that this might be a source
a booms and crashes in prices. Later we model this explicitly by considering agents who only
look finitely far ahead into the future. What might cause house price booms and crashes in the
limited foresight case is quite subtle. We model agents that have limited foresight, but also do not
realise that they have limited foresight - they are naive. Naive limited foresight is not enough to
guarantee that people will pay too much for their housing, however, and the basic case in which
they shall not purchase too much may seem slightly surprising: when wages are rising rapidly,
agents with limited foresight will tend to pay too little for their housing. The reason is relatively
straightforward. In our model the critical feature determining how much is paid for housing is
expected average lifetime income. If wages are rising then an agent with limited foresight will not
be able to see his high future wages, so he will tend to under-estimate his average lifetime
income.
6
Although it might be argued that the recent trend towards ''flexible'' mortgages enables people to pay off their mortgages early, and
hence to reduce the terms for themselves, may be a move along these lines in some broad sense.
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4
Figure 2: Nominal Mortgage Interest Rates and the Burden of Debt
Percentage burden of
monthly payments after…
Annual
Salary today
£30,000
£30,000
£30,000
£30,000
House
Mortgage
Monthly
Average
price
Interest Rate Mortgage Inflation Rate
£100,000
10%
£918
7%
£155,300
5%
£918
7%
£155,300
5%
£918
2%
£100,000
5%
£591
2%
Average Wage
Inflation
9%
9%
4%
4%
1
5
year years
34% 24%
34% 24%
35% 30%
23% 19%
10
years
16%
16%
25%
16%
25
years
4%
4%
14%
9%
Percentage burden of
whole Mortgage (in real
terms)
15%
15%
23%
15%
Monthly mortgage payment as
percentage of income
Burden of debt over the lifetime of a mortgage
40%
35%
30%
25%
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Year of mortgage
Inflation 7%, Mortgage Rates 10%, Mortgage of £100,000
Inflation 2%, Mortgage Rates 5%, Mortgage of £155,300
Inflation 2%, Mortgage Rates 5%, Mortgage of £100,000
The cases in which agents pay ''too much'' for their housing are when their average wages over
the horizon of foresight will be above the lifetime average wage. In practice this scenario (wages
being above trend in the near future) tends to happen late in an economic boom or early in an
economic bust. This would suggest that if agents have limited foresight then house price cycles
will lag cycles in the general economy, as they do in practice - thereby giving some support to the
limited foresight model.
We shall not consider below issues like migration (e.g. from the North of England to the South)
that have been discussed extensively with reference to the housing market. We note in passing
that net movements between regions within a country would indeed be expected to make a
difference to the house prices in those regions, but that the aggregate effect would be expected to
net out. If people move from the North of England to the South, then house prices in the South
will rise and those in the North will fall, but the net effect on national house prices of internal
migration should be zero. The rise in house prices compared with trend illustrated in Figure 1 is
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for overall national house prices, not the house prices in any one region, and hence cannot be
explained by migration from the North of England to the South.7
Our models will focus on the demand side of housing, mainly regarding the supply of housing as
fixed. This is clearly a modelling abstraction. Our focus has been on what might make people
willing to pay too much for their housing. House price cycles could also be related to supply-side
factors. Further work could incorporate a model of the supply-side of the housing market.
7
However, if there were a rise in the proportion of transactions which took place in high-earnings/high-house-prices areas of the
country (for example if people started moving more regularly from one part of London to another, compared with moves between
lower-income areas of the country, than in the past) that could make a difference to the measured house-price to average earnings
ratio. The reason is that the measured ratio relates house sales which actually take place against all earnings (not just the earnings
of the people involved in those sales), so if a higher proportion of the transactions take place at the top end of the market then the
measured ratio will be higher.
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2
THE BASIC MODEL
The world lasts three periods, t = 1,2,3. We consider the decision of a representative household
over its housing purchase.
In each period t households consume goods and services of At. At the beginning of period 1 a
household purchases a housing stock of H at a price p per unit of housing (with the price
expressed in goods-and-services-consumption units), the price being paid in each period (like a
mortgage). Thus housing cost is pH per period. This housing stock then returns a flow of housing
consumption H in each of the three periods.
At the beginning of the problem firms have an endowment of E, and in each period t households
receive wage wt.8 Then total lifetime household wealth is given by
W = E + w1 + w2 + w3
The household consumes all remaining wealth in period 3, so that at the end of the problem there
is nothing left over - i.e.
A3 = W - A1 - A2 - 3pH
We shall abstract from discounting, with utility assumed to take a log form and given by :
U = 3 ln(H) + λ.(ln(A1) + ln(A2) + ln(W - A1 - A2 - 3pH))
where λ is a parameter relating the value of housing consumption to that of goods and services
consumption to the household.
The combination of log utility and the utility from consumption of housing being additively
separable from utility from other forms of consumption implies that housing and other
consumption are complements - i.e. we will enjoy, say, our food more if we eat it in a pretty
kitchen, and we won't be able to appreciate our kitchen at all if we are starving. This seems a
reasonable assumption.
We consider the decision of a representative household. Considering multiple households or
overlapping generations makes the analysis more complicated but does not change the
substance of the results.
8
Though it is not required for the model, for later interpretations it may be best to think of this wage as being relative to the trend path.
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3
SOLUTION WITH FULL FORESIGHT
Our baseline case is a full foresight model in which agents solve the problem optimally. We shall
investigate when, in such a model, agents may pay ''too much'' for their housing.
The decision-problem of the representative household is
maxH, A1, A2 U = 3ln(H)+ λ.ln(A1)+ λ . ln(A2)+ λ.ln(W - A1 - A2 - 3pH)
FOC(H): 3/H = λ.3p/A3
FOC(A1): λ/A1 = λ/A3
FOC(A2): λ/A2 = λ/A3
⇒
A1 = (A2) = (A3)
3λpH = 3A1 = W - 3pH
Assume the housing stock is fixed, so
H = H^
⇒
p = W/(3H^.(1+λ)) = W*/((1+λ)H^)
(1)
where W* is the average wealth over the lifetime.
In this case agents may pay too much for their housing if they are overly-optimistic in their views
about their future wage prospects. If future wages are not going to rise as fast as expected, then
average wealth will be lower, and house prices will become too high.
A related possibility is that people misperceive, in some way, the price they will actually pay for
their housing (they think that pH is, say, lower than it actually is). Such misperceiving might arise
if, for example, agents believed that real mortgage interest rates would be lower than in fact they
would be, for example if they thought that low interest rates would persist but low inflation were a
temporary phenomenon.
The discussion in the Introduction of popular misconceptions about why house prices should be
higher (such as ''affordability'' or internal migration) suggests that even though there may be no
good reason why lower interest rates associated with low inflation should lead to higher house
price to earnings ratios, it may be natural for economic agents to perceive such an effect,
believing that the future cost per period of buying a house (pH) will be lower than it actually will
be. Equation 1 suggests that such a perception might lead agents to purchase more expensive
housing than, in retrospect, they would want.
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3.1
The house prices to earnings ratio
One noteworthy feature of this model is that, if we consider the total amount spent on housing as
the ''price'' of houses (more strictly: the cost of housing) (call this P = 3pH), then we have
P = W/(1+λ) = E/(1+λ) + (w1+w2+w3)/(1+λ)
This means that if endowments are fairly low compared with wages (E close to zero), then the
price of housing will bear an almost constant relationship to average wages, w* = (w1+w2+w3)/3
given by
P = (3/(1+λ)).w*
Typical endowments are of order £10,000 to £100,000, while typical lifetime wages are of order
£1,000,000, suggesting that endowments are a fairly small proportion of lifetime wealth. Hence,
given that house prices are widely thought to follow a trend path of about 3.75 times average fulltime wages9, the relative constancy of house prices to wages in our model is attractive.
Consider Figure 1 again. As that figure illustrates house prices cycle, sometimes being above
trend and sometimes below. The diagram also illustrates that this cycle is correlated with cycles in
GDP growth, but with a lag.
During 2002 house prices rose by over 25 per cent10 so that by the end of 2002 the house prices
to average earnings ratio had reached about 6 (well above even the 4.16 average for 19692002). Our model suggests that, for a reasonably stable stock of housing, the house prices to
average earnings ratio should indeed be fairly stable. Hence significant deviations from trend
(such as the late 1980s or early 2000s) should be monitored by policy-makers and lenders and
may reflect misperceptions. On the other hand, significant deviations from trend could also reflect
correctly expected future changes in wage growth or in the real interest rate. For example, if
future wage growth rates are actually going to be higher than current rates, the house price to
average earnings ratio will rise above trend, with the correction back to trend coming through
higher wages, rather than falls in prices.
The discussion above and our model regard tastes (λ) as stable. But it may be that housing is a
good for which people prefer to spend a higher proportion of their incomes on it as they become
richer. That would mean that the house prices to average earnings ratio might have a fixed value
for each generation, but rise over time. Though such an effect may imply a slight upward trend in
the ratio (perhaps reflecting the change from 3.75 as the long-term ratio to 4.16 over the past
thirty years), it seems most implausible that taste changes could result in significant house price
to average incomes ratio cycles.
9
10
''The spectre of negative equity'', Daily Telegraph, 4/10/2000
25.3 per cent according to the Nationwide Quarterly Review, Winter 2002; 26.4 per cent according to the Halifax house price index see http://www.hbosplc.com/view/housepriceindex/nationalcommentary.asp
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9
Thus a crucial question for policy-makers and for lenders observing deviations from historic
multiples for the house price to average earnings ratio is: “What is the source?” Small rises could
simply be the result of changing tastes in favour of housing as the country becomes richer.
Larger rises might reflect correct or incorrect expectations about higher future wage growth, or
about the servicing costs of mortgages. In Section 5 below we attempt to quantify how much
different sorts of effect might move the house price to average earning ratio.
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4
SOLUTION WITH LIMITED FORESIGHT
Next we shall consider a model in which agents (a) look only a limited distance into the future —
not far enough to solve the problem completely; and (b) do not realise that their foresight is limited
— they are naive. It is often suggested that when households are naive and short-sighted they
may pay too much for their housing. We shall see that this can be so, but is not a necessity, and
consider some conditions under which it will be true.
We assume that the household can now see only one period ahead (so that from the beginning of
the problem it can see only periods 1 and 2, and believes that the world ends at the end of period
2), and also believes (wrongly) that it will be able to consume all its remaining wealth during
period 2. Thus its decision-problem has become
maxH,A1 U^ = 2ln(H)+ λ.ln(A1) + λ.ln(W^ - A1 - 2pH)
where
W^ = E + w1 + w2
FOC(H): 2/H = λ.2p/A2
FOC(A1): λ/A1 = λ/A2
⇒ A1 = A2
2λpH = 2A1 = W^ - 2pH
Assuming again that the housing stock is fixed at H^, we have
p = W^ /(2H^.(1+λ)) = W^*/(H^.(1+λ))
where W^* is the average wealth over periods 1 and 2.
Now this price may be more than that in the full foresight case, but it may not. The crucial
question is whether average lifetime wealth is more or less than average wealth over the first two
periods. In order for it to be less (so that households pay more for their housing when they have
limited foresight), we would need
(E + w1 + w2)/2 > (E + w1 + w2 + w3)/3
⇒ E + w1 + w2 > 2w3
Suppose that the initial endowment is very small, so E is close to zero. Then this condition says
that the period income that can be observed must be greater than average income would be if the
income over periods that cannot be observed were received instead. This would be the case if
income were falling over the period. It would also be the case if income were to rise in period 2,
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11
but then fall back in period 3 (e.g. if we were in a period of unusually tight labour markets, which
would be replaced out beyond the horizon of foresight by a return to normal labour markets).
If endowments are substantial (so E >> 0) then even with quite strong and sustained wage
growth (w3 > w2 > w1) households with limited foresight will over-pay.
On the other hand, if initial endowments are quite low (E close to zero) then if wages are growing
(or below lifetime trend), agents with limited foresight will pay too little for their housing, not too
much.
To summarise, when agents have limited foresight and do not realise it, they may pay too much
for their housing if their wages are relatively high at the moment and fall beyond the horizon of
foresight. Perhaps counter-intuitively, if wages are growing rapidly and will continue to do so
beyond the horizon of foresight then agents with limited foresight will tend to pay too little for their
housing, not too much. On the other hand, if wages are falling currently and will continue to do so
beyond the horizon of foresight then agents will pay too much.
These insights may suggest that agents with limited foresight will tend to pay too much for their
housing in the late stages of a boom (when wages are tending to be above their trend path and
rising in the near-term), and even too much for a while after the economy starts to falter (when
wages are tending to be above their trend path though falling in the near-term), but too little during
the later stages of a bust and in the early stages of a boom - hence we might expect house prices
to be slightly lagged compared with the economic cycle.
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5
MAGNITUDES OF EFFECTS
In this section we try to give some sense of what magnitudes of “over-pricing” of housing different
kinds of errors might lead to. We focus on two of our identified possible sources of error — overoptimism about wage prospects and misperception about future costs.
5.1
Over-optimism about wage prospects
Let us re-calibrate from our three-period model to an n-period model, using equation (1):
p = W*/((1+λ)H^)
(1A)
where, now,
W* = E + w1 + w2 + … + wn
Suppose that n is a forty-year working life, and that the actual growth rate of real wages will be 2
per cent. Then we have
W* = E + (1.0240 - 1).50w1
(2)
Suppose that people believe that their real wages are actually going to grow at 2.5 per cent
throughout their lives instead of 2 per cent. Then their believed lifetime wealth, W* believed will be
given by
W* believed = E + (1.02540 - 1).40w1
(2A)
If we assume that E is small, that suggests that if people believe their wages are going to rise by
2.5 per cent per year when in fact they are going to rise at about 2 per cent per year then W*
believed
/W* equals about 1.12, and so from (1A), (2) and (2A) we can conclude that people would
pay more than 10 per cent more for their housing than they would if they knew all the facts.
Equally, if perceptions about future wages rising faster than in the recent past, this would mean
that a half per cent rise in the future real growth rate of wages might lead to a justified rise in the
house price to average earnings ratio of a little over 10 per cent. Similarly, a justified rise in the
house price to average earnings ratio of about 50 per cent would require correct expectations of a
rise in future real earnings growth from about 2 per cent to about 4 per cent — a huge increase.
5.2
Misperception about future costs
Suppose that people believe that the burden of the debt of their housing depends on the nominal,
rather than real interest rate. Let us suppose that the form this takes is for people to be
concerned about their nominal repayments, rather than the real burden of their debts over the
lifetime of a mortgage. The table in Figure 2 suggests that when nominal mortgage interest rates
fall from 10 per cent to 5 per cent the same nominal burden is achieved when house prices are
about 55 per cent higher, but the same real burden is achieved at the same house price. Hence
this suggests that a mistake of this sort might lead people to pay over 50 per cent too much for
their housing.
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6
CONCLUSION
We have produced a model of house prices and argued that, for a reasonably stable stock of
housing, the house prices to average earnings ratio should be fairly stable — albeit perhaps with
a slight upward trend if higher wealth leads to a greater preference for housing. Hence significant
deviations from trend (such as the late 1980s or early 2000s) should be monitored by policymakers and raise questions.
We have also considered factors that might lead to agents paying more for their housing than
they would, in hindsight, consider optimal. We have identified three main sources of error:
1
Fully rational agents may pay too much for their housing if they are over-optimistic
about their future wage prospects (or, equivalently, about their prospects of
avoiding unemployment). For example, if people believe that real wages will rise in
the future at an average of about 2.5 per cent, while in fact they will rise at only 2
per cent, that would lead people to pay more than 10 per cent too much for their
housing.
2
Agents may over-pay if they misperceive the future cost (for example by confusing
low nominal interest rates with low real interest rates). For example, if people base
their purchasing decisions on the nominal cost of repayments rather than on the
real lifetime burden of their debts, and mortgage interest rates fall from 10 per cent
to 5 per cent purely because of lower inflation, that would lead people to pay more
than 50 per cent too much for their housing.
3
When agents have limited foresight and do not realise it, they may pay too much
for their housing if their wages are relatively high at the moment and fall beyond
the horizon of foresight.
We have focused on the demand side of housing, mainly regarding the supply of housing as
fixed. This is clearly a modelling abstraction. Further work could incorporate a model of the
supply-side of the housing market. Other possible areas for future work include the effects on
house prices of household fission or changes in commuting patterns.
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