2.5 The role of liquidity constraints
2.5.2 Time-varying liquidity constraints
Let now the share of liquidity constrained consumers be determined by (2.12).
While the rule still suggests increasing the interest rate in the face of an increase in expected in‡ation, the ouput gap or house prices, the weights on these
vari-16Sincex2=0, it must hold that 1( 4+ 6)> 2 5.
ables now vary with the share of constrained agents. The share of constrained agents is positively related to the output gap and expected in‡ation and nega-tively related to the nominal interest rate and house prices. Consequently, each weight is a function of expected in‡ation, the output gap, the interest rate and house prices. We will discuss each weight in turn.
2.5.2.1 The optimal weight on expected in‡ation
When liquidity constraints are time-varying the optimal rule (2.11) not only suggests responding to house prices, but also that the optimal weights on the arguments in the rule change with the house price shock. Consider how the optimal weight on expected in‡ation changes with house price movements
df dqt =
0q ( 2 1s1( 2+ 4 + 6))
[ ( 6+ 4+ 2(1 t))]2
Since 0q < 0 the weight on expected in‡ation decreases with house prices if the share of income going to the young s1 is small, s1 < 2
1( 2+ 4+ 6). Higher house prices decrease the share of constrained agents because they allow to extract equity from the house to …nance consumption. Intuitively, the same two e¤ects as above with constant are at work. On the one hand, since an agent doesn’t react to changes in the interest rate when constrained, but does so when unconstrained, a weaker interest rate response is required with more agents unconstrained to bring down expected in‡ation by a given amount.
On the other hand, the constrained consume out of current income and liquid assets, while the unconstrained don’t. An interest rate increase depresses current income and lowers consumption by the constrained, which helps to bring down in‡ation through an indirect channel. When this channel is partially shut down because more agents are unconstrained, it must be compensated by a stronger direct interest rate channel. However, the smaller the share of incomes1 going to the constrained, the weaker is the indirect e¤ect that must be compensated.
If s1 < 2
1( 2+ 4+ 6) the direct e¤ect more than compensates the indirect e¤ect and a weaker response to expected in‡ation is warranted.
Moreover, the weight on expected in‡ation is a function of expected in‡ation itself, via its e¤ect on t.
df
d(Et t+1) =
0 ( 2 1s1( 2+ 4+ 6))
[ ( 6+ 4+ 2(1 t))]2
which is positive if the share of income going to the young s1 is small, s1 <
2
1( 2+ 4+ 6). To recapitulate the intuition, consider now an increase in the share of constrained agents. Higher expected in‡ation increases the share of
constrained agents by raising the optimal level of consumption. Since the con-strained don’t react to changes in the interest rate, a stronger interest rate response is required. On the other hand, the newly constrained consume out of current income and liquid assets. An interest rate increase depresses their income and as such lowers consumption by the constrained. This e¤ect however is smaller the smaller the share of income s1 going to the constrained, calling for a stronger rate increase.
Similarly, the weight on expected in‡ation also increases with the output gap if s1 < 2
1( 2+ 4+ 6): df dyt =
0y ( 2 1s1( 2+ 4+ 6))
[ ( 6+ 4+ 2(1 t))]2
Again with more agents constrained there are two e¤ects: On the one hand a stronger interest rate increase is necessary to bring down expected in‡ation since fewer agents react to interest rate changes. On the other hand, the constrained react indirectly to the interest rate change as far as it a¤ects aggregate income.
If the indirect e¤ect is small, because a small share of aggregate income goes to the constrained, then the …rst e¤ect outweighs the second one and a stronger interest rate response is warranted.
Finally the optimal weight on expected in‡ation decreases with the interest rate if s1 < 2
1( 2+ 4+ 6). df
dit =
0i ( 2 1s1( 2 + 4+ 6))
[ ( 6+ 4+ 2(1 t))]2
The intuition is again the same as in the previous cases. Note that the interest rate e¤ect and the house price e¤ect tend to o¤set the other two, since 0 >0,
0y >0 and 0i <0, 0q<0, and the strength of each e¤ect crucially depends on the sensitivities 0, 0y, 0i and 0i.
2.5.2.2 The optimal weight on the output gap
House price changes have an impact on the share of constrained agents and therefore on the optimal weight on the output gap.
dfy dqt =
0q [ 2 3x3 x2( 1( 4+ 6) 2 5)]
[ 6+ 4+ 2(1 t)]2
The intuition is analogous to the case where either all the young are constrained or no one is constrained. The weight on output decreases with the house price if the share of consumption in aggregate income when middle-aged is small, as
de…ned by x2 < 2 3x3
1( 6+ 4) 2 5
17. Higher house prices reduce the proportion of constrained agents. On the one hand these newly unconstrained respond to interest rate changes, which allows to achieve a given reduction in the output gap with a smaller interest rate increase. On the other hand, the newly unconstrained also react to their expected future consumption, which increases with x2. This e¤ect calls for a stronger interest rate increase to bring down the output gap by a given amount. However, if this e¤ect is small, the …rst e¤ect dominates and a weaker interest rate response is required.
The optimal weight on the output gap reacts to an increase in the output gap as follows.
dfy dyt
=
0y [ 2 3x3 x2( 1( 4+ 6) 2 5)]
[ 6+ 4+ 2(1 t)]2
which is positive if x2 < 2 3x3
1( 6+ 4) 2 5. The intuition is the same as above.
The optimal weight on output changes with respect to expected in‡ation and the interest rate in an analogous manner.
dfy d(Et t+1) =
0 [ 2 3x3 x2( 1( 4+ 6) 2 5)]
[ 6+ 4+ 2(1 t)]2
dfy dit =
0i [ 2 3x3 x2( 1( 4+ 6) 2 5)]
[ 6+ 4+ 2(1 t)]2
2.5.2.3 The optimal weight on house prices
The optimal weight on house prices is a function of house prices themselves.
dfq dqt =
0q 2b( 2+ 4+ 6)
[ 6+ 4 + 2(1 t)]2 <0
This is because with rising house prices fewer agents are constrained, who don’t react to house prices. In this case consumption and the output do increase. After all, becoming unconstrained means that consumption of the young has increased up to or beyond the optimal level of consumption. However, this increase in consumption is now captured by the increase in the output gap. Therefore a separate response to house prices is not warranted. The di¤erence lies in the coe¢ cients on house prices and the output gap. Pressure on the output gap due to a wealth e¤ect from house prices requires a slightly di¤erent response than pressure due to an increase in expected future income and consumption.
Furthermore, the optimal weight on house prices increases with expected
17Remember that sincex2=0, it must hold that 1( 4+ 6)> 2 5.
in‡ation.
dfq d(Et t+1) =
0 2b( 2+ 4+ 6)
[ 6+ 4 + 2(1 t)]2 >0
This is because more agents are constrained, who react to house price increases.
The optimal weight also increases with the output gap dfq
dyt =
0y 2b( 2 + 4+ 6)
[ 6+ 4+ 2(1 t)]2 >0
and decreases with the interest rate. A higher interest rate reduces the propor-tion of constrained agents, who react to changes in house prices.
dfq dit =
0i 2b( 2+ 4+ 6)
[ 6+ 4+ 2(1 t)]2 <0