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Precautionary Saving and Prudence
One behavior under uncertainty that is observed with regularity is the willingness to save
more in the present in response to increased uncertainty in the future. This behavior
is referred to as the precautionary savings motive. One might think that risk aversion
can explain precautionary saving since it is a response to risk, but this is not true. Risk
aversion only explains aversion to contemporary risks. The explanation of precautionary
savings requires a concept that is related to risk aversion, but distinct from risk aversion.
That concept is known as prudence.
This concept was first introduced by Hayne Leland with the following argument.
Consider a two-period model where saving in the first period can be used to finance
consumption in the second period. Assume that income in the first period, y1 , is certain,
but income in the second period, y˜2 , is uncertain. Then the decision maker chooses first
period saving, s, which earns rate of return, r, to solve the following problem:
max U ((1 − s)y1 ) + EU (y˜2 + s(1 + r)y1 )
s
The first-order condition is:
0
0
U ((1 − s)y1 )y1 = E U y˜2 + s(1 + r)y1 (1 + r)y1
⇒
U 0 (c1 ) = E(U 0 (c˜2 ))(1 + r)
(1)
If the future income is certain, the first order condition for optimal saving is:
U 0 (c1 ) = U 0 (c2 )(1 + r)
Call the solution to this first order condition, s̄ and corresponding consumption in
Period 2, c¯2 .
In order to analyze the savings response to increased uncertainty of future income, we
expand both sides of the equation (1) about (c1 , c¯2 ).
U 0 (c1 ) = U 0 (c1 )
1
E(U 0 (c˜2 )) ≈ U 0 (c¯2 ) + U 00 (c¯2 )E(c˜2 − c¯2 ) + U 000 (c¯2 )E(c˜2 − c¯2 )2
2
1
⇒ U 0 (c1 ) ≈ U 0 (c¯2 )(1 + r) + U 000 (c¯2 )σc22 (1 + r)
2
U 0 is a decreasing function, so if U 000 > 0 you must reduce c1 and increase c2 to
restore the equality of the first order condition. In that case, saving in Period 1 is higher
than in the case without uncertainty. We call the extra saving, precautionary saving.
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Risk aversion alone does not explain precautionary saving. We need U 000 > 0 to explain
precautionary saving. This property of the utility function is referred to as prudence.
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Example
In a two period model, suppose the agent’s income are y1 and y2 in period 1 and 2,
respectively. In particular, y1 = y2 . There is no initial wealth endowment. The agent’s
problem is
max U (c1 , c2 ) = u(c1 ) + βu(c2 )
c1 ,c2
s.t. c1 +
c2
y2
= y1 +
1+r
1+r
The FOC is
u0 (c1 ) = (1 + r)βu0 (c2 ).
If r = 0 and β = 1, this condition would be reduced to
u0 (c1 ) = u0 (c2 ).
Then c1 = c2 = y1 = y2 .
Now suppose there is uncertainty in agent’s income in period 2. The agent’s income
σ > 0 and
in Period 2 can be y2h and y2l with equal probability. Moreover, y2h = y1 +q
y2h +hl2
l
y2 = y1 − σ > 0 and then 2 = y2 = y1 . Specifically, we assume that σ = 32 y1
Then the agent’s problem is
max
l
c1 ,ch
2 ,c2
U (c1 , c2 ) = u(c1 ) + β[0.5u(ch2 ) + 0.5u(cl2 )]
s.t. c1 +
ch2
yh
= y1 + 2
1+r
1+r
c1 +
cl2
yl
= y1 + 2
1+r
1+r
The FOC is
u0 (c1 ) = (1 + r)β(0.5u0 (ch2 ) + 0.5u0 (cl2 )).
If r = 0 and β = 1, this condition would be reduced to
u0 (c1 ) = 0.5u0 (ch2 ) + 0.5u0 (cl2 ).
ch +cl
Note that 2 2 2 = y2 + y1 − c1 . If u0 is convex, that is u000 > 0, 0.5u0 (ch2 ) + 0.5u0 (cl2 ) >
u0 (y2 +y1 −c1 ). So u0 (c1 ) > u0 (y1 +y1 −c1 ) and the consumption c1 in this problem would be
lower than that in previous problem. If u0 is concave, that is u000 < 0, 0.5u0 (ch2 )+0.5u0 (cl2 ) <
2
u0 (y2 + y1 − c1 ). So u0 (c1 ) < u0 (y1 + y1 − c1 ) and the consumption c1 in this problem would
be higher than that in previous problem.
If the utility function is log utility ln(c), the optimal consumption in period 1 would
satisfy
2c21 − 6y1 c1 + 3y12 + y2h y2l = 0.
p
The optimal consumption c1 = 41 6y1 − (4y12 + 8σ 2 ) = 0.5y1 < y1 and therefore,
saving in Period 1, s > 0 .
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