LECTURE 5:
EXPECTATIONS, UNCERTAINTY, AND ASSET HOLDINGS
Reading:
• William A. Lord (2002), Household Dynamics: Economic Growth and Policy, Chapters 2 (Section 2.2)
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1. INTRODUCTION AND BACKGROUND
INTRODUCTION AND BACKGROUND
In this lecture we will:
• Introduce economic uncertainty
• Study household consumption and savings decisions under uncertainty • Understand the precautionary savings motive
• Study the relationship between asset prices and risk
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BACKGROUND ON ECONOMIC RISK
• A major feature of economic life is the presence of uncertainty and risk • We all face many different risks to our economic outcomes:
• Unemployment, income, unexpected expenses, health, death, etc
• Households must make consumption and savings decisions taking these risks into account • As we will see, these risks affect both asset holdings and asset prices
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ECONOMIC RISKS: DEATH RISK BY AGE
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ECONOMIC RISKS: UNEMPLOYMENT RISK BY AGE
Source: US Bureau of Labor Statistics, https://www.bls.gov/web/empsit/cpseea10.htm 5
ECONOMIC RISKS: EARNINGS RISK BY INCOME RANK
Source: Guvenen, Schulhoffer-Wohl, Song, and Yogo (2017), Worker Betas: Five Facts About Systematic Earnings Risk
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2. RISK AVERSION AND THE PRECAUTIONARY SAVINGS MOTIVE
CONCEPTS: RISK AVERSION AND PRECAUTIONARY SAVINGS
• Risk aversion: a tendency to prefer economic outcomes with low uncertainty to those with more uncertainty
• Precautionary Savings: an increase in income uncertainty that leaves expected income unchanged reduces current consumption. But savings increase as a form of self insurance against low income states of the world.
• Expected income = probability-weighted average of possible incomes • E.g. E(y) = αy1 +(1−α)y2
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CONCEPTS: RISK AVERSION AND PRECAUTIONARY SAVINGS
• Risk aversion is a consequence of diminishing marginal utility • For utility function u(·), then u′ > 0 and u′′ < 0
• Implies a loss of $x matters more than a gain of $x
• A risk averse agent would turn down a fair bet with even odds of an increase of x or a decrease
in x
• But risk aversion does not tell us how an agent responds to uncertainty or risk
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CONCEPTS: RISK AVERSION AND PRECAUTIONARY SAVINGS
• Precautionary Savings is a result of marginal utility declining at a decreasing rate: • For utility function u(·), then u′′′ > 0
• This feature of utility functions/preferences is sometimes called prudence
• In this case, an increase in income uncertainty (holding expected income constant) raises expected marginal utility
• This means that the value of additional consumption is higher, which means that households save more in order to consume more in the periods of heightened uncertainty
• These additional savings in the face of greater uncertainty are called precautionary savings
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3. THE PRECAUTIONARY SAVINGS MOTIVE:
AN ILLUSTRATIVE MODEL
A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• Let’s begin with the standard model from earlier lectures
• A household makes consumption and savings decisions, subject to known and constant
incomes
• Assume β = 1, return on savings is zero (r = 0), income in each period is Y ̄, utility function
u′ >0,u′′ <0,u′′′ >0
max u(C1) + u(C2) C1 ,C2
s.t. C1+C2=Y ̄+Y ̄
• The first order condition yields the optimality condition (Euler Equation):
u′(C1) = u′(C2) ⇒ C 1 = C 2 = Y ̄
• Label these consumption choices Cc1 and Cc2 for the choices under certainty
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A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• Now suppose there are different states of the world
• These states affect income in period 2, with a chance of a good outcome and a chance of a
bad outcome
Y2 =
Y ̄ + x with probability 0.5 Y ̄ − x with probability 0.5
(Good Outcome) (Bad Outcome)
• The expected value of period 2 income is:
E ( Y 2 ) = 0 . 5 ( Y ̄ + x ) + 0 . 5 ( Y ̄ − x ) = Y ̄
• Which is the same as period 2 income under the model with certainty
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A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• So the household now chooses consumption and savings to maximize expected utility: max u(C1) + E(u(C2))
C1 ,C2 ,S
s.t. C1+S=Y ̄
C2 = Y2 + S
• where consumption in period 2 is uncertain because it depends on the realization of income in period 2:
C2 =
Y ̄ + x + S Y ̄ − x + S
with probability 0.5 with probability 0.5
(Good Outcome) (Bad Outcome)
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A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• The first order condition in this case yields the Expected Euler Equation: u′(C1) = E(u′(C2))
• Where E(u′(C2)) is the expectation over marginal utility of consumption in period 2 • Note that we can compute this as:
E(u′(C2)) = 0.5 × u′(C2(good)) + 0.5 × u′(C2(bad)) = 0 . 5 × u ′ ( Y ̄ + x + S ) + 0 . 5 × u ′ ( Y ̄ − x + S )
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A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• Suppose the household were to choose period 1 consumption the same as under the certainty case: C1 = Cc1 = Y ̄
• Then from the period 1 budget constraint, savings are: S = Y ̄ − Cc1
• And we can write consumption in period two as:
C2 = Y2 + S
= Y 2 + Y ̄ − C c1
Y ̄ + x + Y ̄ − Cc1 with probability 0.5 = Y ̄ − x + Y ̄ − Cc1 with probability 0.5
Cc2 + x with probability 0.5 = Cc2 − x with probability 0.5
• If choosing the certainty consumption in period 1, period 2 consumption is equal to the certainty consumption (Cc2) plus or minus the uncertain component of income x
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A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• Now, write the expected marginal utility of consumption in period 2 as: E(u′(C2)) = 0.5 × u′(Cc2 + x) + 0.5 × u′(Cc2 − x)
≥ u′(Cc2)
• Because u′′′ > 0, the expected marginal utility of consumption in the uncertain case is greater than marginal utility in the certain case
• This means that the value of the certain consumption choice is greater than the value of the uncertain consumption outcomes
• Another way: households prefer certainty to uncertainty, even when the expected value of outcomes is the same in both cases
Technical Note: We can prove this inequality using the properties of convex functions and Jensen’s inequality. This is not required knowledge for this class.
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GRAPHICAL ILLUSTRATION OF THE PRECAUTIONARY SAVINGS MOTIVE
u′, E(u′)
u′(Cc1) = u′(Cc2) (A)
Euler Equation
u′(C2) Cc2 C2
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GRAPHICAL ILLUSTRATION OF THE PRECAUTIONARY SAVINGS MOTIVE
u′, E(u′)
u ′ ( C c2 − x ) u′(Cc) = u′(Cc)
(1)
(A)
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u ′ ( C c2 + x )
C c2 − x C c2 = E ( C 2 ) C c2 + x
u ′ ( C 2 )
C 2
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GRAPHICAL ILLUSTRATION OF THE PRECAUTIONARY SAVINGS MOTIVE
u′, E(u′)
u ′ ( C c2 − x ) E(u′ (C2 ))
(1)
(B)
(A)
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u′(Cc) = u′(Cc) u ′ ( C c2 + x )
C c2 − x C c2 = E ( C 2 ) C c2 + x • Notice that: E(u′(C2)) > u′(Cc2) = u′(Cc1)
Euler Equation
• Therefore: increase C2, decrease C1, higher (precautionary) savings S
u ′ ( C 2 )
C 2
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A MODEL WITH THE PRECAUTIONARY SAVINGS MOTIVE
• Point (A) corresponds to marginal utility of the certain consumption Cc2
• Recall that this is the optimal consumption choice for the certainty case: u′(Cc1) = u′(Cc2)
• Point (B) is the expected marginal utility over consumption in the uncertain case: E(u′(C2)) • Note that E(u′(C2)) > u′(Cc2) = u′(Cc1)
• This means that consumption is too low in period 2 (i.e. marginal utility is too high)
• Therefore, the household should consume less in period 1: Cu1 < Cc1
• This allows household to save more and so consume more in period 2: Cu2 (s) > Cc2 (s)
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PRECAUTIONARY SAVING IN THE MACROECONOMY: EMPIRICAL EVIDENCE
• Does greater uncertainty really lead to more savings?
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PRECAUTIONARY SAVING IN THE MACROECONOMY: EMPIRICAL EVIDENCE
Equity Market
Volatility Tracker
(Bloom et al., 2019; scaled)
Real Consumption Growth
Equity Market
Volatility Tracker
(Bloom et al., 2019; scaled)
Real Savings Growth
15 10 5 0 −5 −10 −15
150
100
50
0
−50
1992 1996 2000 2004
2008 2012
2016 2020
1992 1996 2000
2004 2008 2012 2016
2020 21
4. PRECAUTIONARY SAVINGS AND ASSET PRICES
PRECAUTIONARY SAVINGS AND ASSET PRICES
• The main lesson from the precautionary savings model:
• An increase in uncertainty increases demand for savings/assets
• But if asset demand increases, what happens to asset prices?
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A MODEL OF PRECAUTIONARY SAVINGS AND ASSET PRICES
• Consider our two-period model:
• Where a one period bond B can be purchased at price Pb
• And again:
Y2 =
Y ̄ + x Y ̄ − x
with probability 0.5 with probability 0.5
Pbu′(C1) = βE(u′(C2)) ⇒ Pb = β E(u′(C2))
(Good Outcome) (Bad Outcome)
• The first order condition yields:
max u(C1) + βE(u(C2)) C1 ,C2 ,B
s.t. C1+PbB=Y ̄ C2 = Y2 + B
u′(C1) • This is referred to as an asset pricing equation
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• Asset prices given by the marginal rate of substitution between consumption across periods
A MODEL OF PRECAUTIONARY SAVINGS AND ASSET PRICES
• The first order condition yields the Expected Euler Equation: Pbu′(C1) = βE(u′(C2))
• And rearranging we have:
Pb = β E(u′(C2)) u′(C1)
• This is referred to as an Asset Pricing Equation
• Asset prices determined by the ratio of marginal utilities of consumption in each period
• Or, another way: asset prices are given by the marginal rate of substitution between consumption across periods
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A MODEL OF PRECAUTIONARY SAVINGS AND ASSET PRICES
How does uncertainty affect asset prices?
• Consider an increase from x to x∗:
• Now Y2 = Y ̄ + x∗ with probability 0.5, and Y2 = Y ̄ − x∗ with probability 0.5
• Still the case that E(Y2) = Y ̄
• This is called a mean-preserving spread in Y2
• Uncertainty only affects period 2, so the effect on asset prices comes through E(u′(C2))
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GRAPHICAL ILLUSTRATION: INCREASE IN UNCERTAINTY x → x∗
u′, E(u′)
E(u′(C∗2 )) E(u′ (C2 ))
Cc2 −x∗
Cc2 −x∗
E(C2) Cc2 −x∗
u′(C2) Cc2 +x∗ C2
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A MODEL OF PRECAUTIONARY SAVINGS AND ASSET PRICES
• Since E(u′(C2)) increases, Pb increases also:
↑Pb = β E(u′(C2))↑
u′(C1)
• But C1 also decreases in response to greater uncertainty, which increases u′(C1)
• So what is the overall effect?
• Substitute in the budget constraints:
Pb =βE(u′(Y2 +B)) u′(Y ̄ − PbB)
• Illustrate optimal choices graphically by plotting the left-hand-side and right-hand-side of the asset pricing equation
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GRAPHICAL ILLUSTRATION: INCREASE IN UNCERTAINTY x → x∗
LHS, RHS
Pb
βE(u′(Y∗2 +B)) u′ ( ̄Y−Pb B)
β E(u′ (Y2 +B)) u′ ( ̄Y−Pb B)
Pb P∗b
Pb 28
A MODEL OF PRECAUTIONARY SAVINGS AND ASSET PRICES
• An increase in uncertainty increases the price of assets • This is intuitive:
• Greater uncertainty induces precautionary savings which increases demand for assets • Higher asset demand is associated with higher asset prices
• Recall that the asset return: R = 1 Pb
• So higher asset prices are associated with lower asset returns • Do we observe this empirically?
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UNCERTAINTY AND ASSET PRICES/RETURNS: EMPIRICAL EVIDENCE
Equity Market
Volatility Tracker
(Bloom et al., 2019; scaled)
3 Month Treasury Bond Return
10
8
6
4
2
0
1992 1996 2000 2004
2008 2012 2016 2020
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A MODEL OF PRECAUTIONARY SAVINGS AND ASSET PRICES
• Note, we need to compare risk-free bonds
• For risky assets, demand and prices may increase or decrease depending on the nature of
the asset risk (see next lecture!)
Equity Market
Volatility Tracker
(Bloom et al., 2019; scaled)
Aaa Corporate Bond Spread Over 10-Year Treasury
Baa Corporate Bond Spread Over 10-Year Treasury
10
8
6
4
2
0
1992 1996
2000 2004
2008 2012 2016
2020 31
5. HOW MUCH DOES THE PRECAUTIONARY SAVINGS MOTIVE MATTER?
HOW MUCH DOES THE PRECAUTIONARY SAVINGS MOTIVE MATTER?
• The two motives for asset holding that we have studied so far:
• Life-cycle motive: consumption smoothing across time
• Precautionary savings motive: consumption smoothing across outcomes/states of the world
• Consumption over the Life Cycle (Gourinchas and Parker; 2002) uses a model to distinguish between the two motives
• Finds that precautionary savings matter much more for young households’ asset decisions
• Finds that life-cycle motives matter much more for older households’ asset decisions
• Young households start out with low wealth, need to save to build a precautionary savings buffer
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EMPIRICAL APPLICATION: PRECAUTIONARY VS. LIFE-CYCLE SAVINGS
Source: Gourinchas and Parker (2002) 33