# 程序代写代做代考 Lecture 4: Consumption

Lecture 4: Consumption

Patrick Macnamara

ECON60111: Macroeconomic Analysis University of Manchester

Fall 2020

Introduction

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Keynes Consumption Function

• Behavior of consumption, according to Keynes:1 C = a + bY

where

• a > 0 is an intercept,

• b ∈ (0, 1) is the marginal propensity to consume (MPC) • Y is current disposable income

• Keynes (1936) argued that “the amount of aggregate consumption mainly depends on the amount of aggregate income,” and that this relationship “is a fairly stable function.” He claimed further that “it is also obvious that a higher absolute level of income … will lead, as a rule, to a greater proportion of income being saved.”

1Note the exclusion of the real interest rate.

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MPC vs. APC

• Marginal propensity to consume (MPC): MPC = dC = b

dY

Assumption: 0 < b < 1.
• Average propensity to consume (APC):
APC = C = a + bY = a + b = a + MPC YYYY
1 − APC = fraction of income that is saved.
Since a > 0 ⇒ APC > MPC; APC declines with income.

Hence, Keynes’ claim that higher income will lead to a greater proportion of income saved.

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Illustration of Keynes Consumption Function

C=a+bY MPC

45°

APC

C

YY

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Empirical Evidence For/Against Keynes

• Some evidence was supportive of Keynes:

Across households at a point in time, relationship looks like

what Keynes postulated.2

e.g., a > 0 and 0 < MPC < 1, so that APC declines with Y • Other evidence less supportive:
Simon Kuznets, collecting aggregate data on consumption and income going back to 1869, shows that over time, aggregate C is essentially proportional to aggregate Y .3
e.g.,a≈0andMPC verycloseto1
APC is stable over time, despite large increases in income.
• Permanent Income Hypothesis, due to Friedman (1957), able to reconcile the contradictory evidence.
2For example, see Williams and Zimmerman (1935), Stigler (1954). 3See Kuznets (1952).
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Outline of Today’s Lecture
• Permanent income hypothesis (PIH) without shocks (Friedman, 1957)
• Add uncertainty: the random-walk hypothesis (Hall, 1978) • Overview of some extensions
• The interest rate
• Risk assets
• Precautionary saving • Liquidity constraints
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Consumption Under Certainty
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Simplifying Assumptions
We consider a dynamic optimization model to study consumption and saving.
• Utility is separable over time.
Note: utility does not depend on saving directly.
• Utility is derived from a homogeneous consumption good.
• Labor income is exogenous.
Therefore we do not consider labor supply choices.
• The rate of return on assets is exogenous. Importantly, it does not depend on the net position on that asset or on the total level of wealth held by the consumer.
• The duration of life is certain.
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PIH with No Uncertainty
• Consider an individual/household who lives for T periods
• Household utility, assuming no discounting:
T
U = u(Ct), u′(·) > 0, u′′(·) < 0 t=1
• Individual’s initial wealth is A0
• Labor income Yt for periods t = 1,...,T
(these are known in advance with certainty)
• Individual can borrow and save at the interest rate r = 0.
• Household budget constraint (B.C.):4
TT
Ct ≤ A0 + Yt
t=1 t=1
4Consider the sequence of individual budget constraints,
At+Ct =At−1+Yt fort=1,...,T.Addingalltheseconstraintstogether, and using AT = 0 yields the lifetime budget constraint.
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PIH with No Uncertainty: Solution
• The Lagrangian is
TTT L=u(Ct)+λ A0+Yt−Ct
t=1 t=1 t=1
• The FOC is
Therefore, consumption must be constant.5
u′(Ct)=λ ∀t
• Using that C1 = C2 = ··· = CT, the B.C. implies
1T Ct=T A0+Ys ∀t
s=1 Permanent Income Y P
5Here, we are using the assumption that u(C) is strictly concave.
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Implications of PIH
• An individual’s consumption in a given period is determined by income over his/her entire lifetime. This is the permanent-income hypothesis (PIH).
• Friedman (1957) distinguishes between permanentincomeYP andtransitoryincomeYT:
P1T Y ≡T A0+Ys
s=1
• The time pattern of income is instead critical to saving: St =Yt −Ct =Yt −YP
In words: save when income is temporarily high.
YT ≡Yt −YP
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Empirical Application
• Let’s revisit the Keynesian consumption function. • Let’s make the following modifications:
(1) Index different households by i: Y=YP+YT
it i it
AssumeYT ismeanzero,andYT isuncorrelatedwithYP. it it i
(2) Add i.i.d. mean zero shocks eit:
T
U=u(C−e) ⇒ C=YP+e
t=1
it it it i it
where eit is “transitory consumption”, which generates some
i
variation in consumption over time.
• If the PIH holds, what will happen if we regress Cit on Yit? 13/40
Empirical Application
• Suppose we run the following regression in the cross-section:6 Ci =a+bYi +ui
• In a simple linear regression, we get:
bˆ = Cov(Yi , Ci ) Var(Yi )
Cov(YP +YT,YP +e) bˆ = i i i i
Var(YiP +YiT)
Var(YiP ) + Var(YiT ) • Flatter slope if Var(Y T ) large relative to Var(Y P )
aˆ = C ̄ − bˆY ̄
aˆ = ( 1 − bˆ ) Y ̄ P
ˆ Var(Y P ) b=i
ii
• IfVar(YP)≫Var(YT),aˆ≈0andbˆ≈1. ii
• Interpretation of evidence given at beginning of lecture?
6Time period is the same for all observations, so we drop the t subscript. 14/40
Consumption Under Uncertainty
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Household’s Problem with Uncertainty
• Now suppose there is uncertainty about Yt each period. T
maxE u(C) ts
s=t
s.t. As + Cs = As−1 + Ys
• Assume u(·) is quadratic:7
u(C) = C − aC2
for s = t, . . . , T
2
In this case, Et [u′ (Ct+1)] = u′ (EtCt+1) and we get
certainty equivalence (more on this later) 7Assume marginal utility is always positive.
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Household’s Problem with Uncertainty
• Lagrangian is then: TT
L=E u(C)+Eλ[A +Y−A−C] t s t s s−1 s s s
• FOC:
∂L =u′(Ct)−λt =0
⇒ λt =u′(Ct) (1) ⇒ λt =Etλt+1 (2)
s=t s=t
∂Ct
∂L =−λt +Etλt+1 =0
∂At
• Combining (1) and (2) gives us the Euler Equation: u′(Ct ) = Et [u′(Ct+1)]
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Euler Equation with Uncertainty
• Euler Equation:
u′ (Ct ) = Et [u′ (Ct+1)] (3)
• With quadratic utility, (3) becomes
Ct = Et [Ct+1] (4)
• Mathematically, (4) implies
Ct+1 =Ct +et+1
where et+1 is a mean-zero shock realized in period t + 1.
• Consumption follows a random walk (Hall, 1978). Changes in consumption are unpredictable.8
8More generally, if u(C) is not quadratic, marginal utility u′(C) still follows a random walk, and consumption approximately follows a random walk (with drift).
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Optimal Consumption
• Add all B.C.s together, take expectations, use AT = 0: C1 = A0 + Y1 − A1
C=A+Y−A T T
2 1 2 2 ⇒EC=A+EY
.. 1t0 1t .. t=1 t=1
CT =AT−1+YT −AT
• From Euler equation, C1 = E1C2 = E1C3 = ··· = E1CT.
This implies
1T
C1=T A0+E1Yt (5)
t=1
• Consumption exhibits certainty equivalence: household consumes the same amount it would if it were to receive the expected value of its future incomes, with certainty.
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Optimal Consumption
• Analogously, for period-2 consumption: 1T
C2=T−1 A1+E2Yt t=2
• Re-writeusingA1 =A0+Y1−C1 and(5): A0+Y1+Tt=2E1Yt =TC1
1 T T T
C2 = T −1 A0 +Y1 −C1 +E1Yt + E2Yt −E1Yt t=2 t=2 t=2
1TT C2 = C1 + E2Yt − E1Yt
T−1 t=2 t=2
• Change in consumption caused by update in individual’s estimate of his remaining lifetime resources
(i.e., “news” about future income)
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Intuition for Random-Walk Hypothesis
• If consumption were expected to change, the individual could do a better job of smoothing consumption.
• If consumption is expected to rise:
u′(Ct ) > Et [u′(Ct+1)] 1−aCt >Et [1−aCt+1]

EtCt+1 >Ct

Better off increasing Ct, decreasing EtCt+1

• Thus the individual adjusts her current consumption to the point where consumption is not expected to change.

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Testing the Random-Walk Hypothesis

• Do predictable changes in income produce predictable changes in consumption?

There is excess sensitivity of consumption if consumption responds to predictable income movements.9

• Campbell and Mankiw (1989), using an IV approach, find significant responses to anticipated income changes.

• Consumption increases $0.50 in response to anticipated $1 increase in income.

• But there are grounds to be cautious about the results, as their instruments may not have been good enough.

9Similarly, the hypothesis that consumption responds less than what the PIH predicts to unexpected changes in income is referred to excess smoothness of consumption. See Flavin (1993).

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Testing the Random-Walk Hypothesis

• State of Alaska obtains large revenues from oil royalties, which are distributed to all residents once a year.

• Large & predictable payments (often $1000+ per person)

• Hsieh (2003) finds no/little response to Alaska oil payments

He argues the important distinction is that these payments large and easily predictable.

• However, Kueng (2018) finds significant responses

Average MPC is $0.25. Problem with Hsieh (2003) is that income suffers from measurement error, attenuating the response.

• Usual result in the literature is that consumption responds to predictable changes in income.10

10See Shea (1995), Parker (1999), Souleles (1999), Shapiro and Slemrod (2003), Stephens and Unayama (2011), Parker, Souleles, Johnson, McClelland (2013), Baugh, Ben-David, Park, Parker (2018).

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Extensions

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Adding the Interest Rate

• Assume constant interest rate r. Individual B.C. now: At+Ct=At−1(1+r)+Yt, t=1,…T

• Lifetime budget constraint is now:11

T 1 C t ≤ A 0 + T 1

Y t

• Now assume discount factor 1/(1 + ρ), utility becomes:

(1+r)t (1+r)t t=1 t=1

U=T 1u(Ct) t=1 (1+ρ)t

• Euler equation becomes

u′(Ct)= 1+ru′(Ct+1)

1+ρ

11Note that first period is t = 1, but we are discounting to period 0.

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Effect of the Interest Rate

• With CRRA utility, u(C ) = C1−θ/(1 − θ), Euler Equation tt

implies

Ct+1 1 + r 1/θ C=1+ρ

t

Interest rate important for consumption growth. Consumption grows over time if r > ρ.

• While r will affect ratio of Ct+1/Ct, it is ambiguous what happens to Ct and thus saving At.

• Substitution effect: r ↑ ⇒ Ct ↓ and saving At ↑

• Income effect: depends whether a saver or borrower

If saver: r ↑ ⇒ Ct ↑, if borrower: r ↑ ⇒ Ct ↓

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Interest Rate and the Random Walk Result

• With uncertainty, Euler equation is

u′ (Ct) = 1 + r Et [u′ (Ct+1)]

2

EtCt+1 = 1r −ρ + 1+ρCt a1+r 1+r

Consumption does not follow a random walk if r ̸= ρ.

1+ρ

• With quadratic utility, u(Ct ) = Ct − a Ct2, this would imply

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Risky Assets

• Assume the individual can invest into different assets, and

thegross returnonriskyasseti isRi =1+ri t+1 t+1

, the gross return on risk-free asset is R ̄t+1 = 1 + ̄rt+1.

• Euler equations for each asset:

Riskyasseti: u′(C)= 1 E Ri u′(C ) (6)

t 1+ρt t+1 t+1 Risk-free asset: u′(Ct ) = R ̄t+1 Et [u′(Ct+1)]

(7)

1+ρ • Combine (6) and (7) and re-write:12

E Ri u′(C )=R ̄ E [u′(C )] t t+1 t+1 t+1t t+1

Cov Ri ,u′(C ERi−R ̄=−tt+1 t+1

t t+1 t+1 Et [u′(Ct+1)] 12 Use E [XY ] = E [X ]E [Y ] + Cov(X , Y ).

)

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Risk Premium for Risky Assets

• Risk premium for asset i:

Cov Ri ,u′(C

)

ERi−R ̄=−tt+1 t+1 t t+1 t+1 Et [u′(Ct+1)]

This model of the determination of expected asset returns is known as the consumption capital-asset pricing model, or consumption CAPM.

• Individual not directly concerned with Var (Ri ), but rather

t t+1 • For example, suppose Cov Ri , u′(C ) > 0.

how Ri covaries with u′(C ). t+1 t+1

t t+1 t+1

Ri higher when C is low; useful for hedging risk,

t+1 t+1

hence risk-premium negative.

• For example, suppose Cov Ri , u′(C ) < 0. t t+1 t+1
Ri low when C is low; need positive risk premium to t+1 t+1
compensate investors for taking on the risk of this asset.
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Risky Assets with CRRA utility
• Assume CRRA utility, u(C) = C1−θ/(1 − θ), where θ is the coefficient of relative risk aversion.
• Euler equation is then:
C−θ= 1 E1+ri C−θ
t1+ρt t+1t+1
• Let gc = C /C − 1 and drop time subscripts:
t+1 t+1 t
E 1 + r i (1 + g c )−θ = 1 + ρ
• This implies that the difference between the expected returns on two assets, i and j, satisfies13
Eri−Erj=θCovri −rj,gc 13Use a first-order Taylor approximation around ri = rj = gc = 0.
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Equity Premium Puzzle
Eri−Erj=θCovri −rj,gc (8) Mehra and Prescott (1985): it is difficult to reconcile observed
returns on stocks and bonds with this equation.
• Let ri = returns on stocks, rj = returns on bonds
• Over the period 1890-1979,
E[ri −rj] ≈ 0.06, σgc ≈ 0.036, σri−rj ≈ 0.167, ρri−rj,gc ≈ 0.4
• Substituting these numbers in (8): 0.06=θ×0.40×0.036×0.167 ⇒ θ≈25
θ = 25 is an implausibly high value. This is the equity-premium puzzle.
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Precautionary Saving
• With quadratic utility, we get certainty equivalence.
Household consumes/saves the same amount it would if it were to receive the expected values of its future incomes with certainty.
• Under alternative specifications of utility, however, there will be a precautionary motive for saving.
For example, to protect themselves from future fluctuations in their income, households save more
• To see this, let’s analyze how uncertainty affects the saving and consumption decision.
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Saving Under Uncertainty
• How does uncertainty affect the savings decision?
Shape of marginal utility is important.
• With uncertainty, and constant interest rate r, Euler Equation is
u′(Ct)= 1+rEt [u′(Ct+1)] 1+ρ
• Compare Ct (and At) to what the household would choose if it consumed Et[Ct+1] with certainty tomorrow.14
u′(Ct ) = 1 + r u′ (Et [Ct+1]) (without uncertainty) 1+ρ
u′(Ct ) = 1 + r Et [u′ (Ct+1)] (with uncertainty) 1+ρ
14Think of Ct+1 as a random variable; work backwards to determine Ct.
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Saving Under Uncertainty: Quadratic Utility
• In the case of quadratic utility, u(C) = C − aC2, 2
marginal utility is linear, u′(C) = 1 − aC. Therefore: u′ (Et[Ct+1]) = 1 − aEt[Ct+1]
Et [u′ (Ct+1)] = 1 − aEt[Ct+1]
• As we’ve seen earlier, the agent will choose the same Ct and At with or without uncertainty. Uncertainty has no effect on the savings behavior.
• Risk aversion alone is not enough to affect the savings behavior under uncertainty.
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Saving Under Uncertainty: Quadratic Utility
Example: Assume r = ρ and
suppose Ct+1 = C1 with prob. 0.5 and Ct+1 = C2 with prob. 0.5. Household chooses Ct s.t. u′(Ct ) = Et [u′(Ct+1)]
Consumption tomorrow
Household consumes the same amount (i.e., Ct = Et[Ct+1]), with or without uncertainty.
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Marg. util. tomorrow
Precautionary Savings
• If u′(C) is linear, we get “certainty equivalence.”
The household behaves in the same way as if there was no
uncertainty.
• If u′(C) is strictly convex (i.e., u′′′ > 0), the utility function displays “prudence” and we get “precautionary savings.”

When u′(C) is strictly convex

u′ (Et [Ct+1]) < Et [u′(Ct+1)]
Recall:
u′(Ct ) = 1 + r u′ (Et [Ct+1]) (without uncertainty)
1+ρ
u′(Ct ) = 1 + r Et [u′ (Ct+1)] (with uncertainty)
1+ρ
Households save more with uncertainty.
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Precautionary Savings
Example: Assume r = ρ and
suppose Ct+1 = C1 with prob. 0.5 and Ct+1 = C2 with prob. 0.5. Household chooses Ct s.t. u′(Ct ) = Et [u′(Ct+1)]
Consumption tomorrow
Household consumes less (and saves more) with uncertainty.
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Marg. util. tomorrow
Precautionary Savings
• In general, to get precautionary savings, we need (a) u′ strictly convex (i.e., u′′′ > 0)

OR

(b) quadratic utility plus a borrowing constraint

(that is tighter than the natural borrowing limit)

• Borrowing constraint also generates an incentive to save. Households save more because borrowing constraint might bind in the future.

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Liquidity/Borrowing Constraints

• So far we have assumed that the household can borrow up to some arbitrarily large amount.15

• Now suppose households face a borrowing constraint: At ≥Afort=1,…,T; A≤0

Let μt be the Lagrange multiplier.

• Euler equation, with borrowing constraint:

u′(Ct)=μt +1+rEt[u′(Ct+1)], 1+ρ

μt ≥0,At ≥A,μt(At −A)=0,

Under a binding constraint (μt > 0), u′(Ct) is higher and

Ct is lower than it otherwise would be.

15Only the natural borrowing limit is imposed, which does not bind.

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Liquidity/Borrowing Constraints

• Implication of borrowing constraints:

current income helps predict consumption growth for low-wealth individuals, but not for high-wealth ones.16

• Another possible explanation: if preferences exhibit prudence, current income may predict consumption growth if it contains information about the extent of future income uncertainty.

16For example, Zeldes (1989) concludes borrowing constraints are important for shaping consumption choices, at least for the wealth-poor.

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