# 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
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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• 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 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
• 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. 29/40 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 E􏰉􏰀1+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 E􏰉ri􏰊−E􏰉rj􏰊=θCov􏰀ri −rj,gc􏰁 13Use a first-order Taylor approximation around ri = rj = gc = 0. 30/40 Equity Premium Puzzle E􏰉ri􏰊−E􏰉rj􏰊=θCov􏰀ri −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. 31/40 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. 32/40 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. 33/40 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. 34/40 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. 35/40 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. 36/40 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. 37/40 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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