Topic 2: Consumption · 2019-01-03 · Keynes’ Three Conjectures on the Consumption Function 1...

Topic 2: Consumption

Dudley Cooke

Trinity College Dublin

Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 1 / 48

Reading and Lecture Plan

Reading

1 SWJ Ch. 16 and Bernheim (1987) in NBER Macro Annual.

Plan

1 Review of the Keynesian consumption function

2 Intertemporal Consumption Problem

3 Interest Rates

4 Government Policy and Ricardian Equivalence


Keynes’ Three Conjectures on the Consumption Function

1 Marginal propensity to consume is between zero and one. Theimpact of fiscal policy is determined from the feedback betweenincome and consumption.

2 Average propensity to consume rises as income falls. Idea being thatsaving was a luxury and so rich people save more than poor.

3 Income is the primary determinant of consumption, not the interestrate.


Keynesian Cross

Planned and actual expenditures:

Ep ≡ C (Y , T , ω)︸︷︷︸(i)

+ I p(r , A)︸︷︷︸(ii)

+ G

Y is total income, T is taxation, ω is household wealth, A isautonomous investment and G is government purchases.

(i) The consumption function is based on household behavior.Consumption rises with income, so dC/dY > 0.

(ii) Investment demand is based on firm behavior. Plannedinvestment, I p, falls with the interest rate, so dI p/dr < 0.


Keynesian Cross Analysis

In equilibrium, Ep = Y . On a diagram, the equilibrium, in(Ep, Y )-space, will be on the 45-degree line.

Suppose there is a shock to the economy: ∆r > 0. That impliesEp < Y as I p has fallen. There is disequilibrium in the goodsmarket; planned expenditure is less than total output.

Over time, Ep and Y fall, until we reach a new equilibrium (i.e.Ep = Y ), where both Ep and Y are lower than before.

This can be represented as a shift in the planned expenditure line.


Keynesian Cross

Diagram: Keynesian Cross

Equilibrium: planned expenditure=spending

(income): Ep = Y .

Ep, Planned Expenditure

Y , Output

Y = Ep

Ep(r0)

Ep(r1)

Y0Y1

�

ΔI

Higher r, lowers I, which lowers Ep s.t. EP1 <

Y0. Y decreases from Y0 to Y1 to reach equi-

librium - ΔI ‘faster’ than ΔY .∗

∗We also see that investment is more variable than out-put in the data.


IS Curve: Mathematical Derivation of Slope

The IS curve shows combinations of real output Y and the realinterest rate r such that planned and actual expenditures are equal.Totally differentiating Y = Ep w.r.t. Y and r (holding G , T , A, ωfixed) yields:

∆Y = CY ∆Y + Ir ∆r

or,(∆Y ) /(∆r)|IS = Ir /(1− CY ) < 0

This is the slope of the IS Curve, where 0 < CY < 1 is the MPCand Ir < 0.

IS is steep if the interest sensitivity of planned expenditure (Ir )is high or the marginal propensity to consume out of disposableincome (CY ) is large.


IS CurveDiagram: IS Curve

�

�

r, Interest Rate

Y , OutputY0Y1

r0

r1

IS

Ep > Y

Ep < Y

At points where Ep �= Y the goods market is

not in equilibrium. As before, Δr > 0⇒ Ep <

Y .

A change in r, which shifts the planned expen-

diture curve, produces a movement along the

IS curve.


Shifting the IS Curve

The IS curve shifts with changes in fiscal policy.

The standard multipliers:

1 Government purchases multiplier: 0 < CY ≡ MPC < 1 so1/(1− CY ) > 1. Larger than one because ↑ G ⇒↑ Y ⇒↑ C by∆G · CY , which ⇒↑ Y etc...

2 Tax multiplier: −CY /(1− CY ) < 0.

3 ‘Balanced budget’ multiplier (i.e. when ∆G = ∆T ): unity.1

1Since only part of the money taken away from households would have actually beenused in the economy, the change in consumer expenditure will be smaller than thechange in taxes. Therefore the money which would have been saved by households isinstead injected into the economy, itself becoming part of the multiplier process.


Functional Forms

We usually assume C (·) and I p(·) are linear functions:

C (Y , T , ω) = δY d + ω

I p(r , A) = A− ar

where Y d ≡ Y − T is disposable income.

This gives clear implications for the slope of the IS and the multipliers.

1 The slope:(∆Y ) /(∆r)|IS = −a/(1− δ) < 0

2 The multipliers:

(∆Y ) /(∆G )|r = 1/(1− δ) > 1

(∆Y ) /(∆T )|r = −δ/(1− δ) < 0


Keynesian Consumption Function

Some criticisms of the Keynesian Consumption Function (KCF):

1 Ct = δ(Yt − Tt) + ω is OK. But what about Ct ’s relation toYt+1, Yt+2, etc.? Agents ought to be forward-looking; consumptiontoday should account for changes in future income to maximize utility.

2 Why doesn’t Ct depend on rt in the same way investment does? Andwhat about rt+1, rt+2, etc.? Again, agents ought to beforward-looking.

3 There is no differentiation between permanent and temporary changesin policy.

Conclusion: We need a more developed model to deal with this.Given our criticisms, it ought to include agents who consider thefuture.


Permanent Income and Life Cycle Hypotheses

The Permanent Income Hypothesis, proposed by Milton Friedman,loosely states that households base consumption on average incomelevels, i.e. their permanent income and not merely year-to-yearincome. This implies that households need to know what theirincome is over a long time horizon. This leads to consumptionsmoothing.

The Life Cycle Hypothesis, proposed by Franco Modigliani, suggeststhat households consume a constant percentage of the present valueof lifetime income. The average propensity to consume is high inhouseholds that are young or old. Young households borrow againstfuture income, while old households spend down lifetime savings.


A Two-Period Endowment Economy

Households are ‘alive’ for two periods, t = 1, 2. Call these today andtomorrow.

There is a single good, Yt .

Households borrow and lend at the same market interest rate, r .(There are perfect capital markets.)

There is no uncertainty about the future.


Consumer preferences

Household maximizes lifetime (t = 1, 2) utility, U.

U = u(C1) + βu(C2) (1)

β ∈ (0, 1) is a fixed subjective discount factor, sometimes called theprivate discount factor. It measures the household’s impatience toconsume. Lower β means agents are more impatient.

C is consumption of the good.

u is the period utility function, with u (C )′ > 0 and u (C )′′ < 0.


The Budget Constraint

The following describes the household’s lifetime budget constraint.

C1 +C2

1 + r= (Y1 − T1) + (Y2 − T2)

1

1 + r

where 1/ (1 + r) is the market discount factor for futureconsumption.

Equation (??) tells us that the present value of lifetime consumptionis equal to the present value of lifetime income, minus taxes.2

2We get here by combining the t = 1 and t = 2 period constraints. That is,Wt = Wt−1(1 + r) = Yt − Ct , for t = 1, 2, with W0 = W2 = 0, assumed.


The Household Utility Optimization Problem

Maximize (1) subject to the budget constraint.

maxC1

U = u(C1) + βu(C2)

s.t.C2 = (1 + r)(Y1 − C1 − T1) + (Y2 − T2)

Take derivatives to get the first order condition. The ConsumptionEuler equation.

u′(C1)− β(1 + r)u′(C2) = 0

⇒ βu′(C2)u′(C1)

=1

1 + r

This tells us that the substitution of consumption across time (i.e.,the intertemporal marginal rate of substitution) should be equal tothe intertemporal relative price.


Equilibrium

Suppose taxes are zero. That is, let T1 = T2 = 0.

Recall, the budget constraint is,

C1 +C2

1 + r= Y1 +

Y2

1 + r

Taking the derivative, we have,

∂C2/∂C1 = − (1 + r)


Relation to Microeconomics

In micro, we solve the following problem:

maxCa,Cb

u(Ca, Cb)

such that,

PaCa + PbCb = I

The relative price between the two goods (intratemporal price),Pa/Pb, determines the slope of the budget constraint.

In our example, 1/ (1 + r) is the intertemporal price, that is, therelative price of consumption (of a single good) over time.


Intertemporal Consumption Function (ICF)

We make some functional form assumptions on utility, for thesame reasons we cited when we looked at the Keynesian consumptionfunction.

Suppose:

u(Ct) =C

1− 1σ

t

1− 1σ

; σ > 0, σ 6= 1

= ln Ct ; σ = 1

This is sometimes called power utility.

The parameter σ determines the slope of the indifference curves andhow consumption is substituted across time. Recall that r determinesthe slope of the budget constraint. High σ will be consistent withhigh substitution.


Intertemporal Consumption Function

The consumption Euler equation is now:

C2 = [β(1 + r)]σ C1

Using the budget constraint, we get the consumption function:

C1 =(Y1 − T1) + (Y2 − T2) 1

1+r

1 + (1 + r)σ−1 βσ

We’ll call this the Intertemporal Consumption Function (ICF).We can compute C2 by plugging the two equations together. Recall,the KCF:

Ct = δ(Yt − Tt) + ω ; for t = 1, 2, say


Comparing the KCF and ICF

Important point: in the ICF, consumption depends on r . But rdoesn’t matter in the KCF.

1 Y2 and T2 also matter for C1 in the ICF.

2 Expectations about the future matter for today’s consumption, unlikein the KCF. This is more intuitive.

3 This has policy consequences. For example, there will be differencesbetween permanent and temporary taxation policies.


Comparing ICF and KCF

Let’s compare marginal propensities to consume in the Intertemporaland Keynesian consumption functions.

Let 1/θ ≡ 1 + (1 + r)σ−1 βσ and Yt − Tt = Y dt .

Consumption function:

C1 = θ

(Y d

1 + Y d2 ·

1

1 + r

)where θ ∈ (0, 1) measures the MPC of wealth (here, disposableincome in both periods)

However, δ in the KCF measured the MPC out of current disposableincome alone.



δ versus θ is not a good comparison of the consumption functions.

1 δ is MPC of current disposable income.

2 θ is MPC of wealth.

Let (1 + g e) ≡ Y d2 /Y d

1 . Then:

C1 = θ̃Y d1

θ̃ = θ

[1 +

(1 + g

1 + r

)]Now θ̃ is the MPC of current disposable income and is directlycomparable to δ.



The KCF is:C1 = δY d

1 + ω

The ICF is:

C1 = θ̃Y d1

θ̃ =[

1 +(

1 + g e

1 + r

)]1

1 + (1 + r)σ−1 β

In essence, we have micro-founded δ. We have a solid idea of where θ̃comes from. It depends on preferences and:

g e : expected future disposable income

r : the market rate of return

β : impatience to consume



If ↑ Y d1 , from the KCF, ↑ C1.

The ICF suggests two effects, one implies a direct ↑ C1 and the otherimplies ↓ g e ⇒↓ C1. Overall, ↑ C1. Clearly, C2 also changes.3

This is an example of how intertemporal analysis lets us investigatesubtle effects that may not be immediately intuitive.

3Why? Consumption smoothing.Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 25 / 48

Comparing ICF and KCF: Does ICF Perform Better?

Our new model is more complicated. That is no good if it is notbetter than the old model. We need to check the data.

One idea:

KCF: Richer people have higher average savings than poor people -assumed.ICF: Suppose ge

high

(Y d

2 > Y d1

), like students ⇒ high MPC from

current income, i.e. a high θ̃. Now suppose gelow

(Y d

2 < Y d1

), like

pensioners ⇒ low MPC.

The models are consistent. However, ICF is better. It implies we havedifferent marginal propensities to consume over different parts of thelifecycle. That is, θ̃ changes over time whereas δ does not. This isfar more empirically relevant.


The Interest Rate

The interest rate play a significant role in the intertemporalconsumption function.

A change in r changes the MPC of wealth, θ.

∂θ/∂r > 0 if σ < 1

∂θ/∂r < 0 if σ > 1

This is a result of income, substitution and wealth effects.

When σ = 1, these cancel out and do not affect the MPC of wealth,i.e. ∂θ/∂r = 0.

However, even if σ = 1, ∂θ̃/∂r < 0. That is, higher interest rateslower the MPC of current disposable income.


The Interest Rate: Changes To r

Consider the effect of an ↑ r on C1.

1 Substitution (negative) effect: we swap C2 for C1. We increasesaving when the incentive to save increases.

2 Income (positive) effect: ↑ r allows ↑ C2, for given income. Thisincrease in feasible consumption implies ↑ C1 (lower savings).

Note the opposite effects.

1 When σ > 1, the substitution effect dominates the income effect.When σ < 1, things go the other way.

2 When σ = 1, the fraction of lifetime income spent on C1 does notdepend on r. The income and substitution effects are offsetting.


The Interest Rate: Changes To r

However, when σ = 1, r still affects C1, via a wealth effect.

Wealth (negative) effect: The change in r changes lifetime income,not just the fraction of lifetime income devoted to presentconsumption.

That is, ↑ r lowers the present discounted value of income, reinforcingthe substitution effect.

Which direction is more plausible?

Let’s look at the data. It suggests σ = 2. That is, the substitutioneffect dominates. We increase saving.


The Interest Rate: Mathematical Details

Quick references on income and substitution effects: (i) Romer 3rded. pp. 363-365 [macro] and (ii) Varian ch.8 [micro].

In our example, we can calculate the following (you can verify this athome):

dC1

dr=

(Y1 − C1)− σC21

1+r

(1 + r) + C2/C1≶ 0

C2 = [β(1 + r)]σ C1

When σ = 1:

C1 =Y1 + Y2

11+r

1 + β⇒ dC1

dr< 0

There is no ambiguity in dC1dr as income and substitution effects cancel

out when σ = 1.


Government Policy

We have yet to consider policy options. In the KCF case, consider achange in taxation, T .

(∆Y ) /(∆T )|r = −CY /(1− CY ) < 0.

All else equal, an increase in taxation reduces output.

Points we have missed out:

1 Does the timing of taxation matter?

2 Does the financing of a change in taxation matter? That is,does moving from a situation in which ↓ T =↓ G to deficit financingmake a difference?


Tax Policies: Temporary and Permanent

Recall the basic ICF:

C1 = θ

[(Y1 − T1) + (Y2 − T2)

1

1 + r

]A temporary policy here is a ↓ T1 with T2 = T , fixed.

A permanent policy is T1 = T2 = T , ↓ T .


Tax Policies: Temporary and Permanent

Temporary Policy (T1 6= T2):

∂C1/∂T1 = −θ

Permanent Policy (T1 = T2 = T ):

∂C1/∂T = −θ

(1 +

1

1 + r

)In the first case, ↓ T1 ⇒↑ C1, but as θ < 1, ↑ C2 also. Savings rise ashouseholds engage in ‘consumption smoothing’.

In the second case, the effects are stronger. When β = (1 + r),consumers want perfectly smooth consumption over time

∂C1/∂T = −1


Tax Policies: Financing a Tax Cut

The funds to finance the tax cut have to come from somewhere.

To analyze this, we need to know the government’s lifetime budgetconstraint, in the same way that we need to know the household’sbudget constraint.

Government constraint:

D0 + G1 + G21

1 + r︸︷︷︸govt. consumption

= T1 + T21

1 + r︸︷︷︸govt. income

where D0 is the initial (given) level of government debt.


Constant Government Spending

Suppose G1 = G2 = G . So government spending is constant.

This has immediate implications:

dT1 = −dT2 ·1

1 + r

If taxes fall today, they have to rise tomorrow.

But surely consumers know this? So how will they react?


Implications of the Lifetime Budget Constraint

If in t = 1, the government lowers T1 without a change in G1, thenD1 must rise, i.e. the government issues more debt.

If in t = 2, the government does not subsequently change G2, then ithas to raise T2 to pay for the principal and interest on the extra debtit bore.

Returning to forward-looking households. What if you knew this wasthe case? (i.e. you took Economics)

Well, you know that if ↓ T1 without ↓ G1 or G2, this implies a rise inthe present value of of total taxation by an amount equal andopposite to the ↓ T1.


Implications of the Lifetime Budget Constraint

Formally:

C1 = θ

[(Y1 − T1) + (Y2 − T2)

1

1 + r

]implies,

dC1 = −θ

[dT1 + dT2

1

1 + r

]= 0

So, ↓ T1 without ↓ G1 or G2 must imply no change in consumption.

That is, the financing of the deficit has no implication forconsumption. We can choose either tax or deficit financing. Thepresent value of the tax burden is the same.

This is known as Ricardian Equivalence.4

4In contrast, recall that the balanced-budget multiplier in the KCF was non-zero.Dudley Cooke (Trinity College Dublin) Topic 2: Consumption 37 / 48

Ricardian Equivalence

Ricardian Equivalence is a benchmark result. It breaks down if wechange our model only slightly. In particular, consider any of thefollowing alternative assumptions:

1 Governments and households live for a different no. of periods.

2 Non-lump-sum taxes. There are no associated distortions withlump-sum taxes, but they are very rare in real life. Distortionarytaxation: Anything that affects markets, like VAT or income tax.

3 We don’t borrow and lend at the same rate. Some people are ‘creditconstrained’.

4 However, Barro (1974) provides an interesting analysis when we makebequests.5

5See his paper in the Journal of Political Economy on ‘Government Bonds’ and ‘NetWealth’.


Ricardian Equivalence: Empirical Evidence

Keynesian view: Deficit-financed tax cuts increase aggregatedemand.

Ricardian view: Taxpayers understand that the present discountedvalue of taxes depends on government spending. Tax cuts will haveno effect on aggregate demand.

Context: The mid 1980’s saw a high US government deficit.

Main Finding: There is a short-run relationship between deficits andaggregate consumption, both in cross-country data and time seriestests of the consumption function. So the strict Ricardian viewdoesn’t hold. Problem: We may not really be seeing a test ofRicardian equivalence in this paper.


What the data say, Bernheim (1987)

Keynesian view: deficit financed tax cuts increase aggregatedemand.

Ricardian view: taxpayers understand that the p.d.v. of taxes simplydepends on government spending and tax cuts will have no effect onaggregate demand.

Context: The mid 1980’s saw a high US government deficit.

Main Finding: There is a short-run relationship between deficits andaggregate consumption both in cross country data and times seriestests of the consumption function.


Testing the Consumption Function

Two main types of tests, which are basically equivalent (old papersfrom 1970-1986).

Test of tax discounting hypothesis:

Ct = α0 + α1 (Yt − Tt) + α2Gt + α3Dt + α4Wt

+ α5︸︷︷︸H0:α5=0

· (Tt − Gt − RtDt)︸︷︷︸govt. deficit

+ Xtα + εt

Test of Pure Ricardian equivalence:

Ct = β0 + β1Yt + β2Gt + β3Dt + β4Wt

+ β5︸︷︷︸H0 :β5=0

· (Tt − Gt − RtDt) + Xt β + ηt

Above, H0 is the null hypothesis.


Interpretation

Interpreting regression results:

1 α5 = 0 (β5 = −β1): tax discount hyp. of Ricaridan view.

2 α5 = α1 (β5 = 0): pure Ricardian view.

3 α1 − α5 (β5): measures the effect on current consumption of a 1$ taxdeficit swap.

The estimation method is typically Ordinary Least Squares (OLS).


Problems

If we take some data and run a regression are we actually testingwhat we want?

We definitely test something that looks like an implication of ourmodel.

However, the conditions above look like KCF’s. Thus we are reallytesting the KCF view.

They are not exactly tests of Ricardian Equivalence as we haveformulated it. Our formulation used a Consumption Euler equation.Thus if we run this regression we will get some biased estimates.


Round-up

Keynesian versus intertemporal consumption function

Ricardian equivalence and tax cuts

Confronting the ICF with data


Other Points 1 (not crucial - but FYI): the role of theelas’y of substitution

Everything comes down to 2 equations. Consumption Euler:βu′(C2)/u′(C1) = 1/ (1 + r) and intertemporal budget constraint:C1 (1 + r) + C2 = Y1 (1 + r) + Y2. We then suppose

u(C ) = C 1− 1σ /(1− 1

σ

)so, u′(C ) = C−

1σ . Without the functional form

assumption, taking logs, ln β + ln [u′(C2)]− ln [u′(C1)] = − ln (1 + r).The total deriv. is,[u′′(C2)

][u′(C2)]

−1 dC2 −[u′′(C1)

][u′(C1)]

−1 dC1 = −d [ln (1 + r)],but noting dC1/C1 = d {ln [u′(C2)]}, and defining,

σ (C ) ≡ − [u′(C )] /C[u′′(C )

], the former expression reduces to,

d ln

(C2

C1

)= σd [ln (1 + r)]

such that a high σ implies a strong reaction of relative consumption tochanges in the real interest rate. That is, a high substitution effect, andgently sloped indifference curve.


Other Points 2: Income, Substitution, and Wealth Effects

Given the utility form, consumption is the following.

C1 =1

1 + (1 + r)σ−1 βσ

[Y1 +

Y2

1 + r

]C2 = [β(1 + r)]σ C1

If σ = 1 things simplify alot. Thus, we might suspect σ is critical. Indetermining how consumption reacts to changes in the interest rate.Income: positive; substitution: negative; wealth: negative.


Income, Substitution, and Wealth Effects .....

Call R ≡ (1 + r)−1 and W ≡ Y1 + Y2/ (1 + r). Thus, C1 = C1 (R, W ),which is the Marshallian demand function, as it depends on wealth. NB:Hicksian demands depend on utility and show the pure substitiution effectsbecause we are accounting for the income and wealth effects by holdingutility constant. Obviously, dW /dR = Y2 > 0. However, it is not toodifficult to show the following result:

dC1

dR=

(σ− 1) C1 (β/R)σ

1 + R1−σβσ︸︷︷︸incm and sub fx

+Y2

1 + R1−σβσ︸︷︷︸wealth fx


Income, Substitution, and Wealth Effects .....

Now we can why σ ≷ 1 is important. There is an alternate expression(similar to that derived in the main text) that also helps to clarify things:

dC1

dR= σ

C1 (β/R)σ

1 + R1−σβσ︸︷︷︸sub fx

+Y2 − C2

1 + R1−σβσ︸︷︷︸incm vs. wealth fx

The idea of (Y2 − C2) ≷ 0 is important here (above it was Y1 vs. C1).Suppose r rises. and you are a second period borrower (i.e., a first periodlender). If you are a lender, you get a utility gain. Thus, (Y2 − C2) > 0(that is, (Y1 − C1) < 0) reinforces the sub. fx., that an increase in rreduces C1.


Topic 2: Consumption · 2019-01-03 · Keynes’ Three Conjectures on the Consumption Function 1...

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