micro-foundations: consumption...j. m. keynes’s conjectures about the consumption function 1 0 mpc...

Micro-foundations: Consumption

Instructor: Dmytro Hryshko

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Why Study Consumption?

Consumption is the largest component of GDP (e.g., about 2/3of GDP in the U.S.)

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J. M. Keynes’s Conjectures about theConsumption Function

1 0≤MPC≤1: out of each additional dollar, we spend MPCand save 1−MPC dollars.

2 The average propensity to consume, APC = CY , falls asincome increases. I.e., richer people save a higherproportion of their incomes.

3 Consumption is not responsive to the real interest rate.

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The Keynesian Consumption Function

Summarizing, the Keynesian consumption function can bewritten as:

C = C + c︸︷︷︸=MPC

×Y, C > 0, 0 < c < 1.

APC =C

Y=C

Y+ c,

where Y is (disposable=after-tax) income.

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The Keynesian consumption function

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Successes and Failures

• SuccessesHousehold data: 0 < MPC < 1, APC is smaller forhigher income households.

Aggregate data (in-between the wars, when income waslow): the ratio of C to Y was high; Y was the primarydeterminant of C.

• FailuresFalling APC+rising incomes during the WWII wouldlead to a secular stagnation—a long depression inabsence of changes in G or T . This prediction aboutfalling APC did not hold.

S. Kuznets assembled consumption and income databack to 1869. CY , i.e. the APC was stable (no trend).

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Reconciliation of successes and failures: twoconsumption functions—for the short- and the long-runs

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I. Fisher and the Intertemporal Choice

In reality, consumption responds not only to changes in currentincome but also to changes in (expected) income from futureperiods.

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A Two-Period Model: TheConsumption-Savings Decision and

Credit Markets

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Topics

1 Consumer’s consumption/savings decision—responses ofconsumer to changes in income and interest rates.

2 Government budget deficits and the Ricardian EquivalenceTheorem

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A Consumer’s Consumption/Savings Decision

• Decision involves a tradeoff between current and futureconsumption.

• Dynamic Decision: It has implications over more than oneperiod of time.

By saving, a consumer gives up consumption in exchangefor assets in the present to consume more in the future.By borrowing, a consumer gains more currentconsumption sacrificing future consumption when theloan is repaid. (Borrowing is negative saving.)

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The Consumer’s Current-Period Budget Constraint

Consumption (c) plus savings (s) in the current periodmust equal disposable income in the current period.

Each consumer has real income y and pays lump-sum taxest in the current period.

c+ s = y − t (1)

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The Consumer’s Future-Period Budget Constraint

In the future period, the consumer has disposable incomey′ − t′ and receives the interest and principal on his or hersavings, which totals (1 + r)s.

c′ = y′ − t′ + (1 + r)s (2)

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The Consumer’s Lifetime Budget Constraint

Solve (2) for s:

s =c′ − y′ + t′

1 + r(3)

Substitute (3) in (1) obtaining lifetime budget constraint:

c+c′

1 + r= y − t+ y

′ − t′

1 + r(4)

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Consumer’s Lifetime Wealth

The present value of lifetime disposable income is the quantityof resources that the consumer has available to spend onconsumption, in present-value terms, over his or her lifetime.

we(alth) = y − t+ y′ − t′

1 + r(5)

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Consumer’s Lifetime Budget Constraint

Substitute (5) in (4):

c+c′

1 + r= we (6)

c′ = −(1 + r)︸︷︷︸slope

c+ we(1 + r)︸︷︷︸intercept

(7)

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Consumer’s Lifetime Budget Constraint

8-1© 2011 Pearson Addison-Wesley. All rights reserved.

Figure 8.1Consumer’s Lifetime Budget Constraint

The lifetime budget constraint defines the quantities of currentand future consumption the consumer can acquire, givencurrent and future income and taxes, through borrowing andlending in the credit market.

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Optimization

If the preferences are represented by utility functionU = U(c, c′), then the consumer chooses c and c′ that bring thehighest utility index such that the budget constraint isexhausted.

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A Consumer’s Indifference Curves


Figure 8.2A Consumer’s Indifference Curves

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Consumer Optimization

Marginal condition that holds when the consumer is optimizing:

MRSc,c′ = 1 + r (8)

The marginal rate of substitution of current consumption forfuture consumption is equal to the relative price of currentconsumption in terms of future consumption.

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Optimization

The highest possible utility is achieved at the point (c∗, c′∗),where the slope of the indifference curve is equal to the slope ofthe budget constraint.

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Optimization

The indifference curve: any combination of c and c′ thatbring the same utility index U . The slope of indifferencecurve is defined from: dU = MU1dc+MU2dc

′, or0 = MU1dc+MU2dc

′, i.e.,

The slope of the budget constraint: −(1 + r).

When consumer maximizes utility, on the margin, thebenefit of adjusting his optimal bundle of (c∗, c′∗) should bezero. The period-1 cost of reducing consumption by dc isMU1 · dc, and the period-2 benefit of this reduction isMU2 · dc · (1 + r).

Thus, at the optimum: MU1 · dc = MU2 · dc · (1 + r), or

−MU1MU2

= −(1 + r).

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A Consumer Who Is a Lender


Figure 8.3A Consumer Who Is a Lender

The optimal consumption bundle for the consumer is at pointA. The consumer is a lender, as the consumption bundle chosenimplies positive savings, with E being the endowment point.

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A Consumer Who Is a Borrower

8-1 © 2011 Pearson Addison-Wesley. All rights reserved.

Figure 8.4 A Consumer Who Is a Borrower

The optimal consumption bundle for the consumer is at pointA. Because current consumption exceeds current disposableincome, saving is negative, and so the consumer is a borrower.

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An Increase in Current Income for the Consumer

Because current-period and future consumption are normalgoods, consumption in both periods increases.

Saving increases.

The consumer acts to smooth consumption over time.

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The Effects of an Increase in Current Income for aLender


Figure 8.5 The Effects of an Increase in Current Income for a Lender

An increase in currentincome increaseslifetime wealth formwe1 to we2, shifting thelifetime budgetconstraint to the rightand leaving its slopeunchanged, because thereal interest rate doesnot change. Initially,the consumer choosesA, and he or shechooses B after currentincome increases.

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Observed Consumption-Smoothing Behavior

Aggregate consumption of nondurables and services issmooth relative to aggregate income, but the consumptionof durables is more volatile than income.

This is because durables consumption is economically morelike investment than consumption.

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Percentage Deviations from Trend in Consumption ofDurables and Real GDP


Figure 8.6 Percentage Deviations from Trend in Consumption of Durables and Real GDP

The consumption of durables is economically similar toinvestment expenditures, which is why consumer durablesexpenditure is more volatile than real GDP.

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Percentage Deviations from Trend in Consumption ofNondurables and Services and Real GDP.


Figure 8.7 Percentage Deviations from Trend in Consumption of Nondurables and Services and Real GDP

The consumption of nondurables and services is fairly close to apure flow of consumption services. Consumption of nondurablesand services is much smoother than real GDP, reflecting themotive of consumers to smooth consumption relative to income.

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An Increase in Future Income for the Consumer

Current and future consumption increase.

Saving decreases.

The consumer acts to smooth consumption over time.

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The Effects of an Increase in Future Income


Figure 8.8 An Increase in Future Income

An increase in futureincome increaseslifetime wealth formwe1 to we2, shifting thelifetime budgetconstraint to the rightand leaving its slopeunchanged. Theconsumer initiallychooses point A, andhe or she chooses Bafter the budgetconstraint shifts.

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Temporary and Permanent Increases in Income

A consumer will tend to save most of a purely temporaryincome increase.

As a permanent increase in income will have a larger effecton lifetime wealth than a temporary increase, there will bea larger effect on current consumption.

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Temporary versus Permanent Increases in Income


Figure 8.9 Temporary Versus Permanent Increases in Income

A temporary increase inincome is an increase incurrent income, with thebudget constraintshifting from AB to DEand the optimalconsumption bundlechanging form H to J .

When there is apermanent increase inincome, current andfuture income bothincrease, and the budgetconstraint shifts fromAB to FG, with theoptimal consumptionbundle changing from Hto K.

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Stock Prices and Consumption of Nondurables andServices


Figure 8.10 Stock Price Index and the Consumption of Nondurables and Services

Percentage deviations from trend in stock prices andconsumption are positively correlated, although stock prices aremuch more volatile than consumption.

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Scatter Plot: Consumption of Nondurables and Servicesvs. Stock Price Index


Figure 8.11 Scatter Plot: Consumption of Nondurables and Services vs. Stock Price Index

Positive correlation between stock prices and consumption.

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An Increase in the Real Interest Rate


Figure 8.12 An Increase in the Real Interest Rate

An increase in the real interest rate causes the lifetime budgetconstraint of the consumer to become steeper and to pivotaround the endowment point E.

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An Increase in the Market Real Interest Rate

The relative price of future consumption goods in terms ofcurrent consumption goods decreases—this has income andsubstitution effects for the consumer.

It changes the slope of the budget constraint.

Changes in the real interest rate are an important part ofthe mechanism by which shocks to the economy, fiscalpolicy, and monetary policy affect real activity.

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An Increase in the Real Interest Rate for a Lender


Figure 8.13 An Increase in the Real Interest Rate for a Lender

When the real interestrate increases for alender, the substitutioneffect is the movementfrom A to D, and theincome effect is themovement from D toB. Currentconsumption andsaving may rise or fall,while futureconsumption increases.

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An Increase in the Real Interest Rate for a Borrower


Figure 8.14 An Increase in the Real Interest Rate for a Borrower

When the real interestrate increases for aborrower, thesubstitution effect isthe movement from Ato D, and the incomeeffect is the movementfrom D to B. Currentconsumption decreaseswhile saving increases,and futureconsumption may riseor fall.

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Effects of an Increase in the Real Interest Rate for aLender

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-1

Table 8.2 Effects of an Increase in

the Real Interest Rate for a Lender

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Effects of an Increase in the Real Interest Rate for aBorrower

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-1

Table 8.3 Effects of an Increase in

the Real Interest Rate for a Borrower

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Example: Perfect Complements (1)

With perfect complements, the ratio of future consumption tocurrent consumption is constant.

c′ = ac (9)

The consumer’s budget constraint must hold:

c+c′

1 + r= we (10)

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With perfect complements, we can solve explicitly for currentand future consumption:

c =we(1 + r)

1 + r + a(11)

c′ =awe(1 + r)

1 + r + a(12)

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Substituting (5) in (11) and (12)

c =(y − t)(1 + r) + y′ − t′

1 + r + a(13)

c′ = a

[(y − t)(1 + r) + y′ − t′

1 + r + a

](14)

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Example with Perfect Complements Preferences


Figure 8.15 Example with Perfect Complements Preferences

The consumer desires current and future consumption in fixedproportions, with c′ = ac. With indifference curves representingperfect complementarity between current and futureconsumption, the optimal consumption bundle is at point D onthe lifetime budget constraint AB.

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Government Budget Constraints (1)

The government wishes to purchase G consumption goods,given exogenously.

The aggregate quantity of taxes collected by thegovernment is T . There are N consumers who each pay acurrent tax of t, so that T = Nt.

The government can borrow by issuing bonds B, bearingthe interest rate r.

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Government Budget Constraints (2)

The government’s current-period budget constraint:

G = T +B (15)

The government’s future-period budget constraint:

G′ + (1 + r)B = T ′ (16)

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The Government’t Present-Value Budget Constraint

Solving (16) for B:

B =T ′ −G′

1 + r(17)


G+G′

1 + r= T +

T ′

1 + r(18)

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Competitive Equilibrium

The market in which the N consumers and the governmentinteract is the credit market, they are effectively trading futureconsumption goods for current consumption goods.

In a competitive equilibrium for this two-period economy, threeconditions must hold:

1 Each consumer chooses first- and second-periodconsumption and savings optimally given the real interestrate r.

2 The government present-value budget constraint holds.

3 The credit market clears.

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Credit Market Equilibrium Condition

Total private savings is equal to the quantity of governmentbonds issued in the current period:

Sp = B (19)

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Income-Expenditure Identity

Credit market equilibrium implies that the income-expenditureidentity holds:

Y = C +G (20)

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The Ricardian Equivalence Theorem (1)

This theorem states that a change in the timing of taxes bythe government is neutral.

Neutral means that in equilibrium a change in currenttaxes, exactly offset in present-value terms by an equal andopposite change in future taxes, has no effect on the realinterest rate or on the consumption of individualconsumers.

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Substitute T = Nt and T ′ = Nt′ in (18):

G+G′

1 + r= Nt+

Nt′

1 + r(21)

t+t′

1 + r=

1

N

[G+

G′

1 + r

](22)

Key equation: The consumer’s lifetime tax burden is equal tothe consumer’s share of the present value of governmentspending—the timing of taxation does not matter for theconsumer.

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c+c′

1 + r= y +

y′

1 + r− 1N

[G+

G′

1 + r

](23)

Taxes do not matter in equilibrium for the consumer’s lifetimewealth, just the present value of government spending.

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Ricardian Equivalence with a Cut in Current Taxes fora Borrower


Figure 8.16 Ricardian Equivalence with a Cut in Current Taxes for a Borrower

A current tax cut witha future increase intaxes leaves theconsumer’s lifetimebudget constraintunchanged, and so theconsumer’s optimalconsumption bundleremains at A. Theendowment point shiftsfrom E1 to E2, so thatthere is an increase insaving by the amountof the current tax cut.

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Ricardian Equivalence and Credit Market Equilibrium


Figure 8.17 Ricardian Equivalence and Credit Market Equilibrium

With a decrease incurrent taxes,government debtincreases form B1 toB2, and the creditsupply curve shifts tothe right by the sameamount. Theequilibrium realinterest rate isunchanged, and privatesaving increases by anamount equal to thereduction ingovernment saving.

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Key assumptions behind the Ricardian equivalence

Each individual shares equal burden of taxes

Any debt issued is paid off during the lifetime of the peoplealive when the debt was issued

Taxes are lump-sum

Perfect credit markets (same interest rate for borrowingand lending, no limit on borrowing and lending subject tothe lifetime budget constraint)

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Government Deficits, Taxes, and Government Debt

The government deficit exceeded 6% of GDP in 2009.

Taxes have fallen due to the Bush tax cuts and therecession.

The government debt to GDP ratio has risen to almost80%.

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Total Government Surplus as a Percentage of GDP


Figure 8.21 Total Government Surplus as a Percentage of GDP

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Taxes as a Percentage of GDP


Figure 8.22 Taxes as a Percentage of GDP

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Federal Government Debt as a Percentage of GDP


Figure 8.23 Federal Government Debt as a Percentage of GDP

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Question

What happens if the U.S. runs government deficits of 5% to10% of GDP forever?

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Assumptions

Suppose real GDP grows at its average rate, 3% per year,forever: Yt = Y0(1 + g)

t

Suppose the primary deficit (G+Transfers–T) is a constantfraction of GDP forever: a · Y0(1 + g)t (e.g., a=–5%, or–10%)

Suppose the real interest rate, r, is 1% per year, forever.

Debt at time t is:Bt = (1 + r)Bt−1 − aYt = (1 + r)Bt−1 − a · Y0(1 + g)t.

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Debt to GDP ratio

BtYt

= (1 + r)Bt−1Yt− a

= (1 + r)Bt−1Yt−1

Yt−1Yt− a

=1 + r

1 + g

Bt−1Yt−1

− a

It can be shown that

BtYt

= −a(1 + g)1−

(1+r1+g

)t+1g − r

≈ −a(1 + g)g − r

,

for large t, and g > r. If, e.g., a = −5%, in the long run

BtYt≈ 0.05(1 + 0.03)

0.03− 0.01= 2.575,

or 257.5%.64 / 74

Conclusions

Primary deficit of 5% of GDP forever implies: Debt/GDPratio of 257.5% in the long run (more than 3 times currentratio of 80%), with 2.6% of GDP spent on interestpayments on the government debt per year in the long run.

Primary deficit of 10% of GDP forever implies: Debt/GDPratio of 515% in the long run (more than 6 times thecurrent ratio), with 5.2% of GDP spent on interestpayments on the government debt.

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Liquidity=credit Constraints

If the consumer is a would-be-borrower but cannot borrow, i.e.,is liquidity constrained, then this consumer’s c will be equal toy (current consumption equals current income).

Thus, for liquidity constrained consumers current consumptionis determined by current income.

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Consider first the problem without liquidity constraints. Then,(c∗, c′∗) should satisfy 2 equations. The Euler equation:

MU1(c∗) = (1 + r)MU2(c

′∗),

and the budget constraint:

c∗ +c′∗

1 + r= y +

y′

1 + r.

With liquidity (borrowing) constraints, c cannot be larger thany, that is, c ≤ y. If the unconstrained problem gives you c∗ ≤ y,then we say the constraint is not binding—consumer is a saveranyways. Otherwise, if c∗ > y, the maximum this consumer canhave in period 1 is y. Summarizing, c∗bc = min{c∗, y}; c′∗bc = c′∗ ifc∗bc = c

∗, c′∗bc = y′ otherwise.

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Example

Let the utility function be U(c, c′) = log(c) + β log(c′), where logis the natural logarithm. Let y = 40, y′ = 80, no taxes, r = 0.05,β = 0.90. First, assume that consumer is unconstrained.

For this utility function, MU1 =1c , and MU2 = β

1c′ . The Euler

equation tells us that 1c = β(1 + r)1c′ , or c

′∗ = β(1 + r)c∗. Plugthis result into the budget constraint to obtain,

c∗ + β(1+r)c′∗

1+r = y +y′

1+r , or

c∗ = 11+β

[y + y

′

1+r

]= 11.90 [40 + 80/(1.05)] ≈ 61. Thus,

c′∗ = 0.90 ∗ (1.05) ∗ 61 ≈ 57.8.

If consumer is constrained this implies that c∗bc ≤ y, andc∗bc = min{61, 40} = 40. Also, c′∗bc = y′ = 80. You can also showthat the liquidity constrained consumer will be worse off.

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The Life-Cycle Hypothesis

Idea: want to smooth consumption over the life cycle.Thus, need to save during the working life to supportconsumption during retirement.

Assume consumer prefers a smooth consumption path; will liveT more years; will work for another R years; has wealth of W ;and will receive a sure income Y during the working years.

The lifetime wealth is W +R× Y . Then,

C = (W +R× Y )/T = W/T + (R/T )× Y = α×W + β × Y .

Thus, CY = α× (W/Y ) + β.

In the short run, for a fixed W , an increases in Y leads to afalling APC; in the long run, W and Y grow together, andC/Y is stable.

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Milton Friedman’s Permanent IncomeHypothesis

Milton Friedman postulated the following model ofconsumption:

Income : Y = Y T + Y P ,

Consumption : C = Y P .

Y P is the permanent income (the component of income thatpersists over time); Y T is the transitory income (short-livedcomponents of income such as bonuses, overtime, windfalls fromlottery etc.)

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The Permanent Income Hypothesis (PIH)

Implications:

Consumption changes by the magnitude of a change inpermanent income. Transitory changes in income arepredominantly saved.

APC = CY =Y P

Y . In the household studies, most of

variation in YP

Y comes from the transitory variation inincome. Thus, if Y > Y P , C < Y , and the APC is falling.

In the long time series, most of variation in Y comes fromthe variation in Y P , and so APC will be stable.

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Hall’s Formulation of the PIH

PIH: consumption depends on Y P , and therefore onexpectations of the lifetime income.

Consumption then responds only to the “news” (unexpectedchanges, or “surprises”) about lifetime income.

Robert Hall : if the PIH holds, and consumers form rationalexpectations, then changes in consumption are unpredictable.

Importantly, under the PIH only unexpected changes in policyinfluence consumption.

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Readings

Mankiw and Scarth. Chapter 17 (Consumption), and Section16-4 (Ricardian equivalence).Stephen Williamson. 2013. Macroeconomics. Fourth CanadianEdition. Chapter 9.

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