ConsumptionECON 30020: Intermediate Macroeconomics

Prof. Eric Sims

University of Notre Dame

Fall 2016

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Microeconomics of Macro

I We now move from the long run (decades and longer) to themedium run (several years) and short run (months up toseveral years)

I In long run, we did not explicitly model most economicdecision-making – just assumed rules (e.g. consume aconstant fraction of income)

I Building blocks of the remainder of the course are decisionrules of optimizing agents and a concept of equilibrium

I Will be studying optimal decision rules first

I Framework is dynamic but only two periods (t, the present,and t + 1, the future)

I Consider representative agents: one household and one firm

I Unrealistic but useful abstraction and can be motivated inworld with heterogeneity through insurance markets

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Consumption

I Consumption the largest expenditure category in GDP (60-70percent)

I Study problem of representative household

I Household receives exogenous amount of income in periods tand t + 1

I Must decide how to divide its income in t betweenconsumption and saving/borrowing

I Everything real – think about one good as “fruit”

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Basics

I Representative household earns income of Yt and Yt+1.Future income known with certainty

I Consumes Ct and Ct+1

I Begins life with no wealth, and can save St = Yt − Ct (can benegative, which is borrowing)

I Earns/pays real interest rate rt on saving/borrowing

I Household a price-taker: takes rt as given

I Do not model a financial intermediary (i.e. bank), but assumeexistence of option to borrow/save through this intermediary

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Budget Constraints

I Two flow budget constraints in each period:

Ct + St ≤ Yt

Ct+1 + St+1 − St ≤ Yt+1 + rtSt

I Saving vs. Savings: saving is a flow and savings is a stock.Saving is the change in the stock

I As written, St and St+1 are stocks

I In period t, no distinction between stock and flow because noinitial stock

I St+1 − St is flow saving in period t + 1; St is the stock ofsavings household takes from t to t + 1, and St+1 is the stockit takes from t + 1 to t + 2

I rtSt : income earned on the stock of savings brought into t + 1

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Terminal Condition and the IBC

I Household would not want St+1 > 0. Why? There is not + 2. Don’t want to die with positive assets

I Household would like St+1 < 0 – die in debt. Lender wouldnot allow that

I Hence, St+1 = 0 is a terminal condition (sometimes “noPonzi”)

I Assume budget constraints hold with equality (otherwiseleaving income on the table), and eliminate St , leaving:

Ct +Ct+1

1 + rt= Yt +

Yt+1

1 + rt

I This is called the intertemporal budget constraint (IBC). Saysthat present discounted value of stream of consumptionequals present discounted value of stream of income.

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Preferences

I Household gets utility from how much it consumes

I Utility function: u(Ct). “Maps” consumption into utils

I Assume: u′(Ct) > 0 (positive marginal utility) andu′′(Ct) < 0 (diminishing marginal utility)

I “More is better, but at a decreasing rate”

I Example utility function:

u(Ct) = lnCt

u′(Ct) =1

Ct> 0

u′′(Ct) = −C−2t < 0

I Utility is completely ordinal – no meaning to magnitude ofutility (it can be negative). Only useful to comparealternatives

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Lifetime Utility

I Lifetime utility is a weighted sum of utility from period t andt + 1 consumption:

U = u(Ct) + βu(Ct+1)

I 0 < β < 1 is the discount factor – it is a measure of howimpatient the household is.

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Household Problem

I Technically, household chooses Ct and St in first period. Thiseffectively determines Ct+1

I Think instead about choosing Ct and Ct+1 in period t

maxCt ,Ct+1

U = u(Ct) + βu(Ct+1)

s.t.

Ct +Ct+1

1 + rt= Yt +

Yt+1

1 + rt

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Euler Equation

I First order optimality condition is famous in economics – the“Euler equation” (pronounced “oiler”)

u′(Ct) = β(1 + rt)u′(Ct+1)

I Intuition and example with log utility

I Necessary but not sufficient for optimality

I Doesn’t determine level of consumption. To do that need tocombine with IBC

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Indifference Curve

I Think of Ct and Ct+1 as different goods (different in timedimension)

I Indifference curve: combinations of Ct and Ct+1 yielding fixedoverall level of lifetime utility

I Different indifference curve for each different level of lifetimeutility. Direction of increasing preference is northeast

I Slope of indifference curve at a point is the negative ratio ofmarginal utilities:

slope = − u′(Ct)

βu′(Ct+1)

I Given assumption of u′′(·) < 0, steep near origin and flataway from it

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Budget Line

I Graphical representation of IBC

I Shows combinations of Ct and Ct+1 consistent with IBCholding, given Yt , Yt+1, and rt

I Points inside budget line: do not exhaust resources

I Points outside budget line: infeasible

I By construction, must pass through point Ct = Yt andCt+1 = Yt+1 (“endowment point”)

I Slope of budget line is negative gross real interest rate:

slope = −(1 + rt)

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Optimality GraphicallyI Objective is to choose a consumption bundle on highest

possible indifference curveI At this point, indifference curve and budget line are tangent

(which is same condition as Euler equation)

𝐶𝐶𝑡𝑡+1

𝐶𝐶𝑡𝑡

(1 + 𝑟𝑟𝑡𝑡)𝑌𝑌𝑡𝑡 + 𝑌𝑌𝑡𝑡+1

𝑌𝑌𝑡𝑡 +𝑌𝑌𝑡𝑡+1

1 + 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡

𝑌𝑌𝑡𝑡+1

𝑈𝑈 = 𝑈𝑈0

𝑈𝑈 = 𝑈𝑈1

𝑈𝑈 = 𝑈𝑈2

𝐶𝐶0,𝑡𝑡

𝐶𝐶0,𝑡𝑡+1

𝐶𝐶1,𝑡𝑡

𝐶𝐶1,𝑡𝑡+1

𝐶𝐶2,𝑡𝑡

𝐶𝐶2,𝑡𝑡+1

𝐶𝐶3,𝑡𝑡

𝐶𝐶3,𝑡𝑡+1

(0) (1)

(2)

(3)

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Consumption Function

I What we want is a decision rule that determines Ct as afunction of things which the household takes as given – Yt ,Yt+1, and rt

I Consumption function:

Ct = Cd (Yt ,Yt+1, rt)

I Can use indifference curve - budget line diagram toqualitatively figure out how changes in Yt , Yt+1, and rt affectCt

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Increases in Yt and Yt+1

I An increase in Yt or Yt+1 causes the budget line to shift outhorizontally to the right

I In new optimum, household will locate on a higherindifference curve with higher Ct and Ct+1

I Important result: wants to increase consumption in bothperiods when income increases in either period

I Wants its consumption to be “smooth” relative to its income

I Achieves smoothing its consumption relative to income byadjusting saving behavior: increases St when Yt goes up,reduces St when Yt+1 goes up

I Can conclude that ∂Cd

∂Yt> 0 and ∂Cd

∂Yt+1> 0

I Further, ∂Cd

∂Yt< 1. Call this the marginal propensity to

consume, MPC

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Increase in rt

I A little trickier

I Causes budget line to become steeper, pivoting throughendowment point

I Competing income and substitution effects:I Substitution effect: how would consumption bundle change

when rt increases and income is adjusted so that householdwould locate on unchanged indifference curve?

I Income effect: how does change in rt allow household to locateon a higher/lower indifference curve?

I Substitution effect always to reduce Ct , increase StI Income effect depends on whether initially a borrower

(Ct > Yt , income effect to reduce Ct) or saver (Ct < Yt ,income effect to increase Ct)

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Borrower

𝐶𝐶𝑡𝑡+1

𝐶𝐶𝑡𝑡 𝑌𝑌𝑡𝑡

𝑌𝑌𝑡𝑡+1

Original bundle

New bundle

Hypothetical bundle with new 𝑟𝑟𝑡𝑡 on same indifference curve

𝐶𝐶0,𝑡𝑡

𝐶𝐶0,𝑡𝑡+1

𝐶𝐶1,𝑡𝑡

𝐶𝐶1,𝑡𝑡+1

𝐶𝐶0,𝑡𝑡ℎ

𝐶𝐶0,𝑡𝑡+1ℎ

I Sub effect: ↓ Ct . Income effect: ↓ Ct

I Total effect: ↓ Ct

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Saver

𝐶𝐶𝑡𝑡+1

𝐶𝐶𝑡𝑡 𝑌𝑌𝑡𝑡

𝑌𝑌𝑡𝑡+1

𝐶𝐶0,𝑡𝑡

𝐶𝐶0,𝑡𝑡+1

𝐶𝐶1,𝑡𝑡+1

𝐶𝐶1,𝑡𝑡

𝐶𝐶0,𝑡𝑡ℎ

𝐶𝐶0,𝑡𝑡+1ℎ

Original bundle

New bundle

Hypothetical bundle with new 𝑟𝑟𝑡𝑡 on same indifference curve

I Sub effect: ↓ Ct . Income effect: ↑ Ct

I Total effect: ambiguous

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The Consumption Function

I We will assume that the substitution effect always dominatesfor the interest rate

I Qualitative consumption function (with signs of partialderivatives)

Ct = C (Yt+

,Yt+1+

, rt−).

I Technically, partial derivative itself is a function

I However, we will mostly treat the partial with respect to firstargument as a parameter we call the MPC

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Algebraic Example with Log Utility

I Suppose u(Ct) = lnCt

I Euler equation is:

Ct+1 = β(1 + rt)Ct

I Consumption function is:

Ct =1

1 + β

[Yt +

Yt+1

1 + rt

]

I MPC: 11+β . Go through other partials

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Permanent Income Hypothesis (PIH)

I Our analysis consistent with Friedman (1957) and the PIH

I Consumption ought to be a function of “permanent income”

I Permanent income: present value of lifetime income

I Special case: rt = 0 and β = 1: consumption equal toaverage lifetime income

I Implications:

1. Consumption forward-looking. Consumption should not reactto changes in income that were predictable in the past

2. MPC less than 13. Longer you live, the lower is the MPC

I Important empirical implications for econometric practice ofthe day. Regression of Ct on Yt will not identify MPC (whichis relevant for things like fiscal multiplier) if in historical datachanges in Yt are persistent

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Applications and Extensions

I We will consider several applications / extensions:

1. Wealth2. Permanent vs. transitory changes in income3. Uncertainty4. Random walk

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Wealth

I Allow household to begin life with stock of wealth Ht−1. Realprice of this asset in t is Qt

I Household can accumulate more of this asset or sell it

I Period t constraint:

Ct + St +Qt(Ht −Ht−1) ≤ Yt

I Period t + 1 constraint:

Ct+1 + St+1 +Qt+1(Ht+1 −Ht) ≤ Yt + (1 + rt)St

I Imposing terminal conditions, IBC is:

Ct +Ct+1

1 + rt+QtHt = Yt +

Yt+1

1 + rt+QtHt−1 +

Qt+1Ht

1 + rt

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Simplifying Assumptions and the Consumption Function

I First, assume household must choose Ht = 0. It simply sellsoff the asset in period t at price Qt :

Ct +Ct+1

1 + rt= Yt +

Yt+1

1 + rt+QtHt−1

I Increase in Qt then functions just like increase in Yt

Ct = Cd (Yt+

,Yt+1+

, rt−

,Qt+)

I Empirical application: stock market boom of 1990s (increasein Qt)

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Alternative Simplifying Assumption

I Assume Ht−1 = 0, and assume that household must purchasean exogenous amount of the asset, Ht (e.g. has to buy ahouse)

I IBC:

Ct +Ct+1

1 + rt= Yt +

Yt+1

1 + rt+Ht

(Qt+1

1 + rt−Qt

)

I Increase in Qt+1: functions like increase in Yt+1:

Ct = Cd (Yt+

,Yt+1+

, rt−

,Qt−

,Qt+1+

)

I Empirical applications: house price boom of 2000s(anticipated increase in Qt+1)

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Permanent vs. Transitory Changes in Income

I Go back to standard consumption function:

Ct = Cd (Yt ,Yt+1, rt)

I Take total derivative (differs from partial derivative in allowingeverything to change):

dCt =∂Cd (·)

∂YtdYt +

∂Cd (·)∂Yt+1

dYt+1 +∂Cd (·)

∂rtdrt

I If just dYt 6= 0, then dCtdYt

is equal to partial ∂Cd (·)∂Yt

I But if changes in income are persistent (i.e. dYt > 0 ⇒dYt+1 > 0), then dCt

dYt> ∂Cd (·)

∂Yt

I Implication: consumption reacts more to a change in incomethe more persistent is that change in income

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Application: Tax Cuts

I Suppose household pays taxes, Tt and Tt+1, to governmenteach period, so net income is Yt − Tt and Yt+1 − Tt+1.Consumption function is:

Ct = Cd (Yt − Tt ,Yt+1 − Tt+1, rt)

I A cut in taxes is equivalent to an increase in income

I Implication: tax cuts will have bigger stimulative effects onconsumption the more persistent the tax cuts are

I Empirical studies: Shaprio and Slemrod (2003) and Shapiroand Slemrod (2009)

I Initial installment of Bush tax cuts in 2001 was close topermanent (ten years). Theory predicts consumption ought toreact a lot. It didn’t.

I Tax rebates 2008: known to be only one time. Theory predictsconsumption should react comparatively little. It did.

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Uncertainty

I Suppose that future income is not known with certainty

I Say it can take on two values: Y ht+1 with probability p, and

Y lt+1 with probability 1− p. Expected future income is:

E (Yt+1) = pY ht+1 + (1− p)Y l

t+1

I Household will want to maximize expected lifetime utility.Utility from future consumption is not known, because futureconsumption isn’t known given uncertainty about income

I Euler equation looks the same, but has an expectationoperator:

u′(Ct) = β(1 + rt)E[u′(Ct+1)

]I Key insight: expected value of a non-linear function is not

equal to the function of the expected value.

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Expected Marginal Utility vs. Marginal Utility of ExpectedConsumption

I Assume u′′′(·) > 0. Then E [u′(Ct+1)] > u′(E [Ct+1])

𝐶𝐶𝑡𝑡+1

𝑢𝑢′(𝐶𝐶𝑡𝑡+1)

𝐶𝐶𝑡𝑡+1ℎ 𝐶𝐶𝑡𝑡+1𝑙𝑙 𝐸𝐸[𝐶𝐶𝑡𝑡+1]

𝑢𝑢′(𝐶𝐶𝑡𝑡+1𝑙𝑙 )

𝑢𝑢′(𝐶𝐶𝑡𝑡+1ℎ )

𝐸𝐸[𝑢𝑢′(𝐶𝐶𝑡𝑡+1)]

𝑢𝑢′(𝐸𝐸[𝐶𝐶𝑡𝑡+1])

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Increase in Uncertainty: Mean-Preserving Spread

I Raises E [u′(Ct+1)]

𝐶𝐶𝑡𝑡+1

𝑢𝑢′(𝐶𝐶𝑡𝑡+1)

𝐶𝐶0,𝑡𝑡+1ℎ 𝐶𝐶0,𝑡𝑡+1

𝑙𝑙 𝐸𝐸[𝐶𝐶𝑡𝑡+1]

𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1𝑙𝑙 )

𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1ℎ )

𝐸𝐸[𝑢𝑢′(𝐶𝐶0,𝑡𝑡+1)] 𝑢𝑢′(𝐸𝐸[𝐶𝐶𝑡𝑡+1])

𝐶𝐶1,𝑡𝑡+1𝑙𝑙 𝐶𝐶1,𝑡𝑡+1

ℎ

𝑢𝑢′(𝐶𝐶1,𝑡𝑡+1ℎ )

𝑢𝑢′(𝐶𝐶1,𝑡𝑡+1𝑙𝑙 )

𝐸𝐸[𝑢𝑢′(𝐶𝐶1,𝑡𝑡+1)]

↑ uncertainty → ↑ 𝐸𝐸[𝑢𝑢′(𝐶𝐶𝑡𝑡+1)]

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Precautionary Saving

I Increase in uncertainty raises expected marginal utility ofconsumption E [u′(Ct+1)]

I For the Euler equation to hold, need to adjust currentconsumption to raise current marginal utility, u′(Ct)

I This requires increasing saving – consume less holding currentincome fixed

I Intuition: bad state of the world hurts you more than thegood state helps you, so you adjust behavior in present toeffectively “self-insure” against bad future state

I Empirical application: high uncertainty and weak consumptionduring and in wake of Great Recession

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Random Walk Hypothesis

I Suppose that β(1 + r) = 1

I Euler equation is then implies u′(Ct) = E [u′(Ct+1)]

I Suppose that u′′′(·) = 0 (so no precautionary saving). Thenthis implies that E [Ct+1] = Ct

I In expectation, future consumption ought to equal currentconsumption. This is the “random walk” hypothesis

I Doesn’t mean that future consumption always equals currentconsumption

I But it does imply future changes in consumption ought to notbe predictable, because in expectation future consumptionshould equal current consumption

I Random walk model due to Hall (1978)

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Empirical Tests

I Random walk hypothesis one of the most testedmacroeconomic theories

I Generally fails:I Parker (1999): exploits facts about social security withholding

and predictable changes over course of year. Consumptionreacts to predictable changes in take home pay

I Evans and Moore (2012): look at relationship between receiptof paycheck (which is predictable) and within-month mortalitycycle

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Borrowing ConstraintsI Empirical failures can potentially be accounted for by

borrowing constraintsI Simplest form of a borrowing constraint: you can’t. St ≥ 0.

Introduces kink into budget line

𝐶𝐶𝑡𝑡+1

𝐶𝐶𝑡𝑡

(1 + 𝑟𝑟𝑡𝑡)𝑌𝑌𝑡𝑡 + 𝑌𝑌𝑡𝑡+1

𝑌𝑌𝑡𝑡 +𝑌𝑌𝑡𝑡+1

1 + 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡

𝑌𝑌𝑡𝑡+1

Infeasible if 𝑆𝑆𝑡𝑡 ≥ 0

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Binding Borrowing Constraint

I If borrowing constraint “binds” you locate at the kink in thebudget line (i.e. Euler equation does not hold)

𝐶𝐶𝑡𝑡+1

𝐶𝐶𝑡𝑡 𝐶𝐶0,𝑡𝑡 = 𝑌𝑌0,𝑡𝑡

𝐶𝐶0,𝑡𝑡+1 = 𝑌𝑌0,𝑡𝑡+1

𝐶𝐶0,𝑑𝑑,𝑡𝑡

𝐶𝐶0,𝑑𝑑,𝑡𝑡+1

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Implications of a Binding Borrowing Constraint

I Current consumption equals current income

I Means that if household gets more income, will spend all of it

I Further means that if household expects more income infuture, can’t adjust consumption until the future –consumption will react to anticipated change in income

I Potential resolution of some empirical failures of random walk/ PIH model

I Also has policy implications. Makes sense to targettaxes/transfers to households likely to be borrowingconstrained if objective is to stimulate consumption

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