aaron_cons_ahorro.pdf

Consumption-Savings Decisions and Credit Markets

Economics 3307 - Intermediate Macroeconomics

Aaron Hedlund

Baylor University

Spring 2013

Econ 3307 (Baylor University) Consumption-Savings Decisions Spring 2013 1 / 26

Introduction

So far, we have only looked at static consumer optimization orexogenously specified dynamic behavior.

Now we look at intertemporal decisions.

Just a few issues that we will investigate:I Consumption-savings decisions and investment behavior.

I Ricardian equivalence and fiscal policy.

I The determination of real interest rates.


A Two Period Model

Two period endowment economy with endowments y1 and y2.

Lump sum taxes t1 and t2.

Real interest rate r .

The consumers budget constraints are given by

c1 + s1 = y1 t1c2 = y2 + (1 + r)s1 t2

The lifetime budget constraint combines both constraints to equatelifetime wealth with the present value of consumption.

c1 +c2

1 + r= y1 t1 + y2 t2

1 + r


A Two Period Model

Time separable preferences U(c1, c2) = u(c1) + u(c2).I Standard assumptions: u > 0 and u < 0.

The discount factor (0, 1) measures the households degree ofpatience.

Consumers make consumption/savings decisions to maximize utility,

maxc1,c2

u(c1) + u(c2)

subject to

c1 +c2

1 + r= y1 t1 + y2 t2

1 + r


A Two Period Model

Optimality conditions:

u(c1)u(c2)

= MRSc1,c2 = 1 + r

c1 +c2

1 + r= y1 t1 + y2 t2

1 + r

A lender if c1 < y1 t1 and a borrower if c1 > y1 t1.

c 2

c 2

c2 c2

c1 c1 y1 t1 y1 t1

y2 t2

y2 t2

c1 c1


Comparative Statics: Changes in Current Income

How do c1, s1, and c2 respond to changes in y1?

c1y1

=(1 + r)2u(c2)

> 0

c2y1

=(1 + r)u(c1)

> 0

where = u(c1) (1 + r)2u(c2) > 0.

Lastly,s1y1

= 1 c1y1

=u(c1)

> 0


Comparative Statics: Changes in Current Income

Consumers use part of the increase in y1 to augment consumptionand the rest to increase savings. Thus, c1 < y1.

This behavior is called consumption smoothing.

In the data, households smooth their consumption, but not by asmuch as theory predicts. Consumption exhibits excess variability.

Possible reasons:1 Credit market imperfections.

2 Consumer durables are more like investment.

3 General equilibrium effects.


Consumption Variation

Durable goods consumption is much more volatile than nondurablegoods/services because it is economically a form of investment.


Comparative Statics: Changes in Future Income

How do c1, c2, and s1 respond to changes in y2?

c1y2

=1

1 + r

c1y1

> 0

c2y2

=1

1 + r

c2y1

> 0

s1y2

= c1y2

< 0

Consumers use a portion of the expected increase in future income toaugment consumption today (requiring a decrease in savings) and therest to augment consumption in the future.


The Permanent Income Hypothesis

Consumers respond differently to temporary changes in income thanthey do to permanent changes in income.

I Examples of temporary changes: unemployment, bonuses, etc.

I Examples of permanent changes: occupation changes, disability, etc.

The permanent income hypothesis states that consumers basetheir consumption decisions on their discounted lifetime income.

W = y +y

1 + r+

y

(1 + r)2+ = 1 + r

ry

Wtemp =

(y + y +

y

1 + r+ . . .

)W = y

Wperm =

(y + y +

y + y

1 + r+ . . .

)W = 1 + r

ry


Consumption Smoothing and the Stock Market

Stock price movements are closer to permanent shocks, implying thatconsumption should respond noticeably to such changes.


Comparative Statics: Changes in The Real Interest Rate

How do c1 and c2 respond to changes in r?

c1r

=u(c2) (1 + r)u(c2)(y1 t1 c1)

c2r

=(1 + r)u(c2) u(c1)(y1 t1 c1)

s1r

= c1r

Substitution effects:

c1r

subst

=u(c2)

< 0,

c2r

subst

=(1 + r)u(c2)

> 0



Income effects:

c1r

inc

=c1r c1

r

subst

=(1 + r)u(c2)(y1 t1 c1)

c2r

inc

=c2r c2

r

subst

=u(c1)(y1 t1 c1)

The signs of the income effects depend on the sign of y1 t1 c1.

The income effect of a higher real interest rate is negative forborrowers and positive for lenders.



c1

c 2

c 2

c1

weL(1 + rL)

weH(1 + rH)

weH(1 + rH)

weL(1 + rL)

weL weL weH weH

The effect of an increase in the real interest rate for a lender (left)and a borrower (right).


The Government

Suppose the government levies lump-sum taxes T1 = Nt1 andT2 = Nt2 and issues initial debt B1 (S

g1 = B1) to finance spending

G1 and G2.

The governments budget constraints are

G1 = T1 + B1

G2 + (1 + r)B1 = T2

Substituting out for B1 gives the present-value government budgetconstraint

G1 +G2

1 + r= T1 +

T21 + r


Competitive Equilibrium

A competitive equilibrium in this economy consists of an interest rater , allocations for the household c1 and c2, and allocations for thegovernment G1, G2, T1, T2 such that:

1 Consumption c1 and c2 solve the consumers optimization problem.

2 The governments present-value budget constraint is satisfied:

G1 +G2

1 + r= T1 +

T21 + r

3 The credit market clears in the initial period (total savings S1 = 0):

S1 Sp1 + Sg1 = Sp1 B1 = 0 Sp1 = B1where Sp1 = N(y1 t1 c1) and B1 = G1 T1.

4 The goods market clears:

Y1 = C1 + G1 and Y2 = C2 + G2

where Y1 = Ny1, Y2 = Ny2, C1 = Nc1, and C2 = Nc2.


Walras Law

Conditions (1), (2), and (3) automatically imply (4).

c1 +c2

1 + r= y1 t1 + y2 t2

1 + r C1 + C2

1 + r= Y1 T1 + Y2 T2

1 + r

Thus,

Sp1 = Y1 T1 C1 = (Y2 T2 C2

1 + r

)Also,

G1 +G2

1 + r= T1 +

T21 + r

B1 = G1 T1 = (G2 T2

1 + r

)Credit market clearing Sp1 = B1 implies

Y1 T1 C1 = G1 T1 and Y2 T2 C2 = G2 T2 Y1 = C1 + G1 and Y2 = C2 + G2


Ricardian Equivalence

Suppose the government leaves G1 and G2 unchanged but reducesinitial period taxes by t1 =

T1N per household.

The deficit increases by T1 and the government must raise secondperiod taxes by T2 to pay back the increased debt:

G1 = T1 T1 + (B1 + T1), G2 + (1 + r)(B1 + T1) = T2 + T2 G1 + G2

1 + r= T1 T1 + T2 + T2

1 + r

Before the tax cut,

G1 +G2

1 + r= T1 +

T21 + r

Thus, the required tax increase is T2 = T1(1 + r).



What are the macroeconomic effects of this deficit-financed tax cut?

The consumers intertemporal budget constraint becomes

c1 +c2

1 + r= y1 (t1 t1) + y2 (t2 + t1(1 + r))

1 + r

c1 + c21 + r

= y1 t1 + y2 t21 + r

Unchanged! The consumers original c1 and c2 are still feasible andoptimal.

Private savings becomes Sp1,new = Y1 (T1 T1) C1= Y1 T1 C1 + T1 = Sp1 + T1.

National saving is (Sp1 + T1) (B1 + T1) = Sp1 B1 = 0. Thus,the credit market is still in equilibrium without requiring a change in r .



Summary:I Holding r fixed, increased household saving exactly offsets decreased

public saving, leaving national saving unchanged.

I Equilibrium c1, c2, and r are unchanged.

I The true tax burden is the discounted value of government spending.Deficits are just deferred taxes.

SP + T1

SP

1

1

B1 + T1 B1

r


Assumptions of Ricardian Equivalence

Ricardian equivalence relies on four assumptionsI Taxes change by the same amount for all households, i.e. it is not the

case that certain households receive the initial tax cuts and differenthouseholds pay the future tax increases.

I The increased government debt is paid off during the lifetimes of thepeople alive when the debt was issued.

I Lump-sum taxation.

I Perfect credit markets.

Real world example: the income tax withholding reduction in 1992-93.


Government Deficits and Debt

Government deficits in the U.S. and in Europe.


Government Deficits and Debt

Government debt to GDP ratio in the United States.


Can the Government Run a Perpetual Deficit?

Consider the following scenario:I GDP growth rate g : Yt = Y0(1 + g)

t .

I Permanent primary deficit dt Gt Tt = dYt a fraction d of GDP.

I Constant real interest rate r .

From the governments per period budget constraint,

Gt + (1 + r)Bt1 = Tt + Bt Bt =dt=dY0(1+g)t

(Gt Tt) +(1 + r)Bt1 Bt = dY0(1 + g)t + (1 + r)

{dY0(1 + g)

t1 + (1 + r)[dY0(1 + g)

t2 + . . .]}

Bt = dY0[(1 + g)t + (1 + r)(1 + g)t1 + (1 + r)2(1 + g)t2 + + (1 + r)t]

BtYt

=Bt

Y0(1 + g)t= d

[1 +

1 + r

1 + g+

(1 + r)2

(1 + g)2+ + (1 + r)

t

(1 + g)t

]


The Arithmetic of Perpetual Government Deficits

The debt/GDP ratio is therefore

BtYt

= dt

n=0

(1 + r

1 + g

)n= d

1 + g

g r

[1

(1 + r

1 + g

)t+1]

If r > g , BtYt in the long run, which is impossible.

If r < g , BtYt d1+ggr in the long run.

Example: r = 0.01, g = 0.03, d = 0.05I Long run debt/GDP of d 1+ggr = 2.575.

I Annual (primary deficit + debt service)/GDP of d + rd 1+ggr = 0.07575.

I These calculations assume that the world is willing to purchase all ofthis debt at interest rate r .


European Sovereign Default Crisis

Interest rates in Europe.


HouseholdsThe GovernmentCompetitive EquilibriumRicardian EquivalencePerpetual Government Deficits

aaron_cons_ahorro.pdf

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