Consumption and Saving 2009
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Consumption and Saving
2009
Introduction
• Many decisions involve an intertemporal tradeoff; e.g.,
— consume now or tomorrow (saving), work now or tomorrow (schooling),
tax now or tomorrow (government finance)
• To interpret these types of decisions, we need a dynamic model (a model
with an explicit time dimension)
• We will start with the simplest dynamic model: a two-period model (current period vs. future period)
A 2-Period Model
• There are two periods; = 1 2 (current period and future period)
• There are two goods (1 2); current consumption and future consumption
— interpret as a time-dated good; so (1 2) is a commodity bundle
• People have preferences (1 2) with usual properties (note: forward
looking)
• People have a nonstorable endowment (1 2); an exogenous lifetime earnings stream
Robinson Crusoe
• Choice problem is
max { (1 2) : 1 ≤ 1 2 ≤ 2}
12
• Solution is easy: 1 = 1 and 2 = 2 (consumer demand functions)
• Crusoe is constrained to live within his means on a period-by-period basis
• Note: 1 depends on 1 but does not depend on 2 (expected future
earnings)
An International Bond Market
• Imagine now that Crusoe can trade with other economies
• Crusoe may want to import or export output
• Question: how can Crusoe pay for imports; how can foreigners pay for
exports?
• Answer: by issuing debt for imports; by accepting debt for exports
• Debt: a promise to repay in the future
• Let denote the interest rate for risk-free debt; let ≡ 1 + denote the
gross interest rate
• If Crusoe imports one unit of output today, he must repay units of
output tomorrow (principal plus interest)
• If Crusoe exports one unit of output today, he will be repaid units of
output tomorrow
• Define ≡ 1 − 1; i.e., saving (today)
• Note: 0 (positive saving means exporting) and 0 (negative saving
means importing)
• So Crusoe now faces the following constraint
2 ≤ 2 +
• Combine this with the definition of saving ≡ 1 −1 to derive the lifetime
budget constraint (LBC)
1 + 2 ≤ 1 + 2
• Note: Crusoe no longer constrained to live within his means on a periodby-period basis; but must live within his means on a lifetime basis
• Note: If Crusoe imports today, he must export tomorrow (to pay off debt);
if he exports today, he can import tomorrow (calling in his debt)
• Choice problem is now given by
½
2
2
max (1 2) : 1 +
≤ 1 +
12
¾
• Solution is a pair of consumer-demand functions (1 2 ); as function of
exogenous variables ( 1 2 )
• And a desired saving function ≡ 1 − 1
• Mathematical characterization is
(1 2 ) =
2
1 +
= 1 + 2
• or (1 − 2 + ) =
• Example: let (1 2) = ln(1) + ln(2) where is a preference
parameter called the discount factor (interpret as patience)
• Then is given by 2(1); so
2
2
1 2
= and 1 +
= 1 +
1
• Solution is
Ã
!∙
¸
2
1
1 +
1 =
1+
Ã
!∙
¸
2 =
1 + 2
1+
• Use definition of saving to derive desired saving function
• Or solve directly using
• Solution is
=
1 (1 − )
=
(2 + )
Ã
!
Ã
1
1 −
1+
1+
!µ
• Then compute 1 = 1 − and 2 = 2 +
2
¶
Interpretations
• This is a small open economy (SOE) model (small because is exogenous)
• Can interpret as an individual, or a collection of individuals; e.g., desired
domestic saving is
=
X
(1 2 )
• or using log utility function
=
where =
Ã
!
Ã
1
1 −
1+
1+
P
is GDP at date
!µ
2
¶
NIPA Accounting
• Recall income-expenditure identity ≡ + + +
• In this model economy, = 0 and = 0; therefore, ≡ +
1 ≡ 1 +
2 ≡ 2 −
• is the current period trade balance (net exports)
• − is the future period trade balance
Properties of the Consumption/Saving Function
• A transitory increase in earnings (∆1 0 ∆2 = 0)
— leads to 0 ∆1 ∆1 and 0 ∆ ∆1 (so ∆2 0)
• An anticipated increase in future earnings (∆1 = 0 ∆2 0)
— leads to ∆1 0 and 0 ∆ ∆2 (so ∆2 0)
• A permanent increase in earnings (∆1 = ∆2 = ∆ 0)
— combination of two previous effects (stronger consumption response;
weaker saving response)
Implications
• Consumption expenditure should exhibit less volatility than GDP
• Trade balance should react positively to transitory increase in GDP and
bad news over future GDP
• Trade balance should react negatively to transitory declince in GDP and
good news over future GDP
• There is no logical connection between trade balance and economic welfare
• Capital controls (trade restrictions) are a bad idea
A Wedding
• In this section, we assumed (1 2) exogenous
• Here, we make output endogenous; refer to our earlier theory of the determination of output and employment and view (1 2) exogenous
• So now preferences are given by (1 1 2 2) with production function
= and time-constraint + = 1
• Choice problem is now
½
max (1 1 − 1 2 1 − 2) : 1 + 2 ≤ 11 + 2 2
¾
• Solution is characterized by
1. Equating between and to (for = 1 2)
2. Equating between 2 and 1 to
3. Lifetime budget constraint
• Example: let = ln(1) + ln(1) + [ln(2) + ln(2)] ; so
2
and 21 =
=
1 −
1
• Then solve
1
1 − 1
2
1 − 2
2
1
1 + 2
= 1
= 2
=
= 11 + 2 2
• But no need to solve completely in order to deduce qualitative properties
of solution
• We know, for example, that
1 =
Ã
1
1−
!"
22
11 +
• Now combine first two conditions to derive
(1 − 1 )2
2
=
1
(1 − 2 )1
• And use third condition to derive
(1 − 1 )
2
=
1
(1 − 2 )
• So, relative labor input depends on and 21
#
• An increase in means 1 increases relative to 2
• An increase in 21 means 1 decreases relative to 2
• Note too that a permament increase in productivity (wage rate) leads to
no change in the labor input (consistent with secular observation)
• Labor supply responds to transitory changes in productivity (consistent
with seasonal variation; but perhaps business cycle variation too)
• Collectively, these propositions are known as the intertemporal substitution of labor hypothesis
