2. Three Key Aggregate Markets
2.1 The Labor Market: Productivity, Output and Employment
2.2 The Goods Market: Consumption, Saving and Investment
2.3 The Asset Market: Money and Inflation
2.2 The Goods Market:
Consumption, Saving and Investment
Last chapter: determination of the real wage in labor market
equilibrium
This chapter: determination of the real interest rate in
goods market equilibrium
Consumption and Saving
Investment
Recall the income-expenditure identity:
Y = C + I + G + NX
The total aggregate demand for domestic goods consists of the
sum of
demand for goods for consumption C
demand for goods for investment I
net demand for domestic goods NX
demand for goods for government uses G
The Trade Balance NX
Assumption 1: The economy is closed
No international trade, the trade balance is zero, i.e. NX = 0
No net factor payments, i.e. NFP = 0
Therefore the current account CA = NX + NFP = 0
We will consider the open economy later.
Government Spending G
Assumption 2a: Demand for goods on behalf of the government
is given, G = Ḡ
Assumption 2b: Taxes are given, T = T̄
Government spending G and taxation T are determined by
the political process
We treat them as exogenous variables.
Assume no transfers, TR = 0
Note the government budget deficit: D govt = Ḡ + INT − T̄ .
Consumption and Saving
Under assumptions 1, 2a and 2b:
Private disposable income Y d is given by
Y d = Y + INT − T̄
National saving S is given by
S = Y − C − Ḡ = I
Households allocate Y d for saving or consumption:
Yd
= Y + INT − T̄
Y
d
= S + C + Ḡ + INT − T̄
Y
d
= S priv + C
Consumption and Saving
Y d = S priv + C
Understanding consumption requires understanding the
savings decision.
Trade-off between current consumption and future
consumption
Assumption 3: Trade-off occurs through utility maximization
Utility Maximization
Household chooses Ct and Ct+1 to maximize utility
U(ct , ct+1 ) = u(ct ) + βu(ct+1 )
where
ct is real current consumption
ct+1 is real future consumption
0 < β < 1 is a discount factor that captures impatience
subject to
period t budget constraint Pt ct + St = Ytd
d + (1 + i )S
period t + 1 budget constraint Pt+1 ct+1 = Yt+1
t t
where it is the nominal interest rate.
Equivalent to choosing St to maximize
d
d
Yt+1 + (1 + it )St
Yt − St
u
+ βu
Pt
Pt+1
Optimality requires
−
uc (ct+1 )
uc (ct )
+β
(1 + it ) = 0
Pt
Pt+1
where uc is the first derivative and denotes marginal utility.
Saving one additional dollar more in t, means consuming 1/Pt less
in t which lowers utility by uc (ct )/Pt
Saving one additional dollar more in t, means consuming
(1 + it )/Pt+1 more in t + 1 which increases utility by
t+1 )
β ucP(ct+1
(1 + it ).
marginal cost of saving = marginal benefit of saving
A Specific Example
Suppose that
1
u(c) =
c 1− σ
,σ > 0
1 − σ1
Then
−1
ct σ
Pt
⇔ ct
⇔ ct
⇔ ct
−1
c σ
= β t+1 (1 + it )
Pt+1
−σ
Pt
(1 + it )
= ct+1 β
Pt+1
−σ
1 + it
= ct+1 β
1 + πt+1
= ct+1 (β(1 + rt ))
rt is the real interest rate
1+it
Note that 1 + rt = 1+π
≈ 1 + it − πt+1
t+1
−σ
Consider again the budget constraints
Pt ct + St
= Ytd
d
Pt+1 ct+1 = Yt+1
+ (1 + it )St
We can eliminate St and write
Pt+1 ct+1
1 + it
ct+1
⇔ ct +
1 + rt
Pt ct +
d
Yt+1
1 + it
yd
= ytd + t+1
1 + rt
= Ytd +
d
d /P
where ytd = Ytd /Pt and yt+1
= Yt+1
t+1 denote real disposable
incomes.
Final step: use ct = ct+1 (β(1 + rt ))−σ ⇔ ct+1 = ct (β(1 + rt ))σ
and plug into
ct
⇔ ct
⇔ ct
d
yt+1
ct+1
−
1 + rt
1 + rt
d
y
ct (β(1 + rt ))σ
= ytd + t+1 −
1 + rt
1 + rt
"
#
d
−1
y
= ytd + t+1
1 + β σ (1 + rt )σ−1
1 + rt
= ytd +
This last expression allows us to evaluate the determinants of
consumption
ct
=
−1
yd
ytd + t+1
1 + β σ (1 + rt )σ−1
1 + rt
−1
of the net
Result 1 Consumption is a fraction 1 + β σ (1 + rt )σ−1
h
i
d
yt+1
d
present value (NPV) of lifetime wealth yt + 1+rt
There is consumption smoothing.
Result 2 Current consumption ct increases with current real disposable
income ytd
Result 3 Current consumption ct increases with future real disposable income
d
yt+1
ct
=
−1
yd
ytd + t+1
1 + β σ (1 + rt )σ−1
1 + rt
Result 4 The result of an increase in the real interest rate rt on current
consumption is ambiguous
Substitution effect −:
rt ↑ makes saving more attractive and ct ↓. Depends on σ, the
elasticity of intertemporal substitution.
Income effect +:
Keeping fixed the NPV of lifetime wealth rt ↑ lowers the price
of ct+1 and expands the feasible consumption set, leading to
ct ↑
Wealth effect −:
rt ↑ decreases the NPV of lifetime wealth and ct ↓
Note, in reality πt+1 is unknown in period t and rt is the real
expected interest rate
"
yd
ytd + t+1
1 + rt
#
ct
=
ct
= cta + MPCt ytd
1 + β σ (1 + rt )σ−1
−1
The marginal propensity to consume (MPC ): If current real
disposable income ytd increases, how much does current
consumption ct increase?
MPCt =
1+
1
+ rt )σ−1 )
(β σ (1
0 < MPC < 1
if σ > 1, substitution effect dominates income effect and
−
MPCt (rt )
if σ < 1, income effect dominates substitution effect and
+
MPCt (rt )
Question: Do tax rebates stimulate consumer spending?
A tax rebate of $800 raises ytd by $800/Pt
The rebate must be financed today by increased government
borrowing at an interest rate 1 + it .
But must eventually be paid by higher taxes in the future: so
d
yt+1
decreases by (1 + it )$800/Pt+1 = (1 + rt )$800/Pt
The NPV of lifetime wealth is unchanged:
"
#
d − (1 + r )$800/P
yt+1
yd
t
t
d
yt + $800/Pt +
= ytd + t+1
1 + rt
1 + rt
Ricardian Equivalence: Tax rebates have no effect on
consumption!
Theoretical prediction: most consumers will save their tax
rebates and not spend them (as in rebate of 2001).
Summarizing what we have so far:
Y =P
−
c a (rt )
Y = C + I + Ḡ
+ MPC × y d + I + Ḡ
−
Y = C a (rt ) + MPC × Y d + I + Ḡ
Ḡ exogenous
C depends positively on current disposable income Y d
through MPC
We will assume from now on that σ ≈ 1 and therefore that
MPC is constant and C depends negatively on rt .
We still need to determine what drives investment I .
Investment
Why is investment important?
Investment fluctuates sharply over the business cycle, so we
need to understand investment to understand the business
cycle
Investment plays a crucial role in economic growth:
Kt+1 = (1 − δ)Kt + xt
where xt is real investment and Kt is the real capital stock
and remember
∆yt
∆At
∆Kt
∆Nt
=
+α
+ (1 − α)
yt
At
Kt
Nt
In the previous chapter we derived the demand for capital
K
d
= (Aα)
1
1−α
− 1
1−α
R
N
P
Demand for capital is such that MPK =
R
P
R
P
is real the rental price of capital or the user cost of capital
Demand for capital depends negatively on the user cost of
capital
R
=r +δ
P
foregone real interest rt
depreciation δ
For someone that saves by buying e.g. a government bond:
−
uc (ct )
uc (ct+1 )
+β
(1 + it ) = 0
Pt
Pt+1
For someone that saves by a buying a unit of capital in t and rents it out
to firms in t + 1:
uc (ct+1 ) Rt+1 + Pt+1 (1 − δ)
uc (ct )
+β
=0
−
Pt
Pt+1
Pt
Therefore
(1 + it )
=
Pt
(1 + it )
Pt+1
=
1 + rt
=
⇔
Rt+1
Pt+1
Rt+1 + Pt+1 (1 − δ)
Pt
Rt+1
+ (1 − δ)
Pt+1
Rt+1
+ (1 − δ)
Pt+1
= rt + δ
Rt+1
= rt + δ
Pt+1
Rt+1
Pt+1
If today’s real interest rate is high, tomorrow’s user cost
is high, the desired capital stock for tomorrow is low.
⇒ today’s investment is low
If today’s real interest rate is low, tomorrow’s user cost
low, the desired capital stock for tomorrow is high.
⇒ today’s investment is high
Rt+1
Pt+1
Again note, in reality the future is unknown and rt is the
expected real interest rate.
Bottomline: Demand for investment depends negatively on
the (expected) real interest rate!
is
1
1
− 1−α
d
Kt+1
=
(At+1 α) 1−α (rt + δ)
d
Kt+1
=
(1 − δ)Kt + xt
Nt+1
Result 1 Current investment xt = PItt depends negatively on rt since it
increases the user cost of capital which lowers the desired capital
d
stock Kt+1
.
Result 2 Current investment xt = PItt depends positively on total factor
productivity tomorrow At+1 because it raises the desired capital
d
stock Kt+1
.
Result 3 Current investment xt = PItt depends positively on labor input
d
tomorrow Nt+1 because it raises the desired capital stock Kt+1
.
Summarizing what we have so far:
Y =P
−
c a (rt )
Y = C + I + Ḡ
+ MPC × y d + I (rt ) + Ḡ
−
−
Y = C a (rt ) + MPC × Y d + I (rt ) + Ḡ
Ḡ exogenous
C depends positively on current disposable income Y d
through MPC , and negatively on rt
I depends negatively on rt .
Goods market equilibrium
Recall that in a closed economy:
S = I = S priv − D govt
We took D govt as exogenous
S priv = Y d − C private savings increases with r , therefore
national saving S increases with r
I decreases with the real interest rate
Goods market equilibrium occurs at the real interest rate r
for which savings equal investment.
Saving-Investment Diagram
Shifts in the Saving Curve
e.g. higher current or future disposable income
Shifts in the Investment Curve
e.g. higher future productivity, employment
The Real Interest Rate