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Saving and Investment
Classical Model
Learning Objectives
• Learn how to derive and shift the investment
demand curve
• Learn how to derive and shift the saving supply
curve.
• Understand the role of interest rates in the
classical model.
• Understand the impact of technology on
investment demand.
• Understand the impact of “animal spirits” on
investment demand.
• Understand the impact of social changes on the
supply of saving.
Business Cycles and Investment:
Classical Model
• Classical economists argue that business
fluctuations are caused by a series of shocks to
technology that alter the productivity of labor in a
random way from one year to the next.
• These shocks are transmitted to the capital market
through changes in investment, causing saving,
investment and the interest rate to fluctuate
during the business cycle in an apparently random
way.
• Business cycles are a necessary and unavoidable
feature of market economies.
Investment Volatility: Classical
Model
• Investment spending is volatile.
– Investment fluctuates much more than GDP over
time.
• In classical theory, output and employment are
determined by fundamentals such as technology,
which can also influence investment.
– For example, new technology can cause an
investment boom in new machines that are designed
to exploit the invention.
Investment Volatility: Keynesian
Model
• In Keynesian theory, changes in investment
represent changes in the mass psychology of
investors or “animal sprits.”
– Greenspan’s use of “irrational exuberance” is a
recent example of this Keynesian idea.
• According to Keynesians, business cycles can be
avoided if investment is more efficiently
coordinated.
• Keynesians favor government policies to stabilize
the business cycle.
Consumption
• Unlike investment, consumption fluctuates
less than GDP over time.
• Consumption is smooth because households
borrow and lend in the capital market in an
effort to redistribute their income more evenly
over time.
• Keynesian economists agree that consumption
is smooth, but they argue that it could be even
smoother.
“Smooth” Consumption
• Keynesian economists argue that consumption is
not as smooth as it could be because households
with low incomes do not have easy access to capital
markets.
• If people who prefer to consume more than their
income are credit constrained, their optimal
decision is to consume all their income.
• Therefore, the presence of credit constrained
individuals implies that aggregate consumption will
fluctuate more than otherwise since the credit
constrained individuals’ consumption will fluctuate
equally with GDP.
Theory of Investment
• Assumptions:
– Individuals consume what they produce.
– As time passes, individuals allocate their
produced commodities between consumption
goods and investment goods.
Intertemporal Production
Possibilities Frontier
Future
Income
D
E
The maximum attainable resources
available for consumption and
investment = 0A.
If an individual invests 0B and
consumes BA, he will have 0E
available to divide between
consumption and investment in
the future.
0
Investment
B
Consumption
Current
A Income
The Production Function
• The intertemporal production function slopes
up because the more an individual invests
today, the greater his income in the future.
• The intertemporal production function’s slope
rises at a decreasing rate because of the law of
diminishing returns.
– For every increase in capital, output increases by
smaller amounts.
Investment and Profit Maximization
• Firms invest up to the point where the
output produced by an extra unit of
investment is equal to its cost.
– Firms equate the marginal product of
capital investment with the real rate of
interest.
• At this point, they are investing in the
profit maximizing amount of capital.
Profit: Definition
• A firm’s profit is the value of its produced output
minus the accrued principal and interest on loans
needed to purchase current investment goods.
• Profit = Value of Future Sales – Cost of Borrowing
p =
Y
–
(1+r)I
– Y = value of output tomorrow
– r = market rate of interest
– I = investment of resources today
Investment and Production Function
Yt+1
A
Panel A shows the intertemporal production
function: the amount of output that can be
produced in the future (Yt+1) for any given
investment today (I).
I
0
(1+r)
Panel B shows the relationship between
the real interest factor (1 + r) and the quantity
of capital investment demanded (I).
B
I
0
I
Investment and Production Function
Yt+1
A
(slope =(1+r))
Panel A: As the firm buys additional capital,
the extra output produced by the last unit of
capital decreases.
Y
0
(1+r)
I
I1
The firm invests up to the point where (1+r)
just equals the marginal product of capital
investment. At this point, profits are maximized.
B
(1+r)1
I
0
I1
The additional output is the marginal product
of capital investment = slope of the production
function.
I
Panel B shows the investment schedule. At
higher rates of interest, investment is less and
at lower rates of interest, investment is greater.
Deriving the Investment Schedule:
Math
• Max p
AI – (½)(I)2
(1 + r)I
A
=
AI – (½)(I)2 – (1 + r)I
=
=
=
Total Product in t+1
Total Borrowing Cost
Technology induced shifts in
investment.
Find the derivative of profit with respect to I, set it equal
to zero, and solve for the marginal product of investment.
• dp/dI = A – I – (1 + r) = 0
•
A–I
=
(1 + r)
Households and Saving
• Intertemporal utility theory forms the basis
for most modern explanations of how
income is divided between consumption and
saving.
• This theory argues that, given the choice,
families would prefer that consumption be
evenly distributed over time.
Intertemporal Budget Constraint
• Assumptions:
– Households consume part of its income and
saves part.
– They put their savings into the capital market
by lending to another household or firm and
receive future resources with interest.
• The amount of additional goods that households can
buy in the future grows with the rate of interest.
Present Value
• The capital market can be used to transfer
resources from the present to the future.
• It can also be used to transfer resources
from the future to the present.
– When an individual borrows against future
income, he/she borrows its present value.
Present Value
• Present value tells us how much an
expected future payment is worth today.
– For example, if we expect to inherit $10,000
next year, but wish to spend it today, we can
borrow some amount less than $10,000 today.
– The amount that we can borrow is determined
by the interest rate.
Present Value Formula
• The formula for present value can be found
by rearranging the compounding formula.
FV = PV(1 + i)
Compounding
• Solve for PV
FV/(1 + i) = PV
Present Value
Intertemporal Budget Constraint
• Households can use capital markets to
redistribute resources over time.
• The intertemporal budget constraint places a
bound on the amount of consumption that is
available over a household’s lifetime.
• C1 + C2/(1 + r) <= Y1 + Y2/(1 + r)
Intertemporal Budget Constraint
C1 + C2/(1 + r) <= Y1 + Y2/(1 + r)
C1
C2/(1 + r)
Y1
Y2/(1 + r)
= Present consumption
= Present value of future
consumption
= Current resources
= Present value of future
resources.
Intertemporal Budget Constraint
• C1 + C2/(1 + r) <= Y1 + Y2/(1 + r)
– The left hand side of the constraint represents
the value of the goods that a person consumes
at every point in life, valued in terms of current
consumption goods.
– The right hand side of the constraint represents
the value of the income that a person earns at
every point in life, valued in terms of current
consumption goods.
Deriving Supply of Saving Curve:
Math
•
Household Utility:
Utility
U
•
= Utility of C1 + Utility of C2
= – (2 – C1)2/2 + C2
Two-Part Household Budget Constraint:
1) Future Consumption = Interest + Principal
C2
= (1 + r)S
2) Saving = Income – Current Consumption
S
=
Y – C1
Deriving Saving Supply: Math
• Incorporate the budget constraints into the
utility function.
S = Y – C1
C2 = (1+r)S
implies
– C1 = – Y + S
• Household Utility
Utility = Utility of C1
+
U
= – (2 – Y + S)2/2 +
Utility of C2
(1 + r)S
Deriving Saving Supply: Math
• Maximize Household Utility
• Max U
= – (2 – Y + S)2/2 +
(1 + r)S
• Present utility given S = – (2 – Y + S)2/2
• Future utility given S = (1 + r)S
• Find the derivative of U with respect to S, set it
equal to zero, and solve for S.
Deriving Saving Supply: Math
• Max U = – (2 – Y + S)2/2 + (1 + r)S
• du/dS = – (2 – Y + S) + (1 + r)
Utility lost by saving one more unit
= – (2 – Y + S)
Utility gained by saving one more unit = (1 +r)
• S = Y–1+r
The Saving Supply Curve
r
S
As interest rates rise, individuals
substitute future consumption for
present consumption.
0
S
Saving and Interest Rates
• Substitution Effect
– When the real rate of interest increases, current
consumption becomes relatively more expensive,
causing households to want to substitute consumption
today for consumption tomorrow. Saving increases.
• Wealth Effect
– When the real rate of interest increases, households
may save more or less. Higher rates of interest cause
wealth grow faster, but higher interest rates also reduce
the present value of accumulated wealth.
Capital Market Equilibrium
The upward sloping line represents
saving.
r
S
The downward sloping line represents
investment.
At the intersection of S and I, the
model is in equilibrium
I
0
S,I
Equilibrium: Math
• Previous Results:
– Investment Demand = A – I = (1 + r)
– Saving
= Y–1+ r
Set I = S and solve for equilibrium r.
I = A – (1 +r) = A – 1 – r
S=Y–1+r
A– 1–r = Y–1+r
A – 1 – Y + 1 = 2r
r = (A –Y /2)
Equilibrium: Math
• Previous Results:
r = (A –Y /2)
I=A–1–r
Substitute equilibrium r in I or S to solve for equilibrium
I or S.
I = A – 1 – (A – Y/2)
I = A – 1 – A/2 + Y/2
I = 2A/A – A/2 + Y/2 – 1
I = A/2 + Y/2 – 1
I = (A + Y)/2 - 1
Capital Market Equilibrium
r
S
r3
C
D
r2
r1
A
B
At r1, investors want to borrow more funds
than individuals are willing to save. The
excess demand pushes interest rates up.
At r3, investors want to borrow fewer
funds than savers are willing to save. The
excess supply demand pushes interest
rates down.
At r2, investors want to borrow the same
amount that individuals are willing to save.
I
0
S,I
Say’s Law and the Classical Theory
of Interest
• Output in the classical model is determined
via the production function and depends on
what happens in the labor market.
– Involuntary unemployment is impossible.
• What if there is a glut of goods that cannot
be purchased by a fully employed labor
force?
Say’s Law and the Classical Theory
of Interest
• Say’s Law asserts that supply creates its
own demand or every act of production
constitutes a demand for something.
– There is no problem if we are in a barter
economy, but once money is introduced a
loophole emerges if people decide to save some
of their income rather than spending it all.
Saving and Investment
Income
Firms
Investment
Households
Expenditures
Borrowing Financial Markets
Saving
Say’s Law and the Classical Theory
of Interest
• By making saving and investment a
function of the interest rate, the classical
theory of saving and investment preserves
the integrity of Say’s Law.
– In classical theory, the rate of interest unites
decisions to save by households with decisions
to invest by business firms.
Say’s Law and the Classical Theory
of Interest
• If individuals and households try to save
more out of current income than firms want
to spend on investment at the prevailing rate
of interest, interest rates fall.
– As rates fall, savings fall, consumption rises,
and investment rises.
Say’s Law and the Classical Theory
of Interest
• If individuals and households try to save
less out of current income than firms want
to spend on investment at the prevailing rate
of interest, interest rates rise.
– As rates rise, savings rise, consumption falls,
and investment falls.
Business Cycles and Investment
• Classical Theory
– Changes in fundamentals cause business cycles.
• For example, a new invention makes investment
more productive, leading to an increase in capital
investment.
• Keynesian Theory
– Irrational swings of optimism and pessimism
are more important driving forces in the stock
market than fundamentals.
Productivity and Investment
Y
A
Y2
3
2
(1+r1)
Y2
An increase in productivity shifts
investment demand to the right. Why?
Y1
Y1
1
(1+r1)
0
r
I1(r1) I2(r1)
We begin at point 1. The interest rate is r1,
and the marginal product of investment
equals the slope of a tangent line at
Point 1. Investment equals I1(r1).
I An increase in productivity shifts up the
S
production function from Y1 to Y2.
B
r2
r1
0
If interest rates remain at r1, investment
increases to I2(r1). We move to point 2.
1
2
I1 I2
I1(r1) I2(r1)
At point 2, I>S, causing r to rise to r2.
I Equilibrium occurs at Point 3.
Saving and Investment: Open
Economy
• In an open economy, domestic saving does
not have to equal domestic investment.
– It can be less than domestic investment or more
than domestic investment.
• When I < S, a country may lend abroad.
• When I > S, a country must borrow from abroad.
Saving and Investment: Open
Economy
• Y = CNAT + INAT + NX
• Y – CNAT – INAT = NX
• SNAT – INAT = NX
– If SNAT = INAT, NX =O, trade balance
– If SNAT > INAT, NX >0, trade surplus
– If SNAT < INAT, NX <0, trade deficit
Looking at X - M
• X – M then is income minus consumption
vis a vis the rest of the world.
– If X > M, a country has excess funds to lend to
the ROW, or S > I.
– If X < M, the country’s trading partner has
excess funds to lend to it or domestically S < I.
Saving and Investment: Open
Economy
• Definitions:
– USA demand for capital from the rest of the world
equals the difference between national domestic
investment and national domestic saving for different
values of the world interest rate.
– The world supply of capital equals the amount that
other countries are willing to lend in world capital
markets at different values of the world interest rate.
• The world supply of capital curve slopes up because countries
are willing to lend more as the interest rate rises.
USA Demand for Capital from the
Rest of the World Curve
r
A
r1
0
NB1>0
S1
rw
At r1 in Panel A, national domestic
investment is greater than national
domestic saving.
S
I
S,I
I1
B
The difference in domestic investment
over domestic saving is paid for by
borrowing NB1 in the world capital
market.
SKROW
At r1 in Panel B, net borrowing from the
ROW equals NB1 = I1 – S1.
r1
0
A
DK
NB1
USA
K
Point A represents one point on the USA
demand for ROW capital curve.
r
USA Demand for Capital from the
Rest
of
the
World
Curve
A
NB2<0
r2
r1
0
B
NB1>0
S1 I2
r2
At r2 in Panel A, national domestic
investment is less than national
domestic saving.
S
I
S,I
S2 I1
B
The difference in domestic saving over
domestic investment results in negative
borrowing = NB2 in the world capital
market.
SKROW
At r2 in Panel B, negative net borrowing
from the ROW equals NB2 = I2 – S2.
r1
NB2
0
A
DK
NB1
USA
K
Point B represents another point on the
USA demand for ROW capital curve.
Equilibrium in the World Capital
Market
A
r
S
NB2<0
r2
Equilibrium occurs at the point where USA
demand for capital from the world just
equals the world supply of capital to the USA.
NBE>0
r1
0
B
NB1>0
S1 I2
r2
req
S2 I1
B
0
Point C in Panel B represents equilibrium
S,I in the world capital markets.
SKROW
At this point, USA investment exceeds
domestic savings. The difference is made up
by borrowing NBE from abroad
C
A
r1
NB2
I
DKUSA
NBE
NB1
K