Chapter 11 - IS Curve

Chapter 11: The IS Curve
Ryan W. Herzog
Spring 2020
Ryan W. Herzog (GU) IS Curve Spring 2020 1 / 47

1 Introduction
2 Setting up the Economy
3 Deriving the IS Curve
4 Using the IS Curve
5 Microfoundations of the IS Curve

Introduction
Learning Objectives
The first building block of our short-run model: the IS curve describes
the effect of changes in the real interest rate on output in the short
run.
How shocks to consumption, investment, government purchases, or
net exports (i.e. aggregate demand shocks) can shift the IS curve.
A theory of consumption called the life-cycle/permanent-income
hypothesis.
That investment is the key channel through which changes in real
interest rates affect GDP in the short run.

Introduction
The Federal Reserve exerts a substantial influence on the level of
economic activity in the short run by setting the rate at which people
borrow and lend in financial markets
The basic story is this:
↑ interest rate ⇒↓ investment ⇒↓ output (1)

Introduction
The IS curve
The IS curve captures the relationship between interest rates and
output in the short run.
There is a negative relationship between the interest rate and
short-run output.
An increase in the interest rate will decrease investment, which will
decrease output.
IS stands for “investment equals savings”
(Y − T − C)
| {z }
private saving
+ (T − G)
| {z }
government saving
+ (IM − EX)
| {z }
foreign saving

Introduction
The IS Curve

Setting up the Economy
The national income accounting identity
Implies that the total resources available to the economy equal total
uses
One equation with six unknowns
Yt = Ct + It + Gt + EXt − IMt (2)

Key Equations
We need five additional equations to solve the model:
Ct = acY t (3)
Gt = ag Y t (4)
EXt = aex Y t (5)
IMt = aimY t (6)
It
Y t
= ai − b(Rt − r) (7)
Level of potential output is given exogenously.
Consumption C, government purchases G, exports EX, and imports
IM depend on the economy’s potential output.
Each of these components of GDP is a constant fraction of potential
output.

Potential Output
Potential output is smoother than actual GDP.
A shock to actual GDP will leave potential output unchanged
The equation depends on potential output.
Shocks to income are ÒsmoothedÓ to keep consumption steady.

Investment Equation
It
Y t
= ai − b(Rt − r)
where, ai is the share of potential output that goes to investment.
b is a term weighting the difference between the real interest rate and
the MPK.
Rt is the real interest rate
r is the marginal product of capital (K)

Marginal Product of Capital
The MPK is an exogenous parameter and is time invariant
If the MPK is low relative to the real interest rate firms should save
money and not invest in capital
If the MPK is high relative to the real interest rate firms should
borrow and invest in capital
In the short run, the MPK and the real interest rate can be different.
Installing capital to equate the two takes time

The IS Curve Model
The model has six endogenous variables (Yt, Ct, It, Gt, EXt, IMt)
The model has six equations:
National income identity: Yt = Ct + It + Gt + EXt − IMt
Consumption: Ct = ac Y t
Government: Gt = ag Y t
Exports: EXt = aex Y t
Imports: IMt = aimY t
Investment: It
Y t
= ai − b(Rt − r)
Exogenous parameters: Yt, ac, ai , ag , ag , aex , aim, b, r, Rt

Deriving the IS Curve
Divide the national income accounting identity by potential output.
Yt
Y t
=
Ct
Y t
+
It
Y t
+
Gt
Y t
+
EXt
Y t
−
IMt
Y t
(8)
Substitute the five equations into this equation.
Yt
Y t
= ac + ai − b(Rt − r) + ag + aex − aim (9)

Recall the definition of short-run output. Simplifies the equation for
the IS curve:
Ỹt =
Yt − Y t
Y t
(10)
Subtract one from both sides:
Yt
Y t
− 1
| {z }
Ỹt
= ac + ai + ag + aex − aim − 1
| {z }
a
−b(Rt − r) (11)
Rewriting
Ỹt = a − b(Rt − r) (12)
where ac + ai + ag + aex − aim − 1 = a

The gap between the real interest rate and the MPK is what matters
for output fluctuations.
Firms can always earn the MPK on new investments.
The parameter a:
Is a = ac + ai + ag + aex − aim − 1
Is called the aggregate demand shock
Will equal zero when potential output is equal to actual output

Using the IS Curve
The Basic IS Curve
When the demand shock parameter equals zero, the IS curve has a
short-run output of 0 where the real interest rate is equal to the long-run
value of the MPK.

Using the IS Curve
Basis IS Curve

Using the IS Curve
Interest Rate Change
When the real interest rate changes, the economy will move along the
IS curve.
An increase in the interest rate
causes the economy to move up the IS curve
Causes short-run output to decline
The higher interest rate
raises borrowing costs
reduces demand for investment
reduces output below potential

Using the IS Curve
An Increase in the Interest Rate

Using the IS Curve
Sensitivity Parameter
If the sensitivity to the interest rate were higher
The IS curve would be flatter
Any change in the interest rate would be associated with larger
changes in output

Using the IS Curve
Example
IS Curve:
Ỹt = a − b(Rt − r) (13)
Suppose a = 0, b = 2, R = r = 5%, and R1 = 6%
Initially the economy has no output gap Ỹt = 0 − 2(0) = 0.
If interest rates increase to 6% then: Ỹt = 0 − 2(6 − 5) = −2%

Using the IS Curve
An Aggregate Demand Shock
Suppose that information technology improvements create an
investment boom.
The aggregate demand shock parameter will increase.
Output is higher at every interest rate and the IS curve shifts right.
For any given real interest rate Rt, output is higher when a is higher.

Using the IS Curve
Shift in the IS Curve

Using the IS Curve
Movement or Shift?
A change in R shows up as a movement along the IS curve. The IS
curve is a graph of R versus short-run output.
Any other change in the parameters of the short-run model causes the
IS curve to shift.

Using the IS Curve
Shock to Potential Output
Shocks to potential output change actual output by the same amount
in our setup
Do not change short-run output
Some shocks to potential output may change other parameters.
Earthquake, for example:
Reduces actual and potential output by the same amount
Leads to an increase in short-run output because it also increases the
MPK

Microfoundations of the IS Curve
Microfoundations
The underlying microeconomic behavior that establishes the demands for
C, I, G, EX, and IM.

Consumption
People prefer a smooth path for consumption compared to a path
that involves large movements.
The permanent-income hypothesis: People will base their
consumption on an average of their income over time rather than on
their current income.
The life-cycle model of consumption: Suggests that consumption is
based on average lifetime income rather than on income at any given
age.

The Life-Cycle Model of Consumption
Young people borrow to consume more than their income.
As income rises over a personÕs life consumption rises more slowly
and individuals save more
During retirement, individuals live off their accumulated savings.

Life Cycle / Permanent Income Hypothesis
Implies that people smooth their consumption relative to their income
This is why we set consumption proportional to potential output
rather than actual output.

Life Cycle Model of Consumption

Multiplier Effects
We can modify the consumption equation to include a term that is
proportional to short-run output.
Ct
Y t
= ac + xỸt (14)
Solving for the IS curve:
Ỹt =
1
1 − x
| {z }
multiplier
× (a − b(Rt − r))
| {z }
original IS curve
(15)
Now includes a multiplier on the aggregate demand shock and
interest rate terms:

With a multiplier:
Aggregate demand shocks will increase short-run output by more than
one-for-one.
A shock will ÒmultiplyÓ through the economy and will result in a
larger effect.
If short-run output falls with a multiplier
Consumption falls
Which leads to short-run output falling
Consumption falls again
“Virtuous circle” or “vicious circle”

Investment
At the firm level, investment is determined by the gap between the
real interest rate and MPK.
In a simple model the return on capital is the MPK minus
depreciation.
The richer framework includes:
Corporate income taxes
Investment tax credits
Depreciation allowances

Investment cont.
A second determinant of investment is the firmÕs cash flow or the
amount of internal resources the company has on hand after paying
its expenses
Agency problems occur when one party in a transaction has more
information than the other party
It is more expensive to borrow to finance investment because of this.

Asymmetric Information
Adverse selection
If a firm knows it is particularly vulnerable
It will want to borrow because if the firm does well it can pay back the
loans.
If it fails, the firm cannot pay back the loan but will instead declare
bankruptcy.
Moral hazard
A firm that borrows a large sum of money may undertake riskier
investments
if it does well, it can repay.
if it fails, it can declare bankruptcy.

Investment Equation
The potential output term in the investment equation incorporates
cash flows.
It = ai Y t − b(Rt − r)Y t (16)
Captures cash flow.
If we wish to add short-run output, it would provide additional
justification for a multiplier

Government Purchases
Government purchases can be
A source of short-run fluctuation
An instrument to reduce fluctuations
Discretionary fiscal policy
Includes purchases of additional goods in addition to the use of tax
rates
For example, the government can use the investment tax credit to
encourage investment

Transfer spending often increases when an economy enters into a
recession.
Automatic stabilizers
Programs where additional spending occurs automatically to help
stabilize the economy
Welfare programs and Medicaid are two such stabilizer programs. They
receive additional funding when the economy weakens

Fiscal policyÕs impact depends on two things:
1 The problem of timing
discretionary changes are often put into place with significant delay.
2 The no-free-lunch principle
implies that higher spending today must be paid for today or some
point in the future.
such taxes may offset the impact of the discretionary spending
adjustment.

Consumption
What matters for consumption today?
The permanent-income hypothesis says what matters is the present
discounted value of your lifetime income, after taxes.
Ricardian equivalence says what matters is the present value of what
the government takes from the consumers rather than the specific
timing of the taxes.

Government Policy
An increase in government purchases financed by taxes today
Will have a modest positive impact on the IS curve
Will raise output by a small amount in the short run
An increase in spending today financed by taxes in the future
Will shift the IS curve out by a moderate amount
Perhaps by 75 cents to $1 for each dollar

Case Study: ARRA
Economists had a wide range of opinions about the effectiveness and
costs of the stimulus.
Congressional Budget Office (CBO) gave estimates of unemployment
with and without a stimulus.
Estimated 9 percent peak without a stimulus
Actual unemployment rate with stimulus was above this.

Great Recession

Net Exports
If the trade balance is a deficit (surplus) the economy imports more
(less) than it exports
If Americans demand more imports the IS curve shifts left and
reduces short-run output
If foreigners demand more American exports the IS curve shifts right

A Reduction in Interest Rates

Decline in Consumer Confidence

Improvements in Info Technology

Chapter 11 - IS Curve

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Chapter 11 - IS Curve