30409 - Macroeconomics

Lecture 3
(OB, Ch. 17, par 1 & 2 + GF, Ch 4, Q12)
The 𝐼𝑆 − 𝐿𝑀 with expectations
BEMACS
2024/2025
 Introduction
• AIM: the 𝐼𝑆 − 𝐿𝑀 model with expectations
• 2 periods:
– Today: 𝑡
• Notation: 𝑋𝑡 = 𝑋
– Future: 𝑡 + 1
𝑒
• Notation: 𝑋𝑡+1 = 𝑋 ′𝑒
• 𝐼𝑆 curve: equilibrium in the goods market
▪ 𝑌 =𝐶+𝐼+𝐺
 Consumption
𝐶 = 𝐶(𝑌 − 𝑇, 𝑌 ′𝑒 − 𝑇 ′𝑒 , 𝑊𝐹𝐻)
+ + +
𝑌↑ ⇒ 𝑌−𝑇 ↑
𝑇↓
𝑌 ′𝑒 − 𝑇 ′𝑒 ↑
𝑌 ′𝑒 ↑ ⇒
Π ′𝑒 ↑ ⇒ 𝐷′𝑒 ↑ ⇒ 𝑄 ↑ 𝑪↑
𝑇 ′𝑒 ↓ ⇒ 𝑌 ′𝑒 − 𝑇 ′𝑒 ↑

𝑃𝐵𝑜𝑛𝑑 ↑
𝑟 ↓, 𝑟 ′𝑒 ↓ ⇒
𝑄↑
 Investment
𝐼 = 𝐼(Π, 𝑉 Π )
+ +
𝑌↑ ⇒ Π↑

𝑌 ′𝑒 ↑ ⇒ 𝑉 Π ↑
𝑰↑
𝑟 ↓, 𝑟 ′𝑒 ↓ ⇒ 𝑉 Π ↑
𝛿↓
 The 𝐼𝑆 curve: standard
• No expectations:
𝐼𝑆: 𝑌 = 𝐶 𝑌 − 𝑇 + 𝐼 𝑌, 𝑟 + 𝑥 + 𝐺
• In a more compact way:
𝐼𝑆: 𝑌 = 𝐴 𝑌, 𝑇, 𝑟, 𝑥 + 𝐺

𝐴 𝑌, 𝑇, 𝑟, 𝑥 ≡ 𝐶 𝑌 − 𝑇 + 𝐼(𝑌, 𝑟 + 𝑥)
Aggregate private spending
 The 𝐼𝑆 curve with expectations
• With expectations:

𝑰𝑺: 𝒀 = 𝑨 𝒀, 𝑻, 𝒓, 𝒙, 𝒀′𝒆 , 𝑻′𝒆 , 𝒓′𝒆 + 𝑮

+ − − − + − −

• 𝑌 ↑, 𝑌 ′𝑒 ↑ ⇒ 𝐴 ↑
• 𝑇 ↑, 𝑇 ′𝑒 ↑ ⇒ 𝐴 ↓
• 𝑟 ↑, 𝑟 ′𝑒 ↑, 𝑥 ↑ ⇒ 𝐴 ↓
 𝐼𝑆 with expectations: slope
𝑰𝑺: 𝒀 = 𝑨 𝒀, 𝑻, 𝒓, 𝒙, 𝒀′𝒆 , 𝑻′𝒆 , 𝒓′𝒆 + 𝑮
How is the slope compared to the standard case?
If 𝑟 ↓ ⇒ 𝐴 ↑ ⇒ 𝑍 ↑ ⇒ 𝑌 ↑
But… by how much???
𝐴 depends on 𝑟, but also on 𝑟 ′𝑒 :

Smaller impact of a reduction in

current interest rate only!!! Steeper 𝐼𝑆
 𝐼𝑆 with expectations: Slope
The economy today
𝑟 If 𝑟 ↓ ⇒ 𝐴 ↑ ⇒ 𝑍 ↑ ⇒ 𝑌 ↑
𝐼𝑆𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑡𝑖𝑜𝑛𝑠
But… if we consider
expectations the increase
is smaller

𝑟0 0

𝑟1 1 1

𝐼𝑆𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑

𝑌0 𝑌1 𝑌1 𝑌
 𝐼𝑆 with expectations: Shifts
The economy today
𝑟
𝐼𝑆𝑒𝑥𝑝𝑒𝑐𝑡𝑎𝑡𝑖𝑜𝑛𝑠

𝑇 ↓; 𝐺 ↑; 𝑌 ′𝑒 , ↑ 𝑇 ′𝑒 ↓; 𝑟 ′𝑒 ↓

But… smaller multiplier!

a change in current income, given
unchanged expectations of future
income, is unlikely to have a large effect
on spending

𝑌
 LM curve and expectations?
• The CB sets the interest rate that keeps the
money market in equilibrium
• Money demand:
– Current level of transaction
– Current opportunity cost of money

𝑳𝑴: 𝒓 = 𝒓ത
 Summary
• 𝐼𝑆 curve with expectations
– Slope
– Position
• 𝐼𝑆 − 𝐿𝑀 model
– Present: 𝑡
– Future: 𝑡 + 1 (expectations)
HOW TO USE THE MODEL
 How to use the model
1. Impact of the shock/policy in 𝑡 + 1 (future)
– 𝐼𝑆𝑡+1 : 𝑌𝑡+1 = 𝐴 𝑌𝑡+1 , 𝑇𝑡+1 , 𝑟𝑡+1 + 𝐺
– 𝐿𝑀𝑡+1 : 𝑟ҧ = 𝑟𝑡+1
2. What is the impact today (𝑡) on expectations?
– Δ𝑌 ′𝑒 ; Δ𝑟 ′𝑒 ; Δ𝑇 ′𝑒
3. Impact today (𝑡) of the policy/shock,
incorporating changes in expectations
– 𝐼𝑆𝑡 : 𝑌𝑡 = 𝐴 𝑌𝑡 , 𝑇𝑡 , 𝑟𝑡 , 𝑥, 𝑌 ′𝑒 , 𝑇 ′𝑒 , 𝑟 ′𝑒 + 𝐺
– 𝐿𝑀𝑡 : 𝑟ҧ = 𝑟𝑡
 How to use the model
Present, 𝑡 Future, 𝑡 + 1
𝑟𝑡 𝑟𝑡+1

0 𝐿𝑀𝑡 0 𝐿𝑀𝑡+1
𝑟ҧ 𝑟ҧ

3 1 𝐼𝑆𝑡+1
𝐼𝑆𝑡

𝑌𝑡0 𝑌𝑡 0
𝑌𝑡+1

2
TRANSITORY MONETARY POLICY
 Transitory monetary policy
• 𝑟𝑡 ↓: expansionary monetary policy
– Announced and performed in 𝑡
– Surprise!!! Not expected
– Transitory: no change expected for the future

Impact???
 Transitory monetary policy
Future, 𝑡 + 1
• Impact in the future??? 𝑟𝑡+1

Transitory policy:
no change expected for 𝑟ҧ
0 𝐿𝑀𝑡+1

the future

Δ𝑌 ′𝑒 = 0 𝐼𝑆𝑡+1
Δ𝑟 ′𝑒 = 0 0
𝑌𝑡+1
 Transitory monetary policy
Present, 𝑡
𝑟𝑡 At time 𝑡
• 𝑟𝑡 ↓: 𝐿𝑀𝑡 ↓

0 𝐿𝑀𝑡 𝒀𝒕 ↑
𝑟ҧ

1 𝐿𝑀′𝑡
• Smaller impact with
𝑟’ҧ
expectations:
– Steeper 𝐼𝑆
𝐼𝑆𝑡

𝑌𝑡0 𝑌𝑡1 𝑌𝑡
 Transitory monetary policy
𝑒
• Assume that, before the policy, 𝑟𝑡 = 𝑟𝑡+1 =𝑟
• Draw the yield curve before the policy
𝑒
𝑟𝑡 + 𝑟𝑡+1
𝑟2𝑡 ≈ =𝑟
2
• What happens to the yield curve after the
policy is implemented?
 Transitory monetary policy
𝑟𝑛𝑡
Yield Curve With the policy:
• 𝑟𝑡 ↓
• Δ𝑟 ′𝑒 = 0
𝑟1𝑡 = 𝑟2𝑡 𝑒
𝑟𝑡 + 𝑟𝑡+1
′
𝑟2𝑡 ⇒ 𝑟2𝑡 ≈
2
′ ′
𝑟1𝑡
• 𝑟2𝑡 < 𝑟2𝑡
Δrt
• In fact, Δ𝑟2𝑡 =
2
1𝑦𝑒𝑎𝑟 2𝑦𝑒𝑎𝑟𝑠 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦

Individuals expect higher interest rates for the future

PERMANENT MONETARY POLICY
 Permanent monetary policy
• The CB announces in 𝑡 an expansionary
monetary policy
• The policy is perceived as permanent
– If permanent, expectations about the future are
affected
 Permanent monetary policy
Future, 𝑡 + 1
• Expected monetary 𝑟𝑡+1
policy in the future
𝐿𝑀𝑡+1 ↓
⇒ 𝑟𝑡+1 ↓; 𝑌𝑡+1 ↑
0 𝐿𝑀𝑡+1
𝑟ҧ

1 𝐿𝑀′𝑡+1
• Expectations changes: 𝑟’ҧ
𝑟 ′𝑒 ↓; 𝑌 ′𝑒 ↑
𝐼𝑆𝑡+1

0
𝑌𝑡+1 1
𝑌𝑡+1 𝑌𝑡+1
 Permanent monetary policy
Present, 𝑡
𝑟𝑡 Impact in t
• Expectations change:
𝑟 ′𝑒 ↓; 𝑌 ′𝑒 ↑
𝑟ҧ 0 𝐿𝑀𝑡 ⇒ 𝑪 ↑; 𝑰 ↑ ⇒ 𝑰𝑺 right
𝐿𝑀′𝑡 1
𝑟’ҧ
• Monetary policy in 𝑡:
𝐼𝑆′𝑡
𝐼𝑆𝑡
⇒ 𝑳𝑴 ↓
𝑌𝑡0 𝑌𝑡1 𝑌𝑡

𝒓 ↓ 𝒀 ↑↑
 Permanent monetary policy
𝑒
• Assume that, before the policy, 𝑟𝑡 = 𝑟𝑡+1 =𝑟
• Draw the yield curve before the policy
𝑒
𝑟𝑡 + 𝑟𝑡+1
𝑟2𝑡 ≈ =𝑟
2
• What happens to the yield curve after the
policy is implemented?
 Permanent monetary policy
𝑟𝑛𝑡
Yield Curve With the policy:
′𝑒
• 𝑟𝑡 ↓, 𝑟𝑡+1 ↓
• Δ𝑟 ′𝑒 = Δ𝑟
𝑟1𝑡 = 𝑟2𝑡 𝑒
𝑟𝑡 + 𝑟𝑡+1
⇒ 𝑟2𝑡 ≈
2
′ ′
𝑟1𝑡 = 𝑟2𝑡

Δ𝑟2𝑡 = Δ𝑟
1𝑦𝑒𝑎𝑟 2𝑦𝑒𝑎𝑟𝑠 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦

Individuals expect constant interest rates over time

 Monetary policy and stock prices
𝐷′𝑒 𝐷′𝑒
𝑄= + ′𝑒
+ …
1+𝑟 1+𝑟 1+𝑟

Transitory Permanent
• Δ𝑌 > 0; Δ𝑟 < 0 • Δ𝑌 > 0; Δ𝑟 < 0
• Δ𝑌 ′𝑒 = Δ𝑟 ′𝑒 = 0 • Δ𝑌 ′𝑒 > 0; Δ𝑟 ′𝑒 < 0

Small (if any) positive 𝑸↑

impact on 𝑄
 Summary
• Impact of a monetary policy
– Transitory or permanent?
– Impact on expectations is key to determine the
impact today of the policy
ANNOUNCEMENT OF A MONETARY
POLICY
 Announcement of a policy
• In 𝑡, the CB announces an expansionary
monetary policy
– The policy will be implemented in 𝑡 + 1
Impact???

Is the CB CREDIBLE???
CREDIBLE ANNOUNCEMENT
 Credible announcement
Future, 𝑡 + 1
• Expected monetary 𝑟𝑡+1
policy in the future
𝐿𝑀𝑡+1 ↓
⇒ 𝑟𝑡+1 ↓; 𝑌𝑡+1 ↑
0 𝐿𝑀𝑡+1
𝑟ҧ

1 𝐿𝑀′𝑡+1
• Expectations change: 𝑟’ҧ
𝑟 ′𝑒 ↓; 𝑌 ′𝑒 ↑
𝐼𝑆𝑡+1

0
𝑌𝑡+1 1
𝑌𝑡+1 𝑌𝑡+1
 Credible announcement
Present, 𝑡
𝑟𝑡 Impact in t
• Expectations change:
𝑟 ′𝑒 ↓; 𝑌 ′𝑒 ↑
𝑟ҧ 0 1 𝐿𝑀𝑡 ⇒ 𝑪 ↑; 𝑰 ↑ ⇒ 𝑰𝑺 right

𝐼𝑆′𝑡

𝐼𝑆𝑡 𝒓 =; 𝒀 ↑
𝑌𝑡0 𝑌𝑡
𝑌𝑡1
 Credible announcement
• No policies implemented today
𝒀 ↑, even if no policy has been implemented

• Does the Central Bank need to implement the

policy in 𝑡 + 1?
 Credible announcement
𝑒
• Assume that, before the policy, 𝑟𝑡 = 𝑟𝑡+1 =𝑟
• Draw the yield curve before the policy
𝑒
𝑟𝑡 + 𝑟𝑡+1
𝑟2𝑡 ≈ =𝑟
2
• What happens to the yield curve after the
announcement?
 Credible announcement
𝑟𝑛𝑡
Yield Curve With the policy:
′𝑒
• 𝑟𝑡 = ; 𝑟𝑡+1 ↓
𝑒
𝑟𝑡 + 𝑟𝑡+1
𝑟1𝑡 = 𝑟2𝑡 ⇒ 𝑟2𝑡 ≈
2
′𝑒
′
𝑟2𝑡 Δ𝑟𝑡+1
Δ𝑟2𝑡 =
2
1𝑦𝑒𝑎𝑟 2𝑦𝑒𝑎𝑟𝑠 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦

Agents expect lower future interest rates

 Non-credible announcement
• What happens if the announcement is not
credible???
Nothing!!!

Credibility is crucial in determining the impact of

economic policies!!!
 The role of expectations: example 1
• Assume that a country is experiencing high
inflation
– The target of the CB is a low and stable inflation
rate
– Credible CB:
• Agents expect contractionary policies in the future
– What if the CB is NOT credible in pursuing the
target???
 The role of expectations: example 2
• Assume that a country has a very large public
debt
– The Government promises to reduce public deficit to
reduce the debt
– Credible Government:
• Agents expect contractionary fiscal policies in the future
• Financial markets expect a reduction in public debt: 𝑥 ↓
(lower probability of default)
– What if the Government is not credible?
• Probability of default ↑ ⇒ 𝑥 ↑ !!!
The role of expectations: example 2
 Summary

Key role of expectations in determining

economic outcomes
CHAPTER 4, QUESTION 12, POINT A)
 Question 12, point a) - Text
• In country Macrolandia, there are only one-
year and two-year bonds
– 2 Periods: 𝑡 and 𝑡 + 1
• The current and the expected future inflation
rates are both zero, so that the real and the
nominal interest rates are equal
– 𝑟 = 𝑖 !!!
 Question 12, point a) - Text
• In period 𝑡 + 1 (“the future”), a standard IS-LM
model describes the functioning of the economy
• In particular, in year 𝑡 + 1 consumption depends
only on disposable income at 𝑡 + 1 , and
investment on the interest rate and output at 𝑡 +
1 only.
𝐼𝑆𝑡+1 : 𝑌𝑡+1 = 𝐶 𝑌𝑡+1 − 𝑇𝑡+1 + 𝐼 𝑌𝑡+1 , 𝑖𝑡+1 + 𝐺𝑡+1
– Standard model: the text helps us…!
– Pay attention to different assumption: e.g. exogenous
investment, …
 Question 12, point a) - Text
• Turning now to the current period (𝑡, “the present”),
consumption is increasing in both current and expected
future disposable income, 𝐶𝑡 = 𝐶 (𝑌𝑡 − 𝑇ത𝑡 , 𝑌𝑡+1
𝑒
−
𝑇ത𝑡+1
𝑒 )
, investment depends positively on current and
expected future income levels, and negatively on the
current and expected future levels of short-term interest
rates, the demand for money has the usual functional form,
and 𝐺 and 𝑇 are exogenous as usual.

– 𝐶𝑡 = 𝐶 𝑌𝑡 − 𝑇ത𝑡 , 𝑌𝑡+1 𝑒
− 𝑇ത𝑡+1
𝑒
𝑒 𝑒
– 𝐼𝑡 = 𝐼(𝑌𝑡 , 𝑌𝑡+1 , 𝑖𝑡 , 𝑖𝑡+1 )
– Again, our standard model!!!
– Pay attention to different assumptions
 Question 12, point a) - Text
• Starting from an initial equilibrium, at time 𝑡 , the
government of Macrolandia announces an increase of its
purchases of goods and services at time 𝑡 + 1, ΔGt+1 > 0
– Announcement of a future expansionary fiscal policy
– Is the announcement credible? Is it a surprise?
• At the same time, the central bank announces that it will
adjust the policy rate so as to prevent any change in
equilibrium income that, at time 𝑡 and/or at time 𝑡 +
1, could be caused by the fiscal policy just described
– CB announces policies to keep income constant, both in 𝑡 and
𝑡+1
– Credible?
 Question 12, point a) - Text
• Explain how these announcements, which are
unexpected and credible, affect the time
𝑡 yield to maturity on the two-year bonds
circulating in Macrolandia, 𝑖2𝑡
– Credible!!!
• Strategy???
1. Impact in 𝑡 + 1
2. Change in expectations
3. Impact in 𝑡
 Q12, a): Solution
Future, 𝑡 + 1
• Expected expansionary 𝑟𝑡+1
fiscal policy
𝐺𝑡+1 ↑ ⇒ 𝑍𝑡+1 ↑ ⇒ 𝑌𝑡+1 ↑
𝐿𝑀′𝑡+1
𝑰𝑺𝒕+𝟏 𝐑𝐢𝐠𝐡𝐭 𝑟’ҧ 𝑡+1
1

• CB keeps 𝑌𝑡+1 constant 0 1′ 𝐿𝑀𝑡+1

𝑟𝑡+1
ҧ
– Expected contractionary
monetary policy 𝐼𝑆′𝑡+1
𝑟𝑡+1 ↑∶ 𝐿𝑀𝑡+1 𝑼𝒑 𝐼𝑆𝑡+1
0 𝑌𝑡+1
𝑌𝑡+1

Expectations change:
𝒓′𝒆 ↑; 𝒀′𝒆 =
 Q12, a): Solution
Present, 𝑡
𝑟𝑡 Impact in t
• Expectations change:
𝑟 ′𝑒 ↑; 𝑌 ′𝑒 =
𝑟ഥ𝑡 0 𝐿𝑀𝑡 ⇒ 𝑰𝒕 ↓ ⇒ 𝒀𝒕 ↓
𝐿𝑀′𝑡
𝑰𝑺 left
1
𝑟ഥ𝑡 ’
• Monetary policy in 𝑡:
𝐼𝑆𝑡 ⇒ 𝑳𝑴 ↓
𝐼𝑆′𝑡
𝑌𝑡0 𝑌𝑡

𝒓𝒕 ↓ 𝒀𝒕 =
 Q12, a): Solution
• 𝑟 = 𝑖 by assumption
𝑒
• 𝑖1𝑡 ↓; 𝑖1𝑡+1 ↑
𝒊𝟐𝒕 ? ? ?
𝑒
𝑖1𝑡 +𝑖1𝑡+1
• 𝑖2𝑡 ≈
2
• Two contrasting effects
– Relative magnitude?
We are not able to predict the change in 𝑖2𝑡
CHAPTER 4, QUESTION 12, POINT B)
 Question 12, point b) - Text
• Assume for simplicity that, in Macrolandia, at
time 𝑡 the yield curve is initially horizontal.
• Explain how the position and the slope of this
curve change after the announcement studied
above.
• Represent both curves, the one “before” and the
one “after” the announcement.
• Clarify whether it is possible to determine the
position and the slope of the new yield curve
unambiguously
 Question 12, point b) - Text
𝑒
• 𝑖1𝑡 ↓, 𝑖1𝑡+1 ↑
• We are not able to predict what happens to
𝑖2𝑡 (point a.)
• However,
𝑒
Δ𝑖1𝑡 Δ𝑖1𝑡+1
Δ𝑖2𝑡 = +
2 2
< >
|𝚫𝒊𝟏𝒕 |
⇒ 𝚫𝒊𝟐𝒕 <
𝟐
Question 12, point b) - Text
Yield Curve
𝑖𝑛𝑡

𝑖1𝑡 = 𝑖2𝑡

′
𝑖1𝑡

1𝑦𝑒𝑎𝑟 2𝑦𝑒𝑎𝑟𝑠 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦

The yield curve becomes upward sloping

 Conclusion
• Expectations play a crucial role in determining
the economic impact of shocks/policies
• A model to incorporate changes in
expectations
• Extended vs. standard model