Intermediate
Macroeconomics
Tutorial 4
Chapter 4: Macroeconomic Dynamics II
- Capital Accumulation Decisions
- Dynamic IS-LM and the Structure of Interest Rates
- Unanticipated and Anticipated Shocks
,T4 3.4, 4.1, 4.3(a-c)
3.4 Stability of the IS-LM model
𝑅̇ = 𝜙! [𝑙(𝑌, 𝑅) − 𝑀]
𝑌̇ = 𝜙" [𝑙𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 + 𝑌]
The change in interest rates 𝑅̇ = 𝜙! [𝑙(𝑌, 𝑅) − 𝑀]
- The interest rates rises if there is excess demand 𝑙(𝑌, 𝑅) for money. An increased
demand for money means that individuals and businesses want to hold more money
relative to what is available at the current interest rate (𝑅). This excess demand for
money occurs because people prefer to hold more cash rather than invest it in bonds
or other interest-bearing assets. Consequently, bond prices decrease because there
are fewer buyers for bonds. Bond prices and interest rates have an inverse relationship
because when bond prices fall due to excess supply and lower demand, their yields
(interest rates) increase to attract buyers. So, the price of bonds falls, and the interest
rates rises.
The change in output 𝑌̇ = 𝜙" [𝑙𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 + 𝑌]
- Output rises if there is excess demand for goods and falls if there is excess supply of
goods.
Expect 𝜙! > 𝜙"
One would expect that the financial markets (for money and bonds) adjust much more quickly
than the goods market. The financial markets (for money and bonds) adjust more quickly than
the goods market because they are highly liquid, with prices and interest rates responding
almost instantly to changes in supply and demand. In contrast, the goods market involves
production, inventory adjustments, and consumption decisions, which take longer to respond
to changes in economic conditions. Additionally, financial markets operate continuously with
rapid information flow, while goods market adjustments depend on factors like production
cycles, contracts, and consumer behaviour, leading to slower responses.
1
, T4 3.4, 4.1, 4.3(a-c)
3.4 Stability of the IS-LM model
𝑅̇ = 𝜙! [𝑙(𝑌, 𝑅) − 𝑀]
𝑌̇ = 𝜙" [𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 − 𝑌]
Use a two-dimensional phase diagram in order to derive the stability properties of this model
and the qualitative nature of the transient adjustment paths for output and the interest rate
associated with fiscal expansion.
ISOCLINES
Total differential 𝑑𝑌̇ = 𝜙" 5𝐶($%&) 𝑑𝑌 + 𝐼( 𝑑𝑅 + 𝑑𝐺 − 𝑑𝑌6
𝑑𝑌̇ = 𝜙" [(𝐶$%& − 1) 𝑑𝑌 + 𝐼( 𝑑𝑅 + 𝑑𝐺 ]
The slope of IS
Isocline IS 𝑌̇ = 𝜙" [𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 − 𝑌]
𝑌̇ = 0
𝑌 = 𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺
𝑑𝑌 = 𝐶$%& 𝑑𝑌 + 𝐼( 𝑑𝑅 + 𝑑𝐺
(1 − 𝐶$%& )𝑑𝑌 = 𝐼( 𝑑𝑅 + 𝑑𝐺 for now 𝑑𝐺 = 0
(1 − 𝐶$%& )𝜕𝑌 = 𝐼( 𝜕𝑅
)$ *
)(
= !%+! <0 𝐼( < 0
"#$
)( !%+"#$
)$
= *!
<0
It might be more conventional to compute it as 𝑑𝑅/𝑑𝑌 because 𝑅 is on the vertical axis.
However, the result is the same, the IS curve is downward sloping.
2
Macroeconomics
Tutorial 4
Chapter 4: Macroeconomic Dynamics II
- Capital Accumulation Decisions
- Dynamic IS-LM and the Structure of Interest Rates
- Unanticipated and Anticipated Shocks
,T4 3.4, 4.1, 4.3(a-c)
3.4 Stability of the IS-LM model
𝑅̇ = 𝜙! [𝑙(𝑌, 𝑅) − 𝑀]
𝑌̇ = 𝜙" [𝑙𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 + 𝑌]
The change in interest rates 𝑅̇ = 𝜙! [𝑙(𝑌, 𝑅) − 𝑀]
- The interest rates rises if there is excess demand 𝑙(𝑌, 𝑅) for money. An increased
demand for money means that individuals and businesses want to hold more money
relative to what is available at the current interest rate (𝑅). This excess demand for
money occurs because people prefer to hold more cash rather than invest it in bonds
or other interest-bearing assets. Consequently, bond prices decrease because there
are fewer buyers for bonds. Bond prices and interest rates have an inverse relationship
because when bond prices fall due to excess supply and lower demand, their yields
(interest rates) increase to attract buyers. So, the price of bonds falls, and the interest
rates rises.
The change in output 𝑌̇ = 𝜙" [𝑙𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 + 𝑌]
- Output rises if there is excess demand for goods and falls if there is excess supply of
goods.
Expect 𝜙! > 𝜙"
One would expect that the financial markets (for money and bonds) adjust much more quickly
than the goods market. The financial markets (for money and bonds) adjust more quickly than
the goods market because they are highly liquid, with prices and interest rates responding
almost instantly to changes in supply and demand. In contrast, the goods market involves
production, inventory adjustments, and consumption decisions, which take longer to respond
to changes in economic conditions. Additionally, financial markets operate continuously with
rapid information flow, while goods market adjustments depend on factors like production
cycles, contracts, and consumer behaviour, leading to slower responses.
1
, T4 3.4, 4.1, 4.3(a-c)
3.4 Stability of the IS-LM model
𝑅̇ = 𝜙! [𝑙(𝑌, 𝑅) − 𝑀]
𝑌̇ = 𝜙" [𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 − 𝑌]
Use a two-dimensional phase diagram in order to derive the stability properties of this model
and the qualitative nature of the transient adjustment paths for output and the interest rate
associated with fiscal expansion.
ISOCLINES
Total differential 𝑑𝑌̇ = 𝜙" 5𝐶($%&) 𝑑𝑌 + 𝐼( 𝑑𝑅 + 𝑑𝐺 − 𝑑𝑌6
𝑑𝑌̇ = 𝜙" [(𝐶$%& − 1) 𝑑𝑌 + 𝐼( 𝑑𝑅 + 𝑑𝐺 ]
The slope of IS
Isocline IS 𝑌̇ = 𝜙" [𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺 − 𝑌]
𝑌̇ = 0
𝑌 = 𝐶(𝑌 − 𝑇) + 𝐼(𝑅) + 𝐺
𝑑𝑌 = 𝐶$%& 𝑑𝑌 + 𝐼( 𝑑𝑅 + 𝑑𝐺
(1 − 𝐶$%& )𝑑𝑌 = 𝐼( 𝑑𝑅 + 𝑑𝐺 for now 𝑑𝐺 = 0
(1 − 𝐶$%& )𝜕𝑌 = 𝐼( 𝜕𝑅
)$ *
)(
= !%+! <0 𝐼( < 0
"#$
)( !%+"#$
)$
= *!
<0
It might be more conventional to compute it as 𝑑𝑅/𝑑𝑌 because 𝑅 is on the vertical axis.
However, the result is the same, the IS curve is downward sloping.
2
