Intermediate
Macroeconomics
Tutorial 5
Chapter 5: Macroeconomic Dynamics II
- Capital Accumulation Decisions
- Dynamic IS-LM and the Structure of Interest Rates
- Unanticipated and Anticipated Shocks
,T4 en T5 4.3
4.3 Closed economy with s3cky prices
𝑦 = −𝜎𝑅! + 𝑔
𝑚 − 𝑝 = −𝜆𝑅" + 𝛾𝑦 + 𝛼
𝑝̇ = 𝜙(𝑦 − 𝑦0)
𝑅̇!
𝑅! − = 𝑅"
𝑅!
The last equa+on is the arbitrage equa+on between short bonds and consols (coupon bond).
If we assume that two types of financial instruments are perfect subs+tutes, their respec+ve
rates of return must equalize. For short-term bonds there is no capital gain or loss. The return
on long-term bonds should be the same as the return on a short-term bond.
The yield of a bond is a measure of the return an investor can expect to earn by holding the
bond un+l maturity, expressed as a percentage of the bond's current price.
When we talk about a capital loss in the context of long-term bonds, we are referring to a
decrease in the bond's price. Bond prices and yields are inversely related: when yields rise,
bond prices fall, and vice versa. If 𝑅̇! > 0, it means that the yield on long-term bonds is
expected to increase in the future. This increase in yield will lead to a decrease in the bond's
price, resul+ng in a capital loss for investors who hold the bond.
Suppose the long-term bonds are expected to experience a capital loss (i.e., 𝑅̇! > 0, meaning
yields are rising). To offset this loss, the current yield 𝑅! must be higher than the short-term
rate 𝑅" . This higher yield compensates investors for the an+cipated decrease in the bond's
price.
For investors to be indifferent between holding long-term bonds and short-term bonds, the
total return on long-term bonds must equal the return on short-term bonds. If 𝑅! were not
higher than 𝑅" when 𝑅̇! > 0, investors would prefer short-term bonds, which do not carry the
risk of capital loss. This preference would drive down the price of long-term bonds, increasing
their yield un+l the arbitrage condi+on is sa+sfied.
, T4 en T5 4.3
Demonstrate the effect of an unan0cipated and permanent increase in the natural output level
𝑦0 on: actual output, the long-term and short-term interest rates, the price level, and the real
money supply. Show the effects in a diagram with 0me on the horizontal axis, and use 𝑡# as
the indicator for the 0me at which the shock is announced.
First assemble a dynamic model
𝑝̇ (𝑅! , 𝑝, 𝐸𝑋)
𝑅̇! (𝑅! , 𝑝, 𝐸𝑋)
𝑦 = −𝜎𝑅! + 𝑔
𝑚 − 𝑝 = −𝜆𝑅" + 𝛾𝑦 + 𝛼
𝑝̇ = 𝜙(𝑦 − 𝑦0) where 𝑦 is endogeneous
𝑝̇ = 𝜙(−𝜎𝑅! + 𝑔 − 𝑦0) where 𝑔 and 𝑦0 are exogeneous
$̇!
$!
= 𝑅! − 𝑅" where 𝑅" is endogeneous
&'()*(,*-)
𝑅" = /
subs+tute
$̇! &'()*(,*-)
$!
= 𝑅! − /
where 𝑦 is endogeneous
$̇! $! / &(*0$! (1)()*(,*-)
$!
= /
− /
where 𝑔 and 𝑚 are exogeneous
$̇! $! / 0&$! *&1*)((,*-) $! /(0&$! *&1*)((,*-)
$!
= /
+ /
= /
System of differen3al equa3ons
𝑝̇ = 𝜙(−𝜎𝑅! + 𝑔 − 𝑦0)
𝑅̇! 𝑅! (𝜆 + 𝛾𝜎) − 𝛾𝑔 − 𝛼 + (𝑚 − 𝑝)
=
𝑅! 𝜆
Macroeconomics
Tutorial 5
Chapter 5: Macroeconomic Dynamics II
- Capital Accumulation Decisions
- Dynamic IS-LM and the Structure of Interest Rates
- Unanticipated and Anticipated Shocks
,T4 en T5 4.3
4.3 Closed economy with s3cky prices
𝑦 = −𝜎𝑅! + 𝑔
𝑚 − 𝑝 = −𝜆𝑅" + 𝛾𝑦 + 𝛼
𝑝̇ = 𝜙(𝑦 − 𝑦0)
𝑅̇!
𝑅! − = 𝑅"
𝑅!
The last equa+on is the arbitrage equa+on between short bonds and consols (coupon bond).
If we assume that two types of financial instruments are perfect subs+tutes, their respec+ve
rates of return must equalize. For short-term bonds there is no capital gain or loss. The return
on long-term bonds should be the same as the return on a short-term bond.
The yield of a bond is a measure of the return an investor can expect to earn by holding the
bond un+l maturity, expressed as a percentage of the bond's current price.
When we talk about a capital loss in the context of long-term bonds, we are referring to a
decrease in the bond's price. Bond prices and yields are inversely related: when yields rise,
bond prices fall, and vice versa. If 𝑅̇! > 0, it means that the yield on long-term bonds is
expected to increase in the future. This increase in yield will lead to a decrease in the bond's
price, resul+ng in a capital loss for investors who hold the bond.
Suppose the long-term bonds are expected to experience a capital loss (i.e., 𝑅̇! > 0, meaning
yields are rising). To offset this loss, the current yield 𝑅! must be higher than the short-term
rate 𝑅" . This higher yield compensates investors for the an+cipated decrease in the bond's
price.
For investors to be indifferent between holding long-term bonds and short-term bonds, the
total return on long-term bonds must equal the return on short-term bonds. If 𝑅! were not
higher than 𝑅" when 𝑅̇! > 0, investors would prefer short-term bonds, which do not carry the
risk of capital loss. This preference would drive down the price of long-term bonds, increasing
their yield un+l the arbitrage condi+on is sa+sfied.
, T4 en T5 4.3
Demonstrate the effect of an unan0cipated and permanent increase in the natural output level
𝑦0 on: actual output, the long-term and short-term interest rates, the price level, and the real
money supply. Show the effects in a diagram with 0me on the horizontal axis, and use 𝑡# as
the indicator for the 0me at which the shock is announced.
First assemble a dynamic model
𝑝̇ (𝑅! , 𝑝, 𝐸𝑋)
𝑅̇! (𝑅! , 𝑝, 𝐸𝑋)
𝑦 = −𝜎𝑅! + 𝑔
𝑚 − 𝑝 = −𝜆𝑅" + 𝛾𝑦 + 𝛼
𝑝̇ = 𝜙(𝑦 − 𝑦0) where 𝑦 is endogeneous
𝑝̇ = 𝜙(−𝜎𝑅! + 𝑔 − 𝑦0) where 𝑔 and 𝑦0 are exogeneous
$̇!
$!
= 𝑅! − 𝑅" where 𝑅" is endogeneous
&'()*(,*-)
𝑅" = /
subs+tute
$̇! &'()*(,*-)
$!
= 𝑅! − /
where 𝑦 is endogeneous
$̇! $! / &(*0$! (1)()*(,*-)
$!
= /
− /
where 𝑔 and 𝑚 are exogeneous
$̇! $! / 0&$! *&1*)((,*-) $! /(0&$! *&1*)((,*-)
$!
= /
+ /
= /
System of differen3al equa3ons
𝑝̇ = 𝜙(−𝜎𝑅! + 𝑔 − 𝑦0)
𝑅̇! 𝑅! (𝜆 + 𝛾𝜎) − 𝛾𝑔 − 𝛼 + (𝑚 − 𝑝)
=
𝑅! 𝜆
