# FORMULAS MACRO-2 ```MACROECONOMICS
FORMULAS
EDITION A.A. 2018 - 2019
Written and edited by Amina Costanzo and Valentina Tuveri
1
2
This material has been written by students without any possible intention of substituting the official teaching
materials provided by the University. Therefore, it should be merely seen as an useful tool for the study of the subject.
It does not provide such complete exam preparation as the teaching material Implemented by the University.
FINANCIAL MARKETS AND
EXPECTATIONS
REAL INTEREST RATE
)
𝑟&quot; ≈ 𝑖&quot; − 𝜋&quot;'(
EXPECTED PRESENT DISCOUNTED VALUE (PDV)
(Or present discounted value of the sequence of payments)
1
1
€𝑉&quot; = €𝑧&quot; +
€𝑧 ) +
€𝑧 ) + ⋯ = 𝑓(€𝑧&quot; , €𝑧 ) , 𝑖&quot; , 𝑖 ) )
)
1 + 𝑖&quot; &quot;1( (1 + 𝑖&quot; )(1 + 𝑖(,&quot;1(
) &quot;15
[It is an increasing function of both current and expected future payments and it is a decreasing function of
both current and expected future interest rates]
With
)
)
)
(€𝑧&quot; , €𝑧&quot;1(
, €𝑧&quot;15
, … , €𝑧&quot;19
, )= sequence of current and expected future payments
)
)
(𝑖&quot; , 𝑖&quot;1( , 𝑖&quot;15 , … )= current and expected future interest rates
By which we get
€𝑉&quot; = €𝑃&quot; &times; 𝑉&quot;
BOND PRICES
1. One-year bonds (short-term):
€𝑃(,&quot; =
2. Two-years bonds (long-term)
€𝑃5,&quot; =
100
1 + 𝑖(,&quot;
100
100
=
)
(1
+
𝑖5,&quot; )5
=1 + 𝑖(,&quot; &gt;(1 + 𝑖(,&quot;1( )
3. N-years bonds (long-term):
€𝑃9,&quot; =
100
(1 + 𝑖5,&quot; )9
ARBITRAGE CONDITION
1. With no risk premium (x=0)
1 + 𝑖(,&quot; =
By which we get:
€𝑃5,&quot; =
3
)
€𝑃(,&quot;1(
€𝑃5,&quot;
C
€?@,AB@
(1D@,A
1 + 𝑖(,&quot; + x =
By which we get:
)
€𝑃(,&quot;1(
€𝑃5,&quot;
€?C
€𝑃5,&quot; = (1D@,AB@
1F
@,A
EXPECTED PRICE OF A SHORT-TERM BOND
)
€𝑃(,&quot;1(
=
By which we get:
€𝑃5,&quot; =
100
)
1 + 𝑖(,&quot;1(
100
)
(1 + 𝑖(,&quot; )(1 + 𝑖(,&quot;1(
)
BOND YIELDS TO MATURITY
1. Relation between long-term, short-term and next year's expected one-year interest rates
1
)
𝑖5,&quot; = =𝑖(,&quot; + 𝑖(,&quot;1(
&gt;
2
2. General case (n-years)
9'(
𝑖9,&quot; =
1
1
)
)
)
)
=𝑖(,&quot; + 𝑖(,&quot;1(
+ 𝑖(,&quot;15
+ ⋯ + +𝑖(,&quot;19'(
&gt; = I 𝑖(,&quot;1J
𝑛
𝑛
JKL
STOCK MARKET AND MOVEMENTS IN STOCK PRICES
1. In nominal terms
)
)
)
€𝐷&quot;1(
€𝐷&quot;15
€𝐷&quot;19
€𝑄&quot; =
+
+
⋯
+
)
)
1 + 𝑖(,&quot; + 𝑥 =1 + 𝑖(,&quot; + 𝑥&gt;=1 + 𝑖(,&quot;1(
+ 𝑥&gt;
=1 + 𝑖(,&quot; + 𝑥&gt; … =1 + 𝑖(,&quot;19'(
+ 𝑥&gt;
With
€𝑄&quot; = nominal price of the stock in the current period
)
)
€𝐷&quot;1(
, €𝐷&quot;15
,…= expected future dividends
2. In real terms
)
)
)
𝐷&quot;1(
𝐷&quot;15
𝐷&quot;19
𝑄&quot; =
+
+
⋯
+
)
)
1 + 𝑟(,&quot; + 𝑥 =1 + 𝑟(,&quot; + 𝑥&gt;=1 + 𝑟(,&quot;1(
+ 𝑥&gt;
=1 + 𝑟(,&quot; + 𝑥&gt; … =1 + 𝑟(,&quot;19'(
+ 𝑥&gt;
[the stock price depends positively on expected future dividends and negatively on expected future
ARBITRAGE CONDITION BETWEEN STOCKS AND CONDIZIONE DI
ARBIRTRAGGIO TRA AZIONI E TITOLI ANNUALI
1 + 𝑖(,&quot; + x =
)
)
€𝐷(,&quot;1(
+ €𝑄(,&quot;1(
€𝑄&quot;
Stock dividends = bond yields
4
THE ROLE OF EXPECTATIONS IN THE
REAL ECONOMY: CONSUMPTION AND
INVESTMENT
WHEALTH
1. Non-human wealth (WFI)
𝒇
WFI = 𝑾𝒕 + 𝑾𝒉𝒕
with
𝒇
𝑾𝒕 = financial wealth
𝑾𝒉𝒕 = housing wealth
2. Human wealth (V(𝑦 ) ))
𝑉(𝑦&quot;) ) = 𝑦&quot; +
)
)
𝑦&quot;1(
𝑦&quot;15
+
) )+⋯
1 + 𝑟&quot; (1 + 𝑟&quot; )(1 + 𝑟&quot;1(
3. Total wealth (𝑾&quot;X&quot;
𝒕 )
𝒇
𝑾&quot;X&quot;
= 𝑾𝒕 + 𝑾𝒉𝒕 + 𝑉(𝑦&quot;) )
𝒕
CONSUMPTION BUDGET CONSTRAINT
1. Present consumption
𝐶&quot; + 𝑆 = (𝑌&quot; − 𝑇&quot; )
2. Future consumption
)
)
𝐶&quot; + 𝑆 = 𝑌&quot;1(
− 𝑇&quot;1(
+ (1 + 𝑟)𝑆
With S= private saving
3. Intertemporal consumption
a. If the individual has only human wealth
𝐶&quot; +
By which we get
)
)
𝐶&quot;1(
𝑌&quot;1(
− 𝑇&quot;1(
= 𝑌&quot; − 𝑇&quot; +
1+𝑟
1+𝑟
)
)
𝐶&quot;1( = −(1 + 𝑟)𝐶&quot; + (1 + 𝑟)(𝑌&quot; − 𝑇&quot; ) + (𝑌&quot;1(
− 𝑇&quot;1(
)
b. If the individual has also non-human wealth
𝐶&quot; +
)
)
𝐶&quot;1(
𝑌&quot;1(
− 𝑇&quot;1(
= 𝑌&quot; − 𝑇&quot; +
+ 𝑊𝐹𝐼
1+𝑟
1+𝑟
DEMAND FUNCTION FOR INVESTMENT
𝑉(𝜋&quot;) ), 𝜋&quot;
𝐼&quot; = 𝐼[
]
+
5
With 𝑉(𝜋&quot;) ) = present value of the exected profits (following the investment)
6
𝑉(𝜋&quot;) ) =
)
)
)
(1 − 𝛿)𝜋&quot;15
(1 − 𝛿)5 𝜋&quot;1c
𝜋&quot;1(
+
+
+⋯
) )
) )(1
)
(1 + 𝑟&quot; ) + (1 + 𝑟&quot;1(
1 + 𝑟&quot; (1 + 𝑟&quot; ) + (1 + 𝑟&quot;1(
+ 𝑟&quot;15
)
Where d= depretiation rate
The firm invests in the current period if and only if:
𝑉(𝜋&quot;) ) ≥ 𝑝f
With 𝑝f = real price of the investment
So we get
𝐼&quot; = 𝐼 g
𝑉(𝜋&quot;) ) 𝑝f
𝜋)
, h = 𝐼[
−
+
+
𝑟 𝑟 ) 𝑝f
− − −
d
]
−
OPEN ECONOMY AND GOODS MARKET
REAL EXCHANGE RATE
Price of domestic goods in terms of foreign goods
𝜀=
Where
𝐸𝑃
𝑃∗
P= domestic price level (expressed as domestic currency)
E= nominal exchange rate
P*= general foreign price level (expressed as foreign currency)
UNCOVERED INTEREST PARITY (U.I.P.)
1 + 𝑖&quot; =
The U.I.P can be rewritten as follows:
1 + 𝑖&quot; =
By which we get
1 + 𝑖&quot; =
Where
C 'l
lAB@
A
lA
(1 + 𝑖&quot;∗ )
)
𝐸&quot;1(
+ 𝐸&quot; − 𝐸&quot;
𝐸&quot;
= nominal expected tasso di depreciation/appreciation rate
𝑖&quot; = 𝑖&quot;∗
7
(1 + 𝑖&quot;∗ )
− 𝐸&quot;
+1
𝐸&quot;
)
𝐸&quot;1(
A good approximation is
Where
(1 + 𝑖&quot;∗ )𝐸&quot;
)
𝐸&quot;1(
)
𝐸&quot;1(
− 𝐸&quot;
𝐸&quot;
𝑖&quot; = domestic interest rate
𝑖&quot;∗ =foreign interest rate
𝐸&quot; =current nominal exchange rate
)
𝐸&quot;1(
= expected future exchange rate
[we do not consider transaction costs and insolvency risk]
EQUILIBRIUM CONDITION OF THE GOODS MARKET IN AN OPEN
ECONOMY
Where
𝑌−𝑇
𝑌 𝑅
𝑌∗
𝑌 =𝐶m
n+𝐼m
n+𝐺 +𝑋m
+
+ −
+
𝐼𝑀 𝑌 𝜀
𝜀
n−
(
)
−
𝜀 + +
𝑌∗ 𝜀
𝑋m
n= demand for exports
+ −
st 𝑌
𝜀
(
)= demand for exports
u + +
st 𝑌
𝜀
𝑌∗ 𝜀
𝑁𝑋 = 𝑋 m
n− u (
)= net exports
+ +
+ −
SAVING, INVESTMENT AND THE CURRENT ACCOUNT BALANCE
𝑌 = 𝐶 + 𝐼 + 𝐺 + 𝑁𝑋
By subtracting to both sides C+T
Knowing that S=Y-C-T
We get
𝑌 − 𝐶 − 𝑇 = 𝐶 + 𝐼 + 𝐺 + 𝑁𝑋 − 𝐶 − 𝑇
𝑆 = 𝐼 + 𝐺 − 𝑇 + 𝑁𝑋
𝑁𝑋 = S + 𝑆xyz − 𝐼
&Oslash; NX&gt;0= trade surplus (an excess of savings over investments)
&Oslash; NZ&lt;0= trade deficit (an excess of investments over savings)
8
THE IS-LM IN AN OPEN ECONOMY
UNCOVERED INTEREST PARITY
𝐸&quot;
)
𝐸&quot;1(
Domestic interest rates= foreign interest rates
1+𝑖 )
𝐸=
𝐸{
1 + 𝑖∗
(1 + 𝑖&quot; ) = (1 + 𝑖&quot;∗ )
Where 𝐸{ ) = expected exchange rate
MUNDELL- FLEMING MODEL
1. In a flexible exchange rates regime
IS
LM
𝑌−𝑇
𝑌 𝑅
𝑌∗
𝑌 =𝐶m
n+𝐼m
n + 𝐺 + 𝑁𝑋 m
+
+ −
+
𝑖 = 𝚤̅
UIP
𝐸=
1+𝑖 )
𝐸{
1 + 𝑖∗
By which we get the equilibrium condition
𝑌−𝑇
𝑌 𝑅
𝑌∗
𝑌 =𝐶m
n+𝐼m
n + 𝐺 + 𝑁𝑋 ~
+
+ −
+
2. In a fixed exchange rates regime
IS
LM
𝑌 1 + 𝑖 𝐸{ )
𝑖∗ •
− 1 +−
𝑌−𝑇
𝑌 𝑅
𝑌∗
𝑌 =𝐶m
n+𝐼m
n + 𝐺 + 𝑁𝑋 m
+
+ −
+
𝑌 𝜀
n
− −
𝑀 = 𝑃 &times; 𝐿(𝑌, 𝑖)
By which we get the equilibrium condition
𝑌−𝑇
𝑌 𝑅
𝑌∗
𝑌 =𝐶m
n+𝐼m
n + 𝐺 + 𝑁𝑋 ~
+
+ −
+
9
𝑌 𝐸
n
− −
𝑌 1 + 𝑖 𝐸{ )
𝑖∗ •
− 1 +−
EXCHANGE RATE REGIMES
EXHANGE RATE MOVEMENTS UNDER FLEXIBLE EXCHANGE
RATES
𝐸&quot; =
1 + 𝑖&quot; )
𝐸
1 + 𝑖 ∗ &quot; &quot;1(
Continuing to solve forward in time in the same way we get:
𝐸&quot; = 𝐸 ) &quot;1(
(1 + 𝑖&quot; )(1 + 𝑖&quot; ) ) … . (1 + 𝑖&quot;19 ) )
(1 + 𝑖 ∗ &quot; )(1 + 𝑖&quot; ∗) ) … . (1 + 𝑖&quot;19 ∗) )
10
PUBLIC DEBT
PUBLIC DEFICIT
𝐷&quot; = 𝑟𝐵&quot;'( + 𝐺&quot; − 𝑇&quot;
Where
𝐷&quot; = government deficit at time t
𝐵&quot;'( = government debt at the end of the year t-1
𝑇&quot; =taxes minus transfers during year t
𝐺&quot; = government spending on good and services during year t
𝐺&quot; − 𝑇&quot; = primary deficit
(if 𝐺&quot; &gt; 𝑇&quot; government spending is larger that net taxes income&agrave; deficit)
(if 𝐺&quot; &lt; 𝑇&quot; net taxes income is larger than goverment spending&agrave; surplus)
𝑟𝐵&quot;'( = interest payments
GOVERNEMENT BUDGET CONSTRAINT (in real terms)
In order to fit the budget constraint the government must run a new deficit
On the basis of the intertemporal budget constraint:
𝐵&quot; − 𝐵&quot;'( = 𝐷&quot;
By using the definition of deficit
𝐵&quot; = (1 + 𝑟)𝐵&quot;'( + 𝐺&quot; − 𝑇&quot;
FULL REPAYMENT OF THE DEBT IN YEAR t
𝑇&quot; − 𝐺&quot; = (1 + 𝑟)&quot;'( (−∆𝑇L )
With 𝑇&quot; − 𝐺&quot; = primary deficit
DEBT STABILISATION IN YEAR t
𝑇&quot; − 𝐺&quot; = 𝑟(−∆𝑇L )
THE EVOLUTION OF THE DEBT-TO-GDP RATIO
𝐵&quot;
𝐵&quot;'( 𝐺&quot; − 𝑇&quot;
= (1 + 𝑟)
+
𝑌&quot;
𝑌&quot;'(
𝑌&quot;
With g= real growth rate of output
𝑔=
𝑌&quot; − 𝑌&quot;'(
𝑌&quot;'(
𝑌&quot;
=1+𝑔
𝑌&quot;'(
𝐵&quot;
𝐵&quot;'( 𝐺&quot; − 𝑇&quot;
= (1 + 𝑟 − 𝑔)
+
𝑌&quot;
𝑌&quot;'(
𝑌&quot;
Where
‡A 'ˆA
‰A
11
= primary deficit and GDP ratio=d
1 + 𝑟=interest rate
ŠA
= the evolution of the debt-to-gdp ratio=bt
‰
A
So we get
𝑏&quot; = (1 + 𝑟 − 𝑔)𝑏&quot;'( + 𝑑
THE EVOLUTION OF THE DEBT-TO-GDP RATIO
𝐵&quot; 𝐵ˆ'(
𝐵ˆ'( 𝐺&quot; − 𝑇&quot;
−
= (𝑟 − 𝑔)
+
𝑌&quot;
𝑌&quot;'(
𝑌&quot;'(
𝑌&quot;
If
ŠA
‰A
−
Š•Ž@
‰AŽ@
=0&agrave;
ŠA
‰A
= 𝑏{ (there are no changes of the debt-to-GDP ratio)
𝐺&quot; − 𝑇&quot;
𝑌&quot;
1 𝐺&quot; − 𝑇&quot;
𝑏{ =
𝑔 − 𝑟 𝑌&quot;
𝑏{ = (1 + 𝑟 − 𝑔)𝑏{ +
Where
‡A 'ˆA
‰A
=d