Macroeconomics II (English)

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1 
 
Macroeconomics II 
 
Professor: Caio Machado 
Vicente García Casassus 
vsgarcia@uc.cl 
 
 
 
Index 
Introduction to money 2 
Monetary policy and equilibrium with fixed prices (short run) 28 
Monetary policy and equilibrium in the long (medium) run 46 
Monetary policy 57 
 
 
 
 
 
 
This material is based mainly in the following books: Macroeconomics, Blanchard, 
Macroeconomía: Teoría y Políticas, DeGregorio, Macroeconomics, Mankiw and Monetary Theory 
and Policy, Walsh. 
Underlined words correspond to new concepts. 
More material at: https://drive.google.com/drive/folders/0Bx20jJIsUiAOU1RTUFg2QlU3V2M 
mailto:vsgarcia@uc.cl
https://drive.google.com/drive/folders/0Bx20jJIsUiAOU1RTUFg2QlU3V2M
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Introduction to money 
- Money is an asset which is part of the people’s and company’s financial wealth and is widely 
used for any kind of transaction. Therefore, it is a stock variable and it has three main functions: 
stores value (used to accumulate assets), payment method (transactions) and unit of account 
(the value of goods and services are expressed in a unit of account). 
- There are two types of money: 
1- Fiat Money: Currency without intrinsic value that has been established as money, often by 
government regulation. Fiat money does not have use value, and has value only because a 
government maintains its value, or because agents engaging in exchange agree on its value. 
For example, one dollar is only worth one dollar because we believe that is its value, even if 
the cost of printing one dollar is only a portion of the value. 
2- Commodity Money: Currency with intrinsic value, the market assigns how much it is worth. 
For example, gold. One ounce of gold will be US$1.200 even if people don’t trust in gold. 
Also, because gold can be used for various purposes other than transactions, you can use 
it for jewelry as well. 
- As history has taught us, people bartered (trueque) goods as the first attempt for transactions, 
one good for the other. As civilizations grew and humanity had more access to precious metals, 
we moved from bartering to doing transactions with commodity money. 
- After some time, people realized that it was absurd having always to carry your bad with gold 
or silver coins, so the governments decided that it would be easier if they gave “gold 
certificates” for the corresponding amount of gold. I gave the government 5 gold coins and 
they me a certified paper saying the paper was worth 5 gold coins. 
- When the certificates became a common thing, the government began changing this 
certificates for what we now know as the fiat money. I changed my 5-gold coin certificate for 
a $5 bill. 
- The quantity of money available in an economy is called money supply. In a system of 
commodity money, the money supply is simply the quantity of that commodity. 
- In an economy that uses fiat money, such as most economies today, the government controls 
the supply of money: legal restrictions give the government a monopoly on the printing of 
money. The government’s control over the money supply is called monetary policy. 
- But how can we measure the quantity of money in more complex economies? The answer is not 
obvious, because no single asset is used for all transactions. People can use various assets, such 
as cash or deposits in their checking accounts, to make transactions, although some assets are 
more convenient than others. 
- The most obvious asset to include in the quantity of money is currency, the sum of outstanding 
paper money and coins. Most day-to-day transactions use currency as the medium of exchange. 
https://en.wikipedia.org/wiki/Currency
https://en.wikipedia.org/wiki/Intrinsic_value_(numismatics)
https://en.wikipedia.org/wiki/Money
https://en.wikipedia.org/wiki/Use_value
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- A second type of asset used for transactions is demand deposits or “depósitos a la vista”, the 
funds people hold in their checking accounts. If most sellers accept personal checks, assets in 
a checking account are almost as convenient as currency. In both cases, the assets are in a form 
ready to facilitate a transaction. Demand deposits are therefore added to currency when 
measuring the quantity of money. 
- As discussed previously, money can be used as payment method, but there is also a level of 
arbitrariness because money is constituted by liquid financial assets, which can also be used as 
payment methods. 
- This way is how the most liquid way of money is described, M1, where M1 = C + Dv = currency 
+ demand deposits. On the next level we have M2, where M2 = M1 + Dp = M1 + saving deposits 
+ money market + mutual funds + etc = M1 + “depósitos a plazo”. 
- We usually use the term “money” (M) when speaking of M1 or M2. Moreover, D represents 
deposits in general. If the kind of money we are talking about is specified, M1 or M2, we are 
thinking in terms of Dv or Dv + Dp, respectively. 
- If we lived in a world without banks M = C. If we have banks, then there is a monopoly in the 
emission and storage of money. In this system, the amount of money in circulation exists 
because all of it was emitted by the Central Bank, therefore, there is a monetary base (H). 
M = C + D 
- If the bank keeps the 100% of the deposit (doesn’t lend it to anyone) all deposits are kept as 
reserves (R). But the bank can lend part of its reserves to people who needs loans, in that case 
we can express the relation between of reserves as a percentage of deposits: 
R = θD 
- Then, the monetary base only corresponds to the circulating currency and the reserves in the 
bank: 
H = C + R 
- Every bill and coin the Central Bank has emitted are circulating freely in the economy or are in 
some way deposited in the bank itself. 
- When the Central Bank decides to increase the money supply, the change can be quantified 
as: 
ΔM = ΔC + ΔD 
- Let’s suppose that John receives X dollars from the bank, the amount of currency in the economy 
increases in X. John decides to deposit the X dollars, the amount of currency in the economy 
decreases in X, but the amount of money deposited goes up by X. The bank stores θX, therefore, 
it lends (1 – θ)X to Bob, then currency goes up in (1 – θ)X. 
ΔM = (1 – θ)X + X > X 
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- Now suppose Bob also deposits the (1 – θ)X in the bank and the bank takes (1 – θ)θX for reserve 
and lends (1 – θ)2X and that way on and on: 
ΔM = (1 – θ)X + (1 – θ)2X … + X 
- By solving this sum, we get that the increase in the money supply is 
X
1 − (1 − θ)
 = 
X
θ
. The money 
got multiplied by 
𝟏
𝛉
 times, a money multiplier. 
- If we suppose that the public wishes to have a proportion between circulating money and 
deposits equal to �̅�: 
C = c̅ D 
- If I get $100, and c̅ = 1, then $50 will go to the bank and I will keep $50 for myself. If c̅ = 3, then 
$25 go to the bank and I get to keep $75 as cash. The bigger the �̅� is, the more amount of money 
is kept as currency. 
- Therefore, we can notice that the proportion of circulating and deposited money are: 
 C = c̅ D C = c̅ D 
 C = c̅ (M – C) M – D = c̅D 
 (1 + c̅)C = c̅M M = (1 + c̅) D 
 
C
M
 = 
c̅
(1 + c̅)
 
D
M
 = 
1
(1 + c̅)
 
- Let’s suppose that the Central Bank emits $X again, where only 
Xc̅
(1 + c̅)
 will stay as currency while 
the other 
X
(1 + c̅)
 stays as a deposit. From the deposits, only 
Xθ
(1 + c̅)
 will be turned into reserves, 
while the 
X(1 − θ)
(1 + c̅)
 will be given as a loan to a third party. Basically, the same idea discussedbefore with John and Bob. 
- The “money multiplier” tells us, for every unit of currency emitted, by how much will the money 
supply will increase in. The money multiplier can also be understood as the ratio between 
money supply and the monetary base: 
Money multiplier: M/H 
M
H
 = 
D + C
R + C
 
M
H
 = 
D + C
D
R + C
𝐷
 
M
H
 = 
1 + c̅
θ + c̅
 = μ 
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- The decision of what percentage of the money is to be kept as deposit and which is going to be 
circulating freely will depend in the cost of changing deposits for cash and the use of each in 
different transactions. 
- We can see how the parameters affect the money supply, if the banks get to keep a bigger 
share of the deposits for a reserve, they will have less money to lend, there will be a lower 
money supply. 
- If people have remarkable preference for cash over deposit, bigger �̅�, there will be less money 
destined for deposits, less money for the banks to lend, less money supply. 
- You can also see this logic by calculating the derivative of M with respect of the parameters. 
- In the limit, if people don’t like deposits at all, the money multiplier would be 1. 
lim
c̅→∞
M = 1 lim
c̅→0
M = 
1
θ
 
- How do Central Banks control and change the monetary supply? You can change the 
components, the monetary base or the parameters. Central Banks can change the monetary 
base by: 
1- Open Market Operations: The government buys and sells all kind of financial instruments. 
2- Intern Credit Operations: The most practical way would give loans to banks, so they can 
lend it to the private sector. The other way would be that the Central Bank moves the 
interest rate so banks ask loans or lend loans to the Central Bank itself. 
3- Change Operations: Modify the international reserves (R*) by buying and selling foreign 
currency. 
- Central Banks can change the money multiplier by changing the proportion of the deposits 
that they keep for the reserve. Nevertheless, this is a last resource option. Economies change 
the “lace” (θ) only when there is no other instrument that can provide liquidity. On the other 
hand, you can change �̅� by changing people’s preference with the interest rate. 
- With what we have studied we can establish the Central Bank’s balance sheet: 
 
- What determines the money demand? Let’s assume that every period the agents of an economy 
receive a deposit in their accounts of a determined amount of money, Y. We can also assume 
that every period can be divided by “n” Also, the agents want to smooth consumption, then, 
each “n” period the agents go to the bank and withdraw R = Y/n. The average amount of 
money all time is 
𝐑
𝟐
 = 
𝐘
𝟐𝐧
. 
- We can model de amount of money in the agents account as the following: 
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- Complexity can be added if we consider that the money in the bank generates interests. If the 
money generated interests, we are more likely to go as many times as we can to the bank, so 
each time we have to withdraw less money, and more is kept as a deposit. 
- The opportunity cost of having the money in cash is the nominal interest rate, i. I keep going as 
many times as I can to the bank, every time I withdraw 
Y
n
. This amount of money has an 
opportunity cost, then, the cost of every withdrawal is 
𝐢𝐘
𝐧
. But, as money is spent as time 
passes, the opportunity cost of it decreases, hence, we can describe the average opportunity 
cost of the money as 
𝐢𝐘
𝟐𝐧
. 
- But none of us has an ATM or a bank right next to our house, some must to take the bus, go by 
car or walk to the nearest bank, then there is a cost, Z, of going to the closest bank. The “cost” 
can be either in money or time. 
- Our new cost function can be written as: 
C(n) = 
iY
2n
 + Zn 
- And we want to maximize our utility by minimizing our costs: 
C’(n) = - 
iY
2(n)2
 + Z = 0 
n* = √
iY
2Z
 
- The average money being hold (money demand) is 
MD = 
Y
2n∗
 = 
Y
2√
iY
2Z
 = √YZ
2i
 
- By doing a little modification we get that: 
ln MD = 0,5 (lnY + lnZ – ln2 – lni) 
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- As it can be seen, the income elasticity and the transport cost elasticity are 0,5. While the 
interest rate elasticity is -0,5: 
𝑑lnMD
𝑑lnY
 = 0,5 
𝑑lnMD
𝑑lni
 = -0,5 
- If we define real variables as y = 
𝐘
𝐏
 and z = 
𝐙
𝐏
, then the money demand can be written as: 
MD = P√
yz
2i
 
- Sadly, this model is too basic to take in consideration as a money demand model. Thus, three 
general approaches are created to model the money demand. All three incorporate money into 
general equilibrium: 
1- Preliminaries: Assume that money yields direct utility by incorporating money balances 
into the utility functions of the agents of the mode. 
2- Benchmark with no money: Impose transaction costs of some form that give rise to a 
demand for money, by making asset exchanges costly. Requiring that money be used for 
certain types of transactions. 
3- Money In the Utility function (MIU) model: Treat money like any other asset used to 
transfer resources intertemporally. 
- In the preliminaries there is an infinite horizon, a discount factor (β), discrete time and we 
seek to optimize the representative agent. The economy has only one good that can be used 
for capital (K), which depreciates at a rate δ, or consumption (C). Agents have perfect foresight. 
- The neo-classical production function depends on the capital of the period before: 
Yt = f(Kt-1) with f’(Kt-1) > 0, f’’(Kt-1) < 0 
- The agents decide how much they consume, how much capital and money they hold and how 
much they invest in capital or bonds. 
- The real interest rates, 1 + rt, can be understood as the gross real interest rate between t–1 and 
t: 
How many goods you can buy in t after buying $1 in bonds at t – 1
How many bonds does $1 buys at t−1 
 
- In an economy with uncertainty, the relation between nominal and real rates can be described 
through the Fisher Equation: 
(1 + it) = (1 + rt) (1 + πet+1) 
- If there is no uncertainty, πet = πt, in other words, the expected inflation is equal to the inflation 
itself. 
- A lineal approximation would say that: 
it = rt + πet+1 
- By introducing money into our model, the benchmark with no money, agents will have a 
budget constrain, which determines how much money they can spend, it can be used for 
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consumption (PtCt), capital investment (PtKt), hold it (Mt) for the next period, invest in bonds 
(Bt) and money transferred or withdrawn by the government. 
PtCt + PtKt + Mt + Bt < Wt = Ptf(Kt-1) + (1 – δ)PtKt-1 + Mt-1 + (1 + it-1)Bt-1 + Tt 
- This means that what we consume will depend on the production, yesterday’s capital, how 
much money was held, the returns of the bonds and the transfers of the government. 
- Dividing by Pt will give us the real variables where at = 
At
Pt
: 
Ct + Kt + mt + bt < wt = f(Kt-1) + (1 – δ)Kt-1 + 
Mt−1
Pt
 + (1 + it-1) 
Bt−1
Pt
 + tt 
- We also know that (1 + πt) = 
Pt
Pt−1
, Pt = Pt-1 (1 + πt). Hence: 
Ct + Kt + mt + bt < wt = f(Kt-1) + (1 – δ)Kt-1 + 
Mt−1
Pt−1 (1 + πt)
 + (1 + it-1) 
Bt−1
Pt−1 (1 + πt)
 + tt 
Ct + Kt + mt + bt < wt = f(Kt-1) + (1 – δ)Kt-1 + 
mt−1
(1 + πt)
 + (1 + it-1) 
bt−1
(1 + πt)
 + tt 
- We can apply Fisher’s Equation with no uncertainty: 
Ct + Kt + mt + bt < wt = f(Kt-1) + (1 – δ)Kt-1 + 
mt−1
(1 + πt)
 + (1 + it-1) 
(1 + r−1t)bt−1
(1 + it−1)
 + tt 
 
- This is the budget constraint in real units. As we can see, consumption and capital are always 
expressed in real terms (because they were multiplied by the price at the beginning). So, we can 
see that small cap letters refer to real variables, except consumption and capital. This means 
that, Ct and Kt also are real terms. 
- By definition: 
Mt
Mt−1
 = 
mt
mt−1
 
Pt
Pt−1
 = 
mt
mt−1
(1 + πt) 
- Hence, if mt = mt-1: 
Mt
Mt−1
 = 1 + πt 
- Let’s suppose mt = 0 and agents cannot choose mt. Also, there are no governmenttransferences. 
- Agents have a utility over consumption, U(Ct) with the following characteristics: 
1- U’(C) > 0 
2- U’’(C) < 0 
3- lim
C →0
U′(C) = ∞ 
- The representative households solve their consumption plan by maximizing their utility: 
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MAX ∑ βtU(Ct)
∞
t=0 
- Where β is the patience parameter, β = 
1
1 + 𝑝
. p is the “personal interest rate”. The bigger my 
personal interest rate is, the less patient I am. 
- We also assume we have capital from the previous period (t = -1), and bonds as well. Lim
C →∞
bt = 
0 is also assumed. 
- Utility can be maximized with the Lagrangian, constrained by the budget: 
L = ∑ βt{U(Ct)
∞
t=0 – λt[Ct + Kt + bt – f(Kt-1) – (1 – δ)Kt-1 – (1 + rt-1)bt-1]} 
- The First Order Conditions (FOC’s) are: 
1- [Ct]: βtU’(Ct) – βtλt = 0 → U’(Ct) = λt 
2- [Kt]: – βtλt + βt+1λt+1[f’(Kt) + (1 – δ)] = 0 → 
1
β
λt
λt+1
 = [f’(Kt) + (1 – δ)] 
3- [bt]: – βtλt + βt+1λt+1(1 + rt) = 0 → 
1
β
λt
λt+1
 = (1 + rt) 
- How is that we got the FOC’s for capital and bonds? Remember, the Lagrangian is a sum of 
variables in different periods, one specific term (as capital or bonds) can be found in more than 
one period, for example: 
L = ∑ βt{U(Ct)
∞
t=0 – λt[Ct + Kt + bt – f(Kt-1) – (1 – δ)Kt-1 – (1 + rt-1)bt-1]} 
 
L = βt{U(Ct) – λt[Ct + Kt + bt – f(Kt-1) – (1 – δ)Kt-1 – (1 + rt-1)bt-1]} + 
βt+1{U(Ct+1) – λt+1[Ct+1 + Kt+1 + bt+1 – f(Kt) – (1 – δ)Kt – (1 + rt)bt]} + 
βt+2{U(Ct+2) – λt+2[Ct+2 + Kt+2 + bt+2 – f(Kt+1) – (1 – δ)Kt+1 – (1 + rt+1)bt+1]} + … 
 
- Is it clearer now? As we can see, Kt can be found both on the first line and the second. When 
the differential wants to be calculated, the only thing we have to do is to look at lines one and 
two. 
- By replacing one equation in the other we get that: 
1- f’(Kt) = rt + δ 
2- 
U′(Ct)
βU′(Ct+1)
 = 
λt
βλt−1
 = 1 + rt 
3- U’(Ct) = β(1 + rt)U’(Ct+1) 
- The third equation is known as the Euler Equation. If my personal interest rate is the same as 
the market’s, the marginal utility gained today is the same that would be gained tomorrow. 
- A situation in which every variable is constant in time is known as a steady state (ss) or “the 
long run”. For example: 
Kt = Kt+1 = Kt+2 … = KSS 
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- In this context, and using equation 1 (above) we have that: 
f'(KSS) = rSS + δ 
- As every variable is the steady, we can say that agents consume the same amount of goods in 
every period, therefore, their marginal consumption’s utility is the same for each period. 
Looking at equation 3 we can conclude that 
1
β
 = 1 + rSS and then: 
f'(KSS) = 
1
β
 – (1 – δ) 
- If we are more patient, bigger β, then we will have a bigger KSS. This is because if β increases, 
the marginal rate of production also has to decrease, and the only way of having a lower f’(KSS) 
is by having more capital. It works the other way around with depreciation. 
- As lim
C →∞
bt = 0, b
SS = 0. Hence, by the budget constrain is: 
Css + Kss + bss = f(Kss) + (1 – δ)Kss + (1 + rss)bss 
 
Css + Kss + 0 = f(Kss) + Kss – δKss + (1 + rss)0 
 
CSS = f(KSS) – δKSS 
- Then an increase in the consumption (derivative in KSS) will always be positive because: 
C’SS = f’(KSS) – δ = 
1
β
 – (1 – δ) – δ = 
1
β
 > 1. 
- Now, let’s consider the possibility in which people want to hold money because there is Money 
in the Utility (MIU). Which is the same model as the last one, but now the Central Bank can make 
nominal monetary transfers (Tt = Pttt), agents can hold money (mt ≠ 0) and the utility (Ut[Ct, mt]) 
depends on real money holdings (mt) with: 
1- U’c(C, m) > 0 
2- Concave utility function. 
3- Lim
C →0
Ut[Ct, mt] = Lim
m →0
Ut[Ct, mt] = ∞ 
- The agents maximize their utility by maximizing the following equation: 
 
- The aggregate economywide budget constraint of the household sector takes the form: 
 
- The solution that maximizes the Lagrangian 
L = ∑ βt{U(Ct, mt)
∞
t=0 – λt[Ct + Kt + mt + bt – f(Kt-1) – (1 – δ)Kt-1 – 
mt−1
(1 + πt)
 – (1 + rt-1)bt-1 – tt]} 
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- The First Order Conditions are: 
1- [Ct]: βtU’c(Ct, mt) – βt λt = 0 → U’c(Ct, mt) = λt 
2- [bt]: – βtλt + βt+1λt+1(1 + rt) = 0 → 
λt
βλt+1
 = (1 + rt) 
3- [Kt]: – βtλt + βt+1λt+1[f’(Kt) + (1 – δ)] = 0 → 
λt
βλt+1
 = (1 + rt) = f’(Kt) + (1 – δ) 
4- [mt]: βtU’m(Ct, mt) – βtλt + βt+1 
λt+1
(1 + πt+1)
 = 0 → λt = U’m(Ct, mt) + 
βλt+1
(1 + πt+1)
 
- By replacing one equation in the other we get that: 
1- f’(Kt) = rt + δ 
2- 
Um
′(Ct, mt)
Uc
′(Ct, mt)
 = 1 – 
βλt+1
λt
1
(1 + πt+1)
 = 1 – 
1
(1 + rt)(1 + πt+1)
 = 1 – 
1
(1 + it)
 
3- 
UCt
′(Ct, mt)
UCt+1
′(Ct+1, mt+1)
 = 
β
β
λt
λt+1
 = β(1 + rt) 
- From equation 2 we get that an increase in the nominal interest rate decreases the preference 
of holding money which has some logic, the bigger is the cost of opportunity of currency, the 
less I want to hold. 
- Often, particularly when the focus is on the relationship between money and prices, one might 
be more interested in a steady state in which real quantities such as consumption and the capital 
stock are constant, but the growth rate of money varies over time. Let’s assume, then, that ct = 
css, mt = mss and Kt = Kss for all t. 
- Setting population growth n to zero: 
UCss
′(Css,mss)
UCss
′(Css,mss)
 = β(1 + rt) = 1 = β[f’(Kss) + (1 – δ)] 
f’(Kss) = 
1
β
 – (1 – δ) 
- The budget constrain becomes: 
Css + Kss + mss + bss = f(Kss) + (1 – δ)Kss – 
mss
(1 + πt)
 – (1 + rss)bss 
Css + Kss + mss + 0 = f(Kss) + Kss – δKss + 
mss
(1 + πt)
 + (1 + rss)0 
Css = f(Kss) + tss + 
mss
1 + πss
 – δKss – mss 
- In these models the only option Central Banks have to modify the money supply is through 
money loans, which in this case is known as money transfer. Thus: 
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Mt – Mt-1 = Tt → 
Mt−Mt−1
Pt
 = 
Tt
Pt
 → mt – 
Mt−1
Pt−1
 
Pt−1
Pt
 = tt → mt – 
mss
1 + πss
 = tt 
- Replacing this equation in the budget constrain: 
Css = f(Kss) + (mt – 
mss
1 + πss
) + 
mss
1 + πss
 – δKss – mss 
 
Css = f(Kss) – δKss 
 
- The main conclusion is that money can be catalogued as a super-neutral variable in the MIU 
model. Money being super-neutral means that neither the level of the money supply nor its 
growth rate has effects real variables. Money only affects nominal variables as prices, wages 
or exchange rates. The steady values (xSS) do not depend on the level nor the growth rate of the 
money stock. 
- Now, it remains to determine what is mSS. We can use the equation of the relation on marginal 
utilities: 
Um
′(Ct, mt)
Uc
′(Ct, mt)
 = 1 – 
1
(1 + it)
 = 1 – 
1
(1 + rt)(1 + πt+1)
 
 
- Using the Euler equation with equal marginal utilities of consumption, β(1 + rt) = 1: 
Um
′(Css,mss)
Uc
′(Css, mss)
 = 1 – 
β
1 + π𝑠𝑠
 
 
- This means that if there is an increase in inflation, there will be an decrease in the real money 
supply because it would will produce an increase in 1 – 
𝛃
𝟏 + 𝛑𝒔𝒔
, then, 
𝐔𝐦
′(𝐂𝐬𝐬,𝐦𝐬)
𝐔𝐜
′(𝐂𝐬𝐬,𝐦𝐬𝐬)
, also has to 
increase to keep the relation. 
- As we know, Umm
′′(Css, mss) < 0, if Ucm
′′(Css, mss) > 0, the real money supply has to decrease. 
A steady state only exists if the previous equation has a solution for mss. 
- Inflation does not affect the optimal level of consumption (Css), but it does affect mss. Thus, while 
Um
′(Css, mss) > 0, we should lower πss to increase mss. In the optimal: 
𝐔𝐦
′(𝐂𝐬𝐬, 𝐦𝐬𝐬) = 0 
- Then: 
Um
′(Css,mss)
Uc
′(Css,mss)
 = 0 = 1 – 
β
1 + πss
 
β = 1 + πss 
1 = 1 + iss 
iss = 0 
- The Friedman Rule says that, as r + π = i: 
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πss = - rss 
- As the opportunity cost of money is the nominal interest rate, we can say that, in steady state, 
there is no nominal cost of holding money, but there is a real benefit in holding money 
because πss = - rss. The amount of money that I hold today has more value tomorrow. Then, 
the marginal cost of producing money is zero. 
- The MIU model saysthat the demand for money comes mechanically by adding real money (m) 
in utility. The Cash In Advance (CIA) model tells us that before buying consumption goods you 
need to have money you brought from previous periods. 
- Let’s suppose the period “t” is represented by one month. You get your payment at the end of 
the month, but you need to consume goods during it. Thence, you must have money from last 
month. This is the CIA constraint: 
PtCt < Mt-1 + Tt 
- In real terms: 
Ct < 
Mt−1
Pt−1
Pt−1
Pt
 + tt = 
mt−1
1 + πt
 + tt 
 
- As long as the nominal interest rate is zero, our second constraint will be expressed as an 
equality. This is because there is no opportunity cost of holding money, then, you will only hold 
as much as you need, not more. 
- But we will assume that the nominal interest rate is positive, so there is an opportunity cost. 
The agents maximize their utility constrained by the budget constraint and the cash in advance 
constraint: 
 
- The First Order Conditions are: 
1- [Ct]: βtU’c(Ct) – βt λt – βtμt = 0 → U’c(Ct, mt) = λt + μt 
2- [bt]: – βtλt + βt+1λt+1(1 + rt) = 0 → 
λt
βλt+1
 = (1 + rt) 
3- [Kt]: – βtλt + βt+1λt+1[f’(Kt) + (1 – δ)] = 0 → 
λt
βλt+1
 = f’(Kt) + (1 – δ) = (1 + rt) 
4- [mt]: – βt λt + βt+1 
λt+1
(1 + πt+1)
 + βt+1 
μt+1
(1 + πt+1)
 = 0 → μt+1 + λt+1= 
(1 + πt+1)λt
β
 
- Just as the MIU we get almost the same equations, the only thing we have to do is “move them 
through time”: 
1- f’(Kt) = rt + δ 
2- 
UCt
′(Ct,)
UCt+1
′(Ct+1,)
 = 
λt + μt
λt+1 + μt+1
 = 
𝛽(λt + μt)
(1 + πt+1)λt
 = 
𝛽
1 + πt+1
 + 
𝛽
(1 + πt+1)λt
[
(1 + πt)λt−1 
β
 - λt] = 
VGC 
 
14 
 
𝛽
1 + πt+1
 + 
(1 + πt)λt−1 
(1 + πt+1)λt
 - 
𝛽λt 
(1 + πt+1)λt
 = β(1 + rt-1) 
(1 + it−1)(1 + rt)
(1 + rt−1)(1 + it)
 = β(1 + rt) 
(1 + it−1)
(1 + it)
 
 
- Rearranging: 
 
UCt
′(Ct)
1 + it−1
 = β(1 + rt) 
UCt+1
′(Ct+1)
1 + it
 
- This means that the ratio of my marginal benefit of consumption at “t” over the cost of 
opportunity at “t” has to be the same as the ratio of the marginal benefit of waiting to consume 
at “t+1” over the opportunity cost at “t+1”. 
- If the nominal interest rate changes between every period there is a wedge, the Euler 
equation is different from the benchmark. 
- In the steady state we have bss = 0, marginal benefit of consumption is the same in each period, 
nominal interest rate also stays constant through time, but it still is different to zero. These 
imply that: 
1 + rss = 
1
β
 
- Hence: 
f'(Kss) = 
1
β
 – (1 – δ) 
- The budget constraint wit bss = 0 imply: 
Css = f(Kss) + 
mss
1 + πss
 + tss – δKss – mt 
- But, as in the MIU model: 
Mt – Mt-1 = Tt → 
Mt−Mt−1
Pt
 = 
Tt
Pt
 → mt – 
Mt−1
Pt−1
 
Pt−1
Pt
 = tt → mtss – 
mss
1 + πss
 = ttss 
- Thus, 
Css = f(Kss) – δKss 
 
- The steady level of money supply comes from the CIA constraint: 
Css = 
 m𝑠𝑠
1 + π𝑠𝑠
 + tss 
- Since tss = mtss – 
mss
1 + πss
 
Css = 
 m𝑠𝑠
1 + π𝑠𝑠
 + mtss – 
mss
1 + πss
 = mtss 
- As we can see in the household utility is ∑ βt{U(Css)∞t=0 , the people´s utility is independent of 
πss. Then, any level of inflation is optimal. 
VGC 
 
15 
 
- We have studied money supply and money demand, now we have to move toward the 
monetary equilibrium. How is the price level determined in the long run? 
- The theory quantitative of money explains the relationship between amount of money in 
circulation and the general level of prices, in its simplest formulation. 
- The relationship between the quantity of money and prices can be expressed through the called 
the equation of change. Suppose that in one year, people buy T goods using money, T being 
the number of transactions. The average price of goods is P. The money circulating the 
economy (money supply) is M. Then, we can define the velocity of money (speed of 
circulation, for how many hands the money passes) as: 
 
M x V = P x T 
- A simple example would be the case of an economy in which bread is the only good that is 
produced, and its annual production is 60 kilos. Suppose that the bread price is P = $200 per 
kilo, we also have there were sixty transactions, T = 60 per year. So, T x P = Y = $ 12.000 a year. 
- Suppose that the amount of money in the economy is M = $1.000, then the speed of money is 
12. This means that to make $12.000 in transactions with an offer of $1.000 in the economy 
means that each monetary unit ($) changes hands 12 times. 
- Though, T x P is not always the GDP, because there might be inventories (we don’t sell 
everything we produce) and because second hand transactions wouldn’t be counted. Still, we 
will consider that T x P = Nominal GDP (Y). 
- Then, the Real GDP can be described as T = 
Y
P
 = y. 
- To move from the quantitative equation to the quantitative theory as such it is necessary to 
make some adjustments in mathematics. Given that the Central Bank is the one that decides 
the amount of money that will be in the economy, the M in the equation represents the money 
supply, but in the equilibrium both the supply and the nominal demand for money is the same, 
for what we now have that the equation is: Md x V = P x y. If we clear the real demand for money 
we get the quantitative theory of money: 
Md = 
P x y
V
 
- If there is an increase in the nominal interest rate, the real demand for money decreases then, 
as the nominal production is constant (it is not affected by a change in nominal interest), the 
speed of circulation would increase. 
- If the opportunity cost of money is high, agents seek to quickly get rid of money that is not 
generating income (“money in the pocket"). 
- Let’s see if we can rearrange the equation of change: 
 
M x V = P x y 
ln M + ln V = ln P + ln y 
VGC 
 
16 
 
- By considering different periods: 
ln M t + ln V t = ln P t + ln y t 
ln M t-1 + ln V t-1 = ln P t-1 + ln y t-1 
 
ln M t - ln M t-1 + ln V t - ln V t-1 = ln P t - ln P t-1 + ln y t - ln y t-1 
 
- But, as ln A – ln B = 
A−B
B
 for small values of A and B: 
Mt − Mt−1
Mt−1
 + 
Vt − Vt−1
Vt−1
 = 
Pt − Pt−1
Pt−1
 + 
yt − yt−1
yt−1
 
 
ΔMt
Mt−1
 + 
ΔVt
Vt−1
 = 
ΔPt
Pt−1
 + 
Δyt 
yt−1
 
 
M̂ + V̂= P̂ + ŷ 
 
- Friedman postulates that in the long run the interest rate remains unchanged, then the speed 
of circulation will also be constant and that the product in the economy has constant decay 
rates, that is ŷ = g. 
- More than a theory, it is considered as an identity since it must always be complying. In addition, 
this equation manages to relate the rate of monetary growth (�̂� = μ) with inflation and the 
growth of the country, that is, from the equation of change you can reach the following 
relationship: 
M̂ + V̂= P̂ + ŷ 
 
μ + 0 = π + g 
 
μ – g = π 
- In the long run there is inflation when μ > g. Since g is constant, an increase in the rate of 
monetary growth implies and increase in the inflation in a 1: 1 ratio since inflation is a 
monetary phenomenon in the long term. 
- Now, we depart from a standard money demand: 
𝑀𝑑t
Pt
 = L(yt, it) 
- The money demand increases with the economy’s growth and decreases with the nominal 
interest rate. We have to assume the classical dichotomy; real and nominal variables are 
independent (yt has nothing to do with M) and we also assume that the nominal interest rate is 
exogenous. 
- Then, setting the money demand equal to the money supply (Mdt = MSt = Mt): 
Mt
L(yt, it)
 = Pt 
VGC 
 
17 
 
- Hence, the price level is determined by the money. 
- Now, we drop the assumption that it is exogenous. In order to simplify, we assume that: 
M𝑑t
Pt
 = L(yt, it) = ytα (1 + 𝑖𝑡)−𝜂 
- Let mdt = lnMdt and that pt = lnPt. Then, we have: 
 
mdt – pt = αlnyt - ηln(1 + it) 
 
- But, by the fisher equation with perfect foresight we can write it as: 
 
mdt – pt = αlnyt - η ln(1 + rt) - η(pt+1 - pt) η 
 
- We assume that the output (real GDP)and real interest rate are constant, and we also assume 
that yt and rt are determined on the real side (classical dichotomy). Adding money equilibrium: 
 
 mdt – pt = C - η(pt+1 - pt) (1) 
- With C = αlnyt - ηln(1 + rt). Solving pt: 
pt = 
𝐦𝐭 − 𝑪 + 𝜼𝒑𝒕+𝟏
𝟏+𝛈
 (1,1) 
- Given mt > 0, a solution to this equation is a function, f(m0, m1, m2…) such that pt = f(m0, m1, 
m2…) satisfies the equation. 
- Using (1) forward to replace pt+1 in (1): 
 
pt = 
mt − 𝐶
1+η
 + 
𝜂
1+η
(
mt+1 − 𝐶 + 𝜂𝑝𝑡+2
1+η
) 
- We can rewrite the sum as: 
pt = ∑ (
η
1+η
)𝑖 
− C
1+η
1
0 + ∑ (
η
1+η
)𝑖 
mt+i
1+η
1
0 + (
η
1+η
)2pt+2 
- After n interactions: 
pt = ∑ (
η
1+η
)𝑖 
− C
1+η
n
0 + ∑ (
η
1+η
)𝑖 
mt+i
1+η
n
0 + (
η
1+η
)𝑛+1pt+n+1 
 
- We search for a solution for (1) that satisfies lim
n→∞
(
η
1+η
)
n+1
pt+n+1 = 0. We guess it is true and 
then we will verify. It does if mt is bounded/limited, because in that way pt is bounded as well. 
Consequently, we have: 
pt = 
− C
1+η
 ∑ (
η
1+η
)𝑖 n0 + 
1
1+η
 ∑ (
η
1+η
)𝑖 n0 mt+i 
- Note that: 
1
1+η
 ∑ (
η
1+η
)𝑖 ∞0 = 
1
1+η
 (
1
1 − 
η
1 + η
) = 1 
VGC 
 
18 
 
- Thus: 
pt = - C + 1
1+η
 ∑ (
η
1+η
)𝑖 k0 mt+i (2) 
- This is known as the fundamental solution to (1). 
- The intuition tells us that if there is an increase in the money supply, the price level will also 
increase. 
- If mt+i is constant and equal in every period, then: 
pt = - C + m 
- Is there another solution to (1)? A non-fundamental solution? Yes 
pt = - C + m + b0(
1 + η
η
)t (3) 
- Were b0(
1 + η
η
) is the bubble commitment. What is the bubble commitment? 
- The idea is for this kind of equilibrium is the following: suppose people expect p t to rise in the 
future (b0 > 0). Then, the Fisher equation implies that the nominal interest rate will be high, 
reducing agents incentives to hold money. But then, there is too little money demand for too 
much money supply, and prices have to increase to equate the demand and the supply for real 
balances. But this confirm agents initial expectations (a self-fulfilling prophecy). This is Cagan’s 
model in the end, 𝛈 represents the expected growth of price level each period. 
- Another way to see it is to imagine that we live in Venezuela. From experience we know that 
inflation rises every day, so every day we get up thinking that it will increase inflation (b0) and 
with each passing day, we believe that inflation will grow even more (
𝟏 + 𝛈
𝛈
)t. But, inflation 
does not rise just because people believe it. The problem is that the government must make 
policies taking into account the beliefs of the people, so it makes policies that seek to stop the 
inflation. But, these policies would exist only if there was an actual increase in the price level. 
People see these policies and think "I was right, there will be a more inflation, that's why they 
made new policies". As the whole economy is prepared and self-confirmed the rise in prices, 
the price level goes up. Self-fulfilled prophecy. 
- This is the reason why it is very important to regain the confidence of the people when it comes 
to reversing and eliminating inflation. If they do not believe you, and do not trust your policies, 
inflation will continue. This is the same reason why Argentina cannot stop inflation, nobody 
trusts the government. 
- Now, let us consider an economy with no banks, in other words, all the money in the economy 
is currency. The real revenue obtained by the government when they expand the money supply 
is known as seigniorage (S). Seigniorage works as a tax on real money holdings. Thus, it can be 
written mathematically as: 
VGC 
 
19 
 
St = 
ΔM
P
 = 
Mt – Mt−1
Pt
 = mt – 
mt−1
1 + πt
 
- If we assume that we are in steady state, mt = mt-1 = m: 
S = m – 
m
1 + π
 = 
π
1 + π
 m 
- 
𝛑
𝟏 + 𝛑
 is the “tax rate” and m is the “tax base”. Hence, 
𝛑
𝟏 + 𝝅
 m is an “inflation tax”. 
- With a usual real money demand, 
𝐌𝐝𝐭
𝐏𝐭
 = L(i, y), thence, the nominal money demand would be 
Md = PL(i, y). When we apply total differentials: 
𝚫M = P[Li 𝚫i + Ly 𝚫y] + 𝚫P L 
- Dividing both sides by M = PL: 
ΔM
M
 = Li 
Δi
L
 + Ly 
Δy
L
 + 
ΔP
P
 
 
ΔM
M
 = Li 
i
L
 
Δi
i
 + Ly 
y
L
 
Δy
y
 + π 
 
ΔM
M
 = εL,i Δi
i
 + εL,y Δy
y
 + π 
- Rearranging the definition of S: 
St = 
𝚫𝐌
𝐏
 = 
𝚫𝐌
𝐌
𝐌
𝐏
 = 
𝚫𝐌
𝐌
 L(i, y) = [εL,i 𝚫𝐢
𝐢
 + εL,y 𝚫𝐲
𝐲
 + π] L(i, y) 
- Let’s see how seigniorage responds to inflation in steady state, where there are no changes in 
the nominal interest rate and no changes in the real GDP: 
S = L(i, y) π 
- Assuming perfect foresight, i = r + π, also assuming that the real GDP is exogenous (super-
neutrality) we get: 
S = L(r + π, y) π 
∂S
∂π
 = L + 
∂L
∂π
π = L + 
∂L
∂π
 
π
L
 L 
∂S
∂π
 = L[1 + εL,π] = M
P
 [1 + εL,π] 
- Thus, the derivative will be positive as long as εL,π > -1. In other words, for S to increase with 
inflations increases, the real money demand cannot react too much in π. 
- In general, we have that real money demand is sensitive to inflation for high values of inflation, 
εL,π > - 1. 
VGC 
 
20 
 
- Logically, if the tax rate increases too much, the tax base will begin to decrease. This can be 
seen in the Laffer curve: 
 
- We will have a Laffer curve as long as interest rate affects negatively on the real money 
demand. For example: M/D = L(i, y) α + βi + δy. If β = 0, we would have a linear function, no 
max, no min. If β > 0, we would have a convex parabola, there would be an inflation that 
minimizes the seigniorage instead of maximizing it. Thence, β must be negative. 
- Note that we talk about π as πe, in such case we can prove that the inflation is generated 
exclusively by the monetary growth as long as other variables (output and interest rate) 
remain constant, this is achieved in the steady state. 
- In short, when there is perfect foresight and in the steady state we get: 
< 
𝚫M = P[Li 𝚫i + Ly 𝚫y] + 𝚫P L 
 
𝚫M = 𝚫P L 
 
 
𝚫𝐌
𝐌
 = 
𝚫𝐏𝐋
𝐏𝐋
 = 
𝚫𝐏
𝐏
 = π 
- Now we can discuss several mechanisms by which hyperinflations can occur. The idea is used 
that they are inflations greater than 50% per month, this is approximately 13,000% per year. 
- However, there are those who argue that even without the need to reach such a high level of 
inflation, the central characteristic of hyperinflation is that there is an exponential increase in 
the inflation rate, which has as a counterpart a reduction in the amount of money up to 0. It 
can have a stabilization in before reaching 50% monthly, but also be a process of explosion in 
inflammation. 
- Hyperinflations can be results of: 
1- Expectations: Repetitive self-fullfiling prophecies. 
2- Fiscal needs that lead to high seigniorage. 
- The nominal exchange rate between the domestic currency (the national currency, CLP$) and 
the foreign currency (US$ for example) is how many Chilean pesos I need to buy one American 
dollar (e), in other words, the price of the USD in CLP. 
VGC 
 
21 
 
- Hence, e = 
𝐂𝐋𝐏$
𝑼𝑺$
. If e decreases, it means that the Chilean peso is more appreciated. This is 
because you need less CLP to buy one USD. 
- Now, for simplicity we assume Chile and the US produce a single and identical good. Then we 
can obtain the real exchange between CLP and USD: 
 
q = 
How many CLP I need to buy 1 unit of the good in the US
How many CLP I need to buy 1 unit of the good in Chile
 = 
Price of the US good in CLP
Price of the Chilean good in CLP
 = e 
P∗
P
 
 
- If q > 1 buying domestically is cheaper. When q decreases we can say that there is a real 
appreciation of the domestic currency, that is to say, buying domestic is harder/more expensive 
or that buying abroad is cheaper. 
- When there are many goods the levelof q has no real meaning, but the changes still have: 
log qt = log et + log P*t – log Pt 
- Hence: 
�̂� = �̂� + 𝑷∗̂ – �̂� = �̂� + π* – π 
- If there is no change in the nominal exchange rate, the changes in the real exchange rate will 
only rely on the US and Chilean inflations. 
- The theory of purchase power parity (PPP) holds that the value of goods is the same in all 
parts of the world: 
P = e P* 
- This means that q = 1, this is due because of the relation between the price of the good in the 
same currency. The good has the same price (given a determined currency for both economies) 
in both countries. If the real exchange rate = 1, this means that the changes in the nominal 
exchange rate (e) can only be explained by nominal variables (P and P*). 
- This supposes no cost of transactions nor tariff tax, hence, one good has to have the same price 
because of the law of only price. If Chile sells apples at $2 and the US sells them at $1, everyone 
will buy from the US. The US apple’s price will rise because of the increase in the demand. In the 
long run the prices in both economies will meet. 
- Therefore, the real exchange rate is constant. This is known as the "tiered" version of PPP. 
Undoubtedly this is extreme, because it would be necessary to consider that there are different 
tariffs for the same good between countries, there are transport costs, etc., which mean that 
this relationship is not fulfilled. 
- In its weakest version, or in "variation rates", the PPP theory states that the percentage change 
in the price in a country is equal to the percentage change of the same good abroad. This is 
�̂� + 𝑷∗̂ = �̂� 
- In this case, recognizing that prices may differ in different markets, changes in prices in one 
market are transmitted proportionally to the other. This theory has a strong assumption of 
VGC 
 
22 
 
"nominal neutrality", since all changes in the nominal exchange rate are transmitted one by 
one at prices, and the real exchange rate cannot be altered. Classic dichotomy. 
- This theory fails empirically for reasonable periods. Although in very prolonged periods - until a 
century - it seems that between countries the prices converge, this does not happen in the 
relevant periods for our analysis. This does not mean that this theory is useless. 
- Let us look into the equilibrium. We know that the GDP, by definition, is: 
 
PY = P[C + I + G + X – M] = P[C + I + G + X] – eP*M 
 
- Where eP*M is the total CLP spent on imports. Rearranging we get that: 
 
Y = C + I + G + X – qM = C + I + G + XN 
 
- We assume that X depends on q and in Y*. A depreciation in the domestic currency (increase 
in q), will imply that it is more attractive to buy in our economy and not abroad, hence, there 
will be an increase in the exports. 
- If there is an increase in the foreign real GDP, people abroad will have more money, so they 
will be able to demand more domestic products, exports increase. 
- Looking at the imports we can say that it depends on q as well, but it also depends on the 
domestic real GDP (Y) and the tariff taxes (t). Both q and tariff taxes have negative impact on 
the imports, but the real GDP has a positive impact on the imports. 
- Thus, the net exports [XN (Y*, Y, t, q) = X(q, Y*) – qM(q, Y, t)] will depend on foreign real GDP 
(Y*), domestic real GDP (Y), the tariff taxes (t) and q. The only factor that has opposite effects is 
q. So, to know the impact of q on the net exports we have to see: 
∂XN
∂q
 = 
∂X
∂q
 – [M - q
∂M
∂q
] 
- From this equation we cannot define if the effect is either positive or negative. But we will 
assume that it is positive because of the Marshall-Lerner condition. 
- To the extent that X and M react, the effects of volume would begin to dominate. In fact, there 
are two important concepts that arise from this: 
1- J Curve: Refers to the way that the evolution of the net exports have in time as a result of 
a depreciation of domestic currency (increase in q). At the beginning it deteriorates (the 
decreasing part of the J) as a product of the price effect, but then it meets the Marshall-
Lerner condition and improves as the volumes respond. This is because prices always adjust 
faster than quantities. 
 
VGC 
 
23 
 
 
- When we speak at a macroeconomic level of the flow of production, we can represent several 
aspects of a country. We can obtain different data depending on whether the presence of both 
the government and free trade exists. 
- In order to better understand the following cases, we must bear in mind that everything that is 
not spent (what is saved) is inverted. 
- The most basic model corresponds to a closed economy without government, where GDP is 
clearly composed of the sum of household spending and investment in the country. By clearing 
the investment in the equation, we obtain that this equals the difference between GDP and 
consumption, this subtraction is known as private saving (SP). 
 
Y = C + I 
Sp = I 
 
- Once the government is included in the equation, it is most likely that the GDP does not 
increase. Since the government gets its income from the taxes (T) it charges families and 
businesses, these are considered in the equation as part of the flow. What the government does 
not spend, saves it, giving rise to government savings or public savings (SG). 
 
 (Y – T – C) + (T – G) = I 
SP + SG = I 
 
- When saving is taken into account by the private sector and by the public side, in a general 
aspect we are visualizing the savings of the country or rather the domestic savings (SD). 
- This leaves us with: 
 
SD = SP + SG 
 
- If we consider free trade, we find all the components of GDP. By reordering the equation, we 
obtain that the investment is equal to the domestic savings plus the net imports. 
- The external savings (SE) represents the difference between imports and exports of goods. 
VGC 
 
24 
 
- The rest of the world has income from this economy through the payment that the national 
economy makes for the goods it consumes and are produced abroad, that is, the payment of 
imports. 
- The other source of income is the payment you receive for the assets you have in the country 
(interest, dividends, etc.), net payments of the rest of the world (F). On the other hand, the rest 
of the world pays this economy the goods that they export to the rest of the world. Therefore, 
external savings is: 
 
SE = M + F – X = F - XN 
 
- Now that we have all components: 
 
Y = C + I + G + (X – M) 
(Y – T – C) + (T – G – F) + (M + F – X) = I 
SP + SG + SE = I 
 
- Whatever the rest of the world saves, or un-saves will affect us. We keep the records in the 
current account (CC). The current account is a factor of the balance of payments, it includes 
the real operations (trade in goods and services) and rents that take place between the 
residents of the country and the rest of the world in a given period of time. In short: 
 
-SE = CC = SD – I = (Y – T – C) + (T – G – F) – I 
 
- In fact, we can make a graphic which shows the relation between investment, internal/domestic 
savings and interest rates: 
 
- Moreover, we could see how the balance in CC shifts due to external shocks (considering the 
interest rate as exogenous). For example, let’s consider and increase in consumption and an 
increase in investment. The increase in consumption would cause a decrease in the private 
saving, hence, the domestic saving curve would suffer a reduction. On the other hand, an 
increase in investment would produce a lifting of the investment curve: 
VGC 
 
25 
 
 
- Keeping the real interest rate constant, this would generate a huge deficit in the balance of the 
current account. 
- As we have already seen, net exports correspond to GDP minus domestic absorption, that is, 
the trade surplus is the excess of the product over expenditure. The current account is defined 
as: 
CC = XN – F- Therefore, the deficit in the current account measures the excess of expenditure over income. 
 
 
 
- If we assume that all SE, Y*, Y, F and t are exogenous. Then, the last equation determines 
equilibrium: 
 
CC = XN – F 
 
- You have to be careful when choosing which curve you are going to move, you have to consider 
all the variables. The CC shifts because of changes in q, Y*, Y, t and F. On the other hand, to see 
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26 
 
how -SE moves we have to see the graphic, if there was an increase in the CC deficit, it means 
that the -SE curve shifted to the left (as in the example above). 
- From this we can get different types of conclusions. For example, an increase in government 
spending generates a depreciation of domestic currency. This is because an ↑G, ↓SD, ↓-SE, ↓q, ↓e, 
↑CLP. 
 
- If we take this to the extreme, we would get a twin deficit. A strong fiscal defici (SG < 0) that 
leads to a strong debt in the current account. It represents a potential source of instability for 
the nation and the whole of the international economy. 
- Let’s take investment as exogenous, SE adjusts to ensure I = S. Government increases public 
spending, but the taxes remain constant, then, there is a decrease in public savings. If SP is given, 
SE must increase (or -SE must decrease). This is shown in the graphic above. 
- We can see that the increase in public spending does not require reallocation of resources 
within the economy, there is only an increase in demand for goods produced abroad (↓q). This 
generates a worse state of the CC (now there is more money leaving the country than entering). 
- In extreme cases, public spending will be so high that SG < 0 in some point, if the private saving 
and the overall saving (S) stay the same, SE will have to increase, hence, there will be a huge 
deficit in the currency account. 
- If a Chilean spends CLP$1 in Chilean bonds, it gets CLP($1 + it) after 1 year. If the same Chilean 
spends CLP$1 to buy US bonds its expected return in CLP is: 
1 + i = 
1+i∗
et
 E(et+1) 
- This is the uncovered interest rate parity. Where 
𝟏+𝒊∗
𝒆𝒕
 is the total return in USD and E(et+1) is 
the expected exchange rate. If we have perfect capital mobility (interest rate remains 
unchanged), a risk neutral agent must be indifferent. 
- If we define Δe/et = (E(et+1) – et)/et which corresponds to the expected depreciation rate, we 
can write the previous equation as: 
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27 
 
1 + it = (1 + it*)(1 + 
𝚫𝐞
𝐞
) 
- By solving though lineal aproximation 
it = it*+ 
Δe
e
 
- Doing some few changes we could get (exercise 3, TA Session 5): 
rt = rt* + 
Δqet+1
qt
 
- However, in case there are no restrictions in the financial markets, it is possible to make a risk-
free operation using the futures markets. For this, if someone borrows in pesos at a rate i and 
invests in dollars, he knows that at the end of the period he will have 1+i* for every dollar 
invested. 
- Therefore, you can sell the future USD for CLP to a value of ft + 1 today. That is to say, in t + 1 an 
forward contract (ft + 1) CLP per USD will be paid at the agreed price in t. In t+1 delivery with 
certainty 1 + i* dollars, sells and receives them, without risk, (1 + i *) ft + 1 pesos. 
- Assuming that the instruments in which they are invested are free of risk (i and i* are risk-free 
rates), this operation has no uncertainty. Therefore, with perfect capital mobility, the parity of 
interest covered must be met exactly: 
 
- Or 
𝟏 + 𝐢𝐔𝐒𝐃 = 
(𝟏 + 𝐢𝐂𝐋𝐏)∗(𝐄𝐱𝐜𝐡𝐚𝐧𝐠𝐞 𝐑𝐚𝐭𝐞 𝐓𝐨𝐝𝐚𝐲 
𝐂𝐋𝐏
 𝐔𝐒𝐃
)
 𝐅𝐨𝐫𝐰𝐚𝐫𝐝 𝐂𝐨𝐧𝐭𝐫𝐚𝐜𝐭 ( 
𝐂𝐋𝐏
𝐔𝐒𝐃
 )
 
 
- The most important thing is that, if the Exchange rate is like 
𝐂𝐋𝐏
𝐔𝐒𝐃
, the exchange rate that goes 
in the fraction must be iCLP (as shown above). If the exchange rate is 
𝐔𝐒𝐃
𝐂𝐋𝐏
, then the interest 
rate that goes in the fraction is iUSD. 
 
 
 
 
 
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28 
 
Monetary policy and equilibrium with fixed prices (short run) 
- Let’s go from this discussion to an equation describing the demand for money. Denote the 
amount of money people want to hold (their money demand) by Md. The demand for money in 
the economy as a whole is just the sum of all the individual demands for money by the people 
and firms in the economy. 
- Therefore, it depends on the overall level of transactions in the economy and on the interest 
rate. The overall level of transactions in the economy is hard to measure, but it is likely to be 
roughly proportional to nominal income (income measured in dollars). 
- If nominal income were to increase by 10%, it is reasonable to think that the dollar value of 
transactions in the economy would also increase by 10%. So, we can write the relation between 
the demand for money, nominal income, and the interest rate as: 
 
Md = PY L(i) 
- Where PY is the nominal income. 
- As discussed before, Central Banks can decide how much money supply. Let’s suppose the 
money supply is M, in equilibrium we would have that the money supplied is the same as the 
money demanded. Hence, M = PY L(i). The graphic equilibrium is: 
 
- Now that we have characterized the equilibrium, we can look at how changes in nominal 
income or changes in the money supply by the Central Bank affect the equilibrium interest 
rate. Let’s say, the only good that is exported suffers an increase in its price because of 
international demand. Then, the nominal GDP increases. What happens with the nominal 
interest rate? 
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29 
 
 
- The reason is that at the initial interest rate, the demand for money exceeds the supply. The 
increase in the interest rate decreases the amount of money people want to hold money (𝑐 
decreases) and reestablishes equilibrium. 
- Let’s assume now that the Central Banks just wants to affect nominal interest rate by printing 
more money: 
 
- An increase in the supply of money leads to a decrease in the interest rate. The decrease in the 
interest rate increases the demand for money (from A to A’) so it equals the now larger money 
supply. 
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30 
 
- The main conclusion is that the Central Bank can, by printing/withdrawing any amount of 
money it wants, choose the interest rate. If it wants to increase the interest rate, it decreases 
the amount of Central Bank money. If it wants to decrease the interest rate, it increases the 
amount of Central Bank money. 
- But this conclusion comes with an important warning: The interest rate cannot go below zero, 
a constraint known as the zero lower bound (ZLB). When the interest rate is down to zero, 
monetary policy cannot decrease it further. Monetary policy no longer works, and the 
economy is said to be in a liquidity trap. 
- If i = 0, in the margin, people are indifferent between holding money and having bonds. As the 
interest rate decreases, people want to hold more money (thus, fewer bonds): The demand for 
money increases. 
 
- The liquidity trap begins once i = 0 is reached. Central Banks loses control of monetary policy. 
- Now consider the case where the money supply is Ms′, so the equilibrium is at point B; or the 
case where the money supply is Ms′′, so the equilibrium is given by point C. In either case, the 
initial interest rate is zero. And, in either case, an increase in the money supply has no effect on 
the interest rate, only contractive monetary policies would have impact on the economy. 
- Now let’s focus on the relationship between interest rates and asset markets, also known as 
real output in goods and services market plus money market. And how do we settle this? 
Through the IS-LM model. This model consists in the intersection of the “investment-saving” (IS) 
and “liquidity preference-money supply” (LM) curves which generate a “general equilibrium” in 
both goods and financial markets. 
- For the investment-saving curve, the independent variable is the interest rate and the 
dependentvariable is the level of income. The IS curve is defined by the equation: 
 
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31 
 
Z = c [Y – T] + I + G + XN 
 
Z = c YD + I + G + XN 
 
 
- Where Z represents income, c represents the marginal propensity to consume, thus, c[Y – T(Y)] 
represents consumer spending as an increasing function of disposable income. I represent 
investment, which is exogenous, and we have both government spend and net exports. 
- Investment can be affected through an increase in the nominal GDP (Y). Higher Y means higher 
demand and results in higher levels of investment. 
- Investment is also affected by the nominal interest rate. If interest rates increase, it is more 
expensive to ask for loans, thus, investment decreases. 
- So, we can write the relationship as: 
 
- Graphically, we can describe the equilibrium in the goods market as: 
 
 
- The 45-degree line is the production curve. 
- Let’s now derive what happens if the interest rate changes. Suppose that the demand curve is 
given by Z, and the initial equilibrium is at point A. Suppose now that the interest rate increases 
from its initial value i to a new higher value i′. 
- At any level of output (Y), the higher interest rate leads to lower investment, lower demand. 
The demand curve Z shifts down to Z′. The new equilibrium is at the intersection of the lower 
demand curve Z′ and the 45-degree line, at point A′. The equilibrium level of output is now equal 
to Y′. 
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32 
 
- In other words, the increase in the interest rate decreases investment. The decrease in 
investment leads to a decrease in output, which further decreases consumption and 
investment, through the multiplier effect. 
- The curve created through by the different equilibriums in a graphic of output and interest rate 
is the IS curve. Graphically: 
 
- Once we have the IS curve we can see how it changes when the other variables, such as taxes 
and government spending increase. The IS curve gives the equilibrium level of output as a 
function of the interest rate. It is drawn for given values of taxes and spending. 
- Now consider an increase in taxes, from T to T′. At a given interest rate, say i, disposable income 
decreases, leading to a decrease in consumption, leading in turn to a decrease in the demand 
for goods and a decrease in equilibrium output. 
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33 
 
- The equilibrium level of output decreases from Y to Y′. Put another way, the IS curve shifts to 
the left: At a given interest rate, the equilibrium level of output is lower than it was before the 
increase in taxes. 
 
- More generally, any factor that, for a given interest rate, decreases the equilibrium level of 
output causes the IS curve to shift to the left. We have looked at an increase in taxes. But the 
same would hold for a decrease in government spending, or a decrease in consumer confidence 
(which decreases consumption given disposable income). In short, anything that decreases Z in 
the equation: Z = c [Y – T] + I (Y, i) + G + XN, shifts the IS curve to the left. 
- Symmetrically, any factor that, for a given interest rate, increases the equilibrium level of output 
(a decrease in taxes, an increase in government spending, an increase in consumer confidence, 
etc.) causes the IS curve to shift to the right. 
- Let’s take a step back and recall the equilibrium in the financial market: 
M = PY L(i) 
- Recall that nominal income divided by the price level equals real income, Y. Dividing both sides 
of the equation by the price level P gives: 
M
P
 = Y L(i) 
- Hence, we can restate our equilibrium condition as the condition that the real money supply 
has to be equal to the real money demand, which depends on real income, Y, and the interest 
rate, i. 
- In deriving the IS curve, we took the two policy variables as government spending, G, and taxes, 
T. In deriving the LM curve, we have to decide how we characterize monetary policy, as the 
choice of i (assumption 1), the interest rate, or as the choice of M (assumption 2), the money 
stock. 
- If we think of monetary policy as choosing the nominal money supply, M, and, by implication, 
given the price level which we shall take as fixed in the short run, choosing M/P, the real money 
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34 
 
stock, the previous equation tells us that real money demand must be equal to the given real 
money supply. 
- Thus, if for example, real income increases, increasing money demand, the interest rate must 
increase so as money demand remains equal to the given money supply. In other words, for a 
given money supply, an increase in income automatically leads to an increase in the interest 
rate. 
- This is the traditional way of deriving the LM relation and the resulting LM curve. We assume 
that the Central Bank fixes the interest rate to affect the monetary stock. Although, in the past, 
Central Banks thought of the money supply as the monetary policy variable, they now focus 
directly on the interest rate. 
- In assumption 1 they choose an interest rate, call it 𝑖, and adjust the money supply (money 
printing/withdrawing) so as to achieve it. This will make for an extremely simple LM curve, 
namely, a horizontal line, at the value of the interest rate, 𝑖, chosen by the Central Bank. 
 
dick 4 wo reads 
- Thence, both curves can be described as: 
 
IS relation: Z = C[Y – T] + I(Y, i) + G 
LM relation: i = 𝐢 (A1) 
 
- On the other hand, we can also assume that Central Banks fixes the monetary stock to affect 
the interest rate, in which case the LM curve derivation would be: 
 
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35 
 
 
 
- Thence, both curves can be described as: 
 
IS relation: Z = C[Y – T] + I(Y, i) + G 
LM relation: M/P = Y L(i) (A2) 
- Using assumption 1 we can determine the output. Any point on the downward-sloping IS curve 
corresponds to equilibrium in the goods market. Any point on the horizontal LM curve 
corresponds to equilibrium in financial markets. Only at point A are both equilibrium conditions 
satisfied. 
 
 
 
- That means point A, with the associated level of output Y and interest rate 𝑖 is the overall 
equilibrium—the point at which there is equilibrium in both the goods market and the financial 
markets. 
- When using assumption 2 we get that: 
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36 
 
 
- Let’s see how different shocks affect the IS-LM equilibrium with each of the assumptions. 
- Suppose the government decides to reduce the budget deficit and does so by increasing taxes 
while keeping government spending unchanged. Or, the government decides to decrease 
public spending without modifying the taxes, a contractive fiscal policy or a fiscal contraction. 
What would happen with the first assumption? 
- How do the curves change under A1? As we know, the LM curve depends on the Central Banks 
decisions, in this case the curve can only be affected by changes in the interest rate (i). Thus, 
the LM curve remains unchanged. 
- On the other hand, and as we have seen before, an increase in the tax or a decrease in the 
government spending level will result in a decrease of the IS curve. This is because the 
disposable income decreases, leading to a decrease in consumption, leading in turn to a 
decrease in the demand for goods (Z) and a decrease in equilibrium output. 
- Graphically: 
 
- Assuming financial markets adjust instantly, but goods markets can take some time to adjust. 
How do the interest rate and the output change through time? 
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37 
 
 
- The most difficult part is seeing how the nominal interest rate shifts with different shocks. 
The easiest way of seeing it is, after any kind of shock, always “be” on the LM curve. In this 
case the LM curve didn’t shift, so the equilibrium moved from A to A’ without changing i. 
- Now, let’s go by A2 and make the same process: 
 
- The shifts in output and interest rate would be: 
 
- In this case, as we “are” on the LM curve, the interest rate slowly moves from A to A’, thereare no “jumps”. 
- Now let’s see the case of monetary policy. Suppose the Central Bank wants to decrease the 
interest rate. Recall that, to do so (in assumption 1), it needs to increase the money supply. 
Such change is called a monetary expansion. In assumption 1 we get: 
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38 
 
 
- How does this affect our two variables? 
 
- In this case, the interest rate “jumps” because there is a shift on the LM curve. 
- And how does this monetary policy affect the model under assumption 2? 
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39 
 
 
- In practice, the two policies are often used together. The combination of monetary and fiscal 
policies is known as the monetary-fiscal policy mix, or simply the policy mix. Sometimes, the 
right mix is to use fiscal and monetary policy in the same direction. Suppose for example that 
the economy is in a recession and output is too low. Then, both fiscal and monetary policies can 
be used to increase output while reducing the interest rate. 
 
- We know that the IS relation is: Z = Y = C[Y – T] + I(Y, i) + G. But as far as we know, banks lend 
money, but they don’t lend it to the market’s interest rate, lending money is risky, then, banks 
may ask for risk premiums (x). This leaves us with the IS relation as: 
Z = Y = C[Y – T] + I(Y, i + x) + G 
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40 
 
- If banks are in trouble, if they think the firm they gave the loan to may not pay back (default) 
we can expect an increase in the risk premium, Δ+x, which would discourage investment. 
- Under assumption 1 we have: 
 
- We studied the behavior of aggregate demand in a closed economy and how it reacts to changes 
in fiscal and monetary policies and other shocks. Now, we will extend our analysis of IS-LM to 
the case of an open economy. 
- We use the same assumptions as before; fixed prices, one good, firms supply any amount of 
goods at price and there is no expected inflation (i = r). But now, in open economy we will have 
that P = P* = 1. This means that e = 1. 
- Moreover, we have different assumptions about exchange rate and regimes and capital 
mobility. The Mundell-Fleming Model (MF) supposes fixed and flexible exchange rates 
regimes, but not a partially flexible. It is either one or the other. It also assumes perfect capital 
mobility (interest rate remains unchanged). 
- These assumptions are useful, as long as the exchange rate adjusts instantaneously, there will 
be no expectation of depreciation or appreciation, which assures that at all times i = i*. To see 
this, let us remember that the parity of interest rates, which is fulfilled under the assumption of 
perfect capital mobility, implies: 
 
- Under the Mundell-Fleming model with flexible exchange rate and perfect capital mobility we 
also assume that the exchange rate adjusts instantly to the equilibrium level. Thus, Δe/e = 0. 
Then, i = i*, which is what happens when exchange rates adjust instantly. This leaves us with the 
IS and LM relation as: 
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41 
 
 
- From the second equation we can see that the equilibrium of the money market does not 
depend on the value of the exchange rate, therefore the LM is vertical in a graphic output over 
exchange rate, this new curve is written as LM*. The only dependence would come from the 
fact that the interest rate is nominal changes as a result of expectations of appreciation or 
depreciation, but this does not happen because we have assumed that the exchange rate 
adjusts instantaneously. To this curve we denote LM* to remember that it is a LM for i = i*. 
 
- The slope of the IS* is positive because the exchange rate and net export have a positive 
relation. A depreciation of the exchange rate (e increases) increases net exports and, therefore, 
the product increases with the exchange rate. Yes (only here) a depreciation is graphicly 
described as an increase when it comes to exchange rates. 
- Now, we’ll look at the effect of different shocks. Assuming an initial equilibrium in goods and 
financial markets where i = i* (interest rate parity holds), also, we assume flexible exchange 
rates. Let’s begin by analyzing fiscal policies. 
- Consider that the government increases its spending in the magnitude ΔG. As in the case of 
the closed economy, this increase in government spending shifts the IS and the IS* to the right, 
generating in that way a situation of excess demand for goods. 
 
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42 
 
 
- The shift of the IS to the right generates an upward pressure on the interest rate, in order to 
balance the money market. However, the interest rate cannot go up, because there is perfect 
capital mobility. 
- The pressure on the interest rate is to generate an incipient inflow of capital to appreciate the 
exchange rate until the pressure on the rates and the product disappears. As a result, the 
appreciation of the exchange rate increases imports and reduces exports. Graphically, this last 
phenomenon causes the IS to move back to the left. 
- However, the figure on the right shows that in the end fiscal policy does not increase the 
product, but only generates an appreciation of the exchange rate. Therefore, the higher 
government spending simply crowds out net exports. That is, we have: 
ΔG = - ΔXN 
- Now, let us look at the effects monetary policies have. Suppose that the Central Bank increases 
the amount of money in order to increase the product. 
 
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43 
 
- The monetary expansion generates a shift of the LM to the right, from LM to LM’. This would 
induce a decrease in the interest rate. As there is perfect mobility of capital, the downward 
pressure on the interest rate is not materialized. There is also a shift from LM* to LM*’, this 
generates a depreciation of the exchange rate, an increase benefits the exports, which shifts 
the IS curve to the right. This way, monetary policies are the only effective policy that can affect 
the aggregate demand in a flexible exchange rate and perfect capital mobility. 
- Why does the exchange rate changes? The intuition tells us that, when there is an increase in 
the money supply, but interest rate remains unchanged, the monetary supply gets bigger than 
the demand (you have more money saved than what you would like), domestic residents of 
the economy will start buying foreign bonds. To buy foreign bonds you have to have USD in 
the first place, so there is an increase in the demand for USD. USD appreciates, CLP depreciates, 
exchange rate depreciates. 
- Let’s see what happens when the exchange rate is fixed and there is perfect capital mobility. 
We have to start with the assumption that the Central Bank decides to peg de exchange rate 
in some determined level, 𝐞. How can they do this? Buying/selling reserves of foreign currency 
(R*). 
- If the Central Bank wants to increase the monetary supply, expansive monetary policy, they 
can do this by printing money and buying bonds. Having an increase in the monetary supply, 
keeping the interest rate fixed, will generate an excess of money. People will have more money 
than what they demanded, hence, they will want to buy foreign bonds. 
- To buy bonds, people have to buy dollars, so they go to the Central Bank and ask it the sell 
them part of their foreign currency reserves. This will generate a flow out of the domestic 
currency, as the Central Bank receives CLP, the monetary supply decreases to the initial point. 
 
 
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44 
 
- It can be concluded that monetary policy is inefficient with fixed exchange rate. Also, if there 
is perfect capital mobility, the Central Bank can control the exchange rate or the amount of 
money, but not both. To control the amount of money you must adopt a flexible exchange rate 
regime. This is known as the impossible trinity, you cannot have all three: monetary control, 
the exchange rate you want and perfect capital mobility. 
- If the government decides to increase its spending (expansive fiscal policy), the IS will move to 
the right towardsIS', which is equivalent to the IS* moving to IS*'. As the exchange rate has to 
remain fixed, the Central Bank has no other option that printing more money. 
 
- The intuition tells us that higher production, as a result of higher spending, pushes up the 
interest rate, which induces capital inflows. As the Central Bank wants to maintain the exchange 
rate, and prevent it from appreciating, it must absorb the inflow of capital by buying reserves. 
This causes the expansion of the amount of money, until there is no further upward pressure 
on the interest rate, shifting the LM to LM’ (also LM* to LM*'). 
- The final effect is an increase in the product, unlike the case of a flexible exchange rate where 
fiscal policy is ineffective. Note that the increase in money is not a policy decision but a need 
to maintain the exchange rate which causes an increase in the demand for money as a result 
of the increase in the level of activity. 
- In short, monetary policy is effective only with flexible exchange rate and fiscal policy is 
effective only in fixed exchange rate. This can be shown in a table: 
 
 IS - LM Mundell-Fleming Model 
Autarchy Flexible Exchange Rate Fixed Exchange Rate 
 Y i Y i e Y i e 
 MP+ + - + 0 + 0 0 0 
 FP+ + + 0 0 - + 0 0 
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45 
 
 
- The exchange rate dynamics and Downbosch overshooting assumes perfect capital mobility 
and flexible exchange rate. So far, we assumed the exchange rate adjusted immediately: 
i = i* + 
Δe
e
 
- Now, we drop that assumption because this isn’t necessarily true in the long run. Now, e can 
take time to converge to the new equilibrium, hence, i ≠ i* on transition is possible. We revisit 
a monetary policy shock assuming: goods market adjusts slowly, monetary market adjusts fast 
(always in equilibrium), but i = i* + 
Δe
e
 always hold. 
- With an increase in the monetary supply: 
 
- The “overshooting” is from A* to B*. This is because, as i < i* in transition, i – i* = 
Δe
e
 < 0 in 
transition. This is why e must overshoot. How the exchange rate changes through time can be 
described as: 
 
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46 
 
Monetary policy and equilibrium in the long (medium) run 
- The Phillips curve is a single-equation model that’s describes a historical inverse relationship 
between rates of unemployment and corresponding rates of rises in wages that result within 
an economy. Stated simply, decreased unemployment in an economy will correlate with higher 
rates of wage rises. 
- While there is a short run tradeoff between unemployment and inflation, it has not been 
observed in the long run. In 1968, the evidence showed that the Phillips curve was only 
applicable in the short-run and that in the long-run, inflationary policies would not decrease 
unemployment. 
- The long-run Phillips curve is now seen as a vertical line at the natural rate of unemployment, 
where the rate of inflation has no effect on unemployment. In recent years the slope of the 
Phillips curve appears to have declined and there has been significant questioning of the 
usefulness of the Phillips curve in predicting inflation. 
 
- Using the Phillips curve as a base, Lucas develops a new model in which he considers a trade-
off between inflation and unemployment due to imperfect information (first rigidity of the 
Phillips curve) producers receive of the price level. In this model, the Phillips curve depends on 
inflation. 
- A specific example would be that producers don’t know if the price level of their own goods 
increases or if it is the whole economy who suffers an increase. As they are uncertain, their 
production level won’t be optimal. 
- The model considers a “ʓ” number of firms, each live in a different island. Producers care about 
relative prices and produce according to: 
yt(ʓ) = 𝑦 + cE[Pt(ʓ) – Pt|It(ʓ)] 
- Where: 
https://en.wikipedia.org/wiki/Unemployment
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47 
 
1- yt(ʓ): Amount produced in island ʓ. 
2- Pt(ʓ): Price in island ʓ. 
3- Pt: ∫ 𝑃𝑡(ʓ)dʓ 
1
0
, average price across islands. 
4- It(ʓ) information set of island ʓ. 
- Firms observe the price in their own island (Pt(ʓ)), but they do not observe the average price 
level Pt. The information set at time t is: 
It(z) = {𝐼𝑡−1(ʓ), 𝑃𝑡(ʓ)} 
 
- If agents are making rational forecasts, any error is unpredictable. Hence, at some island ʓ 
forecasts at t-1 are such that: 
 
 Pt = E[Pt | It-1 (ʓ)] + ϵt (1) 
 
- Where ϵt is an error term with variance σ2 and expected value of cero (price level shock). In the 
end, any mistake people did yesterday is an error, it won’t happen today as it was random. We 
also assume σ2 is constant through time. 
- Since Pt is the average price we can write: 
 
 Pt(ʓ) = Pt + ʓt (2) 
 
- Where ʓt is on average zero. We assume the variance of ʓt is constant and equal to τ2. We can 
note that: 
 yt(ʓ) = 𝑦 + cE[Pt(ʓ) – Pt|It(ʓ)] = 𝑦 + cE[ʓt|It(ʓ)] (3) 
 
- Firms need to forecast ʓt with two pieces of information: 
1- Their prior information It-1(ʓ). 
2- The new information they get by observing Pt(ʓ). 
- By comparing (1) and (2) we get: 
 Pt(ʓ) = E[Pt|It-1(ʓ)] + (ϵt + ʓt) (4) 
 
- Since the firm has data of the past, the best linear predictor of ʓ is ʓ�̂� = α + θ(ϵt + ʓt), where: 
 
θ = 
𝐶𝑜𝑣(𝜖𝑡 + ʓ𝑡, ʓ𝑡)
𝑉𝑎𝑟(𝜖𝑡 + ʓ𝑡)
 = 
𝐸[(𝜖𝑡 + ʓ𝑡) ʓ𝑡] − 𝐸( ʓ𝑡)𝐸(𝜖𝑡 + ʓ𝑡)
𝑉𝑎𝑟(𝜖𝑡 + ʓ𝑡)
 = 
𝐸(𝜖𝑡 ʓ𝑡) + 𝐸(ʓ𝑡 ʓ𝑡)
𝜎2 + τ2
 = 
τ2
𝜎2 + τ2
 
 
- If ϵt and ʓt are normally distributed, then the best linear predictor is the best predictor. Hence, 
with normal distributions: 
E[ʓt|ϵt + ʓt] = θ(ϵt + ʓt) 
 
- So, we will assume that they distribute normally from now on. 
- Using (3): 
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48 
 
yt(ʓ) = 𝑦 + c(ϵt + ʓt)
τ2
𝜎2 + τ2
 
- And using (4): 
yt(ʓ) = 𝑦 + cθ (𝑃𝑡(ʓ) – E[Pt|It-1(ʓ)]) 
- Applying integrals: 
∫ 𝑦
𝑡
(ʓ)dʓ 
1
0
= 𝑦 + cθ (𝑃𝑡(ʓ) – ∫ E[Pt|It−1(ʓ)]dʓ 
1
0
) 
 
yt(ʓ) = 𝑦 + cθ (P𝑡 – E[Pt|It-1(ʓ)]) = 𝑦 + cθ (P – P
e) = 𝑦 + cθ ϵt 
 
- Where: 
1- E[Pt|It-1] = ∫ E[Pt|It−1(ʓ)]dʓ 
1
0
is the average believe about Pt. 
2- yt = ∫ 𝑦
𝑡
(ʓ)dʓ 
1
0
 is total output. 
- We can conclude that only unexpected shocks to the price level affect output and if prior 
information about Pt is good (σ is very low) or shocks to relative prices are common (high τ), 
then output responds a lot to unexpected shocks to price level (ϵt). Shocks in price level are 
interpreted as shocks to relative prices. 
- In the end, firms care about the relative price to decide how much to produce. They do not 
perfectly observe the price level. Hence, when the price level increases (inflation), they are 
uncertain if that is an increase in relative prices or an increase in the price level, leading them 
to produce more. 
- Now, let’s see what happens when prices are rigid (second friction). Suppose there is a type of 
company (an α fraction) that has its flexible prices (Pf) and sets them according to the demand 
conditions, which are represented by the product gap (y - �̅�). The greater the gap, the greater 
the demand pressure. Consequently, the relative price (with respect to the general level of 
prices), which these companies fix in period t, is: 
 
Pft – Pt = k(yt - �̅�) (1) 
 
- Where k is a positive parameter, and the variables are expressed in logarithmic terms. In 
addition, we assume that the product of full employment is constant over time, which is 
certainly a simplification for exposure purposes. 
- The second type of company (1 – α) has its fixed (rigid) prices at the beginning of the period, 
and its price (Pr) is fixed in the same way as flexible price companies, but based on the expected 
value of the demand, that is: 
<< 
Prt – Pet = σ(yet - �̅�) 
 
VGC 
 
49 
 
- For these firms the