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Topic 3: Price Surprises
Monetary Economics ECOS3010
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Introduction
The relationship between ináation and unemployment ñthe Phillips curve.
Cross country evidence on the relationship between ináation and output.
In this chapter, we develop a theory to rationalize the empirical observations?
Unanticipated changes in money supply. In previous sections, we consider anticipated increases in money supply.
How do unanticipated áuctuations in money supply a§ect output? Can the government exploit such a relationship?
The Lucas (1972) model and the Lucas critique.
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The Data The Phillips Curve
In 1958, Phillips discovered a signiÖcant statistical link between ináation and unemployment of the United Kingdom over a century. For example, the Phillips curve 1948-1969 in US.
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The Data The Phillips Curve
The original Phillips curve suggests that there is a negative relationship between ináation and unemployment, or there is a positive relationship between ináation and output.
Does it imply that there maybe an exploitable trade-o§ between ináation and unemployment? Can the government reduce unemployment and increase output by increasing ináation?
In the following decades, many governments tried to use monetary policy to stimulate the economy. Suddenly, the Phillips curve, a stable relationship for more than a century, disappeared. Ináation occurs with no gains in output or employment.
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The Data The Phillips Curve
Ináation and unemployment in U.S. from 1970 to present.
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The Data Cross-country Comparisons
The Phillips curve is a time series correlation between ináation and unemployment in di§erent periods of the same country.
If we compare across countries, ináation rates are on average higher in countries with lower average real GDP growth rates. Note: its a cross-section comparison here.
How can there exist seemingly contradictory correlations
Time series of the same country: -ve correlation between ináation and unemployment, that is, +ve correlation between ináation and output. Cross-country comparisons: -ve correlation between ináation and output.
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The Data Cross-country Comparisons
Example: ináation and output across countries 1952-1967
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The Lucas Model Basic Environment
Consider the standard OLG model with money. Now assume that individuals live on two spatially separated islands.
Nt individuals are born in each period. Nt is constant. In each period,
half of the old live on each of the islands;
1/3 of the young live on one island and 2/3 live on the other island; the allocation of the young and the old is random.
the old are randomly distributed across the two islands, regardless of where they lived when young
in any single period, each island has an equal chance of having the large population of young.
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The Lucas Model Basic Environment
Money supply grows at a rate zt in period t, Mt = ztMt1. The new money is distributed to each old person as a lump-sum transfer in every period t worth at units of the consumption good.
at =1 1vtMt. zt N
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The Lucas Model Basic Environment
Informational assumptions: in any period,
the young cannot observe the number of young individuals on their island;
the young cannot observe the size of the transfers to the old;
the nominal stock of money supply is known with a delay of one period; e.g., in period t, individuals know Mt1, but not Mt;
the price of goods on an island is observed but only by the individuals on that island;
no communication between islands within a period.
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The Lucas Model Basic Environment
We assume that individuals are rational.
They may not have complete information, but they can infer whatever they can from the information they have and they make the most correct inference possible given the explicitly speciÖed limits on what they can observe.
The assumption of “rational expectation”, Örst introduced by Muth (1961): people understand the probabilities of outcomes important to their welfare.
In our model, individuals do not observe zt and the population of the young on each island, but they know the prices. They know 1/3 of the young are on one island and the rest 2/3 are on the other island. They will try to infer zt and the distribution of the young population.
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The Lucas Model Basic Environment
A reinterpretation of an individualís problem.
y : individuals are endowed when young with y units of time (instead of goods): think of y as 24 hours;
c1 : consumption of leisure (instead of consumption of goods); y c1 is spent working;
c2 : consumption of goods;
l = y c1 : labor supply by the individual when young; lti = l (pti ): the choice of labor by an individual born in period t for a given price of goods, pti , on island i;
production function: 1 unit of l can be used to produce 1 unit of the consumption goods
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The Lucas Model Basic Environment
Consumption
Generation
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The Lucas Model Basic Environment
In our model, the young are endowed with time and the old are endowed with nothing.
In period t, a young individual on island i chooses between working lti and leisure c1i ,t
c 1i , t + l ti  y ,
where lti units of goods are produced and sold to the old to acquire mti
units of money,
Herevtimti stillrepresentstheindividualísrealdemandformoney.
lti =vtimti.
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The Lucas Model Basic Environment
In period t + 1, the young individual born in period t becomes old and could be on island j, where j may or may not be the same island
as i. His consumption ci,j 2,t+1
government transfers,
ci,j 2,t+1
comes from his own saving and
=vj mi+a t+1 t t+1
= t + 1 l ti + a t + 1 vti
=pj lt+at+1
Monetary Economics (ECOS3010)
Topic 3: Price Surprises

The Lucas Model Basic Environment
Note that when the young individual decides to supply 1 more unit of labor by increasing lti by 1, he will be able to produce 1 more unit of good and acquire 1/vti more units of money. Then he can use the 1/vi units of money to buy vj /vi units of goods when old.
This implies that the rate of return to labor is
vj pi t+1 = t .
vti pj t+1
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The Lucas Model Basic Environment
We assume that an increase in the current price of goods pti , other things being equal, will induce the young to work more, that is, lti increases.
Whenpti increases,therateofreturntolaborincreases.
Substitution e§ect: work more because working is more proÖtable. Wealth e§ect: work less because the higher return from labor means higher income and less need to work.
We are assuming that the substitution e§ect of an increase in price dominates the wealth e§ect.
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The Lucas Model Nonrandom Ináation
Before we examine a random zt , we begin with a constant growth rate of money supply zt = z in all periods. In this case, can rational individuals infer the current stock of money?
Yes, they know Mt1 and z. So they can infer Mt = zMt1.
In period t, money market clearing condition on on island i with Ni
young individuals is:
or equivalently
N i y c 1i , t
= v ti M t , 2
N i l ti = v ti M t = 1 M t . 2 pti 2
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The Lucas Model Nonrandom Ináation
We label the island with 1/3 young individuals as island A and the other island as island B. We have
ptA = 2 = , (1)
ptB = 2 = . (2)
Claim: ptA > ptB ñ the price level is higher on the island with less young individuals.
Why? By contradiction. If ptA  ptB , then the rate of return to labor is lower on island A which implies that ltA  ltB . However, from (1) and (2), ltA  ltB implies that ptA > ptB , which is a contradiction to the assumption of ptA  ptB . So it is only possible that ptA > ptB .
N A ltA 13 NltA
NB ltB 23 NltB
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The Lucas Model Nonrandom Ináation
We Önd that the price of goods is high on the island with relatively less young individuals and is low on the island with relatively more young individuals.
Intuition: when there are less young people, there are less people supplying labor to produce the good. With the same number of old individuals, the demand for goods is relatively high on the island with less young individuals. Therefore, the price is high on the island with less young individuals.
Further implication: since we know that ptA > ptB (and all else equal), the rate of return to labor is high on island A, young individuals work more on island A with less young individuals. That is, ltA > ltB .
These implications depend critically on the assumption that the substitution e§ect dominates the wealth e§ect.
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The Lucas Model Nonrandom Ináation
Prices here signal the true state of the economy: the young can infer that
they are on the island with a smaller population if they observe the high price;
they are on the island with a larger population if they observe the low price.
What else can a§ect the price level? A revisit of the prices: ptA = Mt/2 and ptB = Mt/2.
13 NltA 23 NltB Money supply can also a§ect the price level.
When money supply increases, the price level increases. When money supply decreases, the price level decreases.
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The Lucas Model Nonrandom Ináation
Suppose z = 1. What if there is a permanent (once-and-for-all) increase in the money stock? That is, M increases permanently.
Recall that
p ti = M t / 2 . Nilti
Once M increases permanently, pi will increase but pj
increase. Overall, the rate of return to labor
j i Mt/2 jj
vt+1=pt =Nilti vi j Mt+1/2
=Nlt+1Mt, Nili Mt+1
t pt+1 Njlj t+1
is not a§ected by the level of money supply. Therefore, a permanent increase in money supply does not a§ect employment and output in this economy.
Money is neutral in this economy: a permanent change in M does not a§ect the real economic variables.
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The Lucas Model Nonrandom Ináation
What if there is a permanent increase in z? Now the rate of return to labor is
j i Mt/2 jj jj
vt+1=pt =Nilti vi j Mt+1/2
=Nlt+1Mt =Nlt+11. Nili Mt+1 Nili z
t pt+1 Njlj t+1
An increase in z lowers the rate of return to labor, which discourages working because the money earned from labor is now taxed by the government through ináation. Lower lti leads to lower output.
Money is not superneutral in this economy: a permanent change in z a§ects the real economic variables.
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The Lucas Model Nonrandom Ináation
We can construct a graph plotting output as a function of z.
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The Lucas Model Random Ináation
So far we Önd that ináation reduces employment and output in our economy.
Now consider the following random monetary policy.
Mt = Mt1 with probability θ (zt = 1)
= 2Mt 1 with probability 1 θ (zt = 2)
The realization of zt is kept secrete from the young until the end of period t.
Can the young still infer the current money supply? Maybe not.
As before, we will focus on how the youngís labor supply decisions depend on monetary policy.
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The Lucas Model Random Ináation
Again, using the equation that determines the price level on island i, p ti = M t / 2 .
Notice that individuals do not know Mt and Ni, but they know Mt1.
We can rearrange the price equation as
pti = ztMt1/2.
Individuals know that with probability θ, zt = 1 and with probability
1θ,zt =2.
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The Lucas Model Random Ináation
Letís think about the potential prices. Depending on the values of zt and Ni,
zt = 1 zt = 2
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pta = ptb = ptc = ptd =

The Lucas Model Random Ináation
For any young individual, he does not know the population of the young on his island. He also does not know the current money supply in the economy. Can he still infer zt and Ni from the prices?
If the young individual observes pta, he will know that he is on the island with Ni = 2N/3 and zt = 1.
If the young individual observes ptd , he will know that he is on the islandwithNi =N/3andzt =2.
If the young individual observes ptb, what can he infer? If the young individual observes ptc , what can he infer?
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The Lucas Model Random Ináation
There are two factors that a§ect the price level: Ni and zt.
If Ni = N/3 (island with less young individuals), it contributes to a higher pti . If Ni = 2N/3 (island with more young individuals), it contributes to a lower pti .
If zt = 1, it contributes to a lower pti . If zt = 2, it contributes to a higherpti. a d

Two of the four possible prices are unique: pt , pt . Each can have occurred in only one particular combination of events.
If observing the low price pta, the young would supply labor lta (a low level).
If observing the high price ptd , the young would supply labor ltd (a high level).
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The Lucas Model Random Ináation
If the young observe ptb or ptc , the young cannot infer whether they are on the island with N/3 young and zt = 1 or they are on the island with 2N/3 young and zt = 2. Therefore, the young decide to supply labor l.
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The Lucas Model Random Ináation
What is the relationship between ináation and output in this economy? If we graph output and ináation on two islands separately,
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The Lucas Model Random Ináation
If we graph aggregate output and ináation, we have
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The Lucas Model The
Imagine that an economyís time series plot of ináation and output resembles our previous Ögure. What can you infer?
Does the historical correlation suggest that the government can control aggregate output through its control of the money supply?
If the government wants to achieve a higher level of output, should the government print money to stimulate output in every period?
Will such a policy work?
What happens to output if money supply increases at a constant rate in every period?
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The Lucas Model The
Imagine that an economyís time series plot of ináation and output shows a positive correlation. What can you infer?
Does the historical correlation suggest that the government can control aggregate output through its control of the money supply?
If the government wants to achieve a higher level of output, should the government print money to stimulate output in every period?
Will such a policy work?
What happens to output if money supply increases at a constant speed in every period?
When the growth rate of money supply becomes constant, people can perfectly infer current money supply. A higher growth rate of money supply leads to lower labor supply and lower output. The positive correlation between ináation and output disappears!
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The Lucas Model The
The correlation of money and output or any set of variables results from the reaction of decision makers to the environment they face. An important feature of this environment is government policies.
In our example, the relation between ináation and output depends on the monetary policy being followed.
Random ináation: positive correlation between ináation and output. Steady ináation (nonrandom ináation): negative correlation between ináation and output.
When monetary policy changes from random to nonrandom, the labor supply decisions by the young change as well.
A correlation between variables that is the result of equilibrium interactions of an economy can be called a reduced-form correlation.
In our example, it is the correlation between ináation and output.
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The Lucas Model The
The : these reduced form correlations are subject to change when the government changes its policies.
In our example, the positive correlation between ináation and output disappears when the government changes from random ináation to nonrandom ináation because young individuals change their labor supply decisions.
How can we evaluate policies?
Econometric policy evaluation is useful.
But we also need a theory to help us understand peopleís motives (preferences) and constraints (physical limitations, informational restrictions, and government policies).
It is not su¢ cient just to look at the data.
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The Lucas Model
The – A Simple Example
For a simple example, consider the question of how much Fort Knox should spend on protection. The United States Bullion Depository, often known as Fort Knox, is a fortiÖed vault building located within the United States Army post of Fort Knox, Kentucky, used to store a large portion of United States o¢ cial gold reserves and occasionally other precious items belonging or entrusted to the federal government. It is estimated to have roughly 2.3% of all the gold ever reÖned throughout human history.
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The Lucas Model
The – A Simple Example
Fort Knox has never been robbed. Statistical analysis using high-level, aggregated data would therefore indicate that the probability of a robbery is independent of the resources spent on guards. The policy implication from such analysis would be to eliminate the guards and save those resources. This analysis would, however, be subject to the , and the conclusion would be misleading. In order to properly analyze the trade-o§ between the probability of a robbery and resources spent on guards, the “deep parameters” (preferences, technology and resource constraints) that govern individual behaviour must be taken explicitly into account. In particular, criminalsíincentives to attempt to rob Fort Knox depends on the presence of the guards. In other words, with the heavy security that exists at the fort today, criminals are unlikely to attempt a robbery because they know they are unlikely to succeed.
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The Lucas Model

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