Topic 1: A Simple Model of Money
Monetary Economics ECOS3010
Monetary Economics (ECOS3010) Topic 1: A Simple Model of Money 1 / 32
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Introduction
Money has always been important to people and to the economy. Money has a long history.
Commodity money: shells, beads, cigarettes, silver, gold,…… Fiat money: paper currency ! intrinsically useless
Emoney: debit card, smart card, ecash,……
Are we headed for a cashless society?
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Introduction
Why do we need money?
Lack of double coincidence of wants.
Lack of record-keeping or credit: money is MEMORY!
Functions of money:
a medium of exchange: primary function; a unit of account;
a store of value.
A suitable framework to study issues related to money: the overlapping generations (OLG) model.
highly tractable;
an elegant way to introduce money; dynamic.
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Environment of the Model
The economy begins in period 1 and runs forever: t = 1, 2, …, ∞.
Individuals live for two periods: young and old.
In period t, Nt individuals are born. In the Örst period, N0 initial old.
In each period t, Nt young individuals and Nt 1 old individuals.
One non-storable good. A very important assumption! Why?
Each individual receives y units of goods when young and nothing when old.
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Environment of the Model
Here is a summary of the model so far:
Generation
Individuals of all future generations value consumption both when young and when old.
Initial old value consumption only when old.
Preferences:
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Environment of the Model
Three assumptions about an individualís utility. 1. Utility is increasing with the consumption.
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Environment of the Model
Three assumptions about an individualís utility.
2. Individuals value some consumption in both periods of life: the indi§erence curves never cross either axis.
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Environment of the Model
Three assumptions about an individualís utility. 3. Diminishing marginal rate of substitution.
! a typical indi§erence curve c2
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Centralized Solution: the Golden Rule Allocation
Suppose there is a central planner who can allocate the available goods among the young and the old in each period.
Let (c1,t,c2,t+1) denote the consumption bundle by individuals born in period t.
Resource constraint in period t is
Ntc1,t +Nt 1c2,t Nty.
Suppose for now that for all t,
the population is constant: Nt = N;
we focus on stationary allocations where c1,t = c1 and c2,t = c2. Note: does not necessarily imply c1 = c2.
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Centralized Solution: the Golden Rule Allocation
Resource constraint simpliÖes to
Graphically, the feasible set is
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Centralized Solution: the Golden Rule Allocation
Within the feasible set, which allocation would the planner choose? The combination of (c1,c2) that maximizes an individualís utility.
The golden rule allocation is the allocation within the feasible set that maximizes the utility of future generations. It occurs at the unique point of tangency between the feasible set line and an indi§erence curve
Does the golden rule allocation maximize the utility of the initial old?
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Centralized Solution: the Golden Rule Allocation
Point A: the golden rule allocation; Point E: max utility of the initial old.
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Decentralized Solutions: a Competitive Equilibrium without Money
To achieve the golden rule allocation, the planner needs to redistribute c2 units of goods from each young to each old in every period.
Strong assumptions about the power of central planners. Can we achieve the golden rule allocation without a planner?
When individuals trade among themselves ! a competitive equilibrium.
Individuals maximize their own utilities. Individuals are price takers.
Markets clear.
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Decentralized Solutions: a Competitive Equilibrium without Money
What is the competitive equilibrium allocation?
No trade can occur in this economy ! autarkic allocation: individuals have no economic interaction with others.
Lack of double coincidence of wants: the old would like to have some goods from the young, but they have nothing that the young want. No record-keeping or credit.
Each individualís consumption: c1 = y , c2 = 0. (Goods are non-storable!)
Utility is low: both the future generations and the initial old are worse o§ than almost any other feasible consumption bundle.
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Decentralized Solutions: a Competitive Equilibrium without Money
How can we improve the economy?
Monetary Economics (ECOS3010)
Topic 1: A Simple Model of Money
Decentralized Solutions: a Monetary Equilibrium
How can the economy achieve a better allocation than the autarkic allocation?
One way to allow some trading opportunities is to introduce money. Fiat money:
produced by the government (almost) costlessly; cannot be counterfeited;
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Decentralized Solutions: a Monetary Equilibrium
A monetary equilibrium is a competitive equilibrium in which there is a valued supply of Öat money. That is, the Öat money can be traded for consumption good.
For Öat money to have value, 2 conditions must be satisÖed:
supply of money must be limited. impossible (or very costly) to counterfeit.
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Decentralized Solutions: a Monetary Equilibrium Demand for Money
There is a Öxed stock of money: M units.
Each of the initial old is endowed with M/N0 units money. Are there potential trade opportunities?
At t = 1, the initial old have money and the young (newborn) have goods. Would they trade? Yes.
At t = 2, the old (who were young at t = 1) have money and the young (newborn) have goods. Would they trade? Yes.
At t = 3, 4, ……, the old in each period always have some money and the young always have goods.
Now each individual can consume in both periods of life.
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Decentralized Solutions: a Monetary Equilibrium Demand for Money
Consider an individual who is born at time t.
c1,t : consumption when young;
c2,t+1 : consumption when old;
mt : the number of dollars acquired when young (by giving up some of the endowed consumption good);
vt : the value of money, which implies the price level pt = 1/vt .
The individualís budget constraint in the Örst period of life c1,t +vtmt y
The individualís budget constraint in the second period of life c2,t+1 vt+1mt
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Decentralized Solutions: a Monetary Equilibrium Demand for Money
The individualís life-time budget constraint c1,t+vt c2,t+1y.
vt +1 vt+1/vt : the real return of money
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Decentralized Solutions: a Monetary Equilibrium Demand for Money
Graphically, we depict the budget set:
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Decentralized Solutions: a Monetary Equilibrium Demand for Money
Within the budget set, point A maximizes an individualís utility.
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Decentralized Solutions: a Monetary Equilibrium A Monetary Equilibrium
It remains to Önd vt+1/vt. Recall that in any competitive market, the price (or value) of an object is determined as the price at which the supply of the object equals its demand.
demand for money at time t
Nt (y c1,t ) ;
supply of money at time t vt is determined through
Nt (y c1,t) = vtMt ! vt = Nt (y c1,t). Mt
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Decentralized Solutions: a Monetary Equilibrium A Monetary Equilibrium
From vt and vt+1, we can Önd
Letís further simplify our economy: suppose we focus on
stationary allocations where c1,t = c1 and c2,t+1 = c2; a constant population where Nt = N;
a constant money supply where Mt = M.
for all t. Now, we have
vt+1 =1orvt+1 =vt. vt
The value of money is constant.
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Decentralized Solutions: a Monetary Equilibrium A Monetary Equilibrium
We update the graph that depicts monetary equilibrium.
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Decentralized Solutions: a Monetary Equilibrium A Monetary Equilibrium
Quantity theory of money: the price level is proportional to the quantity of money in the economy. In our economy, the price level is
p=1=M. v N (y c1 )
Neutrality of money: the nominal size (measured in dollars) of the stock of money M has no e§ect on the real (measured in goods) values of consumption (c1,c2) and real money demand y c1.
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Decentralized Solutions: a Monetary Equilibrium A Monetary Equilibrium
Is monetary equilibrium the golden rule?
golden rule
monetary equilibrium
max utility
subject to the resource constraint
max utility
subject to the budget constraint
resource constraint: c1 + c2 y budget constraint: c1 + c2 y ! golden rule = monetary equilibrium
Compared to competitive equilibrium without money, the introduction of money allows all future generations to achieve the golden rule allocation. It also beneÖts the initial old, whose consumption increases from 0 to c2.
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An Example
Supposethatu(c1,c2)=c1c2. Considertheeconomywitha constant population and a constant money supply.
Golden rule allocation
max c1c2 subjectto c1+c2=y.
Substitute c2 with c2 = y c1 , it is equivalent to
max c1(y c1). c1
FOC: y c1+( c1)=0
y!y c1 = 2 and c2 = 2
Monetary Economics (ECOS3010)
Topic 1: A Simple Model of Money
An Example
A Monetary equilibrium
max c1c2 subjectto c1+ vt c2 =y. c1 ,c2 vt +1
We know that vt = vt +1 so that vt /vt +1 = 1. Substitute c2 with c2 =y c1,itisequivalentto
max c1(y c1). c1
FOC: y c!+( c)=0 11
c1 =y2 andc2 =y2 Topic 1: A Simple Model of Money
Monetary Economics (ECOS3010)
A Growing Economy
So far we have learned that the introduction of money opens up trade opportunities and monetary equilibrium coincides with the golden rule allocation. We have assumed a constant money supply and a constant population.
What if we have a growing population? Suppose that Nt = nNt 1 where n > 1. How does a growing population a§ect the golden rule allocation and the monetary equilibrium?
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A Growing Economy
Golden rule allocation: the planner maximizes an individualís utility subject to the resource constraint
N t c 1 + N t 1 c 2 N t y ! c 1 + n1 c 2 y .
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A Growing Economy
A Monetary equilibrium: the individual maximizes his own utility subject to the life-time budget constraint
Recall that
c1+vt c2y. vt +1
vt =Nt(y c1) !vt+1 =Nt+1 =n. M vt Nt
When the population is growing, the value of money is also growing at the same speed. Therefore, the budget constraint simpliÖes to
c 1 + n1 c 2 y .
Notice that the budget constraint is identical to the resource constraint. So the monetary equilibrium coincides with the golden rule allocation. Again, the introduction of money helps the economy achieve the best possible allocation!
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