ECOS3010: Tutorial 2 (Answer Key)
Question 1-5. Answer True, False or Uncertain. Brieáy explain your answer.
1. According to RBAís deÖnition of monetary aggregates, M1 includes more assets than M3 does.
False. According to RBAís deÖnition of monetary aggregates, M3 includes M1 plus all other deposits in the economy. So M3 includes more assets than M1 does.
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2. Suppose that the government increases money supply and gives new money to the old in every period. Comparing monetary equilibrium with the golden rule allocation, we Önd that all future generations achieve a lower level of utility but the initial old enjoys a higher level of utility in the monetary equilibrium.
False. When the government increases money supply and gives new money to the old in every period, the allocation in the monetary equilibrium is not the golden rule allocation. In particular, all future generations in the monetary equilibrium achieve a lower level of utility. The initial old also achieve a lower level of utility because c2 in the monetary equilibrium is lower than c2 at the golden rule allocation.
3. Suppose that money supply grows at a constant rate z (z > 1). Comparing monetary equilibrium and the golden rule allocation, we Önd that individuals consume too much when young and too little when old in the monetary equilibrium.
True. In a monetary equilibrium, ináation makes young individuals trade less goods for money. As a result, the old have less money to purchase goods. Comparing monetary equilibrium with the golden rule allocation, we Önd that individuals consume too much when young and too little when old.
4. Suppose that the population grows at a constant rate n (n > 1) and money supply grows at a constant rate z (z > 1), the value of money falls over time.
Uncertain. When the population grows at a constant rate n and money supply grows at a constant rate z, we can derive moneyís rate of return as vt+1=vt = n=z. If n < z, the value of money falls over time. If n > z, the value of money increases over time. And if n = z, the value of money stays constant.
5. When the population is growing, Öxing the price level is the optimal policy.
False. When the population is growing, moneyís rate of return is given by vt+1=vt = n=z. The allocation in the monetary equilibrium is generally not the golden rule allocation. The optimal policy requires that an individualís budget constraint is identical to the plannerís resource constraint. It implies that we need z = 1 for a monetary equilibrium to achieve the golden rule allocation. In this case, the value of money increases over time and the price level actually falls over time.
(Note that for question 4 and question 5, my explanation uses the expression of vt+1=vt. If you cannot memorize the exact expression of vt+1=vt, it is Öne to explain your answer in words as long as the intuition is correct.)
6. Let Nt = nNt 1 and Mt = zMt 1 for every period t, where z and n are both greater than 1. The money created each period is used to Önance a lump-sum subsidy of at goods to each young individual.
(a) Find the equation for the budget constraint of an individual in the monetary equilib- rium. Graph it. Show an arbitrary indi§erence curve tangent to the budget constraint and indicate the levels of c1 and c2 that would be chosen by an individual in this equilibrium.
An individualís Örst- and second-period budget constraints are
c1 +vtmt y+at
and c2 vt+1mt:
Combining these two period budget constraints together, we derive the lifetime budget
constraint
c 1 + v t c 2 y + a t : vt+1
The value of money is derived from the money market clearing condition (when aggregate money supply equals aggregate money demand),
Nt(y+a c1)=vtMt ! vt=Nt(y+a c1): Mt
It follows that moneyís rate of return is
Nt+1(y+a c1) vt+1= Mt+1
vt Nt(y+a c1) Mt
We can update our budget constraint as
=Nt+1 Mt =n: Nt Mt+1 z
c 1 + nz c 2 y + a :
Graphically, we can draw the budget constraint and label point A as the allocation
in a monetary equilibrium. The consumption bundle (c1;c2) at point A maximizes an
individualís utility subject to the budget constraint.
c* y+a c 11
(b) On the graph you drew in part (a), draw the resource constraint. Take advantage of the fact that the resource constraint goes through the monetary equilibrium (c1;c2). Label your graph carefully, distinguishing between the budget and resource constraints.
We Örst derive the resource constraint faced by the planner,
Ntc1+Nt 1c2Nty ! c1+n1c2y:
We add the resource constraint to the graph and label the golden rule allocation as point B.
ny (y+a)n/z
y y+a c 11
(c) Show that the monetary equilibrium does not maximize the utility of the future generations. Support your assertion with references to the graph you drew of the budget and feasible constraints.
We can clearly see from the graph that there are feasible points that are preferred by the future generations to (c1; c2). One such point is point B which is the golden rule allocation. Point B lies on a higher indi§erence curve than (c1; c2). This shows that given the resource constraint (or feasible constraint), the stationary monetary equilibrium (c1;c2) does not maximize the utility of future generations. Furthermore, the initial old also prefer a point like B since it gives them higher second-period consumption than c2.
(Note that we use the term “feasible constraint” and the term “resource constraint” interchangeably.)
7. Consider an overlapping generations model with the following characteristics: Each generation is composed of 1; 000 individuals. The money supply changes according to Mt = 2Mt 1 . The initial old own a total of 10; 000 units of money (M0 = $10; 000). Each period, the newly printed money is given to the old of that period as a lump-sum transfer. Each individual is endowed with 20 units of the consumption good when born and nothing when old. Preferences are such that individuals wish to save 10 units when young at the equilibrium rate of return on money.
(a) What is the gross real rate of return on money in this economy (vt+1=vt)?
To derive the rate return on money, we Örst need to Önd the value of money. The money market clearing condition is
It follows that
N(y c1)=vtMt ! vt = N(y c1): Mt
vt+1=Mt+1 =Mt=1:
vt N(y c1) Mt+1 2 Mt
The rate of return on money is 1=2. The value of money falls by a half every period.
(b) How many goods does an individual consume when young (c1)?
Individuals allocate their endowments between consumption and saving when young.
The Örst-period budget constraint is
c1 + vtmt y:
When individuals save 10 units when young, it means that vtmt = 10. Given that y = 20;
we have c1 = y vtmt = 10. The consumption when young is 10 units of good.
(c) How many goods does an individual receive as a transfer (a)?
From the government budget constraint, the transfer is from the newly printed money.
In aggregate,
Na = vt (Mt Mt 1) = vt Mt Mt = 1vtMt: 22
To Önd the value of vtMt, we use the money market clearing condition vtMt =N(y c1)=1000(20 10)=10;000:
It follows that the amount of the transfer in real terms is given by
21vtMt 12 10;000
a= N = 1000 =5:
(d) How many goods does an individual consume when old (c2)? From an individualís second-period budget constraint,
c2 vt+1mt + a:
Recall that vtmt = 10. We can derive the value of vt+1mt as
Therefore, we have
vt+1mt = vt+1 vtmt = 1 10 = 5: vt 2
c2 =vt+1mt +a=5+5=10:
The second-period consumption is 10 units of good.
(e) What is the price of the consumption good in period 1 in dollars (p1)? The price is the inverse of the value of money,
p1 = v1 : 1
We will Önd the value of v1. Given that
vt = N (y c1);
v1 = N(y c1) = 1000(20 10) = 1000(20 10) = 1: M1 2M0 210000 2
Finally, the price level in period 1 is
p1 = v1 = 2:
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