ECOS3010: Tutorial 3 (Answer Key)
Question 1-4. Answer True, False or Uncertain. Brieáy explain your answer.
1. Suppose that the government creates new money to Önance its own purchases. Com- paring monetary equilibrium and the golden rule allocation, we Önd that both c1 and c2 are lower in a monetary equilibrium than at the golden rule allocation.
False. In a monetary equilibrium, young individuals would choose to trade less goods for money when money supply is growing. As a result, the old consume less but the young consume more compared to the golden rule allocation. That is, c1 is higher and c2 is lower in a monetary equilibrium.
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2. To Önance the same amount of government purchases, using a lump-sum tax is better than using the ináation tax (money creation).
Uncertain. When using the ináation tax to Önance government purchases, we know that the allocation in the monetary equilibrium achieves a lower level utility than the golden rule allocation. If the government uses a lump-sum tax instead of the ináation tax, it is possible to design the lump-sum tax such that the allocation in the monetary equilibrium can coincide with the golden rule allocation. In particular, it requires that the government keeps a constant money supply and sets the per capita lump-sum tax to the per capita government purchases.
3. In most countries, seigniorage contributes signiÖcantly to total government revenue.
False. In most industrialized countries during normal times, seigniorage contributes to total government revenue, but it accounts for only a small fraction of total government revenue. However, for countries that experience high ináation episodes, seigniorage could contribute signiÖcantly to total government revenue.
4. The government can Önance any amount of government purchases by creating new money.
False. When the government creates new money to Önance its own purchases, a higher growth rate of money supply does not always lead to a higher level of seigniorage in real terms. A higher growth rate of money supply means that a higher fraction of real value of money stock will be collected by the government. However, the real value of the total money stock decreases when the growth rate of money supply increases. Overall, when the growth rate of money supply is low, a higher growth rate leads to more seigniorage. When the growth rate of money supply is high, a higher growth rate leads to less seigniorage.
5. Assume that people face a lump-sum tax of goods when old and a growth rate of money supply of z > 1. The tax and the newly created money are used to Önance govern- ment purchases of g goods per young individual in every period. There are N individuals in every generation.
(a) Find the individualís budget constraints when young and when old. Combine them to form the individualís lifetime budget constraint and graph this constraint.
An individualís Örst-period budget constraint is
c1 + vtmt y: The second-period budget constraint is
c2 vt+1mt : 1
Combine the two period budget constraints to get the lifetime budget constraint:
c1+ vt c2y vt : vt+1 vt+1
To Önd the value of vt=vt+1, we need to Önd the value of money vt Örst. From the money market clearing condition, we have
It follows that
N (y c1) = vtMt:
vt = N (y c1): Mt
We can then derive moneyís rate of return as
vt+1=Mt+1 =Mt=1:
vt N(y c1) Mt+1 z Mt
The individualís budget constraint is updated as
c1 + zc2 y z :
We can graph the budget constraint. (Note that I use t to represent in the Ögures below.)
(b) Find the governmentís budget constraint. The governmentís budget constraint is
g= +(Mt Mt 1)vt = + 1 z1Mtvt: NN
Using the money market clearing condition, we have
g = + 1 z1N(y c1) N
= +1 z1(y c1): (c) Graph the stationary monetary equilibrium.
The stationary monetary equilibrium can be found at the tangency point between the 2
indi§erence curve and the individualís budget constraint. Point A in the Ögure below represents the allocation in the monetary equilibrium
(d) Find the stationary monetary equilibrium when z = 1 and add it to the graph in (c).
When z = 1, the individualís budget constraint is c1 + c2 y :
We graph the budget constraint and Önd the allocation in a monetary equilibrium at point B when z = 1. Please see the attached PDF Öle.
(e) Compare the real balances of Öat money when z > 1 to the value when z = 1.
Comparingc1 whenz>1andwhenz=1,weÖndthatthec1 ishigherwhenz>1 than when z = 1. Since the real money balances held by any individual is
vtmt =y c1;
the real money balances of money is lower when z > 1 than when z = 1. That is, a higher growth rate of money supply reduces the real money balances held by any individual. As a result, the aggregate real money balances in the economy also decline as z increases.
(f) What is the optimal choice of ( ;z) so that individuals achieve the highest level of utility?
To achieve the highest level of utility, the allocation of (c1;c2) should be the golden rule allocation. The golden rule allocation can be found when the planner maximizes an individualís utility subject to the economyís resource constraint. In this economy, the resource constraint is
c1 + c2 y g;
where g is the per capita government purchases. For the optimal policy, the government chooses ( ;z) so that the individualís budget constraint can coincide with the resource constraint. The optimal policy is to let = g and z = 1.
6. Consider an economy with a constant population of N = 1000. Individuals are endowed with y = 20 units of the consumption good when young and nothing when old. All seigniorage revenue is used to Önance government expenditures. There are no subsidies and no taxes other than seigniorage. Suppose that preferences are such that each individual
wishes to hold real balances of Öat money worth
1+ vt vt+1
(a) Use the equality of supply and demand in the money market to Önd the total real
balances of Öat money in a stationary equilibrium as a function of the rate of Öat money creation z.
The total real balances of money is
vtMt: The money market clearing condition implies that
vtMt =N(y c1): The value of money in period t is then
Moneyís rate of return is
vt = N (y c1): Mt
vt+1=Mt+1 =Mt=1:
vt N(y c1) Mt+1 z Mt
The amount of money in real terms that each young individual chooses to hold is
vtmt =y c1 = y : 1+ vt
Using vt+1=vt = 1=z, we have
Therefore, the total real balances of money as a function of z is
vtMt = N(y c1) =Ny
1+z 1000 20
We can see that a higher z leads to a lower vtMt.
(b) Use your answer in part (a) to Önd total seigniorage revenue as a function of z.
Graph this function and explain its shape.
The total seigniorage revenue in real terms is
G = ( M t M t 1 ) v t = 1 z1 M t v t : 4
Using the solution of vtMt that we found in part (a), we have G = 1 120;000
= 20;0001+z:
We can graph G as a function of z. (Note that I use excel to graph the example below.
You can use your preferred software to do this.)
z 1+z 1 z1
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