Consider the following version of the Lucas model

.

The number of young individual

s born on

island

i

in period

t

,

, is random according to the following specification:

=

(

1

/

4

)N with probability 0.5

,

=

(

3

/

4

)N with probability 0.5

,

where N denotes the total number (i.e. across the two islands) o

f young people born in every

period.

Assume that the fiat money stock

grows at the fixed rate

in all periods.

a.

Set up the budget constraints of

an

individual when young and when old in terms of

(where

d

enotes an individual's labour supply)

. Find the lifetime budget constraint

of

an individual

.

b.

S

et up the money market

-

clearing condition.

c.

On which island would you prefer to be born? Explain with reference to the rate of

return to labo

u

r.

d.

Show how the ra

te of return to labo

u

r and the individual's labo

u

r supply depend on

the value of

.

For the

remaining questions

, assume

now

that the growth rate of the fiat money stock

is

random according to:

with probability

,

where 0

1;

with probability 1

-

.

The realization of

is kept secret from the young until all pu

r

chases of goods have

occu

r

red (i.e. individuals do not learn

until period

t

is over). Given these changes in

ass

umption, an

s

wer the following questions:

e.

How many states of the world would agents be able to observe if information about

every variable were perfectly available? Describe those possible states.

f.

How many states of the world are the agents able to dis

tinguish when there is limited

information (i.e. they do not know the value of

)?

g.

Draw a graph of labo

u

r supply and the growth rate of the fiat money stock in each

possible state of the world when there is limited

information. What is the correlation

observed between money creation and output?

h.

Suppose the government wanted to take advantage of the relation between money

creation an

d output. If it always inflates

, will the graph you derived in part f remain the

same

? Explain fully.

f.

How many states of the world are the agents able to dis

tinguish when there is limited

information (i.e. they do not know the value of

)?

g.

Draw a graph of labo

u

r supply and the growth rate of the fiat money stock in each

possible state of the world when there is limited

information. What is the correlation

observed between money creation and output?

h.

Suppose the government wanted to take advantage of the relation between money

creation an

d output. I