Question

Consider the following version of the Lucas model. The number of young individuals 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) of 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 ?????? denotes an individual’s labour supply). Find the lifetime budget constraint of an individual.

b. Set 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 labour.

d. Show how the rate of return to labour and the individual's labour 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 θ, where 0 ≤ θ ≤ 1;

??=?? with probability 1-θ.

The realization of ???? is kept secret from the young until all purchases of goods have occurred (i.e. individuals do not learn ???? until period t is over). Given these changes in assumption, answer 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 distinguish when there is limited information (i.e. they do not know the value of ????)?

g. Draw a graph of labour 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 and output. If it always inflates, will the graph you derived in part f remain the same? Explain fully.

Answer 1

a. Given,

In any single period, each island has an equal chance of having the large population of young.

Fiat money stock grows according to zt = z it means rational individuals can easily determine the current money stock.

The choice of labor will be lit = l(pit)

Young person money demand = lit = l (pit) = vit*mit

According to money stock

Young individuals are endowed when young with time y.

Time can be used in leisure, c1, or as labor.

The choice of labor will be lit = l(pit)

pit is price of goods on island i

Each unit of labor produces one unit of goods.

The budget constraint when young in period t on island i can be written as

C1it + lit
=c1it + vit * zit = y

Money market clearing condition on an island with Ni young people.

In period t, each person demand for fiat money is lit = l(pit) = vit*mit

Because there are Ni Young people on island i, the total demand for fiat money is Ni l(pit)

Because the old people are equally distributed across islands regardless of their island of birth, half of the fiat money stock is brought to each island.

Equating the real supply of money is period t, vit (Mt/2)

These are the market clearing condition:

Ni* l(pit) = vit (Mt/2)

Because the value of fiat money vit is equal to the inverse of the price level pit, we can rewrite the equation,

Ni* l(pit) = Mt/2)/pit

Ni is either (1/3) N or (2/3) N, respectively,

pit = (Mt/2)/(Ni l(pit))

This says that the market clearing condition expresses the price level as a function of the population of the young Ni.

b.

Observing the price of goods pit allows all of the young to infer the number of young on their island.

Assume PA and PB denote the price of goods at T when population is small (NA = 1/3 N) and large (NB=2/3N) respectively,

We can rewrite previous equation,

On island A

PAt = (Mt/2)/NA l(pAt) = (Mt/2)/ ((1/3)N l(pAt)

On island B

PAt = (Mt/2)/NB l(pBt) = (Mt/2)/ ((1/3)N l(pBt)

Assume pAt>PBt, the price of good is high when the population is low.

The price of goods is driven by the scarcity of young people producing goods. As island has less number of young people.

When the population on an island is low, people want to work more because the price of their goods and thus the rate of return on labour is more. So island A would be preferred island.

c.

on assumption that substitution effect dominates wealth effects, young people would work more and therefore rates of return will be higher.

Rate of return when the money stock is higher in both this period and the next:

vj(t+1)/vit = pit/pj(t+1) = (Mt/2)/(Ni l(pit)) / (Mt+1/2)/(Nj l(pj(t+1))) = {(Nj l(pj(t+1) / Ni l(pit)} / {Mt / Mt+1)}

a permanent increase in the money stock raises Mt amd Mt+1 by the same portion and so fails to capture the relative price of goods in any of the period. Therefore, a high current price caused by a permanent increase in the money stock does not affect the rate of return of the labour and thus the desire to work.

We can write, as zt=z

Mt+1 = zMt

Mt/ Mt+1 = 1/z

vj(t+1)/vit = {(Nj l(pj(t+1) / Ni l(pit)} / {1/z}

In the growth of supply rate of flat money stock to work falls is TAX

As z increase, Mt/ Mt+1 falls and the rate of return to work falls, money earned is taxed now.