Constraint set up always begins with the basic pattern we studied in Chapter 1. First of all, we need to know what is available for us. Second of all, we need to know what our choices of bundle are. Once we know what we have, we will know what we can afford; our choices of bundle cannot exceed our resources available. Let us start from the very basic economy model. 1. Consider an economy with a constant population. Individuals are endowed with y units of the consumption good when young and nothing when old. Feasible Set The number of young population at time t is Nt, each of them has endowment y. The number of old population at time t is Nt-1, and none of them has endowment. Total amount of consumption good at time t = Nty. Total young consumption at time t = Ntc1,t. Total old consumption at time t = Nt-1c2,t. N t c1,t + N t −1c 2,t ≤ N t y.

Dividing through both sides of this equation by Nt,

⎡N ⎤ c1,t + ⎢ t −1 ⎥ c 2,t ≤ y. ⎣ Nt ⎦ Noting that population is constant, Nt = Nt−1 = N, feasible set is

c1 + c2 ≤ y. Budget Set First – period constraint: c1,t + υ t mt ≤ y. Second – period constraint: c2 ,t +1 ≤ υ t +1mt . Solving for mt and substituting it back into the first period constraint, we will get the lifetime budget constraint:

⎡υ ⎤ c1,t + ⎢ t ⎥ c 2,t +1 ≤ y. ⎣υ t +1 ⎦

Money-Market Clearing Condition Money stock at any particular time is M. M units of money can acquire υM units of consumption good. Hence, υM is the supply of money in terms of consumption good. A group of people who wants to obtain money at any particular time is young people at that given time. Old people no longer want to have money; instead, they want to have consumption of goods. Therefore, the consumption good available for trade is young people’s endowment minus their first-period consumptions.

υ t M t = N t ( y − c1,t ). At time period t+1: υ t +1 M t +1 = N t +1 ( y − c1,t +1 ). At time period t:

N t +1 ( y − c1,t )

N ( y − c1 )

υ t +1 M t +1 M = = = 1, ( ) ( N y c N y − − c1 ) υt 1,t +1 t M Mt due to stationarity, constant population and constant money stock.

The budget set now becomes

c1 + c2 ≤ y. In this scenario, the budget set coincides with the feasible set.

2. Consider an economy with a growing population in which each person is endowed with y1 when young and y2 when old. Assume that y2 is sufficiently small that everyone wants to consume more than y2 in the second period of life. Feasible Set The number of young population at time t is Nt, each of them has endowment y1. The number of old population at time t is Nt-1, and each of them has endowment y2. Total amount of consumption good at time t = Nty1 + Nt-1 y2. Total young consumption at time t = Ntc1,t. Total old consumption at time t = Nt-1c2,t.

N t c1,t + N t −1c 2,t ≤ N t y1 + N t − 1 y 2 .

Dividing through both sides of this equation by Nt,

⎡N ⎤ ⎡N ⎤ c1,t + ⎢ t −1 ⎥ c 2,t ≤ y1 + ⎢ t −1 ⎥ y 2 . ⎣ Nt ⎦ ⎣ Nt ⎦ Noting that population is growing, Nt = nNt−1, feasible set is ⎡1⎤ ⎡1⎤ c1 + ⎢ ⎥ c 2 ≤ y1 + ⎢ ⎥ y 2 . ⎣n⎦ ⎣n⎦

Budget Set Now, old people do not have only money holding from their first period of life. They also have endowment y2 in their second period of life. First – period constraint: c1,t + υ t mt ≤ y1 . Second – period constraint: c 2,t +1 ≤ υ t +1 mt + y 2 . Solving for mt and substituting it back into the first period constraint, we will get the lifetime budget constraint:

⎡υ ⎤ ⎡υ ⎤ c1,t + ⎢ t ⎥ c 2,t +1 ≤ y1 + ⎢ t ⎥ y 2 . ⎣υ t +1 ⎦ ⎣υ t +1 ⎦ Money-Market Clearing Condition Money stock at any particular time is M. M units of money can acquire υM units of consumption good. Hence, υM is the supply of money in terms of consumption good. A group of people who wants to obtain money at any particular time is young people at that given time. Old people no longer want to have money; instead, they want to have consumption of goods. Therefore, the consumption good available for trade is young people’s endowment minus their first-period consumptions.

υ t M t = N t ( y1 − c1,t ). At time period t+1: υ t +1 M t +1 = N t +1 ( y1 − c1,t +1 ). At time period t:

N t +1 ( y1 − c1,t )

nN t ( y1 − c1 ) υ t +1 M t +1 M = = = n, N t ( y1 − c1,t +1 ) N t ( y1 − c1 ) υt M Mt due to stationarity, constant money stock and growing population.

The budget set now becomes ⎡1⎤ ⎡1⎤ c1 + ⎢ ⎥ c 2 ≤ y1 + ⎢ ⎥ y 2 . ⎣n⎦ ⎣n⎦

In this scenario, the budget set coincides with the feasible set.

3. Consider the following economy. Individuals are endowed with y units of the consumption good when young and nothing when old. The fiat money stock is constant. The population grows at rate n. In each period, the government taxes each young person τ goods. The total proceeds of the tax are then distributed equally among the old who are alive in that period. (The tax is less than the real balances people would choose to hold in the absence of the tax.) Feasible Set The number of young population at time t is Nt, each of them has endowment y. The number of old population at time t is Nt-1, and none of them has endowment. Total amount of consumption good at time t = Nty.. Total young consumption at time t = Ntc1,t. Total old consumption at time t = Nt-1c2,t. N t c1,t + N t −1c 2,t ≤ N t y.

Dividing through both sides of this equation by Nt,

⎡N ⎤ c1,t + ⎢ t −1 ⎥ c 2,t ≤ y. ⎣ Nt ⎦ Noting that population is growing, Nt = nNt−1, feasible set is ⎡1⎤ c1 + ⎢ ⎥ c 2 ≤ y. ⎣n⎦

Budget Set In this scenario, young people are taxed τ units of consumption good. Their resources available are their endowment minus tax expenditure. Old people do not have only money holding from their first period of life. They also have revenue from tax given by the government. At time period t, the government collects τ units of goods from each young (Nt people). The total tax revenues are Nt τ. The government then distributes these revenues to each old (Nt-1 people). The per-old-person transfers will be Nt τ/ Nt-1 = nτ. At time period t + 1, the government collects τ units of goods from each young (Nt+1 people). The total tax revenues are Nt+1 τ. The government then distributes these revenues to each old (Nt people; they are young at period t). The per-oldperson transfers will be Nt+1 τ/ Nt = nτ. Now we are ready to write down individual’s budget constraints. First – period constraint: c1,t + υ t mt ≤ y − τ . Second – period constraint: c 2,t +1 ≤ υ t +1 mt + nτ . Solving for mt and substituting it back into the first period constraint, we will get the lifetime budget constraint:

⎡υ ⎤ ⎡υ ⎤ c1,t + ⎢ t ⎥ c 2,t +1 ≤ y − τ + ⎢ t ⎥ nτ . ⎣υ t +1 ⎦ ⎣υ t +1 ⎦ Money-Market Clearing Condition Money stock at any particular time is M. M units of money can acquire υM units of consumption good. Hence, υM is the supply of money in terms of consumption good. A group of people who wants to obtain money at any particular time is young people at that given time. Old people no longer want to have money; instead, they want to have consumption of goods. Therefore, the consumption good available for trade is young people’s endowment minus their first-period consumptions and tax expenditure.

υ t M t = N t ( y − τ − c1,t ). At time period t+1: υ t +1 M t +1 = N t +1 ( y − τ − c1,t +1 ). At time period t:

N t +1 ( y − τ − c1,t )

nN t ( y − τ − c1 ) υ t +1 M t +1 M = = = n, ( ) ( τ τ − c1 ) N y − − c N y − υt t 1,t +1 t M Mt due to stationarity, constant money stock and population growth.

The budget set now becomes ⎡1 ⎤ ⎡1⎤ c1 + ⎢ ⎥ c 2 ≤ y − τ + ⎢ ⎥ nτ , ⎣n⎦ ⎣n⎦ ⎡1 ⎤ c1 + ⎢ ⎥ c 2 ≤ y. ⎣n⎦

In this scenario, the budget set coincides with the feasible set.

4. Assume that people face a lump-sum tax of τ goods when old and a rate of expansion of the fiat money supply of z > 1. The tax and the expansion of the fiat money stock are used to finance government purchases of g goods per young person in every period. There are N people in every generation. Feasible Set The number of young population at time t is Nt, each of them has endowment y. The number of old population at time t is Nt-1, and none of them has endowment. There exist the government purchases of g goods per young person in every period. Total amount of consumption good at time t = Nty.. Total young consumption at time t = Ntc1,t. Total old consumption at time t = Nt-1c2,t. Total government purchases at time t = Nt g. N t c1,t + N t −1c 2,t + N t g ≤ N t y.

Dividing through both sides of this equation by Nt,

⎡N ⎤ c1,t + ⎢ t −1 ⎥ c 2,t + g ≤ y. ⎣ Nt ⎦ Noting that population is constant, Nt = Nt−1, feasible set is

c1 + c 2 + g ≤ y. c1 + c 2 ≤ y − g. Budget Set In contrast to previous scenario, young people are not taxed. When individuals are young, their resources available are their endowment. Old people are taxed τ units of consumption good. When individuals become old in their second period of life, their resources available will be the amount of their real money balance holding (from first period) minus tax expenditure. Now we are ready to write down individual’s budget constraints. First – period constraint: c1,t + υ t mt ≤ y. Second – period constraint: c 2,t +1 ≤ υ t +1 mt − τ . Solving for mt and substituting it back into the first period constraint, we will get the lifetime budget constraint:

⎡υ ⎤ ⎡υ ⎤ c1,t + ⎢ t ⎥ c 2,t +1 ≤ y − ⎢ t ⎥τ . ⎣υ t +1 ⎦ ⎣υ t +1 ⎦ Money-Market Clearing Condition Money stock at any particular time is M. M units of money can acquire υM units of consumption good. Hence, υM is the supply of money in terms of consumption good. A group of people who wants to obtain money at any particular time is young people at that given time. Old people no longer want to have money; instead, they want to have consumption of goods. Therefore, the consumption good available for trade is young people’s endowment minus their first-period consumptions.

υ t M t = N t ( y − c1,t ). At time period t+1: υ t +1 M t +1 = N t +1 ( y − c1,t +1 ). At time period t:

N t +1 ( y − c1,t )

N ( y − c1 ) υ t +1 M t +1 zM t 1 = = = , N t ( y − c1,t +1 ) N ( y − c1 ) z υt Mt Mt due to stationarity, constant population and growing money stock.

The budget set now becomes

c1 + zc2 ≤ y − zτ . With the presence of money stock growth, the budget set does not coincide with the feasible set.