Question

Consider a 2-period overlapping generations economy with an initial population N0 = 500.

Each generation is n = 1.5 times larger than the previous one. Each individual is endowed

with y = 100 units of the consumption good when young, and nothing when old. The only way

to acquire consumption in old age is by exchanging part of the endowment when young for fiat

money. The initial old are endowed with M0 = 20, 000 units of at money. The money stock

grows at rate z = 1.01 every period. Newly created at money in every period t is distributed

as a lump-sum transfer to old individuals worth a units of the consumption good. Assume

stationary allocations of consumption throughout the exercise.

question:

Assume that a = 0.8. Use a graph that combines the feasible set, the lifetime

budget constraint, and arbitrarily drawn indierence curves to argue whether or not the

monetary equilibrium attains the Golden Rule in this economy. Clearly label all axes and

other relevant elements in the graph. Explain in your own words.

Answer 1

Find the Explanation

v Initial Population N0=500

v Consumption of goods endowed when young ; y1=100 units

v Consumption of goods endowed when young ; y2=0 units

v Fiat money endowed by the initial old; M0=20,000 units

v Money stock growth rate; z = 1.01 every period

v Stationary allocations of consumption throughout the exercise

a) In pt, an individual has an endowment of c goods. The individual can either consume the goods and/or sell them for money. As no one in the future generations is born with fiat money. To gain fiat money, an individual must trade. Let Mt denote the number of dollars acquired by an individual at time t, then the total number of goods sold for money is vt*mt . We denote first- and second-period consumption as c1 and c2 = c1 + c2 = y

Therefore, the budget constraint of an individual in pt is = y,t + vtMt ≤ c.

The left-hand side of Equation is the individual’s total uses of goods i.e. consumption and acquisition of money and the right-hand side of Equation represents the total sources of goods i.e. the individual’s endowment.

N=500

Y= 100

Feasible set = 500?1,? + 500?2,? ≤ 500? = 500(100) ⇒ ?1,? + ?2,? ≤ 100

The per-capita feasible set, the pink triangle, represents the set of possible allocations that can be attained given the resources available in the economy.

B) The Budget constraint

facing the individual in the pt of life is:  c1,t + vtmt ≤ y and tThis means that the constraint facing the individual in the second period of life is c2,t+1 ≤ vt+1mt .

In a monetary equilibrium we can rewrite this constraint as mt ≥ (c2,t+1)/(vt+1) and substitute it into the first-period constraint to obtain:

First period: ?1, ? + ?_t?_t ≤ ?

Second Period: ?2, ?₊₁ ≤ ?_t +1 ?_t

Lifetime:

c1,t+(vt/vt+1)C2,t+1<=Y

c1,t+(vt/vt+1)C2,t+1<=100

The above equation expresses the various combinations of first- and second-period consumption that an individual can afford over a lifetime. It is the individual’s lifetime budget constraint.

C)

The money demand for each person is the quantity of goods they don’t consume at young but rather sell in order to have fiat money, y − ct. Therefore, total money demand in an arbitrary period t will be Nt(y − ct).

The total supply of money is Mt, which is the number of dollars multiplied by the value of each dollar, or vtMt .

Therefore, equality of money supply and money demand therefore:

vtMt = Nt(y-ct)

vt = Nt(y-ct)/Mt

Given, Mt=20,000 and Nt=500 and y=100

vt = 500 (y-c1,t)/20000

Real rate of return for fiat money i.e.

vt+1/vt

vt+1/vt = (500(y-c1, t+1)/20000)/(500(y-c1,t)/20000) = 1