Question

Problem 2. Consider an overlapping generations model with money, where t = 1,2,3,.... So, every period t, a generation of two-period lived agents are born. Their preferences are . They are endowed with ey units of the consumption good, when young, and e° units of the consumption good, when old. Assume that ey > eo > 0. The consumption good is perishable The initial old also have a quantity M of intrinsically valueless fiat money. Trading takes place as described in class, i.e., old agents may purchase goods from young agents, using their money. Let st be the amount of savings, expressed in units of the good, which a young agent will save in period t, using money. There is no other store of value. Let πt=Pt/Pt-1 be the inflation from period t-1 to t, where P is the price in units of money for one unit of the consumption good.

 1. Calculate the optimal savings function s_t =(π_t+1;ey,eo).

Answer 1

USING OVERLAPPING GENERATIONS MODEL WITH MONEY FIND OPTIMAL SAVING FUNCTION:

The overlapping generations (OLG) model, is one of the dominating frameworks of analysis, in the study of macroeconomic dynamics and economic growth. OLG model was inspired by Irving Fisher's monograph 'The Theory of Interest' was first formulated in 1947, in the context of a pure-exchange economy by Maurice Allais and later more rigorously by Paul Samuelson in 1958. It contains agents who are born at different dates and have finite lifetimes, even though the economy goes on forever. This induces a natural heterogeneity across individuals at a point in time as well as nontrivial life-cycle considerations for a given individual across time. These features of the model can also generate differences from models where there is a finite set of time periods and agents, or from models where there is an infinite number of time periods, but agents live forever. A closely related feature of the model is that it has a role for fiat money. The OLG model is a framework of study for: Life-cycle behavior, implications of allocated resources across generations, such as Social Security, on the income per capita in the long-run or to determine economic growth, and the factors that trigger the fertility transition.

Basic assumptions of the OLG model:

• Economy has an infinite horizon.
• Individuals live for two periods; in the first period (of life), they are referred to as the Young and in the second period, they are referred to as the Old.
• N^t_t-1 denotes the number of old people in period t. Since the economy begins in period 1, in period 1 there is a group of people who are already old. They are referred to as the initial old. The number of them can be denoted as N₀.
• The size of the initial old generation is normalized to 1: N⁰₀ = 1.
• Utility of generation t: ut = u (c1t, c2t).
• Population grows at a constant rate n: N^t_{t-1 =} (1 + n).
• Equilibrium with no money.
• No gains from borrowing or lending
• Therefore no one consumes when old.
• Everyone attains u^autarky = u (y, 0). This is probably socially inefficient because old people are starving and would pay a lot to consume if they could.
• In the "pure exchange economy" version of the model, there is only one physical good and it cannot endure for more than one period. Each individual receives a fixed endowment of this good at birth. This endowment is denoted as y.
• In the "production economy" version of the model (see Diamond OLG model below), the physical good can be either consumed or invested to build physical capital. Output is produced from labor and physical capital. Each household is endowed with one unit of time which is inelastically supply on the labor market.

Problem with the basic OLG model is that there is no store of value except money, but this can sometimes be fixed up by adding productive capital. This pure-exchange OLG model was augmented with the introduction of an aggregate neoclassical production by Peter Diamond. The OLG economy under this model is characterized by multiple steady-state equilibria, which can affect the long-run level of income per capita. The economy will have the following characteristics:

• Two generations are alive at any point in time, the young (age 1) and old (age 2).
• The size of the young generation in period t is given by Nt = N0 Et.
• Households work only in the first period of their life and earn Y1,t income. They earn no income in the second period of their life (Y2,t+1 = 0).
• They consume part of their first period income and save the rest to finance their consumption when old.
• In Diamond's version of the model, individuals tend to save more than is socially optimal, leading to dynamic inefficiency.

In an overlapping generations model with money, where t- 1,2,3,.... So, every period t, a generation of two-period lived agents are born. They are endowed with e units of the consumption good, when young, and e units of the consumption good, when old. Assume that e > e° >0. The consumption good is perishable. The initial old also have a quantity M of intrinsically valueless fiat money. Trading takes place as described in class, i.e., old agents may purchase goods from young agents, using their money. Let st be the amount of savings, erpressed in units of the good, which a young agent will save in period t, using money. There is no other store of ualue. Let π,-R/P-1 be the inflation from period t-1 to t, where P is the price in units of money for one unit of the consumption good in order to calculate the optimal savings function.

Here we will allocate the initial generation with an amount of outside money m. There is outside money: money, that is, on net, an asset of the private economy. This includes fiat currency issued by the government and inside money (such as bank deposits) is both an asset as well as a liability of the private sector (in the case of deposits an asset of the deposit holder, a liability to the bank). If m 0, then m can be interpreted as fiat money. If m < 0, one should envision the initial old people having borrowed from some institution (outside the model) and m is the amount to be repaid.

Preferences of individuals are representable by: ut(c) = U(c t t ) + βU(c t t+1)

Preferences of the initial old generation is representable by: u0(c) = U(c 0 1)

We shall assume that U is strictly increasing and twice continuously differential.

An allocation c 0 1 , f(c t t , c t t+1 )g ∞ t=1 is Pareto optimal if it is feasible and if there is no other feasible allocation cˆ 1 0 , f(cˆ t t , cˆ t t+1 )g ∞ t=1 such that: ut(cˆ t t , cˆ t t+1 ) ut(c t t , c t t+1 ) for all t 1 u0(cˆ 0 1 ) u0(c 0 1 ) with strict inequality for at least one t_$\geq$ 0.

In the presence of money (m $\neq$ 0), we will take money to be the numeraire. This is important since we can only normalize the price of one commodity to 1. With money, no further normalization is admissible. Let pt be the price of one unit of the consumption good at period t. Given that the period utility function U is strictly increasing, the budget constraints hold with equality.

Summing the budget constraints of agents: c t t+1 + c t+1 t+1 + s t+1 t+1 = e t t+1 + e t+1 t+1 + (1 + rt+1)s t.

By resource balance: s t+1 t+1 = (1 + rt+1)s t

Doing the same manipulations for generation 0 and 1: s 1 1 = (1 + r1)m

By repeated substitution: s t t = Πt τ=1 (1 + rτ)m

Therefore, the amount of saving (in terms of the period t consumption good) has to equal the value of the outside supply of assets, Πt τ=1 (1 + rτ)m.

This condition should appear in the definition of equilibrium. By Walrasílaw, however, either the asset market or the good market equilibrium condition is redundant. So

For rt+1 > 1, we combine both budget constraints into: c t t + 1 1 + rt+1 c t t+1 = e t t + 1 1 + rt+1 e t t+1

Divide by pt > 0: c t t + pt+1 pt c t t+1 = e t t + pt+1 pt e t t+1

Divide initial old generation by p1 > 0 to obtain: c 0 1 e 0 1 + m

Hence, it looks that 1 + rt+1 = pt pt+1 must play a key role.

From the equivalence, the return on the asset equals: 1 + rt+1 = pt pt+1 = 1 1 + πt+1 (1 + rt+1)(1 + πt+1) = 1 rt+1 πt+1 where πt+1 is the ináation rate from period t to t + 1.

Using: p1 = 1 1 + r1 pt+1 = pt 1 + rt+1 with repeated substitution delivers: pt = 1 Πt τ=1 (1 + rτ) ) Πt τ=1 (1 + rτ) = 1 pt

Now, note that we argued before that s t t = Πt τ=1 (1 + rτ)m Hence: s t t = m pt

The total money supply is no longer constant: in = 1 it is $; in = 2 it is $ + $ = 2 · $; in = 3 it is 2 · $ + $ = 3 · $; and, in general, in period , it is · $. In view of this, the money market equilibrium condition in period holds that · $ = () · %() · − (). Thus, %() = · $ () · − () and %( + 1) = ( + 1) · $ ( + 1) · − ( + 1) . Using the relationship ( + 1) = (1 + ) · (), %() %( + 1) = · $ () · − () ( + 1) · $ (1 + ) · () · − ( + 1) = (1 + ) · + 1 · − ( + 1) − () .

You can think about this last condition both as:
1 An equilibrium condition.
2 A money demand function.