Nominal assets in the short and in the long run

November 27, 2007

Nominal assets in the long run

Two non exclusive types of models

1. The infinite horizon neoclassical growth model

1.1 Model without nominal asset1.2 Is there a role for public debt?

2. The overlapping generations model

2.1 An exchange economy with nominal asset as the only durablegood

2.2 Economies with land2.3 Economies with population growth

The neoclassical growth model: consumption

Discrete time, infinite horizon, all purpose commodity, no nominalasset.

Inelastic labour supply Lt at date t.

The representative consumer maximizes

∞∑t=1

βtU(Ct)

subject to the sequence of budget constraints for t = 1, . . . :

ptCt + ptKt = wtLt + Πt + ρtKt−1 + ptKt−1,

given initial capital stock K0.

Arbitrary numeraire at each date to measure the price pt and wagewt , i.e. a good, storable without cost.

The neoclassical growth model: production

The manager of the (aggregate) firm maximizes his short runprofit:

Πt = ptQt − wtNt − ρtKt−1

subject to the technical constraint (production function) :

Qt = F (Kt−1,Nt)

Profits are immediately distributed to the owners of the capitalstock.

Assumption: The production function is twice continuouslydifferentiable, concave and exhibits constant returns to scale. Also

F (0, L) = 0 limK→0

F ′K (K ,N) =∞ limN→0

F ′N(K ,N) =∞

Net production function (capital stock maintained in originalstate).

The neoclassical growth model: equilibrium

Definition: An equilibrium is a sequence of prices and quantities(wt/pt , ρt/pt ,Qt ,Ct ,Nt , Lt ,Kt) such that, every agent maximizinghis/her objective subject to his/her constraints given prices, thecorresponding allocations satisfy the scarcity constraints:

Ct + Kt = Qt + Kt−1,

Nt = Lt .

First order necessary conditions

Profit maximization at each date:

F ′K =ρ

pF ′N =

w

p.

Intertemporal consumer optimization between t and t + 1 :

βU ′(Ct+1)

U ′(Ct)=

pt+1

pt+1 + ρt+1.

Eliminating prices yields

βU ′(Ct+1)

U ′(Ct)=

1

1 + F ′K ,t+1

.

Stationary equilibria

If the stock of capital per labor unit stays constant, a propertyobserved along any equilibrium of the type just described in theabsence of technical progress, F ′K is constant, which implies that

βU ′(Ct+1)

U ′(Ct)=

1

1 + F ′K

is also constant. If consumption stays constant, at any equilibriumone has

β =1

1 + F ′k.

The marginal productivity of capital is equal to the psychologicaldiscount rate of the consumers: this is the simplest version of thegolden rule. This is only possible when labor supply is constant.

Balanced growth paths

If labor supply L exogenously grows at rate n, consumption and thecapital stock must grow at the same rate, if one looks for aconstant growth equilibrium (again stressing the absence oftechnical progress in the model). In general, one cannot expect thefirst order conditions to hold unless U is a homogenous function ofC . If utility is logarithmic, one again gets the golden rule :

β1

1 + n=

1

1 + F ′k.

These conditions on the marginal rates of substitution determinethe ratio K/N, and as a consequence the initial capital stock thatallows to remain from then on on the constant growth path.Consumption can then be computed from the scarcity constraint.

Role of nominal assets: theory

With a finite number of agents and an infinite number of goods,any competitive equilibrium is Pareto optimal.

Checking the first welfare theoremThe optimum allocation maximizes

∑βtU(Ct) subject to

Ct + Kt = F (Kt−1, Lt) + Kt−1.

First order conditions: let βtλt be the multiplier associated withthe constraint of period t.Derivative with respect to Ct :

U ′(Ct) = λt

Derivative with respect to Kt :

β(1 + F ′Kt+1)λt+1 = λt .

Eliminating λ’s yields

βU ′(Ct+1)

U ′(Ct)=

1

1 + F ′K ,t+1

,

identical to the equilibrium condition.

Adding a nominal asset

Budget constraints:

ptCt + ptKt + Bt = wtLt + Πt + ρtKt−1 + ptKt−1 + Bt−1.

An equilibrium with (positive price) of the nominal asset (and nocentral bank intervention) should have a demand Bt equal to B0

for all t.

This is inconsistent with utility maximization as soon as B0 > 0:the plan which consumes B0 at the first date and keeps a zeromoney balance from then on dominates the reference plan.

Assets and bubbles

Fundamental value of an asset

Money, nominal assets, physical assets

Arbitrage conditions

pτ = pτ+1 + ρτ+1.

If∑∞

τ=t ρτ is infinite, the price of the good in terms of numeraireis infinite: the price of the numeraire in terms of good is zero, andthe nominal asset has zero real value at the equilibrium.

The overlapping generations model

Generations of identical agents with finite lives: here for simplicitylife lasts two periods.

Double infinity of (dated) goods and agents.

Possible inefficiency:the savings needs for life cycle considerations,linked to the labor supply profile are a priori likely to be differentfrom the stock of capital required for the efficiency of production.This structure makes room for a financial system to fill the gapbetween the two.

Exchange economy

Consumers with utility U(C y ,C o), where C y (resp. C o) is theconsumption of physical good when young (resp. old).

Initial endowments Y y and Y o .

The physical good is not storable.

Nominal asset B storable without cost, which serves as numeraireat all dates. Borrowing is allowed.

Program of typical consumer born at t

max U(C y ,C o)ptC y

t + Bt = ptY y

pt+1C ot+1 = pt+1Y o + Bt

At the initial date 1, the old consumer has a quantity B0 ofnominal asset, and consumes all her wealth, i.e. Y o + B0/p1.

Intertemporal equilibrium

Definition 1 : A perfect foresight intertemporal equilibriumwith nominal asset is a sequence (pt ,C

yt ,C

ot ), t = 1, . . . , with

(strictly) positive prices, which satisfies:

1.C y

t + C ot = Y y + Y o ,

2. For t ≥ 1, (C yt ,C

ot+1) maximizes the program of the consumer

born at date t, given (pt , pt+1). C v1 is equal to B0/p1 + Y o .

Walras’ law, nominal assets.

Finding the equilibria

{max U(C y ,C o)

C y +pt+1

ptC o = Y y +

pt+1

ptY o

Let zy and zo be the excess demand functions coming out of theprogram:

zy

(pt

pt+1− 1

)= C y

t − Y y = −B

pt

zo

(pt

pt+1− 1

)= C o

t+1 − Y o = − pt

pt+1zy

(pt

pt+1− 1

).

Supply curve

The range of values of (zy , zo), when the price ratiopt/pt+1 = 1 + ρt+1 varies, is the supply curve of the consumer.

zo goes to infinity when pt/pt+1 tends to 0 (zy is negative, so thatC y ≤ Y y : C o cannot stay bounded while keeping the equality ofthe marginal rate of substitution with the price ratio). Similarly, zy

tends to infinity when pt/pt+1 tends to infinity.

Samuelson case:

U ′o(Y y ,Y o)

U ′y (Y y ,Y o)=

1

1 + ρ> 1.

-

6

E

0

Co − Y o

Cy − Y y

Figure 1: Equilibria in the overlapping generations model

Finding the equilibria: continued

Gross real interest rate: 1 + ρt+1 = pt/pt+1. In the plan (C y ,C v ),−(1 + ρ) is the slope of the budget line.

The equality between supply and demand at date t is

zy (ρt+1) + zo(ρt) = 0.

Finite difference equation with initial condition zo(ρ0) = B0/p1 (ofsame sign as B0).

A priori, there exists a continuum of equilibria, depending on theinitial value ρ0.

Stationary equilibria

Fixed points of the difference equation

zy (ρ∗) + zo(ρ∗) = 0,

at the intersection of the supply curve with the second bissector.

1. Autarky of every generation : zy = zo = 0. The equilibriumgross interest rate is ρ. The real aggregate quantity ofnominal assets is null: B = 0.

2. Transfers between generations, B 6= 0, interest rate ρ∗ = 0,golden rule.

Non stationary equilibria

In the Samuelson case, continuum of indeterminate equilibriaconverging to the autarkic equilibria.

Isolated (determinate) golden rule equilibrium

Learning

Economies with land

How robust is the monetary bubble?

A piece of land produces a (non random) crop at each date, madeof non durable good. A unit of land brings a unit of good. Theavailable land area is constant and equal to T . The price of landmeasured in numeraire is qt .

Land : consumer

Denote by q the money price of land. The consumer programbecomes

max U(C y ,C o)ptC y

t + qtTt + Bt = ptY y

pt+1C ot+1 = pt+1Y o + (qt+1 + pt+1)Tt + Bt

Tt ≥ 0

Land : equilibrium

A perfect foresight intertemporal equilibrium is a sequence of pricesand quantities such that, at each date, consumptions and assetholdings are solutions of the consumers’ programs, given the prices,and the scarcity constraints are satisfied:

C yt + C o

t = Y y + Y o + T ,

Bt = B0,

Tt = T .

Land: nominal assets are excluded

Property 1 :there is no equilibrium with a (strictly) positivequantity B0 of nominal asset.

Proof : absence of arbitrage opportunity

qt ≥ qt+1 + pt+1,

with equality if Tt is strictly positive. Therefore:

qt =t′∑

τ=t+1

pτ + qt′ ≥ lim∞∑

τ=t+1

pτ .

For a finite price of land, pt has to go to zero when t goes toinfinity. But this is inconsistent with the budget constraint of theyoung consumer at date t, which becomes in the limitlim qtT + B = 0.

Equilibria without nominal assets

Take the non durable good as the numeraire. Eliminating landholdings between the two budget constraints yields:{

max U(C y ,C o)(1 + ρt)C y

t + C ot+1 = (1 + ρt)Y y + Y o

where:

1 + ρt =1 + qt+1

qtC y

t + qtTt = Y y .

Same geometric analysis as previously, except for the change in thedifference equation

zy (ρt) + zo(ρt−1) = T .

Equilibria: concluded

There is a single equilibrium, close (for small T ) but different fromthe previous efficient stationary equilibrium.

Positive net real interest rate. The equilibrium is not the efficientin the class of stationary allocations.