This subsection intends to illustrate a case of a simple stationary OLG economy with a single firm. At every date t ≥ 1, there is only one good |Lt| = 1, and one individual per
generation |It| = 1.
Consumers
All consumers are identical except the old consumer at date t = 0. They live two periods, and are described by their consumption set X = R2
+, their initial endowment
vector (ey, eo) and their utility function u : X → R. The consumer at date t = 0 who is old at date t = 1 has consumption set R+.
We consider a standard time-separable utility function, namely the Cobb-Doublas utility function: u(xyt, xot) = a ln xyt + (1 − a) ln xot, 0 ≤ a ≤ 1.
The firm
The firm is described by a production function f : −R+ → R which transforms the good
at date t into the good at date t + 1.
We assume that f is of the form: f (zt) =
−γ(zt− ˆz), if zt≤ ˆz ≤ 0 0, if ˆz ≤ zt≤ 0 , where γ > 1, ˆ
z < 0 and zt ≤ 0 the input used at date t. The output at time t + 1 is then given by:
ζt+1= f (zt).
This technology exhibits strict increasing returns due to the fixed cost ˆz.
Pricing rule
The price vector p is an element of R2+. Define the relative price: πt= ppt+1t . We describe
the behavior of the producer by the average cost pricing: ptzt+ pt+1ζt+1= 0, or in term
of relative prices: πtzt+ ζt+1 = 0.
The consumer’s demand
Each consumer t is maximizing her utility function ut under her budget constraint. Given a price (pt, pt+1), the consumer t’s demand is:
( xyt = awpt t = a(e y+eo πt) xot = (1−a)wp t t+1 = (1 − a)(πte y+ eo)
Existence of equilibrium in OLG models with increasing returns 33
Clearly, if the relative price πt increases, the agent will decrease his consumption when
young and increase it again when old. His lifetime utility is an increasing function of πt ≥ 0. However, we note that at each period t, the relative price πt cannot be too
small. Indeed, if we let πt tend to 0+ at date t, the consumption of young xyt will be
infinitely high while the total resource at each date is finite.
Supply function
Whenever zt≤ ˆz, the firm can decide to produce ζt+1= −γ(zt− ˆz) following an average
cost pricing. Thus we can write: ζt+1= −πtzt and:
• ζt+1= πγ ˆtzπ−γt
• zt= −πγ ˆz
t−γ
Remarks:
i) zt and ζt+1 are well defined whenever πt< γ;
ii) for all t ≥ 0, for πt ∈ (0, γ), ζt+1 > 0 and zt < 0, in addition, ζt+1 is an increasing
function of πt;
iii)limπt→γ−ζt+1= limπt→γ−
γ ˆyπt
πt−γ = +∞
iv) In case of constant returns to scale, that is no fixed cost, it is known that the firm, while maximizing its profit would exhibit a discontinuous supply function. The fixed cost associated to the average cost pricing result in a smooth supply function on the range (0, γ).
Equilibrium
An equilibrium is an element (p∗t, (x∗o0 , (x∗yt , xt∗o)∞t=1), (zt,∗ζt+1∗ )) such that:
a) for all t, (x∗yt , x∗ot ) is a solution to
max u(xtt, xtt+1) s.t π∗tx∗yt + x∗ot ≤ ˆw∗t b) For all t, πt∗z∗t + ζt+1∗ = 0; c) x∗yt + x∗ot−1= ey+ eo+ z∗ t + ζt−1∗ ; where π∗t = p∗t p∗t+1 Characterization of equilibria
The sequence of prices (p∗t) is an equilibrium price system if the sequence of relative prices (πt∗) is a solution to:
(1 − a)ey(πt−1− 1) − γ ˆy πt−1− γ πt−1= aeo(1 − 1 πt ) − γ ˆy πt− γ
Existence of equilibrium in OLG models with increasing returns 34
Figure 2.1: Asymptotic efficiency: when the average-cost prices tend to the marginal cost with no fixed cost
with πt∈ (0, γ) for all t ≥ 1, γ > 1, 0 ≥ ˆy ≥ −ey and 1 ≥ a ≥ 0.
The following figure illustrates this equation in terms of (πt−1, πt) for the particular case
where γ = 4 the initial endowment at each period ω = 2, the initial endowment when old e = 0.8, the propension to consume when young a = 0.02, the fixed cost ˆz = −1. Here, the consumer is ready to invest in the production since his endowment is large enough when young and his preferences put a higher weight when old.
We have multiple steady states equilibria: a low and stable steady state π∗ < 1 and π∗∗ = 1 which is not stable. Suppose that at each date t0, πt0 < 1, then the economy
will display a succession of inflationary equilibria since all the following prices will be below 1 and will be decreasing to the low steady state π∗. If we instead have πt0 > 1,
then the successive price πt0+1 > 1 so are all the following prices. This case will lead
to a non stationary equilibria where the relative prices will increase and tend to the marginal price γ without reaching it, that is: 1 < πt0 < πt0+1 < . . . < γ. Thus from
period t0, the production will be increasing, so will be the welfare of all the successive
generations since they will benefit from a deflation: pt0 > pt0+1 > . . .. Thus we have an
economy which is asymptotically efficient: it converges toward an equilibrium which is associated to an economy without fixed costs, at which the utility is at its maximum. For instance, if u∗, u∗∗ and ¯u are respectively the utility levels at π∗, π∗∗ and γ, then given the same parameters above, we have: u∗ = 0.255 < u∗∗= 0.595 < ¯u = 1.597.
The existence of fixed cost and the average cost pricing make the firm decide to produce at each date. So the fixed cost is not always necessarely a barrier to production. It is
Existence of equilibrium in OLG models with increasing returns 35
also important to remark that a production at the marginal price γ is still possible even though it will generate losses equal to γpt+1z = pˆ tz ≤ 0. Indeed, the demand would be:ˆ
(
xyt = a(ey+ ˆz +eγo) xo
t = (1 − a)(γ(ey+ ˆz) + eo)
The corresponding market clearing equation gives a positive production equals to 0.768, given the same parameters above, and the associated utility at γ is 0.344 which is clearly lower than u∗∗ thus lower than ¯u. This confirms the “superiority of average cost over over marginal cost when πt0 > 1.
Note that the choice of a is primordial and that this result relies on the particular fact that consumers have higher initial endowment but lower incentive to consume when young, they are more focused on future consumption.