Macro monetaria y financiera:
Solving the competitive equilibrium in OLG model.
Hernán D. Seoane∗
Universidad Carlos III de Madrid
February 14, 2016
Intro
Following the notation in chapter 2 of Wallace and McCandless, (1992) “Introduction to Dynamics Macroeconomic Theory. Harvard University Press”.
The competitive equilibrium
Suppose the OLG model is populated by households with logarithmic preferences
max
ct (t),ct (t+1)
ln (ct (t)) + β ln (ct (t + 1))
that face the following first period budget constraint (which assume holds with equality)
ct (t) = ωt (t) − l(t)
and a second budget constraint for the second period
ct (t + 1) = ωt (t + 1) + r(r)l(t)
∗ If
you find any typos please email me.
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To find the solution to the households problem we can merge the 2 budget constraints into one intertemporal
constraint (which we write in present value terms
ct (t) +
ct (t + 1)
ωt (t + 1)
= ωt (t) +
r(t)
r(t)
where we can see that l(t), that was used to merge the constraints, dropped.
Remember we want to find the Competitive Equilibrium, defined as: the set of policy
functions ct (t), ct (t + 1) and l(t) and prices r(t), for all t such that households solution is optimal
and demand is equal to supply in all markets.
Now, construct the Lagrangian
ct (t + 1)
ωt (t + 1)
L = ln (ct (t)) + β ln (ct (t + 1)) − λ ct (t) +
− ωt (t) +
r(t)
r(t)
The first order conditions for this optimization problem are
∂L
1
:
−λ=0
∂ct (t) ct (t)
∂L
1
1
:β
−λ
=0
∂ct (t + 1)
ct (t + 1)
r(t)
∂L
ct (t + 1)
ωt (t + 1)
: ct (t) +
− ωt (t) +
=0
∂λ
r(t)
r(t)
We can re-order these optimality conditions
λ=
β
ct (t) +
1
ct (t)
r(t)
=λ
ct (t + 1)
ct (t + 1)
ωt (t + 1)
= ωt (t) +
r(t)
r(t)
We can find the Euler equation by merging the first 2 equations
β
r(t)
1
=
ct (t + 1)
ct (t)
and using this, we can write the first period consumption in terms of second period’s consumption
ct (t) =
ct (t + 1)
βr(t)
2
Using the budget constraint
ωt (t + 1)
ct (t + 1) ct (t + 1)
+
= ωt (t) +
βr(t)
r(t)
r(t)
We can use this last expression to find ct (t + 1).
ct (t + 1)
1
1
+
βr(t) r(t)
ct (t + 1)
= ωt (t) +
ωt (t + 1)
r(t)
1+β
ωt (t + 1)
= ωt (t) +
βr(t)
r(t)
Hence, ct (t + 1) is:
ct (t + 1) =
βr(t)
1+β
ωt (t + 1)
ωt (t) +
r(t)
Using the Euler equation, we can also find consumption in period 1
1
ct (t) =
1+β
ωt (t + 1)
ωt (t) +
r(t)
In order to find the full solution of the households, we still need to find savings in period 1, l(t). Using the
first period budget constraint,
ct (t) = ωt (t) − l(t)
1
l(t) = ωt (t) −
1+β
ωt (t + 1)
ωt (t) +
r(t)
To find the equilibrium, we still need to find the equilibrium price r(t). This has to be such that there
is equilibrium in the goods market. Recall that trade can only occur between people born in the same
generation. That means that total lending has to be 0
N (t)l(t) = 0
which implies that l(t) = 0 in equilibrium. Then adding the budget constraints of all young agents in period
t:
N (t)ct (t) = N (t)ωt (t) − N (t)l(t)
ct (t) = ωt (t)
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Replacing consumption policy function into this equation:
1
1+β
ωt (t) +
ωt (t + 1)
r(t)
r(t) =
ωt (t + 1)
βωt (t)
= ωt (t)
which we can solve for r(t).
Finally,notice that equilibrium consumption in the first period is the consumption policy function evaluated at the equilibrium interest rate:
ct (t) =
1+β
ωt (t)
1+β
equilibrium consumption in the second period is
ct (t + 1) =
βωt (t + 1)
(ωt (t) + βωt (t))
βωt (t)(1 + β)
or
ct (t + 1) = ωt (t + 1)
So, equilibrium prices are such that in equilibrium agents want to comsume their endowment.
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