Overlapping Generations Model:
Equilibrium and Steady State
Prof. Lutz Hendricks
Econ720
August 23, 2016
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Topics
We study the equilibrium of the OLG production economy
1. Dynamics of capital accumulation
2. Steady state
3. Dynamic efficiency
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Competitive Equilibrium
Recall the equilibrium definition for the production economy:
An allocation: cyt , cot , st , bt , Kt , Lt
Prices: (qt , rt , wt )
That satisfy:
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the household EE and budget constraints (3 equations)
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the firm’s FOCs (2 equations)
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the market clearing conditions (4 equations)
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identity: r = q
d.
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Saving Function and Dynamics
Saving Function and Dynamics
We need to describe how the economy evolves over time.
We derive a difference equation (a law of motion) for the
economy’s state variables.
What are the state variables?
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Variables carried over into the current period from the last
period.
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Variables that are predetermind in the current period.
Here: the state variable is Kt .
More conveniently, we use kt = Kt /Nt as the state variable.
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Saving Function and Dynamics
The evolution is k is characterized by the capital market clearing
condition Kt+1 = Nt st+1 or
Kt+1 /Nt+1 = Nt /Nt+1 · st+1
(1 + n) kt+1 = st+1
(1)
together with the household saving function
st+1 = s (wt , rt+1 )
(2)
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Saving function
Start from the Euler equation
b (1 + rt+1 )u0 (cot+1 ) = u0 (cyt )
Substitute in the budget constraints for both ages:
b (1 + rt+1 )u0 ([1 + rt+1 ]st+1 ) = u0 (wt
st+1 )
This implicitly defines a saving function
st+1 = s(wt , rt+1 )
(3)
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Properties of the saving function
Totally differentiate the Euler Equation to find the derivatives of
the saving function:
b (1 + rt+1 )2 u00 ([1 + rt+1 ]st+1 )dst+1
= u00 (wt
st+1 )(dwt
dst+1 )
Effect of higher endowments:
dst+1
u00 (cyt )
=
>0
dwt
b (1 + rt+1 )2 u00 (cot+1 ) + u00 (cyt )
Simply an income effect.
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Effect of the interest rate
b u0 ([1 + rt+1 ]st+1 )drt+1
+b (1 + rt+1 )u00 (cot+1 )(st+1 drt+1 + [1 + rt+1 ]dst+1 )
=
u00 (wt
st+1 )dst+1
The effect is ambiguous (income vs substitution effects):
∂ st+1
=
∂ rt+1
b u0 (cot+1 ) + b (1 + rt+1 )u00 (cot+1 )st+1
b (1 + rt+1 )2 u00 (cot+1 ) + u00 (cyt )
(4)
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Effect of the interest rate
A simplification:
∂ st+1
=
∂ rt+1
where
⇥
⇤
b u0 (cot+1 )(1 s cot+1 )
b (1 + rt+1 )2 u00 (cot+1 ) + u00 (cyt )
s (c) ⌘ u00 (c)c/u0 (c)
(5)
(6)
is the coefficient of relative risk aversion.
It follows that savings respond negatively to the interest rate, if
s > 1.
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High s ! small substitution effect ! income effect raises cyt /
reduces st+1 .
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Derivation
Use the 2nd period budget constraint to replace (1 + rt+1 )st+1 by
cot+1 .
∂ st+1
=
∂ rt+1
b u0 (cot+1 ) + b u00 (cot+1 ) cot+1
b (1 + rt+1 )2 u00 (cot+1 ) + u00 (cyt )
(7)
“Pull out” u0 cot+1 .
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Effect of a higher interest rate
The figure illustrates the case where income and substitution effect
just cancel.
High s =) strong curvature of indifference curves.
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CRRA Utility
In particular, for the popular CRRA utility function
u(c) = c1
s
/(1
s)
the s (c) is constant (namely s , show this!).
For s = 1, this becomes log utility (and sr = 0).
In the data, s is most likely greater than one, although its value is
highly controversial.
CRRA stands for “constant relative risk aversion.”
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s is the coefficient of relative risk aversion (see discussion of
stochastic economicies).
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1/s is intertemporal elasticity of substitution.
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CRRA Utility
0
−5
−10
−15
Utility
−20
−25
−30
−35
−40
sigma = 1.1
sigma = 2.0
sigma = 4.0
−45
−50
0
0.2
0.4
0.6
Consumption
0.8
1
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Law of motion for capital
Recall (1 + n) kt+1 = s (wt , rt+1 ).
Use the firm FOCs to replace the prices:
(1 + n)kt+1 = s(f (kt )
f 0 (kt )kt , f 0 (kt+1 ))
This is a first order difference equation of the form
kt+1 = f (kt )
Implicitly differentiating yields
dkt+1
sw kt f 00 (kt )
=
dkt
1 + n sr f 00 (kt+1 )
(8)
This completely determines the behavior of the economy.
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Concave law of motion
If f is concave, we get simple dynamics.
From any initial condition (k0 ) the economy converges
monotonically to a unique steady state (k⇤ ).
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Properties of the law of motion
We know:
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f (0) = 0: k = 0 is a steady state.
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The derivative is
dkt+1
sw kt f 00 (kt )
=
dkt
1 + n sr f 00 (kt+1 )
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(9)
A sufficient condition for f 0 > 0 is sr > 0. Intuition: the supply
of capital is upward sloping.
Otherwise, little can be said in general.
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Log utility - Cobb Douglas example
The utility function is u(c) = ln(c).
Then the household saves a constant fraction of his earnings:
cyt = wt /(1 + b )
and therefore
st+1 = wt b /(1 + b )
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Log utility - Cobb Douglas example
Assume further that f (k) = kq . Then
w = (1
q )kq
The law of motion then becomes
(1 + n)kt+1 =
b
(1
1+b
q )ktq
Because sr = 0 and sw is a constant, f inherits the curvature of the
production function.
A unique, stable steady state exists.
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Log utility - Cobb Douglas example
Steady state
⇤
k =

1 q b
1+n 1+b
1/(1 q )
Steady state interest rate:
f 0 (k) = q kq 1
q 1+b
f 0 (k⇤ ) =
(1 + n)
1 q b
r = f 0 (k) d
Note: the steady state interest rate could be very small (low q or
high b ) or very large.
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Log utility - Cobb Douglas example
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The example provides a microfoundation for the Solow model.
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But it is a special case.
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An ill behaved example
The economy osciallates towards the steady state.
Multiple steady states are possible.
An important insight: Even very simple models can have
surprisingly complicated (and unpleasant) dynamics.
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Steady State and Dynamic Efficiency
Steady State
Definition
A steady state is an equilibrium where all (per capita) variables are
constant.
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The Golden Rule
Definition
The Golden Rule capital stock maximizes steady state consumption
(per capita).
Consumption per young household is
cy + co /(1 + n) = f (k) + (1
d )k
(1 + n)k0
Impose the steady state requirement k0 = k and maximize with
respect to k:
f 0 (kGR ) = n + d
(10)
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Dynamic Inefficiency
Definition
An allocation is dynamically efficient, if k < kGR .
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k > kGR implies a Pareto inefficient allocation.
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By running down the capital stock, households at all dates
could eat more.
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Nothing rules out a steady state that is dynamically inefficient.
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Why Is Dynamic Inefficiency Possible?
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Vaguely, the First Welfare Theorem says:
when all markets are competitive and some other conditions
hold, every CE is Pareto Optimal.
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One of the "other conditions" comes in 2 flavors:
1. there is a finite number of goods
2. •
j=1 pj < • where pj are the CE (Arrow-Debreu) prices.
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Both conditions are violated in the OLG model.
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Acemoglu, ch. 9.1.
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Intuition: Dynamic Inefficiency
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A missing market: the old must finance their consumption
out of own saving, even if the rate of return is very low.
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Suppose households value only co .
Then households save all income at rate of return f 0 (k0 )
For high k0 , this can be negative.
d.
An alternative arrangement that makes everyone better off:
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In each period, each young gives up 1 unit of consumption.
Each old gets to eat 1 + n units.
If n > f 0 (k) d , this makes everyone better off.
We will return to this idea in the section on “social security.”
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Final Example: Government Bonds
Demographics: Nt = (1 + n)t . Agents live for 2 periods.
Preferences:
(1
b ) ln(cyt ) + b ln(cot+1 )
Endowments:
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The initial old are endowed with s0 units of capital.
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Each young is endowed with one unit of work time.
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Environment
Technology:
Ct + Kt+1
(1
d )Kt = F(Kt , Lt ) = Kta Lt1
a
Government: The government only rolls over debt from one period
to the next:
Bt+1 = Rt Bt
Markets: for goods, bonds, labor, capital rental.
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Questions
1. Solve the household problem for a saving function.
2. Derive the FOCs for the firm.
3. Define a competitive equilibrium.
4. Derive the law of motion for the capital stock
(bt+1 + kt+1 )(1 + n) = b (1
a)kta
(11)
where b = B/L.
5. Derive the steady state capital stock for b = 0. Why does it
not depend on d ?
6. Derive the steady state capital stock for b > 0.
7. Show that the capital stock is lower in the steady state with
positive debt (crowding out).
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Where Are OLG Models Used?
Two period OLG models:
Mostly used for theoretical “examples”
Galor (2005)
Many period OLG models:
Commonly used for policy analysis (computational)
Pioneered by Auerbach and Kotlikoff (1987)
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Reading
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Acemoglu (2009), ch. 9.
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Krueger, "Macroeconomic Theory," ch. 8
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Ljungqvist and Sargent (2004), ch. 9 (without the monetary
parts).
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McCandless and Wallace (1991)and De La Croix and Michel
(2002) are book-length treatments of overlapping generations
models.
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References I
Acemoglu, D. (2009): Introduction to modern economic growth,
MIT Press.
Auerbach, A. J. and L. J. Kotlikoff (1987): Dynamic fiscal policy,
Cambridge University Press.
De La Croix, D. and P. Michel (2002): A theory of economic
growth: dynamics and policy in overlapping generations,
Cambridge University Press.
Galor, O. (2005): “From Stagnation to Growth: Unified Growth
Theory,” in Handbook of Economic Growth, ed. by P. Aghion
and S. N. Durlauf, Elsevier, vol. 1A, 171–293.
Ljungqvist, L. and T. J. Sargent (2004): Recursive macroeconomic
theory, 2nd ed.
McCandless, G. T. and N. Wallace (1991): Introduction to dynamic
macroeconomic theory: an overlapping generations approach,
Harvard University Press.
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