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Unformatted text preview: EC3332 Money and Banking
National University of Singapore
Semester I, 2009—2010
Second Tutorial (Week of 7 September, 2009)
Solution 1. Consider the 2 — period overlapping generations model as in Chapter 1 with no population growth. Assume that the endowment of the good
is yl when young and y2 when old. Let the consumption when young
and old be written as (cm7 691;“). Suppose the population is growing
at rate n so the feasibility consumption in time period t is nlyi — Cu) + ($92 * 62¢) = 0 (1) In a steady state or stationary allocation, we have: “(Eh — Cl) + (3/2 — C2) = 0 (2) Assume that each generation has preferences over consumption when
young and old given by the utility function u(cl,t,c2,t+1). Consumers
can save Via a ﬁnancial asset (bond) with real rate of return Rt. Their
budget equation is given by: 1
(y:  Cu) + —(m — 02,r+1) = 0. (3)
t
Multiplying by Rt, the budget equation can be rewritten as: Rifyl — Cu) + (1J2 — C2,t+1) : 0. (4) Consumers be maximizing utility at a bundle ct at interest factor, Rt
if any bundle, c: that is preferred to it does not lie inside the budget
set. Thus, we say a sequence of consumption or program (c1,t,02,t+1) is
competitive if there exists a sequence R; such that if any c is preferred
to it, then Rtlyi  61) + (9'2 — 02) < 04 (5) We say a program is an equilibrium program if it is both feasible (1)
and competitive. (a) (b) Look at equilibrium programs that are stationary. Show there are
at most two steady states with either R1 = n or 01 = y1. Draw this in a graph for utility functions that exhibit diminishing
marginal rate of substitution (i.e. indifference curves are convex to
the origin). What must the slope be of the budget line through the
endowment point if there is no trade? Through the equilibrium
allocation if there is trade denoting the allocation is E? This helps
us deﬁne a model to be Classical (Samuelson) if 51 > '91 (El < yl). If the bond is zero net supply, then what must the equilibrium
interest rate, R: be in the two cases? Using this show that a
model is classical (Samuelson) if and only if R, > n (Rt < Now we look at nonstationary programs. The consumers will
maximize utility by maximizing utility subject to Show that
the first order conditions are given by: (91 "" Ci,t)u1 + (3J2 — €2,r+1lu2 K 0, (6) where u, are the ﬁrst—order partial derivatives with regard to i L"
1, 2 or young and old consumption respectively. Now note that the
equations (1) and (6) fully describe the equilibria in the model. Now let u(c1,c2) = 10C] 7 4c? + 4c; ~ 0% for c1 3 5/4, 62 5 2,
and let the endowments be y = (0,2), and n = 1 or there is no
population growth. Write down the (1) and (6) for this economy.
Eliminate cu“ to get the equilibrium law of motion as 2 m 2
C1.t+1 _ 5cm _ 4cm. (7) This is a quadratic equation with two steady states c = (0,2)
and 5 = (1,1). There is also a solution of the following kind: If C1,: = (5 — «El/6, then Cl,t+1 = ((5 + “El/5: Ci,t+2 = i5 “ ﬁl/ﬁa etc, or there is a cycle of period 2. 1 Solution 1. Consider a stationary equilibrium where c is independent of time. Then,
(4) becomes: R(yr  Ci) ‘l’ (3J2 — 432) = 0' (8)
(a) Then subtracting (2) from (8) we have:
(R — “llyl * C1) = 0 (9) Thus, there at most two steady states: either R = n or 01 = y1. Consider the ﬁrst case. If there is any other stationary plan, 6
that is preferred to c, then it must be the case that: Rlyl — E1) — (:92 — 52) < 0. Why, as c is an equilibrium plan so that (5) must hold. Now
consider the second case: Here there is no—trade. Thus, the im—
plication is that a steady state must either be (I) A golden rule
program or (H) An autarky allocation (no—trade). (b)
c Ignore the case Where 01 : y]. As E is strictl preferred to , from
r, y
(5) it must be the case that at interest rate R, that: [HZ/1 — 51) + (92 r 52) < 0 (10)
Subtracting (2) from this gives: (R—nﬂyl —51) < 0. (11) Thus, (R — n) and (91 — 51) must have opposite signs as asserted. (d) Maximizing u(cl,t,02,t+1) subject to From the budget set, we
see that: Rt(y1 — Cm) +312 : 02¢“). Thus, we can consider an un—
constrained maximization problem, of maximizing u(c1,t, R,(y1 —
cm) + yg) by choosing (217$. This gives the ﬁrst order condition: u1(q) —— Rtuﬂcr) = 0. (12) 3 Eliminating Rt between (4) and (12) we have the desired equation.
Note that the slope of (6) is given by Rt. From here one can also study the local dynamics around the steady
state, i.e., if the economy was to move out of the steady state (for
some reason) how would the equilibrium evolve. The equilibrium
must satisfy (1) and The ﬁrst will be a straight line with
slope ~11. The second is a non—linear curve. To see how the
economy would evolve, start with a point on (1) as this gives the
consumption of the old and young in period 1. For the young at
period 1 to maximize utility they must be on So corresponding
to cm ﬁnd egg from For markets to clear, it must be the case
that consumption of young and old must satisfy (1) at period 2. r This gives C13. Then these young must be on their which (6) determines (:23 and so on. Now consider the classical case. At the no trade stationary equi—
librium, the (6) is steeper than From here one can show using
the argument above, that if the economy was to move away from
this equilibrium it would continue to move away, or aurtaky is
unstable. Similarly one can show that as the slope of (l) at the
autarky allocation in the Samuelson case is steeper than the slope
of the (6), it is locally stable. That is any initial allocation close to it over time will converge to it.
In this case1 (1) is:
(31‘; = 2 — 62¢. (13) The (6) becomes:
C1,t(5 _ 4C1't) — —‘ C2,t+1)2 I Note that the equation (13) has cm, 622; in it, and (14) has cm, 02¢“)
in it. Update (13)_one period to get: Cl,t+1 ‘—“ 2 — Cam—1‘ Now substitute into (14) to eliminate c2¢+1 from both by substi—
tution to get an equation in C12t3C1¢+1 which gives the evolution of the consumption of the old across time periods. This is the desired equation: 2 Ci,t+1 = 50m _ 4cit (15) It is called a ﬁrst—order diﬁerence equation. By substitution one
can check that if an = (5 — ﬁye : 0.46, then CW1 : ((5 +
x/EVS = 1.217 c1,t+2 = (5 — z 0.46, etc. Or in other words
the consumption of the generations oscillate between (0.46, 0.79)
and (121,154). or we have a 2—period cycle. In the words of Gale,
This is an amusing example of a “business cycle” which has noth—
ing to do with expectations. There are no em posts and ea: antes.
Everyone has perfect foresight but cycling nevertheless occurs as a
consequence of the equilibrium price mechanism. (Gale, 1973). CL)tﬁ N g0? Wit/diam (AM Local M5595“? 7L
mud‘7ka ‘31 C/Uiﬁl‘cji 034 (/0le 95mm; f}
awﬁm/kj 14‘
SMWMxM 0W ...
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 Spring '11
 Meonk
 Economics, Thermodynamics, Equilibrium, Steady State, Utility, Samuelson

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