ec3332TutWk12SeptSoln (1)

ec3332TutWk12SeptSoln (1) - EC3332 Money and Banking...

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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 re-written 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 non-stationary 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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This note was uploaded on 02/01/2012 for the course ECON 101 taught by Professor Meonk during the Spring '11 term at Abu Dhabi University.

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ec3332TutWk12SeptSoln (1) - EC3332 Money and Banking...

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