AdvMacro_Problemset2_Solution
Problem Set #2Answer Key Economics 808:Macroeconomic Theory
Fall 2004
1Linear di?erence equations
See the Excel ?le di?eq.xls on the web site.
2A nonlinear di?erence equation
a )See the computer ?le.
b )We ?nd a ?xed point by solving the equation x ∞=0.5x 0.5∞for x ∞
.This gives us two ?xed points,x ∞∈{0,0.25}.
c )In general,f (x )=0.25
x 0.5
,so f (0)=∞1and f (0.25)=0.5.Zero is an unstable ?xed point,meaning that if the system starts near zero,it will tend to move away from zero.In contrast,the 0.25is a stable ?xed point,implying that if the system starts near 0.25it will tend to move towards 0.25.In addition,because f (0.25)>0,it will move monotonically towards 0.25.
d )Th
e simulated time series (in the computer ?le)shows x t monotonically converging to 0.25,just as the local analysis predicted.
3Another nonlinear di?erence equation
a )See the computer ?le.
b )We ?nd a ?xed point by solving the equation x ∞=ax ∞(1?x ∞)for x ∞.In general there are two solutions:
x ∞∈{0,a ?1
a }
When a =1.1,the two ?xed points are 0and 0.09.c )In general f (x )=a (1?2x ).This means
f (0)=a f (a ?1a
)=2?a
1
If you prefer,lim x →0f (x )=∞
1
The Central University of Finance and Economics
China Center for Human Capital and Labor Market Research
Advanced Macroeconomics
Problem Set 2
ECON808,Fall20042 In this case,f (0)=1.1and f (0.09)=0.9.Zero is an unstable?xed point and0.09is a stable ?xed point.
d)The simulated time series(in the computer?le)shows that x t converges monotonically to the stable?xed point.
e)See the computer?le.
f)The function f has two?xed points,0and0.655.
g)In this case f (0)=2.9and f (0.655)=?0.9.Zero is an unstable?xed point,and the0.655is stable.Since f (0.655)<0,this implies that starting near0.655,the system will converge to0.655 with oscillations.
h)The simulated time series(in the computer?le)shows that x t moves towards the stable?xed point,but oscillates,as predicted by the local analysis.
i)This plot is in the computer?le.
j)The?xed points of f are zero and0.748.
k)In this case f (0)=3.97and f (0.748)=?1.97.Both?xed points are unstable,implying that starting near either,the system will move away.
l)The simulated time series shows that x t appears to jump around randomly with no clear pattern even though each value is a deterministic function of the previous value.
The time series you see here is an example of a chaotic system.Chaos only appears in systems governed by deeply nonlinear di?erence equations.In fact,the way you get“random”numbers from a computer is through successive application of a nonlinear di?erence equation to some“seed”value(say,the current date in seconds since Jan1,1970).
This particular di?erence equation exhibits chaos when3.94 4The CES production function The purpose of this problem is mostly to give you practice in working with these things under a production function other than the Cobb-Douglas one.As you can see,the CES production function give much messier results than Cobb-Douglas.This problem is quite similar to Romer’s Problem1.3. a)The intensive form is f(k)=F(k,1),so f(k)=[akρ+b]1/ρ b)We know that r t=f (k t),so: r t=akρ?1 t a+bk?ρt 1?ρ ρ c)Since L t=1: w t=b[akρt+b]1?ρρ ECON 808,Fall 2004 3 d )Th e basic capital accumulation equation is: k t +1=(1?δ)k t +s [ak ρ t +b ]1/ρ Subtracting k t and then dividing by k t ,we get: k t +1?k t k t =s [a +bk ?ρt ]1/ρ ?δYou might notice that this growth rate is decreasing in k t . e )k t is at the steady state when the growth rate we found above is equal to zero: s [a +bk ?ρ∞]1/ρ =δ Solving,we get: k ∞= b (δ/s )ρ?a 1/ρ 5Capital ?ows in the Solow model a )First we note that: y C y S =Ak αC Ak αS and we solve for k C k S as a function of y C y S and model parameters.This gives k C k S = y C y S 1/α =103=1000 In other words,in order to produce 10times Scotland’s output,Canada must have 1000times Scotland’s capital.b )We know from the ?rm’s pro?t maximization problem that r =f (k )=αAk α?1 So r S =αAk α?1 S =αA k S k C k C α?1 = k S k C α?1?αAk α?1 C = k C k S 1?α ?r C = 10002/3?0.05 =5.00 Switzerland 's output, China Switzerland ECON808,Fall20044 So, Switzerland Switzerland. So Scotland has a500%net return on capital. c)I don’t know about you but I’d put my money in Scotland. Of course,given that we see tenfold di?erences in output per worker,but don’t see these kinds of returns in less-developed countries,something is going on here.Sure enough,poor countries have much lower TFP in addition to less capital.