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Unformatted text preview: Fall 2009 Truman Bewley
Economics 350a
Answers to Homework #11
(Due Thursday, December 3) Problem: 2) Consider a Diamond model with the price of the produced good equal to one, when
the production function is f( K, L) = 2 [KT
and the utility function of each consumer is
u(x0, x1) = 2ln( x0) + n(x1).
Remember that the only endowment consumers have is one unit of labor in youth.
a) Compute a stationary spot price equilibrium
(Xo(r), X1“), K(r), LU), P. WU). r, G(r),T(r)).
where P :1 and r > —1.
b) Graph the function T(r).
c) How many values of r satisfy the equation T(r)=—1 ? oi) Over what range of values of r is the utility of a typical consumer increasing in r and
over what range is the utility of a typical consumer decreasing in r? e) For what values of r is the equilibrium allocation Pareto optimal? f) Suppose that the lumpsum tax is maintained at level T( T) and that an infinitesimal amount of pay—asyou—go social security is i_ntroduced. Social security changes the
equilibrium interest rate to r. Wiii r exceed r or be less than r , when i)—r=0, ii) 7 =2? 3%
:2
it
2
15 wwwmmwwmvwmmmwﬁwmfw __ MWWW “WWW" ail7:» mqwmzwmmwmmwmwmvzmmm Answer: a) The equation af(K=1) =1+ r
6K t:
3
¥
5
g becomes =1+r, WTWWWWW€WWWKWEWQWWWC mamawe J.
K so that :wwmwmwm r" m K: 1
(t+r)2 and total output is y=f(K,1)=2’_.J_= 2.
' (1+r)2 1+r Output net of capital input is Awm:mumwmmmwcwwawwwmmwaxemwxmmmmmmww Mew“? y—K: 2 _ 1 =2(t+r)—1=1+2rl
1+r (1+r)2 (1+r)2 (1+r)2 'Mmt m wewwnxwmwmmwmm The equation au(x,x) au(x,x)
0 1 0 1 (1+r) 6X ax
1 0 becomes awmnwmm)‘rxmﬁﬁWhmwmxwmavmwwnw so that we obtain X+1+r;,(:1+2r
° 2 ” n+0? sothat
3+ rx _ 1+2?
2 ° (1+r)2
andhence
x: 2(1+2r) ’
0 (3+r)(1+r)2
and
X23+rxJr 1+2r _
"1 2 ° (3+r)(1+r)
SinceP=f,
W=at(K,1)=,j_= 1 6L 1+r The government debt is G: 1 —K= 1+2r __ 1 =1+2r—3—r
1+r (3+r)(1+r)2 (1+r)2 (3+r)(1+r)2 = ~2+ r .
(3+ r)(3 + r)2 The tax is [email protected]=_;t&uL_.
(3+ r)(1+ r)2 In summary, the equilibrium is ((XOU). X1“). KM). LU), 3’: WU), f, G(I).T(r)) =[[ 2(1+2r) . 1+2r ’ 1 ,1}1, 1 ,r,
(3+r)(1+r)2 (3+r)(1+r) (1+r)2 1+r 3 ~2+ r r(—2+r) ].
(3+ r)(1 + r)2 (3 + r)(1+ r)2 if r > —1/2. it —1 < r 3 4/2, then there is no meaningful equilibrium, for output net of
capitai, y  K, is less than or equal to 0. Answer: b) 2 1/2 0 1 2 3 4 6 7 8 Answer: 0) Two. They are r = 0.0545673 and r = 1.428625. Answer: d) The utility of a consumer is
2n[_2_(_1_:£l.._]+ m[__1_t_2r__]
(3+r)(1+r)2 (3+r)(1+r) tn(4) + inf—1%]
(3 + r)2(1+ r)5 II = n(4) + 3n(1+ 2r) 3ln(3 + r) —5ln(1 + r).
'The derivative of this function with respect to r is 6 _ 3 __ 5 :_ 10r(2+r) _
1+2r 3+r 1+r (1+2r)(3+r)(t+r) KTM‘NM mm Mawmem'émwwwmm‘wmmwuWRWlWMWﬁmWminim?»p’ Mmimwwwwmwvm: www mmwtm >ga..ur.wszmm;mwwmo mammime» ml1Wmwwwwwbwmwwmiwvmmw(WrwwWJ'N‘mﬁlr: M'mxmmwmmwwuwnvmtxvnvmﬁgm It follows that the utility of a consumer is increasing if —1/2 < r < 0 and is decreasing if r > 0.
It reaches a maximum at r = 0. Answer: e) For r 2 0. Answer: 1‘) We know that TU) : r(—2+ r) .
(3 + r)(1+ r)2
sothat
em?) : —_r‘+473+1772+67—6
C” (3+ r)2{1+ r)‘ The sign of M is the same as that of its numerator. ’cir
_ — 1 ‘—
(i) If r :0, _d_r_ = [an r) T r_ =0,sothatrdoesnotchange.
dl‘ dr 1+ r (ii) If T = 2, the numerator of ¥L is 90 > 0, so that r > T.
r Probiem: 3) Consider'the Diamond model with utility function u(xo, x1) = log x0 + log X1
and production function is
f( K, L) = 4K1'4L3'4.
a) Compute a spot price equilibrium, (x0(r), X1“). K(r), Mr). P, WU). r, G(r),T(r)), when r20andP=1. b) What is the stationary equiiibrium interest rate when the lumpsum tax on youth, T,
equals 0.15? What are the levels of capital and of consumption in youth and old age? 0) Suppose that a social security program is introduced that taxes youth 0.1 and pays a
benefit of 0.1 to the old. The iumpsum tax of 0.15 on youth is continued after the social
5 . security is introduced, so that youths pay 0.25 in totai taxes. What is the stationary
equilibrium interest rate? What are the leveis of capital and consumption in youth and
otd age? d) Suppose that when the sociat security program of part (c) is introduced, the lump sum tax of 0.15 paid by youth is changed to a new ievet, T, so that the introduction of social security does not change the interest rate. What is this new ievei of the lump
sum tax? ' it
A?
it:
r:
!. Answer: a) The equation 52M 6K becomes K'G”=i + r, so that Gm"WnﬁmwaﬂmﬂmﬁﬁmﬁﬂWWW‘mtﬂmWLMJWWEMWKW%SWW$NWW K = (1 + 0““?
and totat output is y = 4(1+ r)‘“3. «wwwm'w‘xwwmmwwimm 'Output net of capital input is y_K= 4 _ 1 : 3+4r .
(1+ r)1n'3 (1+ r)4n'3 (3+r)41‘3 The equation m v"wWemy.wwwmwwmmwwwwmmwwmw (1 + r) au(xo, x1) : au(xo, x1) 8x 6x
1 0 g‘. becomes so that WWW WWWWWW“ ...i_u.w....m.,m.m.,t my. M m, m A t. it «A n» NW ( WWW W. WW“. W .\,.....m a mo x =(1+r)x.
1 0 The feasibility equation xo+x1=y—K
becomes
(2+ r)x =ﬂﬂ
0 (1+ r)“
sothat
x = 3+4r 0 , (2 + r)(1+r)"”3
and 3+4r _
1 (2+ r)(1+ r)“3 K
II it follows that government debt is a: 1 —K=_§+_‘3r____—__1____;=___1_+§r____.
1+r (2+r)(t+r)‘”3 (1+1')“3 (2+r)(1+r)‘”3
Thetaxis g
T: r(1+3r) _
(2+ r)(1+ r)“3
Thewageis
W=3W“= 3 (1+ r)“3 In summary, the equilibrium is ((XOU). X1“), K(r)), LU). P, Wit), r, G(r),T(f)) =[[ 3+4r 7, 3+4r , 1 ,1}1, _i__.n
(2+r)(1+r)‘”3 (2+r)(1+r)“3 (1+r)‘”3 (1+r)“3 1+3r r(1+3r)
(2+ r)(1+ r)"3' (2+ r)(1+ 04"“ Answer: b)
r = 0.25922.
K = 0.735413. x0 = 1.31407.
x1 = 1.654703. Answer: 0) The equation to be solved for r is T(r) = 0.25 — 0.1/(1 + r), where T(r) = r(1+3r) .
(2 + r)(1+ r)“3 The solution of this equation is r = 0.297068. At this interest rate, 11mm,mwswmwmwwmwwwwwwﬁ:ﬁmﬁmﬁiWAWWAWMMW'WLWIrWMWﬁwmwmwmmdmwmwmm K = 0.706941,
x0 = 1.288974.
x = 1.671886. Answer: d) The new tax level is 0.1 T = 0.15 0.1+ —————————
1.25922 = 0.1294142406. (The calculationsof this problem can be done with a hand held calculator or an Excel spread
sheet.) ’ ...
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