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Unformatted text preview: Fall 2008 Truman Bewley Economics 155a
Answers to Homework #11
(Due Thursday, December 4) Probiem: 2) Consider a Diamond model with the price of the produced good equal to one, when
the production function is f(K, L) =2JE and the utility function of each consumer is u(x0, x1) = 2n( x0) + n(x1). Remember that the only endowment consumers have is one unit of tabor in youth. a) Compute a stationary spot price equilibrium
(XOU). X1(r). Km, W). P. er), 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 d) 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? t) Suppose that the iumpsum tax is maintained at levet T( T) and that an infinitesimal amount of pay—asyougo social security is introduced. Social~ security changes the
equiiibrium interest rate to r. Will r exceed r or be less than r , when i)_r=0, ii) 7 =2? Answer: a) The equation 6K becomes (1+ r)2 and total output is y=f(KI1):2—1__.___= 2 .
“(1+02 1+r Output net of capitai input is y_.K= 2 _ 1 1+r (1+r)2 The equation a u(x0, x1) au( (1+r) 6X
1 becomes 1 2 0
On substitution into the feasibiiity
x0 + x1 = y — K, =2(1+r)—1 :
(1+r)2 XX
0'1) 6x
0 1+2r (1+ r)2l we obtain sothat
3+ rx _ 1+2r
2 ° (1+r)2
andhence
x» : 2(1+2r) ,
0 (3+ r)(1 + r)2
and
x : 1+ i‘x + 1+2r .
1 2 ° (3+r)(1+r)
SinceP=1,
W: af(K,1) =JE= 1 6L T+rl The government debt is G: x1 —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)(1+ r)2 The tax is T=rG= r(—2+ r) .
(3+ r)(1+ r)2 In summary, the equilibrium is ((XOU). X1“), KM), LU). P. WU), r, G(r),T(r)) I =[[ 2(1+2r) 1+2r 1 1}1, 1 r, (3+r)(1+r)2 (3+r)(t+r) (1+r)2 i+r
(3+ r)(1+r)2 (3+r)(1+r)2
if r > —1/2. If —1 < r s —1/2, then there is no meaningful equilibrium, for output net of g
capital, y — K, is less than or equal to O. ' ?
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 utiiity of a consumer is 2in[ﬂ)_._]+ ln[_1_.i2_r__...]
(3+r)(1+r)2 (3+ r)(1+r) II In(4) + n[Mi.—]
(3 + r)2(1+ r)5 ll [11(4) +3n(1 +2r) —3In(3 + r) ~5n(‘i + r). The derivative of this function with respect to r is ‘ 6 _ 3 _. 5 __ 10r(2+r) 1+2r 3+r 1+r (t+2r)(3+r)(i+r) It foilows that the utiiity of a consumer is increasing it —1/2 < r < 0 and is decreasing if r > 0.
It reaches a maximum at r = 0. Answer: e)'For r 2 0. Answer: i) We know that TU) : r(—2 + r) 2'
‘ (3 + r)(1+ r)
sothat
dT( r) : —_r“+4?3+ 17T2+6T—6
dr (3+ r)2(1+ r)4
. cmT) . .
The Sign of —d Is the same as that of its numerator.
r
— .— 1 —
(i) If r :0, dr = [&}r r“ =0,sothattdoesnotchange.
dI‘ dl’ 1+ r 0 (ii) If Y = 2, the numerator of dT; r) is 90 > 0, so that r > T
r Problem: 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(r),K(r),L(r),P,W(r),r,G(r),T(r)), when r20andP='1. b) What is the stationary equilibrium interest rate when the lump—sum tax on youth, T,
equals 0.15? What are the ievels 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 lumpsum tax of 0.15 on youth is continued after the social 5 security is introduced, so that youths pay 0.25 in total taxes. What is the stationary
equilibrium interest rate? What are the levels of capital and consumption in youth and
old age? (:1) Suppose that when the social security program of part (c) is introduced, the lump sum tax of 0.15 paid by youth is changed to a new level, T, so that the introduction of social security does not change the interest rate. What is this new Eevel of the lump
sum tax? Answer: a) The equation 6K becomes so that K‘3’4=1+r, K=(1+ r)“‘“3 and total output is "(1+ r) y = 4(1+ r)‘“3.
Output net of capital input is y_K=’ 4 .. 1 = 3+4r _
(1+ r)1.’3 (1+ r)41'3 (1+ 043 The equation 6U X
( 0, x1) BU(XO, *9 6X 6X
1 0 becomes so that x =(1+r)x.
1 {J The feasibility equation x +x =y—K
0 1 becomes (2 + r)x =LL,
0 (14—0413 sothat x = 3+4r 0 (2 + r)(1+ r)“3
and x : 3+4r ‘ (2 + r)(t + r)“. it foiiows that government debt is G: 1 —K=A_—__1_=__LL_.
1+r (2+r)(1+r)‘“a (1+r)“3 (2+r)(t+r)‘“3
Thetaxis
T=_r(1_+_§r.)_._.__. J
(2+r)(1+r)‘”3
{Thewageis {
W=3K“4= 3 (1+ r)“3 In summary, the equilibrium is ((XDU), X1“), KM), LU), P. W“): V, G(F).T(r)) =[[_3+4r_,_3.+_ir__ ___._L_,1,1—3.r.
(2+ r)(1+ {)4’3 (2 + r)(1+ r)“3 (1 + r)‘”3 1+3r r(‘l+3r)
(2+ r)(1 + 04““ (2+ r)(1+r)‘”3 Answer: b)
r = 0.25922.
K = 0.735413. x 1.31407. 0 ll X 1 .654703. 1 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 soEution of this equation is r = 0.297068. At this interest rate, K = 0.706941,
x0 = 1.288974.
x =1.671886. Answer: d) The new tax level is T = 0.15 —0.1 + L = 0.1294142406. 125922 (The calculations of this problem can be done with a hand held calculator or an Excei spread
sheet.) ...
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