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Unformatted text preview: _ ,_ rExercise 3.2 a. In this problem, the money supply is shrinking. To remove money from , ,
circulation, the government taxes each old person '5 units of the cons N A.
sumption good.”You might want to think of the tax as payable in "ﬂat ' 74.3:}: money, but deﬁned in real terms. When the old pay the implied'tax’wrth .
ﬁat money the government destroys the tax receipts. Note that thererisn’o’ :51; 7': if:
time subscript on the r. This tax will be independent of time in a station'— ' ary equilibrium. This tax must be such that __ vt(Mt—1 " Mt) __ vt(1 — Z)Mt—1 __ vt(1/Z —1)Mt
"— N — N — N ‘ The lifetime budget constraint would be [note that the tax must be sub
tracted from» the endonent] ' 1 Ut+1 2 _ y Ut+1 ' Build this lifetime constraint from the individual period constraints on
your own. The rate of return on ﬁat money will be (in a stationary equi
librium with a constant population) Nt+1‘(2 — 01)
Wu Mt+1 _ Mt _ _1_ 1
vt Nt‘(y ‘ Cl) — Mt+1 .— >
Mt Here, since 2 < 1, the value of money increases over time (alternatively,
the price level decreases over time—deﬂation). Given this information,
the lifetime budget set becomes c1+zcz Sy—zr. b. In this case, the feasible set will be c1 + c2 3 y since there is a constant
population. Figure 3D, below, illustrates the budget set line (thick line)
and feasible set line (thin line) on the same graph. The monetary equi
librium is (CE, 05), point B in the diagram. Clearly, this point does not
maximize the utility of future generations. For example, point A (the
golden rule) is preferred over point B by them. PointA could be achieved
by a policy which maintains a constant stock of ﬁat money (2 = 1) and,
hence, no tax on the old. Note that this implies that deﬂation is inefﬁ cient in this economy. 20 .2” c; yzr _ ‘y
FIGURE3D 0. Although the future generations prefer point A over B, the initial old do
not. Point B gives the initial old a higher level of consumption (a higher
level of c2) than point A. Contrastthis with earlier results where the ini— 
tial old also preferred a constant ﬁat money stock. ' Exercise 3.3 a. In this problem, the number of young people born each period is con
stant (N; = NH => n = 1). In other respects, this problem is like Exercise
3.1, with values for many of the parameters. We could easily show that
the gross real rate of return of ﬁat money will be E11 v: = E = ‘2' = 0.5 (sincez=2andn=1). The value of money will decrease 50% each period (the price level will
”double each period—note that this 1s consistent with the quantity theory) b. In period 1, the ﬁat money stock will be M1 = 2M0 .; (2)($10,000) = $201000}; 1‘: 7
In other words, the ﬁat money stock increases by $10,000 from period 0 to: 7 " " ' period 1. Each young perSOn will receive $10, OOO/N 1 — “$10, 000/ I 009": $10?" —— =v—~ ~~9We need toﬁerive the value 0F money 1
marketclearing condition 5,. hen ynﬁng Ly + (1* ~— 01—  1'0) Since each young person receives $10 " '
nd7_each dollar will buy 0.5 units of the consumption gOOd, the total
, “ ' oithe subsidy m goods 1s 5 units ‘ ’c. The price of the consumption good in period 1' is 171 : 1/01 = 11(05): 2; Exercise 3 4
a. The difﬁcult part here IS to realize that total tax revenues in period t Will
be Nt1: When distributed to the old (N¢_1 1n number), the peroldperson gigtransfer Will be Nit/Nt_1 — n1: Now We are ready to write down the bud
' get constraints. First~period budget constraint: c1; + vtmt S y — 1:
Secondperiod budget. constraint: 'c2,t+1 S vt+1mt 1_ n1: Solve the secondperiod constraint at equality for mt and substitute into
the ﬁrstperiod Constraint to get the lifetime budget constraint: U U
01,, +[— :]c2¢+1 <y — r+[ t]n1'. vt 1 Ut+1 b. Using the marketclearing condition for the money market, it can easily be seen that the rate of return (if ﬁat money in a stationary monetary
equilibrium Will be: vt+1 _ Mt+1 __ Nt+1 _ n
vt_ Nt(y—r—cl) “Nt
Mt If We'Substitute this into/the lifetime budget constraint, 7'1 < I l , +[l],c<
,1Ac1r1 ncz_y—r+ nnr =1 01 n ’2—3’. __ 7 .7. Notice that neither the tax nor transfer end up appearing in the lifetime
17 budget set  ’ ilibnum, we need to derive
ary allocation is ‘ c. To discuss the efﬁc1ency of the‘ihonetafy“
I the feasible set. The feaSibIé set With a stat A 3 Ntcl + N145; —’:1;V£3 => The equation for the feas1ble set is'ident1 r
time budget This means that the golden rule allocat
' “ ‘ ‘aximizes the this monetary equilibrium . Asnoted in part (b), the tax and transfer do not appear in the lifetime
budget set. This 1s the same lifetime budget set that would exist if the tax/transfer policy was abandoned. The pol1cy has no effect (positive or 7
negative) on welfare. e. The lifetime budget constraint in this case would be: c +[—v—‘]c < T+[U¢]ﬂ€
1,t 0+1 2,t+1 y vt+1 2 1 Int 17
61+ ‘E'CZSy7:+ ;~”§’=y—'2— In this case, the budget set would lie within the feasible set. The mone
tary equilibrium could not achieve the golden rule allocation. In this
case, the tax/transfer system (due to its inefﬁciency) would have a nega
tive impact on welfare. It Would be better to eliminate the tax/transfer system Individuals can provide for their oWn second period consump
tion _ attalnable 111" Exercise 3.5
a. n = z = 1. There are no government taxes, transfers, or purchases. b. n = 0.5, 2 = 1. There are no government taxes, transfers, or purchases. c. n : z > 1. The newlyprinted money is used to make a lump—sum transfer to the old each period. Be sure you can explain each of these answers. ,, *~'L1'fet1me budget constraint: c1 t "+ [—t ]~c2,,+1 S y — [ t ]‘17.
W" , ’ lIt+1 Um Using the money marketclearing condition, we can ﬁnd that the rate of return of ﬁat money in a stationary monetary equilibrium Will be 1/2.
substituting this into the lifetime budget set: ’ci+zC2$yzr. You will ﬁnd a graph of this in Figure BE, in part (c) of this problem. b“ #The goyernment’s budget constraint in period t will be (total government
purchases equal tax receipts plus seigniorage) Ng 1': NT + [1 ¥ %]‘UtMt.
c. The feasible set for this economy (stationary allocation) would be: c1+c2+gSy => c17+cgsy—_g. The feasible set and the budget set for the stationary monetary equilib
rium are displayed in Figure 3E, below. ' . . ,, C: gr». \ ViewsM» «, .' ﬁ\’
» .._ .su._,~_ d—a“~fY«NM——5< '... _,,a.aaafi»‘ik’»_. '— .. L money demand for z > 1
FIGURE 3E The feasible set is the thick line. Point A represents the golden rule allo
cation. Point B is the monetary equilibrium It IS clear that this policy
where part of the government purchases is ﬁnanced by seigniorage is
inefﬁcient It would be better to ﬁnance government purchases by lump
Sum taxes, as we see in the next part. ' . If we set 2—  1, then there' is no seigniorage revenue. The entire amount of government purchases must be ﬁnanced by lump sum taxes. In this
case, 1' Will equal g. Furthermore, notice that when 2 = 1 (and, hence, ‘1'
 —g), the lifetime budget set becomes (:1 +' (:2 < y —g The equation for the
budget set and the equation for the feasible set are identical (which 1m
plies that the policy where z= 1 and 1': g is efﬁcient). If we plot the bud
get line for the case where z > 1 along with the budget line for the case where z — 1 on the same graph we have Figure 3 7 of the text, reproduced
below. ...
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 Spring '11
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