447-ans-ch3 - rExercise 3.2 a In this problem the money...

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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 con-s N A. sumption good.”You might want to think of the tax as payable in "flat ' 7-4.3:}: 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; y-zr _ ‘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 market-clearing 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 per-old-person gig-transfer 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: Second-period budget. constraint: 'c2,t+1 S vt+1mt -1_- n1: Solve the second-period constraint at equality for mt and substitute into the first-period Constraint to get the lifetime budget constraint: U U 01,, +[— :]c2¢+1 <y — r+[ t]n1'. vt 1 Ut+1 b. Using the market-clearing 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- n-cz_y-—r+ n-nr =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 . As-noted 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'CZSy-7:+ ;~”§’=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 newly-printed 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 market-clearing 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$y-zr. 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». \ Views-M» «, .' fi\’ » .._ .su._,~_ d—a“~f-Y«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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