Solutions to assignment 2

Solutions to Assignment 2

Question 1

(a)

Period 1 2 3 4 5
Number of young 110 121 133 133 133
Number of old 100 110 121 133 133



(b) Total transfer in period t = Number of old (N(t-1)) times transfer per old person (0.2).
Period 1 2 3 4 5
Total transfer 20 22 24.2 26.6 26.6



(c) All people are identical within a generation and the government balances its budget. In such an economy, there is no trade and each person simply consumes his endowment minus (plus) tax (transfer).

Generation 0: ch0(1) = wh0(1)-th0(1) = 1-(-0.2) = 1.2

Generation 1:

ch1(1) = wh1(1)-th1(1) = 2-0.2727 = 1.7273

ch1(2) = wh1(2)-th1(2) = 1-(-0.2) = 1.2

Generation 2:

ch2(2) = wh2(2)-th2(2) = 2-0.2231 = 1.7769

ch2(3) = wh2(3)-th2(3) = 1-(-0.2) = 1.2

Generation 3:

ch3(3) = wh3(3)-th3(3) = 2-0.1820 = 1.818

ch3(4) = wh3(4)-th3(4) = 1-(-0.2) = 1.2

Generation 4:

ch4(4) = wh4(4)-th4(4) = 2-0.2 = 1.8

ch4(5) = wh4(5)-th4(5) = 1-(-0.2) = 1.2

Generation 5:

ch5(5) = wh5(5)-th5(5) = 2-0.2 = 1.8

ch5(6) = wh5(6)-th5(6) = 1-(-0.2) = 1.2



(d)

r(1) = ch1(2)
ch1(1)
 = 1.2
1.7273
 = 0.6947
r(2) = ch2(3)
ch2(2)
 = 1.2
1.7769
 = 0.6753
r(3) = ch3(4)
ch3(3)
 = 1.2
1.818
 = 0.6601
r(4) = ch4(5)
ch4(4)
 = 1.2
1.8
 = 2
3
r(5) = ch5(6)
ch5(5)
 = 1.2
1.8
 = 2
3



(e) The competitive equilibrium is the set of interest rates

{r(1), r(2), r(3), r(4), ... } = {0.6947, 0.6753, 0.6601, 0.6667, ... },
the consumption allocation
{ch1(1), ch2(2), ch3(3), ... } = { 1.7273, 1.7769, 1.818, 1.8, ... }    "h
{ch0(1), ch1(2), ch2(3), ... } = { 1.2, ... }    "h,
and the government policy
{G(1), G(2), G(3), ... } = {10, 5, 0, ... }
{th1(1), th2(2), th3(3), ... } = { 0.2727, 0.2231, 0.1820, 0.2, ... }    "h
{th0(1), th1(2), th2(3), ... } = { -0.2, ... }    "h.

Question 2

(a)

young: cht(t) = wht(t)-tht(t)-lh(t)-p(t) bh(t)

old: cht(t+1) = wht(t+1)-tht(t+1)+r(t) lh(t)+bh(t)

The easiest way to derive the lifetime budget constraint is to divide the budget constraint when old and add it to the one when young. Anyhow, the constraint is

cht(t)+ cht(t+1)
r(t)
 = wht(t)-tht(t)+ wht(t+1)-tht(t-1)
r(t)
 +bh(t) é
ê
ë 		1
r(t)
 -p(t) ù
ú
û
Imposing the present value condition 1/r(t)-p(t) = 0 we get
cht(t)+ cht(t+1)
r(t)
 = wht(t)-tht(t)+ wht(t+1)-tht(t-1)
r(t)
Where does the present value condition come from? Suppose r(t) < 1/p(t), then the interest rate on private borrowing and lending is less than the return on government bonds. Therefore the agent wants to borrow an infinite amount on the private borrowing/lending market and sets bh(t)®¥ to make a riskless infinite profit. But that cannot be an equilibrium because there is no lender on the private borrowing/lending market. Suppose r(t) > 1/p(t), then the interest rate on private borrowing and lending is greater than the return on government bonds. Therefore the agent wants to lend an infinite amount on the private borrowing/lending market and sets bh(t)® -¥ to make a riskless infinite profit. But we restrict bh(t) to be nonnegative, so the best the agent can do is set bh(t) = 0. So we have a situation where the government wishes to sell bonds but where no private agent wants to buy these bonds. That cannot be an equilibrium. Therefore, the only possible case is r(t) = 1/p(t).



(b)

Time t good market clearing

N(t)
å
 h = 1 
 cht(t)+ N(t-1)
å
 h = 1 
 cht-1(t)+G(t) = N(t)
å
 h = 1 
 wht(t)+ N(t-1)
å
 h = 1 
 wht-1(t)
Summing the time t old budget constraint yields
N(t-1)
å
 h = 1 
 cht-1(t) = N(t-1)
å
 h = 1 
 wht-1(t)- N(t-1)
å
 h = 1 
 tht-1(t)+r(t-1) N(t-1)
å
 h = 1 
 lh(t-1)+ N(t-1)
å
 h = 1 
 bh(t-1)
Substituting in time t good market clearing we have
N(t)
å
 h = 1 
 cht(t)
+ N(t-1)
å
 h = 1 
 wht-1(t)- N(t-1)
å
 h = 1 
 tht-1(t)+r(t-1) N(t-1)
å
 h = 1 
 lh(t-1)+ N(t-1)
å
 h = 1 
 bh(t-1)
+G(t) = N(t)
å
 h = 1 
 wht(t)+ N(t-1)
å
 h = 1 
 wht-1(t)
Imposing market clearing on private borring/lending market, åh = 1N(t-1)lh(t-1) = 0, we have
N(t)
å
 h = 1 
 cht(t)- N(t-1)
å
 h = 1 
 tht-1(t)+ N(t-1)
å
 h = 1 
 bh(t-1)+G(t) = N(t)
å
 h = 1 
 wht(t)
rewriting the government budget constraint as
- N(t-1)
å
 h = 1 
 tht-1(t) = N(t)
å
 h = 1 
 tht(t)+p(t) B(t)-G(t)-B(t-1)
and substituting it in the previous equation we get
N(t)
å
 h = 1 
 cht(t)+ N(t)
å
 h = 1 
 tht(t)+p(t) B(t)-G(t)-B(t-1)+ N(t-1)
å
 h = 1 
 bh(t-1)+G(t) = N(t)
å
 h = 1 
 wht(t)
imposing market clearing on the government bonds market, åh = 1N(t-1)bh(t-1) = B(t-1) we get
N(t)
å
 h = 1 
 cht(t)+ N(t)
å
 h = 1 
 tht(t)+p(t) B(t) = N(t)
å
 h = 1 
 wht(t)
imposing utility maximization, cht(t) = cht(r(t)) we finally get
St(r(t)) = N(t)
å
 h = 1 
 [wht(t)-cht(r(t))-tht(t)] = p(t) B(t)



(c) This is an economy with identical agents within each generation and where the government balances its budget in every period. therefore the methodology of question 1 can be used to solve for the competitive equilibrium. Alternatively, we can impose the conditions S1(r(1)) = 0, S2(r(2)) = 0 and so on to solve for the equilibrium interest rates and then plug those in the consumption and savings functions to get consumption when young and when old. Anyhow, the competitive equilibrium is the set of interest rates

{r(1), r(2), r(3), r(4), ... } = {0.5398, 0.5398, 0.5263, ... },
the consumption allocation
{ch1(1), ch2(2), ch3(3), ... } = { 1.95, 1.95, 2, ... }    "h
{ch0(1), ch1(2), ch2(3), ... } = { 1, ... }    "h,
and the government policy
{G(1), G(2), G(3), ... } = {5, 5, 0, ... }
{th1(1), th2(2), th3(3), ... } = { 0.05, 0.05, 0, ... }    "h
{th0(1), th1(2), th2(3), ... } = { 0, ... }    "h.



(d) The agregate savings function is

St(r(t)) = 48.7179 (2- tht(t)) - 51.2821 (1-tht(t+1))
r(t)
 .

Period 1: S1 = 5, no taxes on members of generation 1.

S1(r(1)) = 5    Þ    97.4358- 51.2821
r(1)
 = 5    Þ    r(1) = 0.5548
The price of a government bond is p(1) = 1/0.5548 = 1.8025. Therefore the government issues B(1) = 5/p(1) = 2.774 bonds in order to collect 5 units of the time 1 good for its own consumption.

Period 2: S2 = 5+2.774, no taxes on members of generation 2.

S2(r(2)) = 7.774    Þ    97.4358- 51.2821
r(2)
 = 7.774    Þ    r(2) = 0.5720
The price of a government bond is p(2) = 1/0.5720 = 1.7484. Therefore the government issues B(2) = 7.774/p(2) = 4.446 bonds in order to collect 5 units of the time 2 good for its own consumption and 2.774 units of the time 2 good to pay off its debt.

Period 3: S3 = 0, no government borrowing, taxes on members of generation 3.

The tax on a member of generation 3 (when young) is B(2)/100. Therefore th3(3) = 0.04446.

S3(r(3)) = 0    Þ    95.2698- 51.2821
r(3)
 = 0    Þ    r(3) = 0.5383

Period 4: S4 = 0, no government borrowing, no taxes on members of generation 4.

S4(r(4)) = 0    Þ    97.4358- 51.2821
r(4)
 = 0    Þ    r(4) = 0.5263
The competitive equilibrium is the set of interest rates
{r(1), r(2), r(3), r(4), ... } = {0.5548, 0.5719, 0.5383, 0.5263 ... },
the consumption allocation
{ch1(1), ch2(2), ch3(3), ... } = { 1.95, 1.9223, 1.9555, 2, ... }    "h
{ch0(1), ch1(2), ch2(3), ... } = { 1, 1.0277, 1.0444, 1, ... }    "h,
and the government policy
{G(1), G(2), G(3), ... } = {5, 5, 0, ... }
{B(1), B(2), B(3), ... } = {2.7739, 4.4463, 0, ... }
{th1(1), th2(2), th3(3), ... } = { 0, 0, 0.04446, 0, ... }    "h
{th0(1), th1(2), th2(3), ... } = { 0, ... }    "h.


· We see that government actions raise the interest rate in periods 1, 2 and 3. In periods 1 and 2, government consumption and borrowing raises the interest rate compared to a situation where there are no government actions (like in period 4). Government consumption leaves less unit available to the private sector in periods 1 and 2. The relative scarcity of time 1 and 2 goods increases r(1) and r(2).

· We can also explain the high interest rates in periods 1 and 2 using a savings story. Since the government wants to sell B(1) bonds in period 1, people have to save more to buy these bonds. By making the interest rate higher, the invisible hand makes sure people are going to save more. Since the government borrows even more in period 2, the invisible hand raises the interest rate even more to generate more savings.

· In period 3, government taxation is responsible for the relatively high interest rate (compared to r(4)). We explain this higher interest rate by decomposing the total effect on savings of a change in taxes. The first effect is the direct effect of taxation on savings holding the interest rate constant. Analytically we have

sht(r(t))
tht(t)
 ê
ê
ê

r(t)  constant 
 = -b
1+b
 = -0.4872 < 0.
So, the direct effect of taxes is to reduce savings. But that is not the end of the story because the aggregate savings are now negative, which does not satisfy the equilibrium condition St = 0. This lower level of savings pushes the interest rate up. Think of the private borrowing/lending market. When people save less that means lenders have fewer units to supply on the private borrowing/lending market and borrowers demand more units. The two effects work together to increase the interest rate. The larger interest rate is such that St = 0.

Numerically we have the following. When there are no taxes (like in period 4), the interest rate is 0.5263 and sht(0.5263) = 0. When we introduce taxes tht(t) = 0.04446 we have

sht(0.5263) ê
ê
r(t)  constant 
 = 0.95 (2-0.044446)
1.95
 - 1-0
1.95 ×0.5263
 = -0.0217
So the tax effect reduces savings by 0.0217. When we let the interest rate adjust, we get the new equilibrium level of savings
sht(0.5383) = 0.95 (2-0.044446)
1.95
 - 1-0
1.95 ×0.5383
 = 0
So the tax effect reduces savings by 0.0217 and the interest rate effect increases savings by 0.0217. The total effect on savings is zero but the interest rate is larger.



(e) In part (c), members of generations 1 and 2 pay taxes to finance government consumption in periods 1 and 2. In part (d), members of generation 3 pay taxes to finance government consumption in periods 1 and 2 and members of generations 1 and 2 do not pay any taxes. Therefore, the present value of total tax liability of members of generations 1, 2 and 3 in part (d) is different from the one in part (c). In such a case, Ricardian equivalence of debt and taxes fails.


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