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Williamson end of Chapter 10 problems Solutions
Advanced Topics in Economic Theory (Brunel University London)
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SOLUTIONS of end of chapter problems
Chapter 10
1. (a) The government intertemporal budget constraint is, assuming both t and t are positive:
bt  (1  b)at 
0
(1  r1 )
(b) We are in a situation where asymmetric information becomes important: the government does
not know who will be able to pay the future taxes. The relevant interest rate is the one
corresponding to the steeper part of the budget constraint, and the new endowment thus moves
the flatter part down. Consumption choices of the first period consumers are thus impacted
through a negative income shock, as they have to pay more taxes to compensate for more unpaid
taxes in the future.
(c) The Ricardian equivalence does not apply as the change in timing of taxes has changed some
consumption patterns. Indeed, households cannot fully adjust their savings, thus any change in
periodic disposable income has an impact at least for some households.
2. (a) The government can only tax in the second period as much as it could get from the collateral:
t  pH
(b) Now that the government has priority on the collateral, the new collateral constraint is:
s (1  r)  pH  t
which implies
c
y  t  pH
(1  r )  t  /(1  r )
(c) As we see from the new collateral constraint, Ricardian equivalence holds, as any shifting of
taxes across periods does not affect the constraint. The consumer makes the same consumption
choice.
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Chapter 10 Credit Market Imperfections: Credit Frictions, Financial Crises and Social Security
89
3. (a) The bank will be lending so that it will be able to get the loan back in expectation. Thus, the new
collateral constraint is

s (1  r)  a pH
which leads to
c
y  t  a pH
.
(1  r )
This is much like Figure 10.5 in the textbook, simply with a lower collateral value.
(b) If the collateral is more likely to be of no value, banks will lend less. Thus, some household will
not be able to borrow as much as they could before, leading for them and in aggregate to a
reduction of current consumption and an increase in future consumption. Thus we have exactly
the same impact as if we had a reduction in p, as in Figure 10.5 of the textbook.
4. If each good borrower chooses to borrow L, then the cost of deposits for the bank is (1+r1)L. The
payoff on each loan to a good borrower is (1+r2)L, and bad borrowers always default, but the
bank can have each borrower post A as collateral, so the payoff to the bank on a loan to a bad
borrower is the smaller of A or (1+r2)L. Suppose first that A< (1+r2)L. Then, the expected profit
for the bank to making a loan is zero in equilibrium, or
(1  r1 ) L  a(1  r2 ) L  (1  a) A  0
Then, if we solve the above equation for r2, we obtain
r2 
1  r1  (1  a)
A
L
a
Therefore, the larger is A, the smaller is r2, the loan interest rate. The larger the quantity of collateral
available, the larger the payoff the bank receives when a bad borrower defaults. This increases bank
profits, but in equilibrium expected profits are zero for the bank, borrowers benefit from having a higher
quantity of collateral with which to secure loans. If A is sufficiently large, i.e. A  (1+r2)L, then the bank’s
payoff on a loan to a bad borrower is (1+r2 )L, and so r1 = r2. Thus, if A is large enough then there is
sufficient collateral to eliminate the credit market friction.
5. In this case, the current-period tax is sufficiently low that, in equilibrium,
1 r 
y'
,
y
the consumption of borrowers is
(c, c ')  (2 y, 0) ,
and the consumption of lenders is
(c, c ')  (0, 2 y ') .
In this case, changing the current tax has no effect at the margin. Thus, if tax policy has been set so that the
borrowing constraint for consumers is relaxed, then Ricardian equivalence holds for any changes in taxes.
6. Since this is case 4, from page 358, we have 1+r=a. Then, when taxes are zero, a borrower’s
consumption bundle is
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90
Williamson • Macroeconomics, Fifth Edition
v


(c, c ')   y  , y ' v  ,
a


and a lender’s consumption bundle is
v


(c, c ')   y  , y ' v  .
a


The optimal tax is
t
v
 y,
a
which will imply that borrowers consume 2y in the current period, and lenders consume 2y’ in the future
period.
7. Social Security.
(a) When the program is first instituted, the current old receive b in benefits and pay nothing.
The effect on the current old is as in Figure 10.12 in the text. The current young receive b in
benefits when they are old. This effect is also captured by the shift from BA to FD in the
text’s Figure 10.13. The current young also lend bN to the government in period T and receive
(1  r )bN in principal and interest when they are old. In per capita terms, these amounts are
bN/(1  n)N  b /(1  n) and (1  r )bN /(1  n)N  (1  r )b /(1  n) respectively. However, this
borrowing and lending are represented in Figure 10.13 as movements along the budget line.
Unless there is a change in the real interest rate, there is no additional shift in the budget line.
Therefore, both these generations unambiguously benefit from the program.
(b) Once the program is running, it is identical to the pay-as-you-go system in the text. This program
benefits a typical cohort as long as n  r, as is depicted in textbook Figure 10.13. A special
circumstance applies to the cohort born in period T  1. These individuals each receive a benefit
per capita of b/(1  r) in present value terms. However, they pay taxes to support two generations’
worth of benefits. They pay taxes to retire the principal and interest on debt incurred in period T.
The per capita share of principal and interest on their grandparents’ benefits is equal to
(1  r)b/(1  n)2. The per capita share of their parents’ benefits is equal to b/(1 + n). This
generation can only benefit if:
1
(1  r ) (1  r )2

(1  n) (1  n)2
This requirement is obviously more stringent than n  r.
8. Under this regime, disposable income for the young is y, but the price of current consumption is
(1  s). This implies that the intertemporal budget constraint of the household is now
(1  s) c  c y  y
b


(1  r )
(1  r ) (1  r )
In equibrium, it must be that sc(1  n)  b. Thus whether there is going to be a positive income effect
is going to depend on n is larger than r. But the is also a substitution effect coming from the change in
relative price between c and c. This substitution effect has no impact on welfare, though.
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Chapter 10 Credit Market Imperfections: Credit Frictions, Financial Crises and Social Security
91
9. (a) Consumers born in T are the first ones not to get benefits. The government finances the benefits
of the last generation, bN, with bonds DT. Thus each consumer born in T buys DT/N bonds, or
b/(1  n) each. In T  1, the government has to reimburse principal and interest, (1  r)DT, thus
each old consumer at that time obtains (1  r)b/(1  n). This means that for the intertemporal
budget constraint in the figure below, the endowment point is shifted b/(1  n) to the left and
(1  r)b/(1  I) up. This is on the same budget constraint as before. However, this household is
also losing the old-age benefits it was expecting, thus the new endowment point shifts an
additional b down. However, this generation does not have to pay, when young, for the benefits
of the previous generation, as they are covered by the debt. Thus, the endowment point shifts
b/(1  n) to the right. As r  n, the new endowment point is now to the right of the old budget
constraint, see the figure below, and this household is better of. Essentially, it just the opposite of
the situation that made it viable to institute a pay-as-you-go system when n  r. The exact same
reasoning applies to all future generations.
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