Queen's University Economics Department

Economics 320A - Macroeconomic Theory II

Fall 1998

Partial solutions



Question 1.6

Let a = 4/5, then uht = 1.342 y0.5 for t ³ 1 and ch0(1) = 0.2 y.

Let a = 1/5, then uht = 1.342 y0.5 for t ³ 1 and ch0(1) = 0.8 y.

Let a = 1/2, then uht = 1.414 y0.5 for t ³ 1 and ch0(1) = 0.5 y.

The allocation with a = 1/2 is Pareto superior to the allocation with

a = 4/5. When a = 1/2, members of generations t ³ 1 enjoy more utility and the old people in period 1 have more consumption of the time 1 good. The allocation a = 1/2 is actually Pareto optimal.

The allocation with a = 1/5 is Pareto superior to the allocation with

a = 4/5. When a = 1/5, members of generations t ³ 1 enjoy as much utility and the old people in period 1 have more consumption of the time 1 good.

The allocation with a = 1/2 is not Pareto comparable to the allocation with

a = 1/5. Members of generations t ³ 1 enjoy more utility when

a = 1/2 whereas the old people in period 1 have more consumption of the time 1 good when a = 1/5.



Question 2.2

sht(r(t)) = b2 r(t) wth(t)
1+b2 r(t)
 - wht(t+1)
r(t) (1+b2 r(t))



Question 2.3

(a) St(r(t)) = 100-50 r(t)

(b) St(r(t)) = 75-50 r(t)

Question 2.4

(a) The competitive equilibrium is the set of interest rates r(t) = 0.5 for t ³ 1 and the consumption allocation

c01(1) = 1       for  h = 1,..., 100
cth(t) = 2       for  h = 1,..., 100       and for  t ³ 1
ct-1h(t) = 1       for  h = 1,..., 100       and for  t ³ 1

(b) The competitive equilibrium is the set of interest rates r(t) = 2 for t ³ 1 and the consumption allocation

c01(1) = 2       for  h = 1,..., 100
cth(t) = 1       for  h = 1,..., 100       and for  t ³ 1
ct-1h(t) = 2       for  h = 1,..., 100       and for  t ³ 1

(c) Same as (a).

(d) The competitive equilibrium is the set of interest rates r(t) = 2/3 for t ³ 1 and the consumption allocation

c01(1) = 1       for  h = 1,..., 100
cth(t) = 1.75       for  h = 2, 4,..., 100       and for t ³ 1
cth(t) = 1.25       for  h = 1, 3,..., 99       and for t ³ 1
ct-1h(t) = 1.1667       for  h = 2, 4,..., 100       and for  t ³ 1
ct-1h(t) = 0.8333       for  h = 1, 3,..., 99       and for  t ³ 1

(e) The competitive equilibrium is the set of interest rates r(t) = 0.625 for t ³ 1 and the consumption allocation

c01(1) = 1       for  h = 1,..., 100
cth(t) = 1.8       for  h = 1,..., 60       and for t ³ 1
cth(t) = 1.3       for  h = 61,..., 100       and for t ³ 1
ct-1h(t) = 1.125       for  h = 1,..., 60       and for  t ³ 1
ct-1h(t) = 0.8125       for  h = 61,..., 100       and for  t ³ 1

(f) The competitive equilibrium is the set of interest rates r(t) = 1 for t odd and r(t) = 1/2 for t even and the consumption allocation

Period 1: ch1(1) = 1 and c0h(1) = 1 for h = 1, ..., 100.

Period 2: ch2(2) = 2 and c1h(2) = 1 for h = 1, ..., 100.

Period 3: ch3(3) = 1 and c2h(3) = 1 for h = 1, ..., 100.

Periods 4, 6, ... : like period 2.

Periods 5, 7, ... : like period 3.

Question 3.1

(a) The competitive equilibrium without scheme is a set of interest rates

{r(1), r(2), r(3), ...} = {1/2, 1/2, ...}
a consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = {2, 2, ...}     h = 1, ..., 100
{ch0(1), ch1(2), ch2(3), ...} = {1, 1, ...}     h = 1, ..., 100
and a government policy
{th1(1), th2(2), th3(3), ...} = {0, 0, ...}     h = 1, ..., 100
{th0(1), th1(2), th2(3), ...} = {0, 0, ...}     h = 1, ..., 100.
A period 1 old (who is member of generation 0) consumes 1 unit (ch0(1) = 1). A typical member of generations t = 1, 2, 3, ... has utility uht = cht(t) cht(t+1) = 2×1 = 2.



The competitive equilibrium with scheme is a set of interest rates

{r(1), r(2), r(3), ...} = {2, 2, ...}
a consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = {1, 1, ...}     h = 1, ..., 100
{ch0(1), ch1(2), ch2(3), ...} = {2, 2, ...}     h = 1, ..., 100
and a government policy
{th1(1), th2(2), th3(3), ...} = {1, 1, ...}     h = 1, ..., 100
{th0(1), th1(2), th2(3), ...} = {-1, -1, ...}     h = 1, ..., 100.
A period 1 old (who is member of generation 0) consumes 2 units (ch0(1) = 2). A typical member of generations t = 1, 2, 3, ... has utility uht = cht(t) cht(t+1) = 1×2 = 2. Members of generation 0 are better off (higher consumption) with the scheme and members of generations 1, 2, ... are neither better off nor worse off (same level of utility). Therefore, the competitive equilibrium with scheme is Pareto superior.



(b) The competitive equilibrium without scheme is a set of interest rates

{r(1), r(2), r(3), ...} = {2/3, 2/3, ...}
a consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = {1.75, 1.75, ...}     h = 2, 4, ..., 100
{ch1(1), ch2(2), ch3(3), ...} = {1.25, 1.25, ...}     h = 1, 3, ..., 99
{ch0(1), ch1(2), ch2(3), ...} = {1, 1.1667, 1.1667 ...}     h = 2, 4, ... 100
{ch0(1), ch1(2), ch2(3), ...} = {1, 0.8333, 0.8333 ...}     h = 1, 3, ... 99
and a government policy
{th1(1), th2(2), th3(3), ...} = {0, 0, ...}     h = 1, ..., 100
{th0(1), th1(2), th2(3), ...} = {0, 0, ...}     h = 1, ..., 100.
A period 1 old (who is member of generation 0) consumes 1 unit (ch0(1) = 1). A typical member of generations t = 1, 2, 3, ... has utility uht = cht(t) cht(t+1) = 1.75×1.1667 = 2.0417 for h even and uht = cht(t) cht(t+1) = 1.25×0.8333 = 1.0416 for h odd.



The competitive equilibrium with scheme is a set of interest rates

{r(1), r(2), r(3), ...} = {4, 4, ...}
a consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = {0.75, 0.75, ...}     h = 2, 4, ..., 100
{ch1(1), ch2(2), ch3(3), ...} = {0.25, 0.25, ...}     h = 1, 2, ..., 99
{ch0(1), ch1(2), ch2(3), ...} = {2, 3, 3 ...}     h = 2, 4, ... 100
{ch0(1), ch1(2), ch2(3), ...} = {2, 1, 1 ...}     h = 1, 3, ... 99
and a government policy
{th1(1), th2(2), th3(3), ...} = {1, 1, ...}     h = 1, ..., 100
{th0(1), th1(2), th2(3), ...} = {-1, -1, ...}     h = 1, ..., 100.
A period 1 old (who is member of generation 0) consumes 2 units (ch0(1) = 2). A typical member of generations t = 1, 2, 3, ... has utility uht = cht(t) cht(t+1) = 0.75×3 = 2.25 for h even and uht = cht(t) cht(t+1) = 0.25×1 = 0.25 for h odd. These allocations are not Pareto comparable since period 1 old (higher consumption) and even members of generations t = 1, 2, ... are better off (higher utility) and odd members of generations t = 1, 2, ... are worse off (lower utility).



(c) The competitive equilibrium without scheme is a set of interest rates

{r(1), r(2), r(3), ...} = {1/2, 1/2, ...}
a consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = {2, 2, ...}     h = 1, ..., 100
{ch0(1), ch1(2), ch2(3), ...} = {1, 1, ...}     h = 1, ..., 100
and a government policy
{th1(1), th2(2), th3(3), ...} = {0, 0, ...}     h = 1, ..., 100
{th0(1), th1(2), th2(3), ...} = {0, 0, ...}     h = 1, ..., 100.
A period 1 old (who is member of generation 0) consumes 1 unit (ch0(1) = 1). A typical member of generations t = 1, 2, 3, ... has utility uht = cht(t) cht(t+1) = 2×1 = 2.



The competitive equilibrium with scheme is a set of interest rates

{r(1), r(2), r(3), ...} = {3, 3, ...}
a consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = {1, 1, ...}     h = 1, ..., 100
{ch0(1), ch1(2), ch2(3), ...} = {3, 3, ...}     h = 1, ..., 100
and a government policy
{th1(1), th2(2), th3(3), ...} = {1, 1, ...}     h = 1, ..., 100
{th0(1), th1(2), th2(3), ...} = {-2, -2, ...}     h = 1, ..., 100.
A period 1 old (who is member of generation 0) consumes 3 units (ch0(1) = 3). A typical member of generations t = 1, 2, 3, ... has utility uht = cht(t) cht(t+1) = 1×3 = 3. Members of generation 0 are better off (higher consumption) with the scheme and members of generations 1, 2, ... are also better off (higher utility). Therefore, the competitive equilibrium with scheme is Pareto superior.





(c) Let's take the example of the Canadian Pension Plan. Suppose the Canadian government commits to transfer 2 units of good to any old person in any period. During the first 10 periods, the number of young (workers) paying taxes is always twice as large as the number of old (retired) receiving transfers. So in periods 1 to 10, we have an equilibrium as in part (c) and members of generations 1 to 10 enjoy a utility level of 3. However, in periods 10 and following, the generation size stops growing and the ratio of young to old is now 1:1. The young must pay alot more taxes to support this relatively large number of old. The effect is that members of generations 11, 12 and so on have lower utility (uht = 0) than members of generations 1 to 10. As the ratio of young to old Canadian people decreases, the federal government might have to reduce benefits for CPP so that it does not impose too high a tax burden on the working young.




Question 5.3

The time t temporary equilibrium is the interest rate r(t) = 3/2, the bond price pk(t) = 2/3 and members of generation t consumption cht(t) = 4/3    "h and cht(t+1) = 2    "h.



Question 5.4

The time t temporary equilibrium is the interest rate r(t) = 1, the bond price pk(t) = 1/2 and members of generation t consumption cht(t) = 1.5    "h and cht(t+1) = 1.5    "h.



Question 5.6

The perfect foresight competitive equilibrium for this economy is the set of interest rates

{r(1), r(2), ... } = {0.8333, 0.75, 0.5714, 0.5, ... },
the set of bond prices
{p2(1), p1(2), p0(3)} = {1.6, 1.3333, 1},
the consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = { 1.6, 1.6667, 1.75, 2, ... }    "h
{ch0(1), ch1(2), ch2(3), ...} = { 1.4, 1.3333, 1.25, 1, ... }    "h
and the government policy
{ G(1), G(2), G(3),... } = {0, ... }
{th1(1), th2(2), ... } = {0, 0, 0.25, 0, ... }    "h
{th0(1), th1(2), ... } = {-0.4, 0, 0, 0, ... }    "h
{ B2(1), B1(2) } = { 25, 25 }.



Question 5.7

The perfect foresight competitive equilibrium for this economy is the set of interest rates

{r(1), r(2), ... } = {0.9, 0.625, 0.5, ... },
the set of bond prices
{p2(1), p1(2), p0(3)} = {1.7778, 1.6, 1},
the consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = { 1.5556, 1.6, 2, ... }    "h
{ch0(1), ch1(2), ch2(3), ...} = { 1.4445, 1.4, 1.25, 1, ... }    "h
and the government policy
{ G(1), G(2), G(3),... } = {0, ... }
{th1(1), th2(2), ... } = {0, 0, 0.25, 0, ... }    "h
{th0(1), th1(2), ... } = {-0.4445, 0, 0, 0, ... }    "h
{ B2(1), B1(2) } = { 25, 25 }.
Comparing the policies in question 5.6 and 5.7, Ricardian Equivalence does not hold because different generations pay off the debt.



Question 5.8

The perfect foresight competitive equilibrium for this economy is the set of interest rates

{r(1), r(2), ... } = {0.6667, 0.6, 0.5263, 5, ... },
the set of bond prices
{p2(1), p1(2), p0(3)} = {2.5, 1.6667, 1},
the consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = { 1.75, 1.8333, 1.9, 2 ... }    "h
{ch0(1), ch1(2), ch2(3), ...} = { 1.25, 1.6667, 1.1, 1, ... }    "h
and the government policy
{ G(1), G(2), G(3),... } = {0, ... }
{th1(1), th2(2), ... } = {0, 0, 0.10, 0, ... }    "h
{th0(1), th1(2), ... } = {-0.25, 0, 0, 0, ... }    "h
{ B2(1), B1(2) } = { 10, 10 }.



Question 5.11

The perfect foresight competitive equilibrium for this economy is the set of interest rates

{r(1), r(2), ... } = {0.6667, 0.6, 0.5263, 5, ... },
the set of bond prices
{p(1), p(2) } = {1.5, 1.6667 },
the consumption allocation
{ch1(1), ch2(2), ch3(3), ...} = { 1.75, 1.8333, 1.9, 2 ... }    "h
{ch0(1), ch1(2), ch2(3), ...} = { 1.25, 1.6667, 1.1, 1, ... }    "h
and the government policy
{ G(1), G(2), G(3),... } = {0, ... }
{th1(1), th2(2), ... } = {0, 0, 0.10, 0, ... }    "h
{th0(1), th1(2), ... } = {-0.25, 0, 0, 0, ... }    "h
{ B(1), B(2) } = { 16.6667, 10 }.
Comparing 5.11 and 5.8, we have different patterns of government borrowing but same interest rates, consumption allocation and taxes/transfers. This is an example of irrelevance of maturity composition.


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