Problem Set 6: Solutions

Problem Set 6: Solutions

ECON 301: Intermediate Microeconomics

Prof. Marek Weretka

Problem 1 (Annuity and Perpetuity)

(a) A perpetuity gives amount x in each period, and hence its present value is given by

PV =

x

x

x

+

+

+ ¡¤¡¤¡¤

2

1 + r (1 + r)

(1 + r)3

we can rewrite this as





x

1

x

x

x

PV =

+

+

+

+ ¡¤¡¤¡¤ .

1 + r 1 + r 1 + r (1 + r)2 (1 + r)3

The sum of the elements in the bracket is equal to the present value of the perpetuity and

so

1

x

+

[P V ] .

PV =

1+r 1+r

Solving for P V gives

1

x

PV ?

PV =

1+r

1+r

which gives

x

P V perp = .

r

(b) The cash flow of an annuity differs from that of a perpetuity in that there are no

payments x after terminal period T .

The present value at time T of the future payment left in a perpetuity is P VTperp = xr . These

payments will be missing from the perpetuity. The present value in period one of P VTperp


T


T x


1

1

P VTperp = 1+r

. We subtract this amount off from the value of the

is P V = 1+r

r

perpetuity to get the value of the annuity:

"



T  



T #

1

x

x

1

P V ann = P V perp ?

=

1?

.

1+r

r

r

1+r

1

Problem 2 (Present Value, use a calculator)

(a) To decide whether to buy or rent (forever), we will compare the present value of our

500

=

perpetual rent payment of $500 with r = .01 to $600,000. Since P V rent = xr = .001

500, 000 < 600, 000. You would be better off renting the apartment (which in present value

will cost you $500,000) rather than purchasing the apartment today for $600,000.

(b) The present value of the payment must coincide with the size of the loan, hence



36 !

1

x

1?

=? x = $121.69 .

$4, 000 = P V =

.005

1.005

(c) The present value of the bond described is

$100

PV =

.1



1?

1

1.1

9 !

+

$1, 000

= $961.45 .

(1.1)10

Because the present value of the bond is greater than the price, it is a good idea to purchase

the bond. (Note: If we had assumed that the bond also pays the coupon amount c in the

last period, as is often the case in finance, the present value of the bond would be $1,000.

The Varian textbook does not make this assumption.)

(d) We want to save such that P V (S) = P V (C). The present value of the consumption of

$40,000 every year for 20 year beginning 40 years from today is (rounding)

40 





20 !



$40, 000

1

1

1?

= $70, 808 .

P V (C) =

1.05

.05

1.05

The present value of saving S for 40 years beginning today is



40 !

 

1

$S

1?

= $17, 159 ¡Á S .

P V (S) =

.05

1.05

Equating P V (S) = P V (C) and solving for S gives S = $4, 127 must be be the yearly savings

amount with r = 5%.

(e) Now we have that the present value of consumption is



P V (C) =

1

1.05

40 

$C

.05





1?

2

1

1.05

20 !

= $1, 770 ¡Á C .

The present value of saving $20,000 for 40 years beginning today is







40 !

$20, 000

1

P V (S) =

1?

= $343, 180 .

.05

1.05

Equating P V (S) = P V (C) and solving for C gives C = $193, 870 could be consumed every

year with S = $20, 000 and r = 5% annually.

Problem 3 (Life-Cycle Problem)

(a) With r = 5%, we want to find the C over 60 years that satisfies P V (C) = P V (income)

where income $200,000 is earned annually for 40 years:



60 !

 

1

$C

1?

= $18, 929 ¡Á C

P V (C) =

.05

1.05



and P V (income) =

$200, 000

.05





1?

1

1.05

40 !

= $3, 431, 800 .

Solving for C from the two equations we get C = $181, 300. The level of savings then over

the 40 working years is St = mt ? C ¡Ö $19, 000 for t = 21, 22, . . . , 60 and after retirement

St ¡Ö ?$181, 000 for t = 61, 62, . . . , 80.

(b) We still want to find the C that satisfies P V (C) = P V (income). The calculation of

P V (C) is unchanged, but now for the present value of income we have





40 !



1

$200, 000

1?

+ $1, 000, 000 = $4, 431, 800 .

P V (income) =

.05

1.05

Solving P V (C) = P V (income) this time we get C = 234, 130. The level of savings over the

40 working years is St = mt ? C ¡Ö ?$34, 000 for t = 21, 22, . . . , 60 and after retirement

St ¡Ö ?$234, 000 for t = 61, 62, . . . , 80.

(c) To solve this, we can either add the present value of the bequest to the present value

of consumption or subtract it from the present value of income. If we subtract it from the

present value of income, then P V (C) is unchanged and P V (income) becomes:







40 !



60

$200, 000

1

1

+ $1mil ? $1mil

P V (income) =

1?

= $4, 378, 300 .

.05

1.05

1.05

Solving P V (C) = P V (income) we now have C = 231, 300 The level of savings over the

40 working years is St = mt ? C ¡Ö ?$31, 000 for t = 21, 22, . . . , 60 and after retirement

St ¡Ö ?$231, 000 for t = 61, 62, . . . , 80.

3

Problem 4 (Insurance)

(a) Ben¡¯s affordable bundle if there is no insurance market is his endowment:

(cF , cN F ) = (50,000, 500,000).

cN F

500, 000

cF

50, 000

(b) Letting $x be the amount of insurance coverage Ben purchases at a cost/premium of

0.1x, note that in the state of the world in which there is a flood, his consumption is

cF ¡Ü 50,000 ? 0.1x + x

(1)

and in the state of the world in which the house does not flood, his consumption is

cN F ¡Ü 500,000 ? 0.1x .

(2)

(We use ¡Ü for the budget constraint and = for the budget line. Of course, we know that

consumption will end up being on the budget line, so using = for these equations will for the

most part not hurt even though when we¡¯re talking about the budget constraint we should

technically be using ¡Ü.)

The budget line will be in terms of cF and cN F (we will be determining what x is later),

so to eliminate x, solve at equalty one of the equations for x and plug it for x in the other

equation.

From (1), at equality we get x = 10

(cF ? 50,000). Plugging this into equation (2) we get

9





10

1

cN F ¡Ü 500,000 ? 0.1

(cF ? 50,000) =? cN F ¡Ü 505,556 ? cF (approximately).

9

9

His budget line is shown below:

(c) Ben is risk averse. We can think about this from three different but consistent perspectives:

4

cN F

500, 000

slope =

1

=

.1

1

.1

cF

50, 000

? (Analytically) Here, his Bernoulli utility function, u(c) =

second derivative is negative).

¡Ì

c is concave over c (i.e., its

? (Economically)

The utility he gets from a riskless, definite amount of c? , which is

¡Ì

?

?

u(c ) = c , is greater than the expected utility of some lottery that gives an ex?

?

pected value of

¡Ì c , for

¡Ì instance a 50-50 lottery where he wins either $0 or $2c where

1

1

EU (lottery) 2 0 + 2 2c? , as shown below.

? (Graphically) The points along a straight line connecting two points on u(c) lie under

u(c).

u(c)

u(c) =

p

c

u(c? )

EU (lottery)

c?

c

(d) Ben¡¯s MRS is

?1

M RS(cF , cN F ) = ?

0.1( 12 )cF 2

?1

0.9( 12 )cN F2

1

=?

9



cN F

cF

 12

1

and at the endowment point (50,000, 500,000), M RS(50,000, 500,000) = ? 91 ¡¤ 10 2 . Notice

that this is not optimal since M RS 6= ? 91 .

5

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