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Problem Set 8 ECON 3113 Microeconomic Theory I 2024 Due date: Monday, May 6
- Irma has an initial wealth of $50; 000 and a car that is valued at $100; 000. She faces a probability of 5% of the car being stolen. She can purchase insurance to cover her potential loss. The insurance term is as follows. If she wants to get a coverage of $q (i.e., compensation in case her car is being stolen), she has to pay a (upfront) premium of 0 : 1 q. Irma has a von-Neumann-Morgenstern utility function u (c) = 4 ln c.
(a) Denote Irmaís wealth in the good state and bad state by cg and cb respectively. What is the set of state-wealth combinations (cg; cb) that Irma can choose from? Solution: By purchasing a coverage of q, Irmaís wealth in the good and the bad state are respectively cg = 150000 0 : 1 q cb = 50000 0 : 1 q + q = 50000 + 0: 9 q Substituting q = 10 (150000 cg) into the equation for cb gives: cb = 50000 + 0: 9 10 (150000 cg) , cb = 1400000 9 cg , 9 cg + cb = 1400000 Moreover, q can only range from 0 (zero coverage) to 100000 (full coverage). Therefore, the combo of attainable state-wealth is the line segment with end points zero coverage: (cg; cb) = (150000; 50000) full coverage: (cg; cb) = (150000 (0:1) (100000) ; 150000 (0:1) (100000)) = (140000; 140000) In sum, the set of attainable state-wealth combinations (cg; cb) is given by f(cg; cb) : 9cg + cb = 1400000 and 140000 cg 150000 and 50000 cg 140000 g.
(b) How much insurance coverage would Irma purchase? Solution: Method 1: Equating MRS with price ratio, the FOC gives:
M RS =
4 (cg) ^1 4 (cb) ^1
cb cg
Substituting this into the budget line: 9 cg + cb = 1400000:
9 cg +
cg = 1400000 , cg =
and cb =
Using the fact that cb = 50000 + 0: 9 x, this suggests that the coverage purchase q is
q =
Method 2: Irmaís expected utility of purchasing q unit of coverage is
E [u (c)] = 0 : 95 (4 ln (150000 0 : 1 q)) + 0: 05 (4 ln (50000 + q 0 : 1 q)) = 3 :8 ln (150000 0 : 1 q) + 0:2 ln (50000 + 0: 9 q).
The FOC of expected utility maximization gives @E [u (c)] @q = 0 , 3 :8 ( 0 :1) (150000 0 : 1 q) ^1 + 0:2 (50000 + 0: 9 q) ^1 (0:9) = 0 ) x = 22222.
(c) Suppose Irma can install a burglar alarm, which costs $5000 and lowers the risk of car theft to 1%. Suppose the premium rate stays Öxed at 0 : 1. How much insurance coverage would Irma purchase after installing the burglar alarm? Solution: Method 1: The price ratio stays Öxed at 9. On the other hand, the MRS is now
M RS =
4 (cg) ^1 4 (cb) ^1
99 cb cg
With no insurance coverage, (cg; cb) = (150000 5000 ; 50000 5000) = (145000; 45000). The MRS over the attainable state-wealth combinations thus has
M RS = 99 cb cg
With MRS>price ratio at all attainable state-wealth combinations, Irma would opt for the corner solution of zero coverage q = 0. Method 2: By installing the burglar alarm and purchasing a coverage of $q, Irma has wealth 150000 5000 0 : 1 q with probability 99%, and a payo§ of 50000 5000 + q 0 : 1 q with probability 1%. Her expected utility is thus
E [u (c)] = 0 : 99 (4 ln (150000 5000 0 : 1 q)) + 0: 01 (4 ln (50000 5000 + q 0 : 1 q)) = 3 : 96 ln (145000 0 : 1 q) + 0: 04 ln (45000 + 0: 9 q)
Finally, rewriting the bad-state equation into s = (2700 cb) = 0 : 6 , and substituting it into the good-state equation, we get
cg = 2700 +
(2700 cb)
, cg +
cb = 6750.
(b) Suppose Amy has a von-Neumann Morgenstern utility function of uA (c) = 2 c ^ (^12)
. What is her optimal state-wealth combination? Solution: Her MRS is 0 : 5 0 : 5
u^0 A (cg) u^0 A (cb)
c (^32) g c (^32) b
cb cg
^32
Equating it with the price ratio of state-wealth gives the FOC: cb cg
^32
cb cg
^23
Substituting this into her budget constraint:
cg +
cg | {z } cb
^13!
cg = 6750 , cg =
2
^13 ^3147 :^3.
Using the FOC again,
cb =
^23
cg =
^23
2
^13 ^2401 :^8.
(c) What portfolio, i.e., stock and bond holdings, would Amy choose? Solution: Recall portfolio (b; s) is related to state-consumption by
cg = 500 + 1: 1 b + 2s; cb = 500 + 1: 1 b + 0: 5 s.
Substituting (cg; cb) = (3147: 3 ; 2401 :8) into the system above and solving gives:
b 1503 and s 496 : 97.
