Qualifying Exam Proposed Answer Key Sherman Hanna, March 24, 1996
2.5 Occupation A, income increases from $15,000 at age 25, at 3%/year until 65th birthday.
Occupation B, income increases from $20,000 at age 25, at 1%/year until 65th birthday.
(a) What is difference in PV between occupations.
Use Table T-4, PV of annuity
Interest rate = 5%/year. However, use approximation that discount rate = r-g, where g =
growth rate of income stream.
Therefore, appropriate discount rate for Occ. A = 5%-3% = 2%.
PV of 15000/year for 40 years = 15000 x 27.355 = 410,325
Discount rate for Occ. B = 5%-1% = 4%.
PV of 15000/year for 40 years = 20000 x 19.793 = 395,860
PV(A) – PV(B) = 14,465
(b) Appropriate discount rate for Occ. A = 9%-3% = 6%.
PV of 15000/year for 40 years = 15000 x 15.046 = 225,900
Discount rate for Occ. B = 9%-1% = 8%.
PV of 15000/year for 40 years = 20000 x 11.925 = 238,500
PV(A) – PV(B) = -12,600
(c) If occupation A requires greater investment in human capital, what would you expect to be the impact
of interest rates on investment in human capital?
The lower the interest rate, the greater the investment in human capital.
(d) In terms of the Life Cycle Model, …, what considerations might be important for an individual’s
choice between A and B?
Present versus future orientation, which might be affected by probability of death each year
(or life expectancy,) family composition changes over time, and the intertemporal utility
function (which could be specified separately from the pres. versus future orientation.)
Example: Someone who has four children at age 25 may choose occupation B, not just
because he/she “needs” the money, but because the utility of a dollar of income at age 25 is
much higher than the utility of a dollar of income at age 50. For someone with no children,
the difference based on age might be much smaller.
Example: all other things equal, we should expect women who plan to be single to be more
likely to choose A than men who plan to be single, because of the longer life expectancy of
(e) If the distribution of jobs in the United States shifts from B to A, what would you expect to be the
long-run impact on the overall distribution of income?
The income distribution should become LESS EQUAL, because the population is composed
of households of all ages. Occupation A has a bigger difference in income (3 to 1) than
occupation B (1.5 to 1)
2.6 Assume that you offer the following choices to a sample of people:
(i) 100% chance of losing $1000, or
(ii) 50% chance of losing $2000 and 50% chance of losing nothing
(a) Based on empirical research, which would most people choose?
(ii) want to have chance of no loss. (although outcome may depend on how alternatives are
(b) What SHOULD a rational, risk averse people choose? (i)
Ex. Loss of (i) = 1000
Note that if we want to use utility functions, we MUST (according to Hanna) specify initial
Expected utility of (i) = U(W-1000)
Ex. loss of (ii) = .5(2000) + .5(0) = 1000
Expected utility of (ii) = .5 U(W-2000) + .5 (U(W)
Therefore, a risk NEUTRAL person should be indifferent between (i) and (ii)
However, a rational, risk averse person should choose (i) because the loss in utility from
losing $2000 will be more than twice as much as the loss in utility from losing $1000.
example with square root utility function (just a teensy bit risk averse, almost risk neutral)
Expected Utility of (i) = sq. root(9000) = 94.8683298
Expected Utility of (ii) = .5 [sq. root(8000)] + .5 [sq. root(10000)] =
44.72135954 + 50 = 94.72135954
Therefore, rational, risk averse person should choose (i)
Implication: people are not rational (at least by economist’s definition using Expected Utility
Note that it is implausible that anyone would have a consistently risk-seeking utility function,
because such people would consistently choose unfair gambles, and most would end up broke
or dead. The majority opinion today seems to be that most individuals are should be risk
averse to some degree. I teach that the rational person is risk averse.