Problem set 7, solutions

1. True or false?

1. If G(x) first-order stochastically dominates F (x), the expected valueof x under G is not smaller that the expected value of x under F .

2. If the expected value of x under G is greater than the expected valueof x under F , then G(x) first-order stochastically dominates F (x).

3. If G(x) second-order stochastically dominates F (x) and they have thesame mean, the variance of x under G is not greater than the varianceof x under F . [Hint: Note that the ‘non-decreasing’ requirement fromthe definition of sosd can be dropped without altering the definition.]

4. If G and F have the same mean and G has a smaller variance than F ,then G(x) second order stochastically dominates F (x).

For each statement, provide a proof or a counterexample.

solution: 1. True. The definition of fosd requires that for any non-decreasing u : R≥0 → R,

∫R≥0

u(x)dG(x) ≥∫R≥0

u(x)dF (x). Substituting

u(x) = x into the inequality, it becomes EG[x] ≥ EF [x].

solution: 2. False. Consider X = {1, 2, 3} and two lotteries LG = {0, 1, 0}and LF = {0.4, 0, 0.6}. F has a higher mean, but the two lotteries are notfosd ordered. To see this, the figure below shows the two cdfs.

P

1 2 3

0.5

1 G

P

1 2 3

0.5

1 F

solution: 3. True. The definition of sosd requires that for any non-decreasing concave u : R≥0 → R,

∫R≥0

u(x)dG(x) ≥∫R≥0

u(x)dF (x). Using

the hint we can drop the ‘non-decreasing’ part and use u(x) = −x2 in theinequality to get EG[x2] ≤ EF [x2]. Because the distributions have the samemean, the inequality rewrites as varG = EG[x2]−EG[x2] ≤ EF [x2]−EF [x]2 =varF .

1

Problem set 7, solutions

solution: 4. False. Consider X = {1, 2, 3, 4, 5} and two lotteries LG ={0.01, 0, 0.98, 0, 0.01} and LF = {0, 0.5, 0, 0.5, 0}. The two lotteries obvi-ously have the same mean. EG[x2] = 0.01(1 + 25) + 0.98 · 4 = 4.18 andEF [x2] = 0.5(4 + 16) = 10 so that varG < varF . The two lotteries are notsosd ordered. To see this, the figure below shows the two cdfs.

P

1 2 3 4 5

0.5

1 G

P

1 2 3 4 5

0.5

1 F

Clearly,∫ 20 G(x)dx = 0.01 >

∫ 20 F (x)dx = 0 but

∫ 30 G(x)dx = 0.02 <∫ 3

0 F (x)dx = 0.5.

2. Elasticity of intertemporal substitution is the elasticity of the growthrate of consumption with respect to changes in the interest rate. Recall theinfinite horizon consumption problem we have analysed in one of the previ-ous problem sets. The first order condition pinning down the growth rateof consumption in that model is u′(ct) = u′(ct+1)δR, where R is the (gross)interest rate. Derive the intertemporal elasticity of substitution assumingthat u(x) = x1−a

1−a .

solution: Using u(x) = x1−a

1−a in u′(ct) = u′(ct+1)δR gives ct+1

ct= (δR)1/a.

The elasticity of ct+1

ctwith respect to R is

∂ct+1ct∂R

Rct+1ct

= 1a(δR)

1a−1 δR

(δR)1a

= 1a .

The point of this exercise is to see that with CRRA utility, a capturesnot only the degree of risk aversion, but also determines the intertemporalelasticity of substitution.

3. Suppose a consumer has CARA utility u(x) = 1 − exp (−bx) whereb > 0. Suppose that X and Y are two continuous random variables withpdf fX and fY respectively. Suppose the consumer has wealth w ≥ 0.

1. Show that EY [u(w+y)] = u(w+ cY ) where cY is the certainty equiva-lence of facing Y by a consumer with w = 0, that is u(cY ) = EY [u(y)].

2. Show that EX,Y [u(x + y)] = u(cX + cY ), where cX is the certaintyequivalence of facing X by a consumer with w = 0.

2

Problem set 7, solutions

3. Define cX,w as the certainty equivalence of facing X by a consumerwith w. Show that cX,w − w is constant in w.

solution: 1. We have EY [u(w + y)] =∫R u(w + y)fY (y)dy. Substituting

the functional form for u, this becomes∫R

(1− exp (−b(w + y)))fY (y)dy = 1−∫R

exp (−b(w + y))fY (y)dy

= 1−∫R

exp (−by) exp (−bw)fY (y)dy

= 1− exp (−bw)

∫R

exp (−by)fY (y)dy.

From u(cY ) = 1 − exp (−bcY ) = EY [u(y)] = 1 −∫R exp (−by)fY (y)dy we

know exp (−bcY ) =∫R exp (−by)fY (y)dy. Hence

EX [u(w + y)] = 1− exp (−bw)

∫R

exp (−by)fY (y)dy

= 1− exp (−bw)exp(−bcY )

= 1− exp (−b(w + cY )) = u(w + cY ).

solution: 2. We have EX,Y [u(x+y)] =∫R∫R 1−exp (−b(x+ y))fY (y)fX(x)dydx.

This rewrites as ∫R

∫R

(1− exp (−b(x+ y)))fY (y)fX(x)dydx =

1−∫R

∫R

exp (−bx) exp (−by)fY (y)fX(x)dydx =

1−∫R

exp (−bx)

[∫R

exp (−by)fY (y)dy

]fX(x)dx =

1−[∫

Rexp (−bx)fX(x)dx

] [∫R

exp (−by)fY (y)dy

]=

1− exp (−bcY ) exp (−bcX) = u(cX + cY ).

solution: 3. We have u(cX,w) = EX [u(w + x)]. From the first part,EX [u(w+x)] = u(w+ cX) so that cX,w = cX +w and hence cX,w−w = cX .

4. Suppose a consumer faces a good state, s∗, with probability 1 − p anda bad state, s∗, with probability p. The consumer has Bernoulli utility thatis a function of x. In the bad state, x takes on value of x∗ and in the goodstate it takes on value of x∗. Suppose the consumer can change the actualoutcomes by a factor of α to x∗ + α in a bad state and to x∗ − α in a goodstate.

3

Problem set 7, solutions

1. If the Bernoulli utility of the consumer is given by u(x) = −(x − b)2and x∗ < b < x∗, what value of α maximizes her expected utility? Ifx∗ − b = b− x∗, what is the intuition behind this optimal α.

2. If the Bernoulli utility of the consumer is state dependent of the formus(x) = −(x − b(s))2 where b(s∗) = b∗ and b(s∗) = b∗ and if b∗ <x∗ < x∗ < b∗, what value of α maximizes her expected utility? Ifb∗ − x∗ = x∗ − b∗, what is the intuition behind this optimal α.

solution: 1. The expected utility the consumer maximizes by choosing αis

eu(α) = −p(x∗ + α− b)2 − (1− p)(x∗ − α− b)2.

It is easy to check that eu(α) is strictly concave in α, hence maxα eu(α) willhave unique solution. The necessary (and sufficient in this case) first ordercondition gives

α∗ = (1− p)(x∗ − b) + p(b− x∗) =|x∗−b=x∗−b=t = t.

The intuition behind α∗ in general is that the consumer sets α such that thetwo outcomes, x∗ − α and x∗ + α are as close as possible to b. Notice thatunder the conditions of the problem, α∗ > 0. If x∗ − b = b − x∗ = t, thenthere exists α such that the two outcomes are equal to b irrespective of thestate, that is, the two outcomes are constant across states.

solution: 2. The expected utility the consumer maximizes by choosing αis

eu(α) = −p(x∗ + α− b∗)2 − (1− p)(x∗ − α− b∗)2.

It is easy to check that eu(α) is strictly concave in α, hence maxα eu(α) willhave unique solution. The necessary (and sufficient in this case) first ordercondition gives

α∗ = (1− p)(x∗ − b∗) + p(b∗ − x∗) =|b∗−x∗=x∗−b∗=t = −t.

The intuition behind α∗ in general is that the consumer sets α such that thegood state outcome is as close as possible to b∗ and the bad state outcomeis as close as possible to b∗. By assumption b∗ < x∗ < x∗ < b∗ and henceα∗ < 0, which you can check, makes sense. Note that this means that,relative to (x∗, x

∗), the consumer by choosing α∗ increases the variance of theoutcomes across the states. This seemingly contradicts the received wisdomthat consumption smoothing (lower variance) is optimal. This intuition restson the assumption of state independent preferences (utility function). Whenpreferences vary between periods, it is in fact optimal for the outcomes tovary.

4

Problem set 7, solutions

5. Consider a consumer with wealth w > 0, who faces a potential loss ofD > 0 with probability π. She has an access to insurance. One unit ofinsurance costs q and pays 1 if a loss occurs. The consumer has CARABernoulli utility u(x) = 1− exp (−ax) where a > 0.

1. What is the economically meaningful restriction on q that makes theinsurance really an insurance?

2. Suppose the consumer buys α ≥ 0 units of insurance. Write down herwealth if loss occurs (wl) and her wealth if no loss occurs (wnl).

3. Draw a space with wnl on the horizontal and wl on the vertical axis.Draw all the possible combinations of (wnl, wl) as you vary α ∈ [0, D].Calculate the slope of this ‘budget curve’. [Hint: Isolating α, writedown wl as a function of wnl.]

4. Calculate the expected utility of the consumer who bought α unitsof insurance. Draw the same space as in the previous part and drawindifference curves of the consumer in this space. What is the slope ofthe indifference curves on the wnl = wl ray? Show that the indifferencecurves have constant slope on any straight line with slope equal tounity.

5. Show that the expected utility of the consumer is strictly concave func-tion of α. Show that this does not depend on the CARA assumption.

6. Calculate the optimal amount of insurance the consumer takes up.What is the intuition behind the fact that this optimal amount doesnot depend on w?

solution: 1. If q > 1, then it would cost q (which is spent in every stateof the world) to cover 1 of damage (which happens only in the ‘loss’ state ofthe world). Clearly, this would not make sense (noone would demand suchan insurance). Hence, say, q ∈ (0, 1).

Another economically relevant argument is that if q < π, then the in-surance policy has revenues q, while the cost of covering the damages if1 · π. Hence any insurance company offering such a policy would make loss.Hence, say, q ∈ (π, 1). Below we will see that q < π leads to consumer-optimal over-insurance.

solution: 2. A consumer who purchased α ≥ 0 units of insurance hasstate dependent wealth

wnl = w − qαwl = w −D + α− qα.

5

Problem set 7, solutions

solution: 3. The figure below shows the setup. The end points of the‘budget curve’ are where α ∈ {0, D}. Any α < 0 would not make sense hereand any α > D means over-insurance. α = D is called full-insurance as thisgives wealth w−qD in both states of the world (and hence corresponds to thepoint on the 45◦ line). The slope of the ‘budget curve’ is −1−q

q . To see this,

from above α = wnl−w−q so that wl = w−D+(1−q)α = w−D+(1−q)wnl−w

−q .

wl

wnlw

w −D

w − qD

w − qD

w−qα∗

w −D + α∗ − qα∗

solution: 4. The expected utility of a consumer who bought α units ofinsurance is

eu(α) = π · u(w −D + α− qα) + (1− π) · u(w − qα).

A representative indifference curves of this consumer are drawn above. Tosee the slope of the indifference curves, these are given by the followingequation

π · u(wl) + (1− π) · u(wnl) = k.

Using standard arguments, the slope of the indifference curve at (wnl, wl)can be calculated from πu′(wl)dwl + (1 − π)u′(wnl)dwnl = 0 and is thus∂wl∂wnl

= −1−ππ

u′(wnl)u′(wl)

. On the 45◦ line, wnl = wl and hence the slope of the

indifference curve is −1−ππ . Substituting the functional form for u, u

′(wnl)u′(wl)

=exp (−awnl)exp (−awl)

= exp (−a(wnl − wl)). Hence the slope of the indifference curve

is constant for any (wnl, wl) such that wnl − wl = k, or wl = wnl − k.

6

Problem set 7, solutions

solution: 5. A general expression for the expected utility is above. Takingthe derivative w.r.t. α gives

∂eu(α)

∂α= πu′(w −D + α(1− q))(1− q)− (1− π)u′(w − qα)q

∂2eu(α)

∂2α= πu′′(w −D + α(1− q))(1− q)2 + (1− π)u′′(w − qα)q2

and clearly ∂2eu(α)∂2α

< 0 whenever u is strictly concave.

solution: 6. Substituting the functional form for u and using the firstorder condition, after some algebra and ignoring α ∈ [0, D], gives

α∗ = D +1

alog

(π

1− π1− qq

)Let me denote π

1−π1−qq = c. If q < π, then c > 1 and hence the consumer

would like to over-insure. If she is prevented from doing so, then α has tobe constrained to lie in [0, D]. In this case α∗ = D.

If q = π, then α∗ = D. An insurance with q = π is called actuarially fair[My online dictionary claims that ‘actuary’ is a ‘statistician who computesinsurance risks and premiums’, hence this is not a typo from ‘actually’.] Inother words, it is an insurance policy which breaks even, in terms of revenuesand costs. Any strictly risk averse decision-maker fully insures when she hasan access to actuarially fair insurance.

If q > π, then c < 1 and the consumer will not fully insure. On the otherhand, her α∗ will be positive for a range of q (whenever q ≤ π

exp (−aD)(1−π)+πto be more precise, you can check that π

exp (−aD)(1−π)+π > π).The optimal amount of insurance the consumer purchases is independent

of w since her u is CARA and the loss involves absolute amount of incomeD. If u was decreasing-ARA, α∗ would be decreasing in w and if u wasincreasing-ARA, α∗ would be increasing in w (modulo the corner solutionsat α∗ ∈ {0, D}).

6. Consider a consumer with wealth w > 0, who faces a potential loss oftw, where t ∈ (0, 1), with probability π. She has an access to insurance.One unit of insurance costs her qw and reduces her loss to (t − 1)w. The

consumer has CRRA utility u(x) = x1−a

1−a where a > 0.

1. Suppose the consumer buys α ≥ 0 units of insurance. Write down herwealth if loss occurs (wl) and her wealth if no loss occurs (wnl).

2. Draw a space with wnl on the horizontal and wl on the vertical axis.Draw all the possible combinations of (wnl, wl) as you vary α ∈ [0, t].Calculate the slope of this ‘budget curve’. [Hint: Isolating α, writedown wl as a function of wnl.]

7

Problem set 7, solutions

3. Calculate the expected utility of the consumer who bought α unitsof insurance. Draw the same space as in the previous part and drawindifference curves of the consumer in this space. What is the slope ofthe indifference curves on the wnl = wl ray? Show that the indifferencecurves have constant slope on any ray departing from origin.

4. Show that the expected utility of the consumer is strictly concave func-tion of α. Show that this does not depend on the CRRA assumption.

5. Calculate the optimal amount of insurance the consumer takes up.What is the intuition behind the fact that this optimal amount doesnot depend on w?

solution: 1. A consumer who purchased α ≥ 0 units of insurance hasstate dependent wealth

wnl = w(1− qα)

wl = w(1− (t− α)− qα).

solution: 2. The figure below shows the setup. The end points of the‘budget curve’ are where α ∈ {0, t}. Any α < 0 would not make sense hereand any α > t means over-insurance. α = t is called full-insurance as thisgives wealth w(1 − tq) in both states of the world (and hence correspondsto the point on the 45◦ line). The slope of the ‘budget curve’ is −1−q

q . To

see this, from above α = 1q (1 − wnl

w ) so that wl = w(1 − t + α(1 − q)) =

w(

1− t+ 1−qq (1− wnl

w ))

.

wl

wnlw

w(1− t)

w(1− tq)

w(1− tq)

w(1− qα∗)

w(1− t+ α∗(1− q))

8

Problem set 7, solutions

solution: 3. The expected utility of a consumer who bought α units ofinsurance is

eu(α) = π · u(w(1− t+ α(1− q))) + (1− π) · u(w(1− qα)).

A representative indifference curves of this consumer are drawn above. Tosee the slope of the indifference curves, these are given by the followingequation

π · u(wl) + (1− π) · u(wnl) = k.

Using standard arguments, the slope of the indifference curve at (wnl, wl)can be calculated from πu′(wl)dwl + (1 − π)u′(wnl)dwnl = 0 and is thus∂wl∂wnl

= −1−ππ

u′(wnl)u′(wl)

. On the 45◦ line, wnl = wl and hence the slope of

the indifference curve is −1−ππ . Substituting the functional form for u,

u′(wnl)u′(wl)

=(wnlwl

)−a. Hence the slope of the indifference curve is constant

for any (wnl, wl) such that wnlwl

= k, or wl = wnlk .

solution: 4. A general expression for the expected utility is above. Takingthe derivative w.r.t. α gives

∂eu(α)

∂α= πu′(w(1− t+ α(1− q))w(1− q)− (1− π)u′(w(1− qα))wq

∂2eu(α)

∂2α= πu′′(w(1− t+ α(1− q))w2(1− q)2 + (1− π)u′′(w(1− qα))w2q2

and clearly ∂2eu(α)∂2α

< 0 whenever u is strictly concave.

solution: 5. Substituting the functional form for u and using the firstorder condition, after some algebra and ignoring α ∈ [0, t], gives

α∗ =1−

[1−ππ

q1−q

] 1a

(1− t)

q +[1−ππ

q1−q

] 1a

(1− q)

Let me denote[1−ππ

q1−q

] 1a

= c. If q < π, then c < 1 and hence the consumer

would like to over-insure (you should check this). If she is prevented fromdoing so, then α has to be constrained to lie in [0, t]. In this case α∗ = t.

If q = π, then α∗ = t. Again, insurance with q = π is actuarially fairand the consumer optimally purchases full insurance.

If q > π, then c > 1 and the consumer will not fully insure. On the otherhand, her α∗ will be positive for a range of q (whenever c ≤ 1

1−t to be more

precise). Note that because limq→1 c =∞ and because ∂c∂q > 0, there exists

q∗ such that the consumer will purchase no insurance when q ≥ q∗.

9

Problem set 7, solutions

The optimal amount of insurance the consumer purchases is independentof w since her u is CRRA and the loss involves relative amounts of incomewt. If u was decreasing-RRA, α∗ would be decreasing in w and if u wasincreasing-RRA, α∗ would be increasing in w (modulo the corner solutionsat α∗ ∈ {0, t}).

10