Microeconomics problem set (ECO601)

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Microeconomic Theory Yong Wang (SITE, UIBE)

Microeconomic Theory

1 Consumption Theory

Question 1.1. You are given the following information about a consumer’s purchase. He consumes only two goods.

Year 1 Year 2 Quantity Price Quantity Price

Good 1 100 100 120 100 Good 2 100 100 y 80

Over what range of y would you conclude:

a. That his behavior is inconsistent, or in contradiction with WARP?

b. Suppose that WARP is satisfied.

(1) That the consumer’s consumption bundle in year 1 is revealed preferred to that in year 2?

(2) That the consumer’s consumption bundle in year 2 is revealed preferred to that in year 1?

(3) That good 1 is an inferior good (at some price) for this consumer?

(4) That good 2 is an inferior good (at some price) for this consumer?

Solution. Denote P 1 , (100, 100), X1 , (100, 100), and P 2 , (100, 80), X2 , (120, y), and W 1 , P 1 ·X1 = 20, 000, and W 2 , P 2 ·X2 = 12, 000 + 80y.

a. Inconsistency is present if following two inequalities hold simultaneously:

P 1 ·X2 ≤W 1 ⇔ y ≤ 80

P 2 ·X1 ≤W 2 ⇔ y ≥ 75,

Thus, if y ∈ [75, 80], WARP is violated.

b. Suppose that WARP is satisfied.

(1) If y < 75, then P 1 · X2 ≤ W 1 and P 2 · X1 > W 2. The second inequality implies that X1 is unaffordable under (P 2,W 2), while the first inequality implies X1 %∗ X2

because when X1 and X2 are both affordable, X1 is chosen.

(2) Similarly, if y > 80, X2 %∗ X1 because P 1 ·X2 > W 1 and P 2 ·X1 < W 2.

(3) Let y < 75 and consider the situation from year 1 to year 2. The relative price for good 1 becomes higher (since good 2 gets cheaper), substitution effect leads to lower consumption on good 1; the consumer gets worse off (from part 1), or “real” wealth (or purchasing power) becomes less. Total effect is positive while the substitution effect is negative, therefore wealth effect must be positive and be strong enough. Combing with reduced real wealth, good 1 is an inferior good at some price.

(4) Let 80 < y < 100, good 2 is an inferior good. The reasons are similar: the (relative) price for good 2 becomes lower, substitution effect leads to higher consumption on good 2. Total effect is negative and substitution effect is positive, thus wealth effect must be negative enough. From part (2), we know that the consumer gets better off from year 1 to year 2, so good 2 is an inferior good at some price.

Question 1.2. ( B, c(·)

) is a choice structure that satisfies WARP.

1. Prove such a statement:

“For any sets B1, B2 ∈ B, if x ∈ B1, B1 ⊆ B2, and x ∈ c(B2), then x ∈ c(B1).”

In words, if you choose x when faced with B2, then you should continue to choose x if some of the alternatives in B2 are removed.

2. The statement above is a form of Independence of Irrelevant Alternatives (IIA). However, some voting theorists believe that the simple majority rule is incompatible with IIA.

Consider a voting game with 18 voters and 3 candidates A,B,C, and:

7 votes for A � B � C 6 votes for B � C � A 5 votes for C � A � B

Show that if candidate B drops out of the race, the winner should be C, instead of A. Therefore, IIA doesn’t hold.

Solution. 1. Suppose y is chosen from B1. Since B1 ⊆ B2, therefore, y ∈ B2. In words, y is also available given budget set B2. Since x is chosen when y is available given B2, thus when y is chosen and x is available, x must be chosen, that is, x ∈ c(B1).

2. After the remove of candidate B, the preferences become:

7 votes for A � C, 6 votes for C � A, and 5 votes for C � A

Thus, 6 + 5 = 11 of 18, votes for C � A while only 7 votes for A � C. C defeats A.

Question 1.3. Consider the consumption of a consumer in two different dates, time 0 and t. Suppose that he consumes only two goods x1 and x2. Denote the price vector by pt = (pt1, p

t 2)

for time t and p0 = (p01, p 0 2) for time 0, and the consumption bundle by xt = (xt1, x

t 2) and

x0 = (x01, x 0 2) for time t and time 0, respectively.

To measure the consumer’s welfare change, define a quantity index by Iq = p·xt p·x0 where p is a

non-negative price vector. Two nature choices for the price vector are p0 and pt. The Laspeyres quantity index is defined as follows:

Lq = p0 · xt

p0 · x0 = p01x

t 1 + p02x

t 2

p01x 0 1 + p02x

0 2

Similarly, define the Paasche quantity index by pt:

Pq = pt · xt

pt · x0 = pt1x

t 1 + pt2x

t 2

pt1x 0 1 + pt2x

0 2

Finally, define an index of the change in total expenditure by:

M = pt · xt

p0 · x0 = pt1x

t 1 + pt2x

t 2

p01x 0 1 + p02x

0 2

Suppose the consumer’s behavior is consistent with WARP. Show that:

1. If Lq ≤ 1, the consumer has a revealed preference for x0 over xt.

2. If Pq ≥ 1, the consumer has a revealed preference for xt over x0.

3. No revealed preference is implied by either M > 1 or M < 1.

Solution. 1. Lq ≤ 1 ⇔ p01x t 1 + p02x

t 2 ≤ p01x

0 1 + p02x

0 2, that is, the consumption bundle xt is

affordable at (p0, w0) where w0 = p0 ·x0, but is not chosen (instead, x0 is chosen), therefore, the consumer prefer to x0 over xt.

2. Pq ≥ 1⇔ pt1x t 1 + pt2x

t 2 ≥ pt1x01 + pt2x

0 2. The consumption bundle x0 is affordable at (pt, wt)

where wt = pt · xt, but is not chosen (instead, xt is chosen), therefore, the consumer prefer to xt

over x0.

3. M ≶ 1 ⇔ pt1x t 1 + pt2x

t 2 ≶ p01x

0 1 + p02x

0 2, or wt ≶ w0. The only thing we can say is that

expenditure at t is higher or lower, no conclusion can be drawn about the preference between xt

and x0. For example, let pt = 3p0 and xt = 1 2x

0. Even though wt > w0, the consumer prefers to x0 over xt.

Question 1.4. Consider a three goods setting in which the consumer has a Stone-Geary utility function

u(x) = (x1 − b1)α(x2 − b2)β(x3 − b3)γ ,

where b1, b2, b3 > 0 represent the minimal amounts of goods 1, 2, and 3 that this individual must consume in order to remain alive. And for simplicity, assume that α + β + γ = 1, and α, β, and γ are all strictly positive.

1. Find the consumer’s Walrasian demand and indirect utility function.

2. Verify that the Walrasian demand functions x(p, w) obtained in part (1) satisfy homogene- ity of degree zero in prices and wealth.

Solution. The budget constraint is specified by (holding with equality)

p1x1 + p2x2 + p3x3 = w, or p1(x1 − b1) + p2(x2 − b2) + p3(x3 − b3) = ŵ,

where ŵ = w − ( p1b1 + p2b2 + p3b3

) .

Take a log transformation on the utility function, and set up the lagrangian function:

L = α ln(x1− b1)+β ln(x2− b2)+γ ln(x3− b3)+λ [ ŵ−

( p1(x1− b1)+p2(x2− b2)+p3(x3− b3)

)] Solving it gives:

x1(p, w) = b1 + α ŵ

p1 , x2(p, w) = b2 + β

ŵ

p2 , and x3(p, w) = b3 + γ

ŵ

p3 .

And the indirect utility function is

v ( p, w

) = ααββγγŵ

pα1 p β 2p

γ 3

It’s easy to show that x(p, w) is homog (0). Here we just check x1(p, w) for example.

x1(tp, tw) = b1 + α tw −

( tp1b1 + tp2b2 + tp3b3

) tp1

= x1(p, w) ∀t > 0

Question 1.5. Suppose that utility function u(x) is homogeneous of degree one and strictly concave.

1. Show that the Walrasian demand function x(p, w) and the indirect utility function v(p, w) are homogeneous of degree one in w, therefore can be rewritten in the form of x(p, w) = x̃(p)w and v(p, w) = ṽ(p)w.

2. Show that the Hicksian demand function h(p, u) and the expenditure function e(p, u) are homogeneous of degree one in u, therefore can be rewritten in the form of h(p, u) = h̃(p)u and e(p, u) = ẽ(p)u.

3. Show that all the wealth elasticities of demand are equal to one, that is, dxi(p,w) dw

w xi(p,w)

= 1,∀i.

Solution. 1. We prove x(p, tw) = tx(p, w) by contradiction. Suppose NOT, that is, there exists a consumption bundle, name it x̂ , such that (following equalities come from utility function’s homogeneity of degree one)

p · x̂ ≤ tw ⇔ p · (

t x̂

) ≤ w

and u(x̂) > u ( tx(p, w)

) = tu(x(p, w))⇒ u(x(p, w)) <

t u(x̂) = u(

t x̂)

The first inequality implies that the consumption bundle 1 t x̂ is affordable under (p, w). The

second (strict) inequality implies the consumption bundle 1 t x̂ is strictly better than x(p, w).

This contradicts with the optimality of x(p, w) under (p, w).

Therefore, x(p, tw) = tx(p, w). Let t = 1/w, one can find that x(p, 1) = 1 wx(p, w) ⇒

x(p, w) = wx(p, 1). Denote x(p, 1) by x̃(p) (demand function when wealth is normalized to 1), then x(p, w) = x̃(p)w.

And v(p, tw) = u ( x(p, tw)

) = u

( tx(p, w)

) = tu

( x(p, w)

) = tv(p, w). Let v(p, 1) = ṽ(p), then

v(p, w) = ṽ(p)w.

2. We also prove h(p, tu) = th(p, u) by contradiction. Suppose NOT, that is, there exists a consumption bundle, name it ĥ , such that

p · ĥ < p · (th(p, u))⇔ p · (

t ĥ

) < p · h(p, u)

and u(ĥ) ≥ tu⇒ 1

t u(ĥ) ≥ u, or u

( 1

t ĥ

) ≥ u

The first inequality implies that expenditure on 1 t x̂ is less than that on h(p, u). The second

inequality implies the consumption bundle 1 t ĥ reaches the utility level u. This contradicts with

the optimality of h(p, u) under (p, u).

Therefore, h(p, tu) = th(p, u). Let t = 1/u ⇒ h(p, 1) = 1 uh(p, u) ⇒ h(p, u) = uh(p, 1),

homogeneous of degree one in u. Denote h(p, 1) by h̃(p) (demand function when utility level is normalized to 1), then h(p, u) = h̃(p)u.

And e(p, tu) = p · h(p, tu) = p · ( th(p, u)

) = t ( p · h(p, u)

) = te(p, u),homogeneous of degree

one in u. Similarly, e(p, u) = ẽ(p)u.

3. Given that xi(p, w) = x̃i(p)w, taking differentiation with respect to w gives dxi(p,w) dw =

x̃i(p). Multiply w xi(p,w)

on both sides, dxi(p,w) dw

w xi(p,w)

= x̃i(p) w

x̃i(p)w = 1.

Question 1.6. Suppose that the Walrasian demand function x(p, w) is differentiable, homoge- nous of degree zero, and satisfies Walras’ law. Let S(p, w) denote the Slutsky matrix. Show that:

1. p·S(p, w) = 0 and S(p, w)p = 0 for any (p, w). (Or you can prove that ∑L

l=1 plslk(p, w) = 0

for all k, and ∑L

k=1 slk(p, w)pk = 0 for all l, where slk(p, w) = ∂xl(p,w) ∂pk

+ ∂xl(p,w) ∂w xk(p, w).)

2. For L = 2, S(p, w) is always symmetric, that is, s12(p, w) = s21(p, w).

Solution. Homog (0) in prices and wealth: xl(tp, tw) = xl(p, w), ∀t > 0 and ∀l

Differentiating w.r.t t gives

L∑ l=1

∂xl(p, w)

∂pk pk +

∂xl(p, w)

∂w w = 0, ∀l. (1)

Walras’ law: p · x(p, w) = w

Differentiating w.r.t pk and w, respectively

xk(p, w) + L∑ l=1

pl ∂xl(p, w)

∂pk = 0,∀k (2)

L∑ l=1

pl ∂xl(p, w)

∂w = 1 (3)

1. To prove ∑L

l=1 plslk(p, w) = 0, or prove ∑L

l=1 pl ∂xl(p,w) ∂pk

+ ∑L

l=1 pl ∂xl(p,w) ∂w xk(p, w) = 0 for

all k.

Note that ∑L

l=1 pl ∂xl(p,w) ∂w = 1 (By Eq.(3)), it’s equivalent to prove

L∑ l=1

pl ∂xl(p, w)

∂pk + xk(p, w) = 0,

which is exactly Eq. (2).

To prove ∑L

k=1 slk(p, w)pk = 0, or ∑L

k=1 ∂xl(p,w) ∂pk

pk + ∑L

k=1 ∂xl(p,w) ∂w xk(p, w)pk = 0 for all l.

Note that ∑L

k=1 ∂xl(p,w) ∂w xk(p, w)pk = ∂xl(p,w)

∂w

∑L k=1 xk(p, w)pk = ∂xl(p,w)

∂w w, so it’s equivalent to prove

L∑ k=1

∂xl(p, w)

∂pk pk +

∂xl(p, w)

∂w w = 0,

which is exactly Eq. (1).

2. For L = 2, ∑L

l=1 plslk(p, w) = 0⇒ p1s11 + p2s21 = 0(k = 1) and ∑L

k=1 slk(p, w)pk = 0⇒ s11p1 + s12p2 = 0(l = 1). Taking these two equations together:

p1s11 + p2s21 = s11p1 + s12p2 = 0⇒ s21 = s12,

i.e., S(p, w) is symmetric.

Question 1.7. Consider an individual with Cobb-Douglas preferences u(x1, x2) = x0.51 x0.52 , where x1 and x2 denote the amounts consumed of goods 1 and 2, respectively. The prices of these goods are p1 > 0 and p2 > 0, respectively, and this individual’s wealth is w > 0. The government needs to collect a large amount of money to finance a new health care plan, and contemplates two options: (a) introduce an income tax equivalent to 40 percent of individuals wealth, or (b) charge a sales tax over the price of good 1 which would imply an increase in the price of good 1 from p1 to p1(1 + t), collecting the same dollar amount as with the income tax. Using the indirect utility function of this individual under option a (income tax) and option b (sales tax), explain which tax produces a smaller utility reduction.

Solution. For utility function u(x1, x2) = x0.51 x0.52 , we know that demand functions and indirect utility function are:

xi = w

2pi , i ∈ {1, 2} and v(p, w) =

2p0.51 p0.52

For the case of income tax, ŵ = 0.6w ⇒ va = v(p, 0.6w) = 0.6w

2p0.51 p0.52

= 3w

10p0.51 p0.52

. For the

case of sales tax, to collect the same dollar amount tax, or tp1x̂1 = tp1× w 2p1(1+t)

= 0.4w ⇒ t = 4

(not 4%). Then, vb = v(5p1, p2, w) = w

2(5p1)0.5p0.52

√ 5w

10p0.51 p0.52

Since √

5 < 3, va > vb, income tax leads to a smaller utility reduction.

Question 1.8. Suppose L = 2. Consider an indirect utility function defined by

v ( p, w

) = exp

( − bp1/p2

) [w p2

+ 1

( a p1 p2

+ a

b + c

)] ,

where a, b and c are constants.

1. Find the demand function for the first good.

2. Find the expenditure function.

3. Find the Hicksian demand function for the first good.

Solution.

x1(p, w) = a p1 p2

+ b w

p2 + c.

e(p, u) = p2u exp(bp1/p2)− 1

( ap1 +

b p2 + cp2

) .

h1(p, u) = ub exp(bp1/p2)− a

b .

2 Prodution Theory

Question 2.1. Consider a Cobb-Douglas technology with two inputs and one output, the pro- duction function is f(z1, z2) = Azα1 z

β 2 , where A,α, β > 0, and α+ β ≤ 1.

1. Find the factor demand functions.

2. Find the supply function, and the profit function.

3. Verify Hotelling’s Lemma.

Solution. Setup Lagrangian:

L = pAzα1 z β 2 − w1z1 − w2z2

Differentiating with respect to each input zk, and the first order conditions are necessary and sufficient:

pAαzα−11 zβ2 = w1, and pAβzα1 z β−1 2 = w2.

Solving for z2:

z2 = βw1

αw2 z1

Plugging into FOC, and then solving for z1

pAzα+β−11 α1−βwβ1

( β

)β = w1

• If α+ β = 1, no solution

• If α+ β < 1, the optimal demand for input 1 is:

z∗1 =

[ pA

( α

)1−β ( β

)β]1/(1−α−β)

Inserting z∗1 into z2 = βw1

αw2 z1, the optimal demand for input 2 is:

z∗2 =

[ pA

( α

)α( β

)1−α ]1/(1−α−β)

Inserting z∗1 and z∗2 into f(z1, z2) = Azα1 z β 2 ,

y∗ =

[ A

( αp

)α(βp w2

)β]1/(1−α−β)

The profit function is

π∗ = (1− α− β)

[ pA

( α

)α( β

)β]1/(1−α−β)

To verify Hotelling’s Lemma,

∂π∗

∂p = p(α+β)/(1−α−β)

[ A

( α

)α( β

)β]1/(1−α−β) = y∗ (output)

∂π∗

∂w1 = −

[ pA

( α

)1−β ( β

)β]1/(1−α−β) = −z∗1 (input)

∂π∗

∂w2 = −

[ pA

( α

)α( β

)1−α ]1/(1−α−β)

= −z∗2 (input)

Note also that the law of supply holds:

dy∗

dp =

α+ β

1− α− β p(2(α+β)−1)/(1−α−β)

[ A

( α

)α( β

)β]1/(1−α−β) > 0

Question 2.2. Consider a Cobb-Douglas technology with two inputs and one output, the pro- duction function is f(z1, z2) = Azα1 z

β 2 , where A,α, β > 0, and α+ β ≤ 1.

1. Find the conditional factor demand functions.

2. Find the cost function.

Solution. Setup Lagrangian:

min x1,x2

w1z1 + w2z2, s.t. Azα1 z β 2 ≥ q

Lagrangian L = w1z1 + w2z2 + λ ( q −Azα1 z

β 2

) FOCs: w1 = αAλzα−11 xβ2

w2 = βAλzα1 x β−1 2 , and z2 =

βw1

αw2 z1

Plugging into q = Azα1 z β 2 :

z1(w1, w2, q) = q 1

α+β

( αw2

βw1

) β α+β

, z2(w1, w2, q) = q 1

α+β

( βw1

αw2

) α α+β

Therefore, the cost function is

c(w1, w2, q) = w1z1(w1, w2, q) + w2z2(w1, w2, q) = Bq 1

α+βw α

α+β

1 w β

α+β

where B = (α/β) β

α+β + (β/α) α

α+β is a constant.

Note that if α+ β = 1 (CRS), the solution for CMP still exists:

z1(w1, w2, q) =

( αw2

βw1

)β q, z2(w1, w2, q) =

( βw1

αw2

)α q, and c(w1, w2, q) = (w1/α)α(w2/β)βq.

Question 2.3. Suppose that there are two different observations with the same output level q for a cost-minimization firm:

prices for inputs and firm’s choices at time 0 :(w0, z0)

prices for inputs and firm’s choices at time 1 :(w1, z1)

where wi (i = 0, 1) is the price vector of input, and zi is the input vector. Denote ∆w = w1−w0

the price changes, and ∆z = z1 − z0 the change of demands for inputs.

Show that the law of (conditional factor) demand holds:

∆w ·∆z ≤ 0.

Solution. Given the price wi, the cost of input combination zi is lower than that of zj . There- fore,

w0 · z0 ≤ w0 · z1 ⇔ w0 ·∆z ≥ 0

and w1 · z1 ≤ w1 · z0 ⇔ w1 ·∆z ≤ 0

⇒ ∆w ·∆z ≤ 0.

Suppose for simplicity that w0 −i = w1

−i and w0 i > w1

i , the inequality above becomes:

(w1 i − w0

i )(z 1 i − z0i ) ≤ 0

The demand for input and price move with different directions.

Question 2.4. Suppose that a firm owns two plants, each producing the same good. Every plant j′s average cost is given by

ACj(qj) = α+ βjqj , ∀qj ≥ 0,

where j ∈ {1, 2} where coefficient βj may differ from plant to plant. Assume that you are asked to determine the cost-minimizing distribution of aggregate output q = q1 + q2, among the two plants (i.e., for a given aggregate output q, how much q1 to produce in plant 1 and how much q2 to produce in plant 2). For simplicity, consider that aggregate output q satisfies q < α/maxj |βj |.

1. If βj > 0 for every plant j, how should output be located among the two plants? What’s the case if there is n plants?

2. If βj < 0 for every plant j, how should output be located among the two plants?

3. If β1 > 0 while β2 < 0, how should output be located among the two plants?

Solution. Total cost TC = AC1(q1)q1 +AC2(q2)q2 with qj ≥ 0, therefore the cost minimization problem can be formulated as follows:

min q1≥0,q2≥0

(α+ β1q1)q1 + (α+ β2q2)q2 s.t. q1 + q2 = q,

or equivalently (substituting q2 by q − q1)

min 0≤q1≤q

(α+ β1q1)q1 + (α+ β2(q − q1))(q − q1).

Taking the first and second order derivative with respect to q1:

dTC

dq1 = 2(β1 + β2)q1 − 2β2q

d2TC

dq21 = 2β1 + 2β2

1. If βj > 0 for every plant j, d2TC/dq21 > 0, total cost is convex in q1. Thus, the first order condition is sufficient and necessary for cost minimization. Let dTC/dq1 = 0, and then:

q∗1 = β2

β1 + β2 q, and q∗2 =

β1 β1 + β2

For the case of n plants, setup the Lagrangian:

L = ∑ j

(α+ βjqj)qj + λ(q − ∑ j

qj)

Given each βj > 0, L is convex is qj , FOCs are sufficient and necessary for minimization:

2βjqj = λ⇒ qj = λ/(2βj),∀j

Summation over j: q = ∑

j qj = Θλ/2, where Θ = ∑

j θj , θj = 1/βj . Thus, λ = 2q/Θ, and

qj = θj Θ q.

You can verify that the average cost at any plant is α+ q/Θ, equal across each plant.

2. If βj < 0 for every plant j, d2TC/dq21 < 0, total cost is concave in q1, the first order condition is just for cost maximization, instead of minimization. For such a situation, corner solution exists. Since 0 ≤ q1 ≤ q, we just compare the cost under the case of q1 = 0 with that of q0 = q. Readily, q1 = 0⇒ TC = α+ β2q

2, and q1 = q, TC = α+ β1q 2. So if

β1 < β2, TC ∗ = α+ β1q

2 at optimum, otherwise TC∗ = α+ β2q 2.

In words, the firm optimally chooses the plant with lower (lowest if many plants) average cost.

3. If β1 > 0 while β2 < 0, dTC/dq1 = 2β1q1 + (−2β2)(q − q1) is always strictly positive, implying TC is always (strictly) increasing in q1, so q∗1 = 0, q∗2 = q − q∗1 = q at optimum.

This result is intuitive: average cost decreases with production for plant 2, while average cost increases with production for plant 1, the firm choose such plant 2 for production as many as possible to reduce total cost.

For plant j, if there exist  such AC < ACj , one can verify that q∗j = 0 at optimum.

Suppose NOT, i.e., q∗j > 0. The associated total cost TC = ACq+ACjq ∗ j + ∑

i 6=,i 6=j ACiqi. Consider an adjustment of production plan: other things equal, plant  produce more while plant j produce less. Mathematically, q̂ = q+∆, q̂j = q∗j−∆, where ∆ is a small and positive number. This leads to a change of total cost: ∆TC = −(ACj −AC)∆ < 0. This is a contradiction.

Question 2.5. A firm has a production function described as follows.

Weekly output is the square root of the minimum of the number of units of capital and the number of units of labor employed per week.

Suppose that in the short run this firm must use 9 units of capital but can vary its amount of labor freely.

a. Write down a formula that describes the marginal product of labor in the short run as a function of the amount of labor used. (Be careful at the boundaries.)

b. If the wage is w = 1 and the price of output is p = 4, how much labor will the firm demand in the short run?

c. Redo part (b) when w = 1 and p = 10.

d. Write down an equation for the firm’s short run demand for the labor as function of w and p.

3 Choices under Uncertainty

Question 3.1. Suppose that there are only two states, 1 and 2, with associated probabilities π ∈ (0, 1) and 1− π. Consider a decision-maker who is strictly risk averse and expected utility maximizer:

max x1≥0,x2≥0

πu(x1) + (1− π)u(x2)

s.t. p1x1 + p2x2 ≤ w,

where u(x) is the Bernoulli utility function, and p1, p2, w > 0.

Suppose that inner solution exists, the optimal allocation ( x∗1, x

∗ 2

) is determined by

1− π u′(x∗1)

u′(x∗2) = p1 p2 .

Show that if the decision-maker becomes more risk averse, she will choose a utility-maximizing pair (x̂1, x̂2) that lies closer to the 45-degree line.

Hint: (1) you can assume some conditions such that x∗1 ≤ x∗2 and then prove that x̂1 ≥ x∗1 and x̂2 ≤ x∗2.

(2) Represent “more risk aversion” by “a more concave utility function”.

Solution. Let the new utility function after the change of more risk aversion is v(x) = ϕ ( u(x)

) ,

where ϕ′(·) > 0 and ϕ′′(·) < 0.

Let ∆ , p1 p2

1−π π . Suppose without loss of generality ∆ ≥ 1, then FOC implies u′(x∗1) ≥

u′(x∗2)⇔ x∗1 ≤ x∗2 (If ∆ < 1⇒ x∗1 > x∗2).

FOC after increase in risk aversion:

v′(x̂1)

v′(x̂2) = u′(x̂1)

u′(x̂2)

ϕ′ ( u(x̂1)

) ϕ′ ( u(x̂2)

) = ∆

Note first that ∆ ≥ 1 ⇒ v′(x̂1) ≥ v′(x̂2) ⇒ x̂1 ≤ x̂2 ⇒ u(x̂1) ≤ u(x̂2), the last inequality comes from the monotonicity of u(·). Secondly, ϕ′′(·) < 0 ⇒ ϕ′(·) is decreasing, thus u(x̂1) ≤ u(x̂2)⇒ ϕ′

( u(x̂1)

) ≥ ϕ′

( u(x̂2)

) ⇒ u′(x̂1)

u′(x̂2) ≤ ∆.

u′(x̂1)

u′(x̂2) ≤ ∆ =

u′(x∗1)

u′(x∗2) ⇔ u′(x̂1)

u′(x∗1) ≤ u′(x̂2)

u′(x∗2) .

Suppose that x̂1 < x∗1. On the one hand, budget constraint implies x̂2 > x∗2; on the other hand, u′(x̂1) > u′(x∗1)(diminishing marginal utility) ⇒ u′(x̂2) > u′(x∗2) ⇒ x̂2 < x∗2. This is a contradiction, so x̂1 ≥ x∗1 and x̂2 ≤ x∗2.

Question 3.2. A farmer uses his own labor, x, to produce wheat, q, with linear production function q = f(x) = x. At the end of the harvesting season, the farmer sells all production at the given price of wheat.

The price he receives is a random variable, in particular, has two possible realizations: pH with probability γ, pL with probability 1− γ, where pH > pL, 0 < γ < 1.

The farmer’s preferences are given by utility function u(wk)− f(x), where wk represents the farmer’s income when the realization of the random variable is k, where k ∈ {H,L}. Suppose that u(·) is strictly increasing and concave. Assume that the cost of effort is strictly increasing and convex, and denote by w0 the farmer’s initial income.

The government has just created a price-guarantee program that ensures that the farmer will receive a price pG, where pH > pG > pL, for the proportion of his production that he includes in this program at the beginning of the season, that is, before knowing the particular price for that year. Let λ ∈ [0, 1] denote the proportion of the farmer’s production that he chooses to include in the price-guarantee program.

1. Set up the farmer’s expected utility maximization problem. For simplicity, assume inner solutions. Take the first-order conditions with respect to the two choice variables of the farmer: his own labor, x, and the proportion of his production that he includes in the price- guarantee program, λ. Characterize the solution (note that the solution will be implicit, since we do not have a functional form for the farmer’s utility function).

2. Consider now that the price-guarantee program produces zero expected profits for the government. Explain how your implicit solution for x and λ in above question is affected.

Solution. The farmer maximizes his expected utility:

max x,λ

γu(wH) + (1− γ)u(wL)− f(x)

where wH = w0 + pH(1− λ)x+ pGλx,wL = w0 + pL(1− λ)x+ pGλx

FOC:

x : γu′(wH) ( pH(1− λ) + λpG)

) + (1− γ)u′(wL)(pL(1− λ) + λpG)− f ′(x) = 0

λ : γu′(wH)x(pH − pG)− (1− γ)u′(wL)x(pG − pL) = 0⇔ γ

1− γ u′(wH)

u′(wL) = pG − pL pH − pG

Substituting the second condition into the first one, γu′(wH)pH + (1− γ)u′(wL)pL = f ′(x).

If the price-guarantee program produces zero expected profits, i.e., pG = γpH + (1 − γ)pL, then wH = wL ⇒ λ = 1, fully insured. And x∗ is determined by u′(w0 + p̄x)p̄ = f ′(x), where p̄ = γpH + (1− γ)pL.

Question 3.3. Consider a decision-maker who lives for exactly two periods, t = 0, 1. Let ct ≥ 0 denote her consumption in period t, the expected utility function is

U(c0, c1) = u(c0) + δEu(c̃1),

where δ > 0 is a discount factor, u(·) is an increasing and strictly concave utility function, and the E operator denotes her expectation (at t = 0) concerning events in period t = 1. For simplicity, you can also assume that the marginal utility of consumption is convex, that is, u′′′ > 0.

Suppose that there is initially no uncertainty. Let w0 ≥ 0 be her income in period 0 and let w1 ≥ 0 denote her income in period 1. Let s ∈ R denote her saving (borrowing if s < 0), and let ρ denote the gross return on saving (i.e., ρ = 1 + r where r ≥ 0 is the riskless interest rate). Thus her consumption in period 0 is w0 − s, and consumption in period 1 is w1 + ρs. Assume inner solutions throughout the exercise.

1. Write down necessary and sufficient conditions for the choice of saving, s∗, to be strictly positive.

2. Suppose that w1 = 0. Find a condition on the coefficient of relative risk aversion that is necessary and sufficient for s∗ to be (locally) increasing in ρ.

3. Now suppose that there is uncertainty over her period 1 income. Specifically, suppose that her period 1 income is given by w1 + x̃, where the random variable x̃ exhibits E(x̃) = 0. Let s∗∗ denote the new optimal saving in this context. Show that s∗∗ > s∗.

Hint: Suppose that s∗∗ = s∗ and compare the first-order conditions using Jensen’s inequal- ity.

Solution. 1. Consider the case without uncertainty. c0 = w0−s, c1 = w1+ρs, so U(c0, c1) = u(w0 − s) + δu(c1) Taking the first-order derivative w.r.t. s

∂U(c0, c1)

∂s = −u′(c0) + δρu′(c1)

For s∗ to be strictly positive, the necessary and sufficient condition is

∂U(c0, c1)

∂s

∣∣ s=0

> 0⇔ δρu′(w1) > u′(w0)

2. Suppose that w1 = 0. s∗ is determined by FOC

u′(w0 − s∗) = δρu′(ρs∗).

Denote w0 − s∗ by c∗0 and ρs∗ by c∗1. Total differentiation gives:

−u′′(w0 − s∗) ds∗

dρ = δu′(ρs∗) + δρu′′(ρs∗)

( s∗ + ρ

ds∗

dρ

) Divided by FOC:

ρrA(c∗0) ds∗

dρ = 1− rR(c∗1)− ρ2rA(c∗1)

ds∗

dρ ⇒ ds∗

dρ =

1− rR(c∗1)

ρrA(c∗0) + ρ2rA(c∗1)

For ds∗/dρ > 0, the necessary and sufficient condition is rR(c∗1) < 1.

3. Let v(·) = −u′(·), then v′(·) = −u′′(·) > 0 and v′′(·) = −u′′′(·) < 0, i.e., v function is increasing and strictly concave.

FOC can be rewritten as:

v(c∗0) = δρv(c∗1) (4)

v(c∗∗0 ) = δρEv(c̃1) (5)

where c∗0 = w0−s∗, c∗1 = w1+ρs∗ and c∗∗0 = w0−s∗∗, c̃1 = w1+ x̃+ρs∗∗. To prove s∗∗ > s∗, Suppose NOT, i.e., s∗∗ ≤ s∗. Note first that v(·) is strictly concave, therefore

v(c∗∗0 ) = δρEv(c̃1) < δρv ( E(̃c1)

) = δρv

( w1 + ρs∗∗

) On the one hand, s∗∗ ≤ s∗ ⇒ c∗∗0 ≥ c∗0 ⇒ v(c∗∗0 ) ≥ v(c∗0) (by the monotonicity of v(·)), that is, the LHS of Eq.(5) is bigger than that of Eq.(4).

On the other hand, s∗∗ ≤ s∗ ⇒ w1 + ρs∗∗ ≤ c∗1 ⇒ v ( w1 + ρs∗∗

) ≤ v(c∗1) (again, by the

monotonicity of v(·)), the RHS of Eq.(5) is strictly smaller than δρv ( w1 +ρs∗∗

) ≤ δρv(c∗1),

just the RHS of Eq.(4).

This is a contradiction, thus s∗∗ > s∗. This is precautionary saving when faced by uncer- tainty.

Question 3.4. Consider an individual with the following utility function

u(C,H) = lnC − α

H ,

where C is his expenditure in consumption goods and H is his expenditure on health insurance. Parameter α denotes his monetary loss if he becomes sick where, for simplicity, let α = 1 if he is sick, and α = 0 otherwise.

The probability of getting sick is given by γ ∈ (0, 1), and this individual’s wealth is given by m > 0, where m = C +H.

1. Find the first-order conditions for the optimal amount of consumption goods C∗, and health insurance, H∗.

2. Is health insurance a normal good, or inferior good?

Solution. The expected utility function is

Eu = γ

( lnC − 1

) + (1− γ)

( lnC − 0

) = lnC − γ

H ,

The budget constraint is given by C = m−H, therefore the FOC is

− 1

m−H +

H2 = 0⇒ H∗ =

√ γ2 + 4γm− γ

Readily, ∂H∗/∂m > 0, health insurance is a normal good.

Question 3.5. Assume that there are only two assets. The first is a riskless asset that pays $1, and the second pays amounts a and b (b > a) dollars with probabilities of π ∈ (0, 1) and 1− π, respectively. Denote the demand for the two assets by

( x1, x2

) .

Consider a risk averse decision-maker with initial wealth $1. Prices of both assets are equal to 1, and short-selling is prohibited for both assets. Thus budget constraint is

x1 + x2 = 1, x1, x2 ∈ [0, 1]

a. Give a simple condition (involving a and b only) for the demand for the riskless asset to be strictly positive.

b. Give a simple condition (involving a, π and b only) for the demand for the risky asset to be strictly positive.

Following suppose that the conditions obtained in (a) and (b) are satisfied.

c. Write down the first order conditions for expected utility maximization in the asset demand problem.

d. Assume that a < 1. Show by analyzing the first order conditions that dx1 da

< 0.

Question 3.6. Consider a strictly risk-averse decision maker who has an initial wealth of w but who runs a risk of a loss of D dollars. The probability of the loss is θ. It is possible, however, for the decision maker to buy some insurance. One unit of insurance costs q dollars and pays 1 dollar if the loss occurs. What is the optimal level of insurance? For simplicity you can take the assumption q = θ.

Question 3.7. Assume that there are only two assets. The first is a riskless asset that pays $1, and the second pays amounts a and b (b > a) dollars with probabilities of π ∈ (0, 1) and 1− π, respectively. Denote the demand for the two assets by

( x1, x2

) .

Consider a risk averse decision-maker with initial wealth $1. Prices of both assets are equal to 1, and short-selling is prohibited for both assets. Thus budget constraint is

x1 + x2 = 1, x1, x2 ∈ [0, 1]

1. Give a necessary condition (involving a and b only) for the demand for the riskless asset to be strictly positive.

2. Give a necessary condition (involving a, π and b only) for the demand for the risky asset to be strictly positive.

Following suppose that the conditions obtained in (1) and (2) are satisfied.

1. Write down the first order conditions for expected utility maximization in the asset demand problem.

2. Assume that a < 1. Show by analyzing the first order conditions that dx1 da

< 0.

Question 3.8. Consider an optimal portfolio problem. Suppose there are two assets: a safe one and a risky one. Without loss of generality, suppose that the net rate of return on the safe asset is 0. The rate of (random) return on the risky asset is denoted by a random variable z̃. The average return on the risky asset is assumed to be higher than that on the safe asset, that is, E(z̃) > 0.

Consider an investor with concave (Bernoulli) utility function u(x) who has initial wealth w to invest. Let α ≥ 0 and β = w−α ≥ 0 denote the amounts of wealth invested in the risky and safe asset.

1. Show that the optimal demand for the risky asset, denoted by α∗, is strictly positive.

Then, we take the interior solution following, that is, α∗ ∈ (0, w), therefore FOC is sufficient and necessary for optimality.

2. If the investor exhibits DARA, show that α∗ increases with w. That is, the risky asset is a normal good.

3. Denote γ∗ = α∗/w. Show that γ∗ decrease with w if the investor exhibits IRRA. (This result implies that the elasticity of the demand for risky asset to wealth is smaller than 1, therefore is a necessary good.)

4. Now consider another investor, named Frank, who is more risk averse in the sense of more concave utility function. Show that Frank demands less risky asset than α∗.

Question 3.9. Consider a taxpayer with exogenous income y > 0 who faces a tax rate t, where 0 < t < 1. She is asked to report an income level x to the government and pay taxes according to the report, i.e., tx. If the taxpayer is honest, she will report x = y, but she may cheat by reporting a lower income 0 < x < y. Let z = y − x represent the amount by which income is understated. The government does not know the true income y and must enforce compliance through a system of audits and penalties. Assume that the enforcement policy, known by the taxpayer, is to audit reports with probability p, where p ∈ (0, 1) is constant. If there is an audit, we assume that the government always ends up learning the true income of this individual y. If the taxpayer is caught cheating, she must pay a penalty θ on each dollar of income evaded, θz, in addition to the evaded tax. Assume that the taxpayer is risk averse, meaning her utility u(x) is increasing and concave in income, and that she maximizes expected utility.

1. For any z, where 0 ≤ z < y, write the income the taxpayer will receive in each one of the two possible situations, i.e., there is an audit and if there is no audit (notice that the choice variable for the consumer is z).

2. Calculate the minimum value of t such that she does choose to cheat.

3. Suppose that an inner solution exists, and denote it by z∗ > 0. Prove that z∗ decreases in the probability of being audited, p, and in being fined, θ. What’s the effect of the tax rate t on z∗?

Question 3.10. Suppose that there are five states of nature denoted by ωn, n = 1, 2, . . . , 5, all of which are of equal probability. Consider two risky assets with rates of returns r̃A and r̃B as follows:

state ω1 ω2 ω3 ω4 ω5

r̃A 0.5 0.5 0.7 0.7 0.7 r̃B 0.9 0.8 0.4 0.3 0.7

Which asset a risk averse investor will prefer to?