課程名稱︰資訊經濟
課程性質︰資管系/所選修
課程教師︰孔令傑
開課學院：管理學院
開課系所︰資管系
考試日期（年月日）︰110.05.17
考試時限（分鐘）：180
試題 :
註：以下部分數學符號與式子以LaTeX語法表示。
1. (25 points) A retailer sells a product through a salesperson who may privat-
ely exert sales effort a with the cost a^2/2. The salesperson also privately o-
bserve the market condition \theta \in {\theta_L, \theta_H}, where 0 < \theta_L
< \theta_H < 1. The retailer believes that Pr(\theta = \theta_L) = \beta = 1 -
Pr(\theta = \theta_H). THe sales quantity x \in {0,1} follows a Bernoulli dist-
ribution such that Pr(x=1 | a) = \theta a = 1 - Pr(x=0 | a). The procurement c-
ost is c \in [0,1), and the retail price is 1. The retailer may procure the pr-
oduct after the product is sold. The retailer offers a menu of contract {(u_L,
v_L),(u_H,v_H)} for the salesperson to self-select. If a contract (u,v) is sig-
ned, the retailer and salesperson earns (1-v)x - u and vx + u, respectively. T-
he sequence of events is (1) the salesperson privately observe \theta, (2) the
retailer offers a menu, (3) the salesperson chooses a contract and the effort
level. Everyone earns nothing if the salesperson rejects both contracts.
(a) (5 points) Formulate and solve the retailer's first-best problem by assume-
ing that \theta is publicly observable.
(b) (5 points) Formulate the retailer's second-best problem.
(c) (10 points) Solve the retailer's second-best problem. Show that downward d-
istortion occurs due to infromation asymmetry.
(d) (5 points) Finds the salesperson's equilibrium expected profit under infor-
mation asymmetry.
2. (30 points) A seller (she) sells a product with a warranty plan to a consum-
er (he). The consumer's expected utility of buying the product at price p with
warranty protection probability w (which has one-to-one corresponding with war-
ranty length) is r\theta + (1-r)w\theta - p, where \theta > 0 is the benefit of
using the product and r is the probability for the product to be functional. T-
he seller's expected profit is p - (1-r)wc, where c > 0 is the cost of fixing
the product. The consumer privately observes his type \theta \in {\theta_L,
\theta_H}, where c < \theta_L < \theta_H. The seller believes that Pr(\theta =
\theta_L) = \beta = 1 - Pr(\theta = \theta_H). The seller offers a menu of con0
tracts {(p_L,w_L),(p_H,w_H)} for the consumer to self-select. While p_L and p_H
may be positive or zero, w_L and w_H must be within 0 and 1. The sequence of e-
vents is (1) the consumer privately observes \theta, (2) the seller offers a m-
enu, (3) the consumer chooses a contract. Everyone earns nothing if the consum-
er rejects both contracts.
(a) (5 points) Formulate and solve the seller's first-best problem by assuming
that \theta is publicly observable.
(b) (5 points) Formulate the seller's second-best problem.
(c) (10 points) Solve the seller's second-best problem.
(d) (10 points) Derive a condition under which downward distortion occurs due
to information asymmetry. Determine how \beta affects the likelihood for
downward distortion to occur. Provide economic intuition for that with no
more than 100 words.
3. (35 points) A seller (she) sells a product with a warranty plan to a consum-
er (he). The consumer's expected utility of buying the product at price p with
warranty protection probability w (which has one-to-one corresponding with war-
ranty length) is r\theta + (1-r)w\theta - p, where \theta > 0 is the benefit of
using the product and r is the probability for the product to be functional. T-
he seller's expected profit is p - (1-r)wc, where c > \theta is the cost of fi-
xing the product. The seller privately observes her type r \in {r_L, r_H}, whe-
re 0 \leq r_L \leq r_H \leq 1. The consumer's prior belief is that Pr(r = r_L)
= \beta = 1 - Pr(r = r_H). The type-i seller offers a contract (p_i, w_i) for
the consumer to take it or leave it, i \in {L, H}. While p_i may be positive or
zero, w must be within 0 and 1. The sequence of events is (1) the seller priva-
tely observes r, (2) the seller offers a contract, (3) the consumer updates her
belief on the seller's type according to his observation on the contract, (4)
the consumer chooses a contract. Everyone earns nothing if the consumer rejects
both contracts.
(a) (5 points) Formulate and solve the seller's first-best problem by assuming
that r is publicly observable.
(b) (5 points) Is it the low-type seller (whose r = r_L) having incentive to m-
imic the high-type one (whose r = r_H) or the opposite? Intuitively explain
why with no more than 100 words.
(c) (5 points) Formulate the high-type seller's contract design problem under
separation by assuming that the low-type seller offers her first-best cont-
ract.
(d) (5 points) Is it possible for the high-type seller to costlessly separate
from the low-type one? Show it analytically.
(e) (15 points) Solve the high-type seller's contract design problem under sep-
aration by assuming that the low-type seller offers her first-best contract
4. (10 points) Say something to the instructing team to earn ten free points!