Demand for a good is more elastic among students than it is among non-students.

Question

Demand for a good is more elastic among students than it is among non-students. Knowing this, the producer of the good is trying to figure out the optimal pricing schedule for the good. He estimates that the demand for students is given by qs = 18-5p and he estimates that the demand for non-students is given by qn = 10-2p. The marginal cost for the good is $3 and there are no fixed costs.

A) If the producer cannot determine which consumers are students and which are non-students, what is the optimal uniform price for all consumers?

B) Assume that students can buy the good at a discounted price, PS with a student ID and non-students can buy it for the normal price, PN. What price should be set for each group?

C) How does the price discrimination affect the producer's profits?

D) Calculate the consumer surplus under uniform pricing.

E) Calculate the consumer surplus under price discrimination.

F) Does price discrimination increase or decrease the pareto efficiency of thie market?

Explanation / Answer

(A) In this case, market quantity (Q) = qs + qn = 18 - 5p + 10 - 2p = 28 - 7p

7p = 28 - Q

p = (28 - Q) / 7

Profit is maximized by equating Marginal revenue (MR) with MC.

Total revenue (TR) = p x Q = (28Q - Q2) / 7

MR = dTR / dQ = (28 - 2Q) / 7

Equating with MC,

(28 - 2Q) / 7 = 3

28 - 2Q = 21

2Q = 7

Q = 3.5

p = (28 - 3.5) / 7 = 24.5 / 7 = $3.5

(B) Profit is maximized when MRS = MC & MRN = MC

For Students,

qs = 18 - 5PS

5PS = 18 - qs

PS = 3.6 - 0.2qs

Total revenue (TRS) = PS x qs = 3.6qs - 0.2qs2

MRS = dTRS / dqs = 3.6 - 0.4qs

Equating with MC,

3.6 - 0.4qs = 3

0.4qs = 0.6

qs = 1.5

PS = 3.6 - (0.2 x 1.5) = 3.6 - 0.3 = $3.3

For Non-Students,

qn = 10 - 2PN

2PN = 10 - qn

PN = 5 - 0.5qn

TRN = PN x qn = 5qn - 0.5qn2

MRN = dTRN / dqn = 5 - qn

Equating with MC,

5 - qn = 3

qn = 2

PN = 5 - (0.5 x 2) = 5 - 1 = $4

(C) Profit (Z) = q x (P - MC)

With single price, Z = 3.5 x $(3.5 - 3) = 3.5 x $0.5 = $1.75

With price discrimination,

For Students, Profit (Zs) = qs x (PS - MC) = 1.5 x $(3.3 - 3) = 1.5 x $0.3 = $0.45

For Non-students, Profit (Zn) = qn x (PN - MC) = 2 x $(4 - 3) = 2 x $1 = $2

Total profit = Zs + Zn = $(0.45 + 2) = $2.45

Therefore price discrimination increases profit by $0.7 (= $2.45 - $1.75).

(D) From single-price demand function, when Q = 0, p = 28/7 = $4 (Reservation price)

Consumer surplus = Area between demnd curve & equilibrium price = (1/2) x $(4 - 3.5) x 3.5 = (1/2) x $0.5 x 3.5

= $0.875

NOTE: As per Chegg answering policy, first 4 parts are answered.

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