Question

A manufacturer of flash drives has a profit function a = t - 3q2 where t is the price charged for a flash drive and q2 is the cost of producing a drive whose capacity is a gigabytes. A consumer of type 0 has a utility function u = Oq – t, where 0 takes on a value of 13 for H-type consumers, or 8 for L-type consumers. There are 10 consumers of each type. A consumer gets zero utility if she does not buy. Answer the following. If rounding is needed, round to 3 decimal points. a) (0.25 point) Suppose (@l, ÎL) is the optimal (profit maximising) capacity-price bundle for L-type consumer under complete information. What is the value of Û L? b) (0.25 point) Suppose (@h, ĉh) is the optimal (profit maximising) capacity-price bundle for H-type consumer under complete information. What is the value of ĉ H? c) (0.5 points) What is the seller's overall profit under complete information?

Answer 1

Ans)- Given Profit function-

Π = t – 3q²

Utility function, u = Øq – t

1. For L-type consumer put Ø=8 in utility function

So, u = 8q – t

Since seller has complete information, the utility of consumer will be 0.

So,

u = 8q -t

put u=0

8q – t =0

q_L = t/8 ---------    (1)

now profit function-

π = t – 3(t/8)²

Differentiate w.r.t. t

dπ/dt = 1 – 6t/64

put dπ/dt=0 [first order condition]

1 = 3t/32

[t_L = 10.667] ------------------------------required profit-maximising value of ‘t_L hat’.

Put t=10.667 in equation 1

[q_L = 10.667/8 = 1.333 ]

2. For H-type consumer put Ø=13 in utility function

So, u = 13q – t

Since seller has complete information, the utility of consumer will be 0.

So,

u = 13q -t

put u=0

13q – t =0

q_H = t/13 ---------    (2)

now profit function-

π = t – 3(t/13)²

Differentiate w.r.t. t

dπ/dt = 1 – 6t/169

put dπ/dt=0 [first order condition]

1 = 6t/169

[t_H = 28.166] ------------------------------required profit-maximising value of ‘t_H hat’.

Put t=28.166 in equation 2

[q_H = 28.166/13 = 2.166]

3. Overall profit-

Π = t – 3q²

Put t = t_H = 28.166

And q =q_H =2.166

Π_H = 28.166 – 3(2.166)²

Π_H= 14.092 ----------------------profit from 1 H-type consumer.

Multiply by 10

Total profit from 10 H-type consumer = 10*14.092 = 140.92

Now, Put t = t_L = 10.667

And q =q_L =1.333

Π_L = 10.667 – 3(1.133)²

Π_L= 6.816 ----------------------profit from 1 L-type consumer.

Multiply by 10

Total profit from 10 H-type consumer = 10*6.816 = 68.16

Hence, required overall profit of the seller from 10 consumer of each type will be = 140.92 + 68.16 = 209.08