A lawyer has two clients. She charges a yearly retainer plus a separate fee for

Question

A lawyer has two clients. She charges a yearly retainer plus a separate fee for each consultation. The retainer is a fee each client pays just to secure the services of the lawyer for the following year, and the consultation fee is the price the lawyer charges to do one legal job for the client. The clients' annual demand curves for legal jobs are Client 1: Q20- P Client 2: Q2 -17-2P where P is the consultation fee in thousands of dollars for one legal job and Q, is the number of legal jobs done for client i The lawyer's marginal cost for completing one legal job is 3.5 (also in thousands of dollars). The lawyer wants to determine the fee schedule (retainer and consultation fee) that will maximize her profit. She feels she cannot use a different fee schedule for each client, so she must set one retainer and one consultation fee that apply to both clients She should charge a consultation fee of $ thousand per legal job Enter your response rounded to two decimal places. She should charge a retainer of $thousand. (Enter your response rounded to two decimal places.)

Explanation / Answer

Consultation fee must be such that the MR = MC for the low type customers:

Q1 = 20-P

P = 20-Q1

Q2 = 17-2P

2P = 17-Q2

P = 8.5-0.5Q2

The second type are the low type with demand Q2 and the high type are with demand Q1:

TR for low type:

TR2 = (8.5-0.5Q2)Q2

MR2 = 8.5-Q2

MR2 = MC

8.5-2Q2 = 3.5

5 = 2Q2

Q2 = 5/2 = 2.5

P = 8.5-0.5*2.5

= 8.5-5

Consultation fee of $7.25 thousand

Charge a retainer equal to the consumer surplus at P = 7.25

Q2 = 2.5

CS 2 = ½*(8.5-7.25)*2.5

Retainer of $1.5625 thousand

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