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 service of the lawyer for the following...

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 service of the lawyer for the following year, and the consultation fee is the price that the lawyer charges to do one legal job for the client. The client's annual demand curves for legal jobs are:

Client 1: Q1 = 24 - P

Client 2: Q2 = 21 - 2P

where P is the consultation fee in thousands of dollars for one legal job and Qi is the number of legal jobs done for client i. The lawyers marginal cost for completing one legal job is 2.7 (also in thousands of dollars). The lawyer wants to determine the fee schedule (retainer and consultation fee) that will maximize her profit. She feels that 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. (All responses rounded to two decimal places)

She should charge a consultation fee of $__ thousand per legal job. She should charge a retainer of $__ thousand. (All responses rounded to two decimal places)

 I do not understand what I am doing wrong. Here is my work shown below. 

The total demand curve is the summation of the two clients which is given as

Q= 24-P +(21-2P)  = 24 + 21 –P-2P = 45 – 3P

Q=45-3P

The supply curve of the lawyer is given by the marginal cost oc completing one legal job = 2.7

The price is determined by equating the demand and supply curves

45-3P = 2.7

45-2.7 = 3P

42.3/3 =P

P=14.1

The consultation fee is $14.1 thousand for one legal jon and she should change $45 as retainer.

2 Answers | Add Yours

mathsworkmusic's profile pic

mathsworkmusic | (Level 2) Educator

Posted on

This is a two-part tariff with two consumers problem. The consumers have two different demand functions, but we require that the pricing structure is fair so that the retainer fee is the same for both consumer/client and the price per consultation is the same also. At the same time we wish to maximise profit given these constraints on pricing structure.,

We are told that the demand functions for the two clients are as follows:

Client 1: Q1 = 24 - P1

Client 2: Q2 = 21 - 2*P2

where Pi is the consultation fee for a job for client i and Qi is the number of legal jobs done for client i. The lawyers marginal cost for completing one legal job is 2.7.

Assuming P is equal for the two clients (ie P1 = P2), the total demand curve is the summation of the two clients which is given as

Q=  Q1 + Q2 = (24 - P) + (21 - 2P)  = 24 + 21 – P - 2P = 45 – 3P

Q=45 - 3P   (yes good you have this part right)

This also implies, rearranging, that

P = 15 - Q/3    (1)

The supply curve of the lawyer is given by the marginal cost of completing one legal job

MC = 2.7  (yes good)

The price is determined by equating the demand and supply curves. No, the price P is determined by equating the marginal revenue and marginal cost curves. The marginal revenue MR is the derivative of the total revenue TR = P*Q with respect to demand Q.

Therefore, set MR = MC  where MR = d(TR)/dQ = d(P*Q)dQ 

so that we require (from equation (1) )

d/dQ (15Q - Q^2/3) = 2.7

15 - 2/3 Q = 2.7

Q = (15 - 2.7)*3/2 = 18.45

Since this isn't a whole number, assume that the demand at maximum profit is Q = 18 sessions (number of sessions must be an integer).

Substituting this back into equation (1) gives

P = 15 - 18/3 = $9

Now, the retainer fee, or cover charge, should be set at the surplus value on the demand for the customer with the least demand - here, customer two. (see the web references below for diagrams on how to set the retainer fee)

That is, if the retainer fee is T

T = Q2 * (Intercept of demand curve for consumer 2 - P2)/2       (2)

The intercept of the demand curve is the price which the demand is zero for the consumer. For consumer 2 this is 21/2 (substitute for P in demand curve for consumer two, and this gives zero demand).

Also, Q2 = 21 - 2P2Rearranging gives P2 = (21 - Q2)/2

TR = P2*Q2 = 21/2Q2 - Q2^2/2

MR = 21/2 - Q2

Setting MR = 2.7 gives Q2 = 7.8 (assume Q2 = 7 as number of consultations must be a whole number) and hence P2 = $7. This is what the price for consumer 2 would be in isolation if consumer 1 were not under consideration.

Finally then we have that the retainer fee T should be set at the surplus on the demand for consumer 2 so that, from equation (2)

T = Q2 * (Intercept on consumer two's demand curve - P2)/2 = 7 * (21/2 - 7)/2 = $12.25

The price per consultation should be P = $9 and the retainer fee or cover charge should be T = $12.25

Sources:
shri21's profile pic

shri21 | eNotes Newbie

Posted on

The given question says that the lawyer wants to maximize her profits.

The condition for maximizing the profit is MC = MR i.e. marginal cost equals to marginal revenue.

Now Given things are

MC = 2.7 ............................... (a)

Demand curve can be calculated as Q = Q1+Q2 = 24-P + 21 -2P

i.e. Q = 45-3P

or in terms of P,

P = 15 - (Q/3) ....................... (b)

Now Total revenue (TR) is P*Q

therefore multiplying equation(b) with Q gives us TR curve

TR = P*Q = Q*(15 - (Q/3)) = 15Q - (Q^2/3)

Now differentiating TR curve with respect to Q will give MR curve

d(TR)/dQ = MR = 15 - (2Q/3) ....................... (c)

Now for maximzing profit MC = MR , therefore equating  (a) and (c) we have

15 - (2Q/3) = 2.7 

solving it we get Q = 18.45

but since consultations can't be in fractions, it must be a whole number. 

Therefore we take Q = 18

Now putting this value in demand equation we get 

P = 15 - (18/3) = 15 - 6 = 9 

i.e. P = 9

Therefore she should charge $9 thousand as consultation per legal advice and $45 thousand as retention fee.

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