# A psychiatrist is contemplating to set different prices for two different groups of clients: males (M) and females (F). The marginal cost MC of providing counselling sessions is given by MC = 5 +...

A psychiatrist is contemplating to set different prices for two different groups of clients: males (M) and females (F). The marginal cost MC of providing counselling sessions is given by

MC = 5 + 0.002Q,

where Q is quantity of sessions provided. The demand functions of males and females are respectively:

QM = 500 – 25PM,

QF = 900 – 30PF,

where subscript M represents male, subscript F represents female, QM is quantity demanded by males, QF is quantity demanded by females, PM is price charged to males, and PF is price charged to females.

Suppose the male and female markets can be segregated.Compare the steps in solving the two prices PM and PF. Solve PM and PF.

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Without additional equations or values for some of the variables, we will only be able to solve for Pm and Pf in terms of their relationship to other variables, such as MC.

Given MC = 5 + 0.002Qx

and Qm = 500 - 25Pm

and Qf = 900 - 30Pf

we can substitute the values of Qm and Qf for Qx in the equation for MC, and rearrange to determine the values of Pm and Pf.

First, rearranging the equations (since both can use the same final format)

MC = 5 + 0.002(X -YPx)

MC = 5 + 0.002X - 0.002YPx

0.002YPx = 5 + 0.002X - MC

YPx = 2500 + X - 500MC

Px = (2500 + X - 500MC)/Y

Substituting values:

Pm = (2500 + 500 - 500MC)/25

**Pm = 120 - 20MC**

Pf = (2500 + 900 - 500MC)/30

**Pf = (340/3) + (50/3)MC**

Rearrange the demand functions:

PM = 20 - 0.04QM and

PF = 30 - QF/30

Calculate Total Revenue, TR, for both (Use TR = P*Q):

TRM = PM*QM = 20QM - 0.04(QM^2) and

TRF = PF*QF = 30PF - (QF^2)/30

Calculate Marginal Revenue, MR, from TR. (Use MR = d(TR)/dQ)

MRM = 20 - 0.08QM and

MRF = 30 - QF/15

The Marginal Revenue in each group is equal to Marginal Cost of total. Hence,

(MRM =) 20 - 0.08QM = 5 + 0.002(QM + QF)

i.e, QF = (15 - 0.082QM)/0.002 ............. [Eqn 1] and

(MRF =) 30 - QF/30 = 5 + 0.002(QM +QF)

i.e, QF = (25 - 0.002QM)/0.069 ............. [Eqn 2]

Solving Eqn 1 and Eqn 2 for approx values of QF and QM,

QM = 174 and

QF = 366

Substituting QM and QF in the demand functions:

PM = 13 and

PF = 5.6

To find the Price charged for Males or Females, you would take the equation for QM and solve for PM.

QM = 500 - 25PM (Add 25PM to both sides of the equation) (for Males)

QF = 900 - 30PF (Add 30PF to both sides of the equation) (for Females)

QM + 25PM = 500 - 25PM + 25PM

QF + 30PF = 900 - 30PF + 30PF

QM + 25PM = 500 (Now subtract QM from both sides)

QF + 30PF = 900 (Now subtract QF from both sides)

QM - QM + 25PM = 500 - QM

QF - QF + 30PF = 900 -QF

25PM = 500 - QM (Next divide both sides by 25)

30PF = 900 - QF (Next divide both sides by 30)

25PM/25 = (500 - QM)/25

30PF/30 = (900 - QF)/30

PM = (500 - QM)/25

PF = (900 - QF)/30

Now solve the equation for Marginal Cost for Q.

MC = 5 + 0.002Q (Subtract 5 from both sides)

MC - 5 = 5 - 5 + 0.002Q

MC - 5 = 0.002Q (Next divide both sides by 0.002)

(MC - 5)/0.002 = 0.002Q/0.002

(MC - 5)/0.002 = Q (When you divide by 0.002, you are mulitplying by 500)

500(MC - 5) = Q

500MC - 2500 = Q

Finally substitute this equation for QM and QF and you have the equations for PM and PF

PM = [500 - (500MC - 2500)]/25

PF = [900 - (500MC - 2500)]/30

PM = [3000 - 500MC]/25

PF = [3400 - 500MC]/30

PM = 120 - 20MC

PF = 113.33 - 16.67MC