P 120Q where p is the highprie chared on the first units Q f

P 120-Q where p is the highprie chared on the first units Q, (fest block) and Pa is a lower price charged on the next (o2-Q1) units and Py is the lowest price charged on the (Q-Q2) remaining units. Q3 is the total umber af units actually purchased, and m- $40 is the firm\'s constant marginal and average cost. Using calculus, determine the profit-maximizing values for Py.P2. and p, and the firm\'s profits The value for (round your answers to the nearest penny

Solution

Consider the given problem here the demand curve of the monopolist is given by, “P = 120 – Q”.

=> P1 = 120 – Q1, P2 = 120 – Q2 and P3 = 120 – Q3”.

So, now the profit function of the monopolist is given below.

=> = P1*Q1 + P2*(Q2-Q1) + P3*(Q3-Q2) – 40*Q3, as we have given the “AC=MC=40”.

Now, substituting the values of “P1”, “P2” and “P3” into the profit function, the profit function will look like.

=> = P1*Q1 + P2*(Q2-Q1) + P3*(Q3-Q2) – 40*Q3.

=> = (120 – Q1)*Q1 + (120 – Q2)*(Q2-Q1) + (120 – Q3)*(Q3-Q2) – 40*Q3.

=> = (120*Q1 – Q1^2) + (120*Q2 – Q2^2 – 120*Q1 + Q2*Q1) + (120*Q3 – Q3^2 – 120*Q2 + Q2*Q3) – 40*Q3.

So, now the FOC is given by.

=> /Q1 = /Q2 = /Q3 = 0.

=> /Q1 = 0, => (120 – 2*Q1) + ( – 120 + Q2) = 0,

=> Q2 = 2*Q1……………….(1)

=> /Q2 = 0, => (120 – 2*Q2 + Q1) + ( – 120 + Q3) = 0.

=> Q1 + Q3 = 2*Q2…………….(2)

Now, finally.

=> /Q3 = 0, => (120 – 2*Q3 + Q2) – 40 = 0, => 120 – 40 + Q2 = 2*Q3.

=> 80 + Q2 = 2*Q3 ……………..(3).

Now by solving these 3 equations simultaneously we can find the optimum “Q1”, “Q2” and “Q3”.

So, by putting “1” into “2”, we will get the relationship between “Q1” and “Q3”, which is given by, “Q3 = 3*Q1”………(4).

So, by substituting “1” and “4” into “3” we will get the optimum value of “Q”, which is “Q1=20”.

=> Q2 = 2*20 = 40 and Q3 = 3*Q1 = 3*20 = 60.

=> So, “P1=120-20 = 100”, “P2=120-40=80” and “P3=120-60=60”.

So, the profit of the monopolist is given below.

=> = P1*Q1 + P2*(Q2-Q1) + P3*(Q3-Q2) – 40*Q3 = 100*20 + 80*20 + 60*20 – 40*60 = 4800 - 2400.

=> = 2400.

So, here the optimum price for “Q1” is “P1=100”, for “Q1 < Q < Q2, P2=80” and for “Q2 < Q, “P3=60” and corresponding profit is “2400”.