Question

2. A single firm produces widgets, with a cost function and inverse demand function as follows, C(q) = 150+2q P(Qd) = 10−0.08Qd

   1. (a) Calculate the monopolist’s profit-maximizing price, quantity, and profit if he can charge a single price in the market (single price monopolist).

   2. (b) Suppose the firm wants sell units after your answer to (a) at a lower price in a later time period (2nd-degree price discrimination). What quantity will be sold for what price in this second-tier market? Calculate the monopolist’s profit.

   3. (c) Suppose the firm can perfectly price discriminate; calculate the monopolist’s profit.

Answer 1

a.

Single price:

In this case the stage of profit-maximizing is (MR = MC).

P = 10 – 0.08q ……. (price function)

TR = P × q = 10q – 0.08q^2

MR = Derivatives of TR with respect to q.

       = 10q^(1 – 1) – (0.08 × 2)q^(2 – 1)

       = 10q^0 – 0.16q

       = 10 – 0.16q

Now given, TC = 150 + 2q

MC = Derivatives of TC with respect to q

       = 0 + 2q^(1 – 1)

       = 2q^0

       = 2

For quantity:

MR = MC

10 – 0.16q = 2

10 – 2 = 0.16q

8 = 0.16q

8/0.16 = q

q = 50

For price:

By putting the value of q in the price function.

P = 10 – 0.08q ……. (price function)

   = 10 – 0.08 × 50

   = 10 – 4

   = 6

For profit:

Profit = TR – TC

            = [10q – 0.08q^2] – [150 + 2q]

            = [10 × 50 – 0.08 × 50^2] – [150 + 2 × 50]

            = [500 – 200] – [150 + 100]

            = 300 – 250

            = 50

Answers:

Price = 6

Quantity = 50

Profit = 50

b.

2^nd degree price discrimination:

Given, lower price is in the later time. This could only be avail if there is (P = MC).

For price:

P = MC = 2

For quantity:

By putting the value of P in the price function.

P = 10 – 0.08q ……. (price function)

2 = 10 – 0.08q

2 – 10 = - 0.08q

-8 = - 0.08q

8/0.08 = q

q = 100

For profit:

Profit = TR – TC

            = [10q – 0.08q^2] – [150 + 2q]

            = [10 × 100 – 0.08 × 100^2] – [150 + 2 × 100]

            = [1,000 – 800] – [150 + 200]

            = 200 – 350

            = -150

Answers:

Price = 2

Quantity = 100

Profit = -150

c.

Perfectly price discrimination:

Consumer surplus (CS) in case of “single price” method should be calculated first, where (P = 6; q = 50)

P = 10 – 0.08q ……. (price function)

If (q = 0), P = 10 – 0.08 × 0 = 10.

Therefore, CS = 0.5 × Difference in price × Difference in quantity

                        = 0.5 × (10 – 6) × (50 – 0)

                        = 0.5 × 4 × 50

                        = 100

Note: 0.5 is the part of formula.

Profit = Profit at “single price” system + CS

            = 50 + 100

            = 150 (Answer)

This is so because a price discriminating firm will capture all the CS.