Answer

Double Marginalization

Question 3 (Double Marginalization)

Suppose that Michelin is the only producer of tires and Toyota the only producer of cars. The demand function for cars is given by Q = 40 − 4P. Michelin’s (constant) cost of production for a set of five tires is $3. The production of one car requires a set of five tires plus a bundle of inputs. Toyota can obtain this bundle of inputs at a (constant) cost of $6.

Suppose first that Michelin and Toyota are just two departments within the same vertically integrated monopoly (Firm V).

1. Given that the two companies, Since Michelin and Toyota are parts of one organization, their reduced function would be represented with one equation;

Toyota and Michelin:

Total cost of production of a car = Michelin’s Tire Production cost + Toyota’s production cost

= $6 + $3

=$9

Now, we than the demand function is given by the formula;

Q = 40 – 4p

Q = 40 – 4 (9)

Q = 40 – 36

Q = 4

The company would charge $9 for the cars and it would produce 4 cars.

1. In case Michelin sets a price w for the tires, then the prices for the Toyota cars would be 3w, therefore;

Q = 40 – (4w)

Q = 40 -12w

12w = 40

W = 40/12

W = 3 cars

1. In case the two companies two separate units, then they generate 3 cars and whenever they work as a single unit, they produce 4 cars.

Therefore, is a better option for Toyota and Michelin to become an integrated company.

Suppose now that Michelin and Toyota are separate firms, each a monopoly in its own industry. Michelin quotes a price $w for a set of five tires and Toyota decides how many sets to buy at that price. Now, Toyota’s costs per car includes buying a set of tires from Michelin at $w plus the bundle of other inputs at $6.

We will view this as a two-stage game, where Michelin decides on $w in Stage 1 and in Stage 2 Toyota decides how many cars to sell (and, hence, how many sets of tires to buy from Michelin) as a function of $w. Remember that this type of an extensive form game is solved using backwards induction. However, since the strategies are not a discrete set of choices, we will not use a game tree. Instead let us solve for the equilibrium using the steps listed in the sub-parts below. Broadly, Toyota chooses profit-maximizing quantity of cars to sell as a function of w and Michelin sets w to maximize its own profits, given Toyota’s demand for tires.

Given that Q = a + b

Where a is the identical companies

The cost function for every company is given by the function: C = F + cv

If two firms merge, ten we have:

C = F¹ + C¹V, (F, < F¹< 2F)

1. Every firm’s market shares before and after the merger

Q = a – p

Therefore, p = a – Q

And as such we have; Q = q₁ + q₂ + q₃

Before the merger, the cost function is given by C = F + Cq

Therefore, the company profit function, π_i= P(Q)q_i – c_i

Δπi ÷ Δq₁ = 0

Therefore, a – 2q₁ – q₂ –q₃ – c = 0

We have, q1 = (a – q₂ – q₃ – c) ÷ s…………… equation 1

Similarly, we obtain the following; q₂ = (a – c – q₁ – q₃) ÷ 2…………… equation 2

Again, q₂ = (a – c – q₁ – q₂) ÷ 2…………… equation 3

Finding solutions to equations 1, 2 and 3, we get

q1 = q2 = q3 = (a-c) ÷ 4

Q = 3 (a – c)/ 4

Therefore, the market share for every company is given by q_i / a

Therefore, the market share for the three companies are 1/3, 1/3, 1/3

(As can be observed, they assume symmetric form)

After the cost function is merged c = F¹ – c¹q

The profit function is π1 = (a – p) q¹ – F cq¹

Firm 4 = merger of firms 2 and 3 = π4 = (a – p) _q4 – F¹ –C¹q₄)

Δπ₁ / Δq₁ = 0 = q₁ = (a – c –q₄) /2 …………… equation 4

Δπ₄ / Δq₄ = 0 = q₄ = (a – c¹ –q₁) /2 …………… equation 5

Solving equations 4 and 5 we get;

q1 = (a – c – q₄) / 2

q1 = {a – c – (a –c¹ –q₁) / 2} / 2

q1 = (2a – 2c – a + c¹ +q₁) / 4

therefore;

4q₁ = a – 2c + c¹ +q₁

3q₁ = a – 2c +c¹

Given that, q1 = (a-2c +c¹) / 3

And q4 = {(a – c1 – (a – 2c +c1)/3)} / 2

q4 = (3a – 3c¹ – a + 2c – c¹)/ 6)

therefore,

6q4 = (2a – 4c¹ +2c)

Note that q₄ = (a – 2c¹ + c)/3

Therefore,

Q = q1 +q4 = (a – 2c¹ + c)/3 + (a – 2c¹ + c)/3

Therefore,

Q = (2a – c –c¹) / 3

S1 Firm 1 market share = q1/ Q = (a + c¹ – 2c) / (2a – c – c¹)

S2 Firm 1 and 2 Merger market share = q4 / Q = (a + c – 2c¹) / (2a – c –c¹)

1. If a = 10, c = 3, then Herfindahl Hirschman Index (HHI) after the merger when
2. i) c1 = 2;

S₁ = (a + c1 -2c) / (2a – c – c1)

= (10 + 2 – 6) / (20 – 3 – 2)

= (12 – 6) / (20 – 5)

= 6/15

= 0.5

S₁ = 40%