Good year tire company sells identical radical tires under the firms own brand n

Question

Good year tire company sells identical radical tires under the firms own brand name and to discount stores for private labeling. Marginal cost is constant at $10 per tire, regardless of the market its sold in. The firm has estimated the demand functions for each of it market to be:

Pb= 70- 0.0005 Qb (b=brand name)

Pg= 20- 0.0002 Qg (g=generic)

1) Write an equation expressing this firms total profits... Please write out the steps...

2) Determine the profit maximizing price and quantity in brand name market.

I got ...70-0.001 Qb=10 (why is it 0.0001) for the quantity and 70-0.0005 (60,000) for the price equation?

3) Determine the price elasticity of demand in each market

I got ...Eb= -2000 (40/60,000)=-80/60=-1/333 (where does the -2000 come from?)

...Eg= -5000 (15/25000)=-3 (where does the -5000 come from)

Explanation / Answer

Brand name revenue(BR) = Pb*Qb= 70Qb- 0.0005 Qb^2

Generic revenue(GR) = Pg*Qg= 20Qg- 0.0002 Qg^2

Total Revenue = BR+GR

Total Cost = 10(Qb+Qg)

Total Profits = TR-TC = (70Qb+20Qg-0.0005 Qb^2-0.0002 Qg^2) - 10(Qb+Qg)

In brand name market, MC=MR

MR = dBR/dQ = 70 - 0.001 Qb

70 - .001Qb = 10 (by differentiating, you get 0.001)

Qb = 60/.001 = 60000

Pb = 70 - 0.0005*60000 = 40

Generic

20 - 0.0004Qg = 10

Qg = 10/0.0004 = 25000

Pg = 20- 0.0002*25000 = 15

Brand name

Qb = (70-Pb)/.0005

dQb/dPb = -1/.0005 = -2000

Price elasticity of demand = (dQ/Q)/(dp/P) = -dQb/dPb*Pb/Qb

= -2000*(40/60000) = -80/60=-1/333

Generic

Qg = (20-Pg)/0.0002

dQg/dPg = -1/0.0002 = -5000

Price elasticity of demand = (dQg/Qg)/(dpg/Pg) = -dQg/dPg*Pg/Qg

=-5000 (15/25000)=-3

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