American Tire and Rubber Company sells identical radial tires under the firm's own brand name and private
label tires to discount stores. The radial tires sold in both sub-markets are identical, and the marginal cost is
constant at $10 per tire for both types. The firm has estimated the following demand curves for each of the
markets.
PB = 70 - 0.0005QB (brand name)
PP = 20 - 0.0002QP (private label).
Quantities are measured in thousands per month and price refers to the wholesale price. American currently
sells brand name tires at a wholesale price of $28.50 and private label tires for a price of $17. Are these prices
optimal for the firm?
Your school subject is Economics.
To determine if the prices set by American Tire and Rubber Company (ATRC) are optimal, we need to compare them to the optimal prices that maximize the firm's profits.
To find the optimal prices, we need to calculate the marginal revenue (MR) for each market. MR is the additional revenue generated from selling one additional unit.
In this case, the demand curves are given as:
PB = 70 - 0.0005QB (brand name)
PP = 20 - 0.0002QP (private label)
To find MR, we need to take the derivative of the respective demand curves with respect to quantity. The derivative of a linear quantity-demand function is the coefficient in front of the quantity term. Thus:
MRB = -0.0005 (for brand name tires)
MRP = -0.0002 (for private label tires)
Next, we calculate the optimal quantity by setting MR equal to the marginal cost (MC) of $10:
MRB = MC
-0.0005 QB = 10
QB = 10 / (-0.0005) = 20,000 (brand name tires)
MRP = MC
-0.0002 QP = 10
QP = 10 / (-0.0002) = 50,000 (private label tires)
Now that we know the optimal quantities, we can substitute them back into the demand equations to find the corresponding optimal prices.
For brand name tires:
PB = 70 - 0.0005QB
PB = 70 - 0.0005(20,000)
PB = 70 - 10
PB = 60
For private label tires:
PP = 20 - 0.0002QP
PP = 20 - 0.0002(50,000)
PP = 20 - 10
PP = 10
The optimal prices for brand name tires and private label tires are $60 and $10, respectively.
Comparing these optimal prices to the prices currently set by ATRC ($28.50 for brand name tires and $17 for private label tires), we can conclude that these prices are not optimal. The firm should adjust its prices to $60 and $10 for maximum profitability.