Question

Computer Company assembles personal computers and sells them in the retail marketplace. The company is organized into two profit centers: the assembly division and the distribution division. The demand curve facing the company (and the distribution division) is P=3,500 – 10Q. The marginal cost for assembly (which includes purchasing the parts) is constant at $450. The distribution division faces constant marginal distribution costs of $50 per unit.

A. What is the profit-maximizing retail price and output for the firm as a whole?
B. If the assembly division has monopoly power to set the transfer price, what transfer price will it select (assuming it knows all the information above)? Calculate the profits for the two divisions in this case.

Answer 1

The profit-maximizing price and output for the firm as a whole is $3,050 and 45 units, respectively. If the assembly division has monopoly power to set the transfer price, it will choose a transfer price of $500. The profit for the assembly division is $2,250, while the profit for the distribution division is $135,000.

Step-by-step explanation:

To find the profit-maximizing price and output for the firm as a whole, we need to determine the quantity at which marginal cost equals marginal revenue. Given the demand curve P=3,500 – 10Q, we can calculate marginal revenue as the derivative of the demand curve, which is -10. Setting marginal cost equal to marginal revenue, we have $450 = -10. Solving for Q, we get Q = 45. Substituting this value into the demand curve, we find P = 3,500 – 10(45) = 3,050.

If the assembly division has monopoly power to set the transfer price, it will choose a price that maximizes its own profit. Since the distribution division faces constant marginal distribution costs of $50 per unit, the profit-maximizing transfer price will be the marginal cost of assembly plus the marginal distribution cost. Therefore, the transfer price will be $450 + $50 = $500.

To calculate the profits for the two divisions, we can subtract the cost from the revenue for each division. For the assembly division, the revenue is the transfer price multiplied by the quantity, which is $500 x 45 = $22,500. The cost is the marginal cost of assembly multiplied by the quantity, which is $450 x 45 = $20,250. Therefore, the profit for the assembly division is $22,500 - $20,250 = $2,250. For the distribution division, the revenue is the retail price multiplied by the quantity, which is $3,050 x 45 = $137,250. The cost is the marginal distribution cost multiplied by the quantity, which is $50 x 45 = $2,250. Therefore, the profit for the distribution division is $137,250 - $2,250 = $135,000.

Answer 2

To find the profit-maximizing price and output, we equate Marginal Revenue (MR) to the marginal cost (MC). The profit-maximizing quantity (Q) is 153 units, leading to a retail price of $1,980. If the assembly division has monopoly power, it will choose a transfer price of $450, resulting in zero profit for the assembly and all profit for the distribution division.

Step-by-step explanation:

To determine the profit-maximizing retail price and output for the firm as a whole, we need to use the marginal cost (MC) and the demand curve to find the point where Marginal Revenue (MR) equals MC. The firm's demand curve is P = 3,500 - 10Q. We know that in a perfectly competitive market, MR equals the price, but for a profit center like Computer Company, we need to calculate MR by differentiating the total revenue (TR = P*Q) with respect to quantity Q.

First, find the TR function:
TR = (3,500 - 10Q) * Q = 3,500Q - 10Q^2.
Then calculate MR by differentiating TR:
MR = d(TR)/dQ = 3,500 - 20Q.
Now, set MR equal to marginal cost for assembly and solve for Q:
3,500 - 20Q = 450.
Q = 153.

Next, plug Q back into the demand equation to find the price:
P = 3,500 - 10*153 = $1,980.

For part B, if the assembly division sets the transfer price, it will do so where its marginal cost equals the marginal revenue of the distribution division. Since the assembly marginal cost is $450, it will set the transfer price at $450 because the distribution marginal cost is constant at $50, which would not affect the assembly division's profit-maximizing decision.

The profits for each division would then be:
Assembly profit = (Transfer Price - Assembly MC) * Quantity
Assembly profit = ($450 - $450) * 153 = $0.
Distribution profit = (Retail Price - Transfer Price - Distribution MC) * Quantity
Distribution profit = ($1,980 - $450 - $50) * 153