Question

Suppose that you can sell as much of a product (in integer units) as you like at $60 per unit. Your marginal cost (MC) for producing the qth unit is given by: MC=7q This means that each unit costs more to produce than the previous one (e.g., the first unit costs 7*1, the second unit (by itself) costs 7*2, etc.). If fixed costs are $80, what is the profit at the optimal integer output level?

Answer 1

Answer:

Profit at the optimal integer output level is $176.50.

Explanation:

This can be determined as follows:

Step 1: Calculation of optimal integer output level

At the optimal integer output level, profit is maximized where marginal revenue (MR) is equal to the marginal cost (MC), i.e. where;

MR = MC ................................ (1)

For any product, the MR is equal to the price per unit of the product. Therefore, we have:

MR = Price per unit = $60

Also given,

MC = 7q

Substituting for MR and MC in equation (1) and solve for q, we have:

$60 = 7q

q = $60 / 7

q = 9 units

Therefore, the optimal integer output level is 9 units.

Step 2: Calculation of total revenue at optimal integer output level

Total revenue = Price per unit * q = $60 * 9 = $540

Step 3: Calculation of total cost at optimal integer output level

Since MC = 7q, the total cost (C) can be obtained by taking the integral of the MC as follows:

C = ∫(MC)dq = ∫[7q]dq = (7 / 2)q^2 + F = 3.5q^2 + F ........... (2)

Where F is Fixed cost which is given as $80.

We then substitute F = $80 and q = 9 into equation (2) to have:

C = 3.5(9^2) + 80

C = (3.5 * 81) + 80

C = 283.50 + 80

C = $363.50

Therefore, total cost at the optimal integer output level is $363.50.

Step 4: Calculation of profit at optimal integer output level

Profit = Total revenue - Total cost ...................... (3)

Where;

Total revenue = $540; from step 2 above.

Total cost = $363.50; form step 3 above.

Substituting the values into equation (3), we have:

Profit = $540 - $363.50 = $176.50

Therefore, profit at the optimal integer output level is $176.50.