Suppose you are the manager of a watch-making firm operating in a comptetive market.You cost of production is given by C=100+Q^2,Where Q is the level of output and C is total cost.
a)If the price of watches is birr 60,how many watches should you produce to maximize profit?
b)What will your profit level be?
c)At what minimum price will you produce a posetive output?
revenue R = 60q
profit P = R-C = 60q-(100+q^2)
so find q such that dP/dq = 0
full answer
To answer these questions, we need to understand the concepts of profit, cost, and output maximization in a competitive market.
a) To maximize profit, we need to find the level of output at which the difference between total revenue and total cost is the highest. In a competitive market, the total revenue (TR) is calculated as the price of the product multiplied by the quantity sold, while the total cost (TC) is given by the cost function C.
Total Revenue (TR) = Price (P) x Quantity (Q)
Total Cost (TC) = C = 100 + Q^2
Profit (π) is calculated as the difference between TR and TC:
Profit (π) = TR - TC
In this case, the price of watches is given as birr 60. We can substitute this value in the profit equation:
Profit (π) = (P x Q) - (100 + Q^2)
Profit (π) = (60 x Q) - (100 + Q^2)
To maximize profit, we take the derivative of the profit equation with respect to Q, set it equal to zero, and solve for Q:
d(π)/dQ = 60 - 2Q = 0
60 = 2Q
Q = 30
Therefore, you should produce 30 watches to maximize profit.
b) To determine the profit level, we substitute the value of Q into the profit equation:
Profit (π) = (60 x 30) - (100 + 30^2)
Profit (π) = (1800) - (100 + 900)
Profit (π) = 800 birr
Your profit level will be 800 birr.
c) To find the minimum price at which you produce a positive output, we need to determine the level of output (Q) for which the total cost is greater than zero.
Total Cost (TC) = C = 100 + Q^2
Setting TC greater than zero:
100 + Q^2 > 0
Q^2 > -100
Since Q^2 is always non-negative, any value of Q will result in a positive Q^2.
Hence, no minimum price is necessary to produce a positive output.