Answer:
To maximize profit, a firm can either increase revenue or decrease expenses, or both. Revenue can be increased by adjusting the price, volume, or value proposition of the product or service. Expenses can be decreased by reducing the cost of goods sold, overhead, or other variable costs.
In your scenario, the factory inputs its goods from two different plants with different costs. The price function in the market is decided as p(x,y)=100-x-y where x and y are the demand functions and 0 < x,y.
To maximize profit in this scenario, we need to find the optimal values of x and y that will maximize the profit function. The profit function is given by:
Profit = Revenue - Cost
Revenue = Price * Quantity
Cost = Total Cost of Production
Total Cost of Production = Cost of Plant A * Quantity from Plant A + Cost of Plant B * Quantity from Plant B
Substituting the given price function into the revenue equation gives:
Revenue = (100 - x - y) * (x + y)
Substituting the given costs into the cost equation gives:
Total Cost of Production = 4x + 8y
Therefore, the profit function is:
Profit = (100 - x - y) * (x + y) - (4x + 8y)
Expanding this equation gives:
Profit = 100x + 100y - x^2 - xy - y^2 - 4x - 8y
To find the optimal values of x and y that will maximize profit, we need to take partial derivatives of the profit function with respect to x and y and set them equal to zero:
dProfit/dx = 100 - 2x - y - 4 = 0
dProfit/dy = 100 - x - 2y - 8 = 0
Solving these equations simultaneously gives:
x = 38
y = 31
Therefore, when x=38 and y=31, the factory can attain maximum profit.