h) A monopolist faces the demand function given by x=10-0.5p where x is the number of units produced and sold at price p dollars per unit. The total cos incurred consists of a fixed overhead of 28 dollars plus production cost of 2 dollars per unit of x. i Write down the monopolist's profit as a function of x (2 Marks) ii Find the maximum profit realized by the monopolist in producing x units of the commodity (2 Marks)

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h) A monopolist faces the demand function given by x=10-0.5p where x is the number of units produced and sold at price p dollars per unit. The total cos incurred consists of a fixed overhead of 28 dollars plus production cost of 2 dollars per unit of x. i Write down the monopolist's profit as a function of x (2 Marks) ii Find the maximum profit realized by the monopolist in producing x units of the commodity (2 Marks)

Answer 1

i) To write down the monopolist's profit as a function of x, we need to consider the revenue and cost.Revenue is calculated by multiplying the price per unit (p) by the number of units sold (x). From the given demand function, we can express p as a function of x: p = 10 - 0.5x.Revenue = p * x = (10 - 0.5x) * x = 10x - 0.5x^2The total cost incurred by the monopolist consists of a fixed overhead of 28 dollars plus a production cost of 2 dollars per unit of x.Total cost = Fixed overhead + Production cost = 28 + 2xNow, we can write the profit function as the difference between revenue and total cost:Profit = Revenue - Total cost = (10x - 0.5x^2) - (28 + 2x) = 10x - 0.5x^2 - 28 - 2x = 8x - 0.5x^2 - 28So, the monopolist's profit as a function of x is: Profit = 8x - 0.5x^2 - 28ii) To find the maximum profit realized by the monopolist, we need to find the value of x that maximizes the profit function.To do this, we can take the derivative of the profit function with respect to x and set it equal to zero:d(P)/dx = 8 - x = 0Solving for x, we get x = 8.Now, we can substitute x = 8 into the profit function to find the maximum profit:Profit = 8(8) - 0.5(8)^2 - 28 = 64 - 32 - 28 = 4Therefore, the maximum profit realized by the monopolist in producing 8 units of the commodity is 4 dollars.

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