demand function of a monopolist is given as Q=50 - 0.5p while the cost function is given as C= 50 + 40q. calculate equilibrium quantity and profit maximizing output.
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To calculate the equilibrium quantity and profit-maximizing output, we need to consider the monopolist's demand function and cost function.
The demand function given is Q = 50 - 0.5p, where Q represents the quantity demanded and p represents the price set by the monopolist.
The cost function is given as C = 50 + 40q, where C represents the total cost and q represents the quantity produced by the monopolist.
To find the equilibrium quantity, we need to equate the quantity demanded and the quantity produced, so we set Q = q.
50 - 0.5p = q
To calculate the profit-maximizing output, we need to find the quantity that maximizes the monopolist's profit. This can be done by maximizing the monopolist's total revenue and subtracting the total cost.
The total revenue (TR) is calculated by multiplying the quantity (q) by the price (p), so TR = p * q.
The total cost (TC) is given by C = 50 + 40q.
The monopolist's profit (π) is then calculated by subtracting the total cost from the total revenue, so π = TR - TC.
To maximize profit, we take the derivative of the profit function with respect to q and set it equal to zero.
dπ/dq = d(TR - TC)/dq = 0
Solving this equation will give us the profit-maximizing quantity.