The demand function for a monopolist’s product is p=800-4q and the average cost per unit for producing q units is c=q+50+ 500q, where p = price, and q = quantity demand.
Description of the Project
Monopoly means that firm plays a role of the only seller on the market. In practice, a monopolist firm may have the highest market share significantly exceeding those of its competitors. It, in turn, constitutes that such a company has the power to independently influence prices on the market without regard to actions of its competitors. It’s an ideal situation that any company strives for.
The main goal of a monopolist is to maximize its profits. To achieve this, a company should follow basic rules. First of all, it should compare its marginal benefits and marginal costs. If the former exceed the latter, the company is to increase its output. Otherwise, the company must think of cutting down its production. If, however, the marginal benefits equate the marginal costs, a monopolist firm is to sustain its current level of output.
Thus, the main purpose of this project is to find and calculate the profit-maximizing prices and quantity of output as well as determine the demand elasticity so as to optimize the performance of the firm on the market and to ensure that there is a steady demand for its products that is not affected by incorrect managerial decisions.
Problem Statement
Taking into account everything mentioned above, it is crucial for a firm to find equilibrium of price and quantity that would maximize its profits and create a constant demand for its products. Hence, for the company to maximize its profits and optimize its output, several issues have to be solved:
Find the demand function.
Determine marginal revenue and marginal costs.
Determine maximum revenue.
Determine the price and quantity that would maximize profits.
Calculate maximum profits.
Determine the demand elasticity.
Proposed Solution Methodology
The demand function will look q=800-p4=200-14p.
The marginal cost can be calculated by taking a derivative from the total cost function.
The total cost equals: C=C×q;
C=q+50+ 500q×q=q2+50q+500;
R=q×p;
R=q800-4q=800q-4q2;
q=100.
R''=-8;
R''100=-8<0, thus, revenues will be maximized at q=100.
p=800-4q=800-400=400;
p=400;
R=400×100=40000;
Therefore, the maximum revenue will be 40000.
π=R-C, where R = revenues, C = costs.
π =q×800-4q-q2+50q+500=800q-4q2-q2-50q-500;
π=-5q2+750q-500;
π'=-10q+750=0;
q=75;
π''=-10;
π''75=-10. Thus, profit will be maximized at q=75.
p=800-300=500.
π=37500-752+50×75+500=37500-5625+3750+500=37500-9875=27625;
Thus, the maximum profit gained with profit-maximizing price and quantity will stand at 27625.
The demand elasticity should be determined from the following formula:
Ep=pq×dqdp
q=200-14p
Ep=4p800-p×-14=-p800-p
p=500
E(p)=-500300=123
E(p)>1. Hence, the demand is elastic, i.e. a small increase in prices will lead to a decrease in revenues.
