Answer to Question #257280 in Economics of Enterprise for mony

Question

A company estimated that the relationship between the unit price and demand per month for a potential new product is approximated by P = $ 100 – $ 0.1D. The company can produce the product by increasing fixed costs $ 17,500 per month, and the estimated variable costs is $ 40 per unit. What is the optimal demand, D*, and based on this demand, should the company produce new product? Why?

a) Work out the complete solution by differential calculus, starting with formula for profit or loss per month

b) Solve graphically for an approximate answer

Answer 1

a) The demand function is given as,

$P=100−0.1D$

where P is the price and D is the quantity demanded. The fixed cost is $17500 and the variable cost is $40 per unit. Therefore the total cost function can be written as,

$Total \space cost=Fixed\space cost+Variable\space cost\\TC=17500+40D$

Profit is the difference between total revenue and total cost. Total revenue is the product of price and quantity. Therefore the profit function can be written as,

$Profit=Total \space revenue−Total \space cost\\=(100−0.1D)D−17500−40D\\ π(D)=100D−0.1D^2−17500−40D$

The profit is maximized when the first derivative of the profit function is equal to zero.

$\frac{∂π}{∂D}=0\\100−0.2D−40=0\\60=0.2D\\D=\frac{60}{0.2}\\=300\\\frac{∂^2π}{∂D^2}=−0.2<0$

Therefore the profit is maximized when the quantity of output is 300 units. The maximum profit is calculated as,

$Profit=100D−0.1D^2−17500−40D\\=30000−9000−17500−12000\\=−8500$

The maximum profit that can be earned is negative. Therefore the firms should not be producing the new product.

b)The graph of the total revenue function, total cost function, profit function, marginal cost function, and marginal revenue function is given below.

The figure shows that the profit is maximum at 300 units of output. At this level of output, the difference between total cost and total revenue is minimum and the marginal cost curve intersects the marginal revenue curve.