Your Initials mpuny produces and sells a consumer product an

Your Initials: mpuny produces and sells a consumer product and is able to control the demand for the product by varying the selling price. The approximate relationship between price and demand is: 1. A co p = 38 + (27001) for D2I Where p is the price per unit in dollars and D is the demand per month. The company is seeking to maximize its profit. The total cost equation is $1000+D (20 points) a) Determine the value of D per month that maximizes profir? (5 pts) b) What is the associated maximum profit per month for D found in (a)? (5 pts) c) What is the unit price for the value found in (a)? (5 pts) d) Find the break even points? (5 pts)

Solution

The given inverse demand function is p=38+(2700/D)

The total cost equation is C=1000+D^2

(a) Total revenue (TR) =p*D=(38+(2700/D))*D=38D+2700

Marginal revenue (MR)= d(TR)/dD=38

Marginal cost (MC) = dC/dD=2D

For profit maximization MR=MC

38=2D. Thus the profit maximizing output is D=19

(b) Total revenue when D=19 is found by substituting D=19 in TR=38D+2700

So total revenue=3422

Total cost when D=19 is found by substituting D=19 in  C=1000+D^2

so taotal cost= 1361

Thus profit= Total revenue-total cost=3422-1361=2061

(c) The unit price is found by substituting D=19 in the inverse demand function  p=38+(2700/D)

Thus unit price p= 38+(2700/19)=180.1

(d) The break even point is the point at which total revenue=total cost

38D+2700=1000+D^2

Solving the above quadratic equation we get the value D=64