Question

Professor Bong has just written the first textbook in Punk Economics. It is called Up Your Isoquant. 

Market research suggests that the demand curve for this book will be 

D(p) = 2,000 - 100p, 

where p is its price. 

(a) Notice that the demand curve given above has quantity as a function of price. So, begin by rearranging that equation to get price as a function of quantity. (This is known as the inverse demand function.) 

(b) The total revenue function for Professor Bong's book is _______  and the marginal revenue function is MR(y)= _______ - y/50 

Now, some information on the costs of production. It will cost $1,000 to set the book in type. This setup cost is necessary before any copies can be printed. In addition to the setup cost, there is a marginal cost of $4 per book for every book printed. 

(c) The total cost function for producing Professor Bong's book is C(y) = 4y +_______  and the marginal cost function is MC = _______ 

(d) Look at the marginal revenue function you found it part (b), and the marginal cost function you found in part (c). Set marginal revenue equal to marginal cost to find the profit-maximizing quantity of books for professor Bong to sell:

Answer 1

D = 2000 - 100p (Here D is a function of p, that is demand is a function of price, we take quantity demanded denoted by y)

y = 2000 - 100p

100p = 2000-y

p = 20-y/100

Total revenue is the product of price and quantity demanded so TR = p*y

TR = 20y - y^2/100

Marginal Revenue function is the first order derivative of the total revenue function

MR=dTR/dy = 20 - 2y/100 or 20 - y/50

MR = 20 - y/50

Cost is the addition of variable and fixed costs so

C(y) = 4y + 1000

MC or marginal cost is the first order derivative of cost or total cost function

MC = dC/dy = 4

Equating MR and MC

MR=MC

20 - y/50 = 4

y/50 = 20-4 = 16

y = 16*50

y* = 800