Find the marginal demand to estimate the change in demand when price increases by one dollar

Question

Suppose the demand for a certain item is given by D(p)= -4p²+3p+8, where p = price of the item in dollars. Currently p=20 dollars. Use the marginal demand to estimate the change in demand when the price is increased by one dollar.

Answer 1

"best" asnwer: 161 "purchases" lost for a dollar increase in price when price =$20

marginal change in quantity demanded = -161

D(p) = -4p^2 + 3p + 8

D(20) = -4(20)^2 + 3(20) + 8 = -4(400) + 60 + 8 = -1532 = no one is buying, it's negative quantity demanded, as if consumers are returning 1,532 products for a refund

D(21) = -4(21)^2 +3(21) + 8 = -4(441) +63 + 8 =-1764+71 = -1693, people bought 161 less as price rose $1, if this problem makes any sense as they aren't buying anything to begin with, just bringing back more products for more refunds, 161 more

D'(p) = -8p +3 = 0

p= 3/8 = $.375 = the profit or revenue maximizing price= a price of about 38 cents

any price over $1.84 and consumers stop buying, $20 price is over the top, not a possible price, $21 even more impossible

D(.38) = -4(.38)^2 +3(.38) + 8 = -.578 + 1.14 + 8 = 3.36

Profit or Revenue =xp = .38(3.36) = $1.28 = maximum revenue

If there are costs involved, even that revenue would become negative at P=$20 or at any price. It's a losing business about to go into bankruptcy

D'(20)= -8(20) + 3 = -157 would probably be the calculus solution, if it made any economic sense

but that is an instantaneous rate of change, not the per unit change from P=20 to P=21

D'(21) =-8(21) + 3 = -165

an average of D'(20) and D'(21) = (-157-161)/2 = -161, exactly the same answer in this first part of this post, without using calculus, if the problem as written made any economic sense

-161 is the "marginal quantity demanded" over the price increase from $20 to $21, not -157

also "marginal demand" is not a standard economics term.

"marginal revenue" would be what you seem to mean, but that refers to change in revenue, not change in quantity demanded.

"marginal demand" also confuses demand with quantity demanded, worth points off in an economics course.

but this is likely a math course trying to use a "realistic" business problem, oblivious to economic theory

the demand is the demand curve which never changes when price along changes. price chanes are a movement along the demand curve, not a change in the demand curve

so "technically" the "marginal demand" =0 as demand does not change.

it may sound techncial, but it's a major point in any economics course dealing with the most basic law of supply and demand

possibly the problem was mis-copied or

the original problem had a typo in it

change the coefficients in the origianl D(P) function and it could be very realistic.

question is does the textbook answer have the "real" answer to the original realistic problem or not?

Answer 2

Take the derivative of the given Price-Demand equation:

D ' (p) = - 8p + 3

Let p = 20:

D ' (20) = - 8(20) + 3

D ' (20) = - 160 + 3

D ' (20) = -157

ANSWER: D ' (20) = - 157