For the demand function ,find the following.q=D(p)=500(p+7)2

Question

a) The elasticity

b) The elasticity at p,stating whether the demand is elastic, inelastic or has unit elasticity=2

c) The value(s) of p for which total revenue is a maximum (assume that p is in dollars)

Answer 1

something is very odd about this "Demand Curve" It looks more like a Supply Curve

as price rises, quantity demanded goes up and total revenue infinity..

This might be the case with a "Giffen good" a good so inferior that consumers buy more of it, the higher the price, as they switch out of every other "luxury" good to buy more of the most basic necessity. Ireland's potato famine is the usual example. When the Irish were so poor that as potato prices rose, they bought more potatoes and less of everything else. But in reality, there's a limit, as purchases are constrained by income at some price level. So, this "demand curve" can only explain part of the actual demand.

another example is the upward sloping supply of labor curve. As wages rise, you work less because of the wealth effect. You're getting read to retire, or the illegal alien sending home money to Mexico, at some point he's earned enough to go home and live in relative luxury given lower cost of living in Mexico compared to the US.

q = 500(p+7)^2

total revenue = pq = 500p(p+7)^2 = 500p(p^2 +14p + 49)= 500p^3 +7000p^2 + 24500

which has no maximum real number, just a surreal number, infinity as price rises

elasticity of demand is percentage change in quantity demanded over percentage change in price

or point elasticity of demand is dq/q over dp/p = (dq)p/(dp)q

take the derivative to solve for point elasticity

dq/dp = 1000(p+7) =1000p + 7000 > 1, highly elastic

unit elastic only at a negative price of -$6.999 when dq/dp =1

in elastic only as the negative price gets even more negative, ie paying people more than $7 to take the product off your hands, that's sort of like garbage.

Odds are you left out a sign somewhere as the normal demand curve has price and quantity inversely related.

maybe you meant q =- 500(7-p)^2

then total revenue = pq= 500p(7-p)^2 =500p(49-14p+p^2) = 3495p -7000p^2 +500p^3

R' = 3495 - 14000p + 1500p^2 =0

= 699 - 2800p + 300p^2

p=2800/1398 + or - (1/1398)sqr(2800^2 -4(699)(300))=about 3.90 = revenue maximizing price, probably $3.90

or maybe $390 depending on the product.