Suppose that the quantity supplied S and quantity demanded D of T-shirts at a concert are given by the following functions, where p is the price of a T-shirt.
S(p) = -320 + 30p
D(p) = 1150 = 50p
(a) Find the equilibrium price for the T-shirts at this concert.
(b) What is the equilibrium quantity?
(c) Determine the prices for which quantity demanded is greater than quantity supplied
(d) What will eventually happen to the price of the T-shirts if the quantity demanded is greater than the quantity supplied?
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(a) The equilibrium price is found by setting the quantity supplied equal to the quantity demanded and solving for p:
-320 + 30p = 1150 - 50p
80p = 1470
p = 1470 / 80
p = $18.375
(b) The equilibrium quantity is found by substitifying the equilibrium price into either the supply or demand function:
S(18.375) = -320 + 30*18.375 = 232.25
or
D(18.375) = 1150 - 50*18.375 = 232.25
So, the equilibrium quantity is 232.25 T-shirts.
(c) The quantity demanded is greater than the quantity supplied when D(p) > S(p), or:
1150 - 50p > -320 + 30p
1470 > 80p
p < 1470 / 80
p < $18.375
So, the quantity demanded is greater than the quantity supplied when the price is less than $18.375.
(d) If the quantity demanded is greater than the quantity supplied, there is a shortage of T-shirts. This will cause the price to increase until it reaches the equilibrium price.
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