Question

Suppose there are only two individuals in the market for some product. Individual A’s inverse demand equation is P = 8 - 0.5 QD and B’s inverse demand equation is P = 10 - QD

a) Derive the equation showing A’s quantity demanded as a function of price. Ie: A’s demand equation

b) Derive B’s demand equation

c) Derive the aggregate (market) inverse demand equation. Ie: the equation showing price as a function of aggregate demand

d) Derive the aggregate (market) demand equation

Answer 1

a) A's demand equation

$P = 8 - 0.5QD.$

$P-8=-0.5QD$

$\frac{P-8}{-0.5}=QD$

$-2P+16=QD$

Therefore, A's demand equation is $QD=16-2P$

b) B's demand equation

$P = 10 - QD$

$P-10=-QD$

$-P+10=QD$

Thus, B's demand equation is $QD=10-P$

c) since demand cannot be negative, A enters the market when price is less than 8 and B when price is less than 10. Therefore, when price is between 8 and 10 the aggregate inverse demand is given by $P=10-QD$ (B's inverse demand equation). But when price is less than 8, the aggregate inverse demand equation is given by;

$\frac{P-8}{-0.5}=QD+P-10$

$P-8=-0.5QD-0.5P+5$

$1.5P=13-0.5QD$

$P=\frac{26-QD}{3}$

d) the aggregate demand when price is between 8 and 10 is given by $QD=10-P$ (B's demand equation). But when price is less than 8, the aggregate demand is given by $QD=10-P+16-2P$

$QD=26-3p$