Question

A group of retailers will buy 120 televisions from a wholesaler if the price is $375 and 160 if the price is $325. The wholesaler is willing to supply 88 if the price is $320 and 168 if the price is $410. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market. Find (q,p).

Answer 1

Let the demand and the supply equations be y = d(p) = ap+b and y = s(p) = cp+d respectively where $ p is the price of a television.

Since there is a demand for 120 televisions from a wholesaler if the price is $375 and 160 if the price is $325, hence a = (160-120)/(325-375) = -40/50 = -4/5. Hence d(p) = -4p/5 +b. Further, on substituting p = 375 and d(p) = 120, we get 120 = (-4/5)*375 +b or, b = 300+120 = 420. Hence, d(p) = -4p/5 +420.

Also, since there is a supply of 88 televisions from a wholesaler if the price is $320 and 168 if the price is $410, hence c = (168-88)/(410-320) = 80/90 = 8/9. Hence, s(p) = 8p/9+d. Further, on substituting p = 320 and s(p) =88, we get 88 = (8/9)*320 +d or, d = 88-2560/9 = -1768/9. Hence, s(p) = 8p/9- 1768/9.

The equilibrium point for the market will occur when d(p) = s(p). A graph of d(p) ( in red) and s(p) ( in blue) is attached. The 2 lines intersect at (365,128) ( the exact coordinates may be seen only in the original Desmos graph and not in an image).

Hence the equilibrium price is $ 365 and the equilibrium quantity is 168.