Question

Considering the following Demand curves y1 = 300-3p1 + 5p2 and y2 = 400-10p1 + 8p2 corresponding to differentiated but related goods and whose choice is simultaneous.

a) Write the equilibrium price equation.
b) Calculate the equilibrium price and the quantities sold of good 1 and good.

Answer 1

From the mathematical standpoint the system is indeterminate - there are four variables to solve for, and only two equations. However with the assumption of a closed system and some economic intuition, we may arrive at a tentative solution.

First, in a closed system with two commodities it is the relative price ratio (p1 / p2) that matters, not the individual prices. We can set the p2 = 1, i.e. assume the second commodity is a numeraire good i.e. an item in terms of which other prices are quoted. This yields

y1 = 305 - 3p || y2 = 408 - 10p

=> (y2 - y1) = 103 - 7p

Thus the equilibrium price equation is: [ S2(p) - S1(p) = 103 - 7p ]

Still indeterminate, unless we consider the information about differentiated but related goods. In the absence of better information, we can assume the supply functions of the commodities (which is a really a proxy for firms' cost functions) to be identical i.e. their production cost structures are identical.

So at a given price (or price ratio) p, y1 = S1(p) = S2(p) = y2

=> ( y2 - y1) = 0 => [ p = 103/7 ] => p1 = 103 and p2 = 7 (not necessarily as p1/p2 = 103/7 defines an infinite number of 2-place price vectors but any one is as good as the other)

Equilibrium quantities are equal with y1 = y2 = 1826/7