Consider a market with two firms, A and B, producing two varieties of the same product with different qualities. In particular, the variety produced by firm A has quality Va = 2, while the variety produced by firm B has quality Vb = 10. Firm A costs are TCa(qa) = 1/2*qa, while firm B costs are TCb(qb) = 2*qb Consumers value quality differently, with each individual i valuing quality bi , a number between 0 and 1. Consumers are distributed uniformly between [0,1]. (As considered in class: that is, for any two values b1 and b2 > b1 in [0,1] the number (or mass) of individuals with bi ∈ [b1, b2] is equal to b2 − b1.) a) Draw a graph illustrating the utility of different individuals (as a function of bi on the horizontal axis) for the following given value of (Pa, Pb) = (2/3 , 5) . Indicate which individuals do not buy anything, which individuals buy the variety from firm A, and which individuals buy the variety from firm B. b) What is the value of bi for which an individual is indifferent between buying from firm A or not buying at all? Show this at the prices (Pa, Pb) = (2/3 , 5) and generally as a function of (Pa, Pb) c) What is the value of bi for which an individual is indifferent between buying from firm B or buying from firm A? Show this at the prices (Pa, Pb) = (2/3 , 5) and generally as a function of (Pa, Pb).
Consider a market with two firms, A and B, producing two varieties of the same product with
different qualities. In particular, the variety produced by firm A has quality Va = 2, while the
variety produced by firm B has quality Vb = 10.
Firm A costs are TCa(qa) = 1/2*qa, while firm B costs are TCb(qb) = 2*qb
Consumers value quality differently, with each individual i valuing quality bi , a number
between 0 and 1. Consumers are distributed uniformly between [0,1]. (As considered in class:
that is, for any two values b1 and b2 > b1 in [0,1] the number (or mass) of individuals with
bi ∈ [b1, b2] is equal to b2 − b1.)
a) Draw a graph illustrating the utility of different individuals (as a function of bi on the
horizontal axis) for the following given value of (Pa, Pb) = (2/3 , 5) . Indicate which
individuals do not buy anything, which individuals buy the variety from firm A, and
which individuals buy the variety from firm B.
b) What is the value of bi for which an individual is indifferent between buying from firm
A or not buying at all? Show this at the
function of (Pa, Pb)
B or buying from firm A? Show this at the prices (Pa, Pb) = (2/3 , 5) and generally as a
function of (Pa, Pb).
d) Show that demand for the two varieties is (you can assume both are positive)
Qa(Pa, Pb) = Pb/8 − 5/8 *Pa
Qb(Pa, Pb) = 1 + Pa/8 − 1/8*Pb
e) Derive the Nash
fractions or approximate numbers to 2 decimal digits)
f) What would be the equilibrium prices and quantities if the firms merged (keeping the
two varieties separate, but choosing prices to maximize joint profits)?
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