As discussed in class, public goods are non-rival. If I "consume" some knowledge, then you too can consume it at the same time!

We have 2 students in a class. Everytime someone asks a question in class, each student gets a benefit ofb.

(a) The marginal individual benefit (to each student) isb(as stated above). A

student faces a cost

C(x) =x^2/2

if she asksxquestions. (Assume questions are infinitely divisible - you can ask a fraction of a question.) Suppose student A asksxAnumber of questions, andBasksxBnumber of questions. How many questions should each student ask if she/he just cares about her/his own benefit from asking a question? What is the total number of questions asked in class?

(b) The total social benefit (to the class as a whole) from 1 question (from either student) is2b. Do you agree? What is the total social benefit from a total ofx number of questions, wherex=x_A+x_B

(c) What is the total (social) benefit net of total costs ifAaskedx_Aquestions and Baskedx_Bquestions?

(d) What is the socially optimal number of questions? You'll have to maximize the expression in part (c) by jointly choosingx_Aandx_B. (If it helps, the mathematical structure of this problem is similar to a competitive firm with 2 plants selling in 1 market - the last question in PS 3.)

(e) Is there a different between the total number of questions asked (if the students behaved selfishly) and the socially optimal number of questions? Why?