# optimization problem for two individuals

Two flat mates 1 and 2, rent a flat and play their own music on the only CD player owned by flat-owner. They both like their own music, but dislike the music played by the other. Given the timing constraints each one must play her own music when the other person is also present. Let $$m_i$$ denote the amount of music played by i, a nd $$Y_i$$ denote her amount of money holding. Individual i's utility function is

$$U_i(m_1,m_2,Y_i)=8m_i-2m_i^2-3m_j^2/2+Y_i$$, i,j=1,2.

a)how much music would each individual play? What is the efficient amount of music for each individual? is the amount of music actually played more or less than the efficient level? Explain the economic intuition for your answer.

b)Suppose that individual 2 is considering to gift a headphone to her flat-mate on her birthday. Assume that she doesn't get any utility from just gifting. What is the maximum price that she is willing to pay for the headphone?

c)Suppose that price of headphone is 11\$. Does it make sense for the two flat-mates to jointly buy a headphone, sharing the price equally and making a binding commitment that they would each listen to their own music only via the headphone?

d) Now suppose that the CD player is owned by individual 1 so that she can prevent individual 2 from playing any music at all. Suppose that individual can offer a take it or leave it contract that looks as follows:

"I shall play music at level $$m_1^-$$ and you can play music at level $$m_2^-$$ in return for a sum of T dollars".

In case the offered contract is rejected individual 1 selects $$m_1$$ unilaterally and individual 2 cannot play any music of her choice. Solve for the optimum levels of $$m_1^-$$, $$m_2^-$$ and T. Discuss the economic intuition for your answer.

my attempt:

max $$U_1=8m_1-2m_1^2-3m_2^2/2+Y_1$$

such that $$m_1+m_2=T$$, where T is total time.

this gives $$m_1^*=(8+3T/7)$$ and $$m_2^*=(4T-8/7)$$

As for efficient amount of music I suppose i've to maximize total utility $$U=U_1+U_2$$, but that makes me realize things don't add up, i may have taken a wrong constraint where i should have allowed for (i don't know, maybe) work hours (otherwise m_1+m_2=T and then what do i differentiate U with respect to?), but i'm not sure how to do that.

as for part b) I suppose i've to calculate the monetary value of dis-utility from other individual's music, but how do i do that?