computer science

Question $2[36$ marks$]$

A mass of consumers is uniformly distributed along the interval $0,1.$ Two firms, A and B$,$ are located at points $0$ and $1$ respectively. We denote by $p_i$ the price of firm iinA,$B.$ A consumer located at point xin$[0,1]$ obtains utility $U_A(x)=u-p_A-t{x}^2$ if he consumes from firm $A,$ and $U_B(x)=u-p_B-t(1-x{)}^2$ if he consumes from firm B$.$ In the following, we assume that the gross utility $u$ is sufficiently high, so that the market will be covered and all consumers will get positive utility in equilibrium. Both firms have a cost function equal to $T_i(q_i)=(1+x)q_i,$ where you should substitute $x$ for the last number of your student ID number.

$($a$)$ Find the demand function for both firms.

marks$]$

$($b$)$ Assume firms set their prices simultaneously. Solve for the Nash equilibrium prices, and compute the equilibrium profits.

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