$($a$)$ If the long$-$run cost function is c$($y$)=$ y$2+$ y $+1,$ what is the long$-$run supply curve of the firm?

$($b$)$ A taxi operator appears to be making positive profits in the long run after

carefully accounting for the operating and labor costs. Does this violate the competitive model? Why or why not?

$($c$)$ Show mathematically that a monopolist always sets its price above marginal cost. Explain thoroughly.

$($d$)$ If pMP $1>$ w$1,$ then should the firm increase or decrease the amount of factor $1$ in order to increase profits? Why? Explain thorougly.