The demand function of a good is given by Q $=$ a $$ bP $$ cPA $+$ dY where P is the price of the good, PA is the price of an alternative good, Y is income and the coecients a$,$ b$,$ c and d are all positive. It is known that P $=50,$ PA $=30,$ Y $=1000$ and Q $=5000.($a$)$ Is the alternative good substitutable or complementary? Give a reason for your answer. $($b$)$ Find expressions, in terms of b$,$ c and d$,$ for the $($i$)$ price elasticity of demand; $($ii$)$ cross$-$price elasticity of demand; $($iii$)$ income elasticity of demand. $($c$)$ The cross$-$price elasticity is $0\mathrm{.}012.$ The income elasticity is four times the magnitude of the price elasticity. When income increases by $10\%,$ the demand increases by $2\%.$ Determine the values of a$,$ b$,$ c and d$.$