The demand for good $x$ is given by

$Q_x^d=1200-\frac{1}{2}{P}_x+\frac{1}{4}{P}_y-8P_z-\frac{1}{10}M$

Research shows that the prices of related goods are given by $P_y=$$$5,900$ and $P_z=$$$90,$ while the average income of individuals consuming this product is $M=$$$55,000.$

a$.$ Indicate whether goods Y and Z are substitutes or complements for good X $.$

b$.$ Is X an inferior or a normal good?

c$.$ How many units of good $x$ will be purchased when $P_x=$$$4,910?$

d$.$ Determine the demand function and inverse demand function for good X $.$ Graph the demand curve for good $x.$

$2.$ The demand curve for product $x$ is given by $Q_x^d=460-4Px.$

a$.$ Find the inverse demand curve

b$.$ How much consumer surplus do consumers receive when $Px=$$$35?$

c$.$ How much consumer surplus do consumers receive when $Px=$$$25?$

d$.$ In general, what happens to the level of consumer surplus as the price of a good falls?

$3.$ Suppose demand and supply are given by $Q_d=50-P$ and $Q_s=\frac{1}{2}P-10.$

a$.$ What are the equilibrium quantity and price in this market?

b$.$ Determine the quantity demanded, the quantity supplied, and the magnitude of the surplus if a price floor of $$42$ is imposed in this market.

c$.$ Determine the quantity demanded, the quantity supplied, and the magnitude of the shortage if a price ceiling of $$30$ is imposed in this market.

Also, determine the full economic price paid by consumers.