## An equilibrium supply and demand question via implicit differentiation

I don't entirely remember how these concepts go; the information is muddled by a few other topics. The question is listed below, word for word.

The demand and supply for a good are given by the linear functions

$D=30-2p-y$, $S=p$,

where $D$ is the quantity demanded, $S$ is the quantity supplied, $p$ is the price, and $y$ is the aggregate consumers' income, which is exogenous. Denote the equilibrium quantity by $Q$ and the equilibrium price by $P$.

(i) Find the effect of a change in income on equilibrium price and quantity, that is, $\frac{dQ}{dy}$ and $\frac{dP}{dy}$, by the method of implicit differentiation.

(ii) Check the result by first solving for the equilibrium price and quantity explicitly and then finding $\frac{dQ}{dy}$ and $\frac{dP}{dy}$.

(iii) Repeat (i) and (ii) for the following demand and supply functions

$D=a-bp+cy$, $S=\alpha+\beta p$,

with $a>0, b>0, c>0, \alpha>0, \beta>0, \alpha.

I'm doing my best to try and remember how this all works, but some help on the subject would be appreciated.