6. What is the formula for price elasticity of demand? What the one for the income elasticity of demand? Finally, what is the cross price elasticity of demand?

\(D: Q(P)\)
Price Elasticity of Demand Formula
\(Q=D(P)\)
\(\epsilon_P=\frac{\Delta Q}{Q}=\frac{D(P+\Delta P)-D(P)}{D(P)}\)
\(\frac{\Delta Q}{Q}\frac{P}{\Delta P}=\frac{D(P+\Delta P)-D(P)}{D(P)}\frac{P}{\Delta P}=\frac{D(P+\Delta P)-D(P)}{\Delta P}\frac{P}{D(P)}\)
Income Elasticity of Demand:
\(\epsilon_R=R^{'}(P)\frac{P}{R(P)}\)
\(R^{'}(P)=\frac{d (P\cdot Q(P))}{dP}=Q(P)+PQ^{'}(P)\)
 \(\epsilon_R=\frac{Q(P)}{Q(P)}+Q^{'}(P)\frac{P}{Q(P)}\)
 \(\epsilon_R=1+\epsilon_P\)
Cross Price Elasticity of Demand
\(\frac{dQ}{dP'}\cdot\frac{P'}{Q}\)