Final answer:
The price elasticity of demand when P = $1 is -0.2, showing an inelastic demand, whereas at P = $2, it is -0.5, indicating more elastic demand. Furthermore, the cross-price elasticity of demand is 1, implying that the products are substitutes.
Step-by-step explanation:
To determine the price elasticity of demand for a product, we use the formula:
Elasticity of demand (E) = (% Change in Quantity Demanded) / (% Change in Price)
For part a. when P = $1.00:
The initial quantity demanded (Q) is calculated as Q = 10 - 2P + Ps. With P = $1 and Ps = $2, Q = 10 - 2(1) + 2 = 10.
If P increases by a small amount ΔP, then the new quantity demanded Q' = 10 - 2(P + ΔP) + 2. The change in quantity demanded is then ΔQ = -2ΔP.
The price elasticity of demand at P = $1 is E = (ΔQ/Q) / (ΔP/P) = (-2ΔP/10) / (ΔP/1) = -2/10 = -0.2. Therefore, the demand is inelastic at this price point.
For part b. when P = $2:
Q = 10 - 2(2) + 2 = 8. Again, if P increases by ΔP, then Q' = 8 - 2ΔP.
The price elasticity of demand at P = $2 is E = (ΔQ/Q) / (ΔP/P) = (-2ΔP/8) / (ΔP/2) = -0.5, indicating demand is more elastic than at the $1 price point.
To find the cross-price elasticity of demand, which measures how the quantity demanded of one good responds to a change in the price of another good, the formula used is:
Cross-price elasticity of demand (Ec) = (% Change in Quantity Demanded of Good 1) / (% Change in Price of Good 2)
The cross-price elasticity of demand when P goes to $2 is given by the coefficient of Ps in the demand equation, which is 1. Thus, Ec = 1, indicating that the goods are substitutes and the quantity demanded of the product increases when the price of the substitute increases.