Suppose that the price elasticity of demand for cereal is -0.75 and the crossprice elasticity of demand between cereal and the price of milk is -0.9. If the price of milk rises by 10%, what would have to happen to the price of cereal to exactly offset the rise in the price of milk and leave the quantity of cereal demanded unchanged?

 To find out what would have to happen to the price of cereal to offset the rise in the price of milk and leave the quantity of cereal demanded unchanged, we need to use the concept of price elasticity of demand and cross-price elasticity of demand.

The price elasticity of demand measures the responsiveness of the quantity demanded of a good to a change in its price.

The formula for the price elasticity of demand is: \[ Elasticity = \frac{\text{% Change in Quantity Demanded}}{\text{% Change in Price}} \]

Rearranging the formula to find the % Change in Quantity Demanded, we get: \[ \text{% Change in Quantity Demanded} = \text{Elasticity} \times \text{% Change in Price} \]

The cross-price elasticity of demand measures how the quantity demanded of one good (cereal) responds to a change in the price of another good (milk).

For cereal, we are given: - Price Elasticity of Demand for cereal = -0.75 - Cross-price Elasticity of Demand between cereal and the price of milk = -0.9 - The price of milk rises by 10%

First, let's calculate the change in the quantity demanded of cereal due to the change in the price of milk using the cross-price elasticity of demand.

\[ \text{% Change in Quantity of Cereal Demanded} = \text{Cross-price Elasticity of Demand} \times \text{% Change in Price of Milk} \] \[ \text{% Change in Quantity of Cereal Demanded} = (-0.9) \times (10\%) \] \[ \text{% Change in Quantity of Cereal Demanded} = -9\% \]

This means that with the price of milk increasing by 10%, the quantity demanded for cereal would decrease by 9%, if there were no other changes in the market.

However, we want the quantity demanded for cereal to be unchanged. So, we must adjust the price of cereal in such a way that it offsets this 9% decrease.

Now, we'll calculate the % Change in Price of cereal that would exactly offset the 9% decrease in quantity demanded using the price elasticity of demand for cereal.

Using the price elasticity of demand formula: \[ \text{% Change in Quantity Demanded of Cereal} = \text{Price Elasticity of Demand for Cereal} \times \text{% Change in Price of Cereal} \]

Since we want the quantity demanded to remain the same, we want to find out the % Change in Price of Cereal that would lead to a 9% increase in quantity demanded (to cancel out the -9% from before).

Let's set the % Change in Quantity Demanded of Cereal to 9% and solve for the % Change in Price of Cereal: \[ 9\% = -0.75 \times \text{% Change in Price of Cereal} \] \[ \text{% Change in Price of Cereal} = \frac{9\%}{-0.75} \] \[ \text{% Change in Price of Cereal} \approx -12\% \]

This means that to offset the 10% increase in the price of milk and leave the quantity of cereal demanded unchanged, the price of cereal would have to decrease by approximately 12%.