Alright, let's break down this question step by step.
### Part a:
We need to determine by what percentage the quantity demanded of genetic testing will increase if consumers' income increases by 5 percent.
1. Income Elasticity of Demand: This measures how sensitive the quantity demanded of a good is to a change in consumers' income. We are given that the income elasticity of demand is 0.1.
2. Income Increase: Consumers' income increases by 5 percent.
To find the percentage change in the quantity demanded, we use the formula:
[tex]\[ \text{Percentage Change in Quantity Demanded} = \text{Income Elasticity of Demand} \times \text{Percentage Change in Income} \][/tex]
Plugging in the given values:
[tex]\[ \text{Percentage Change in Quantity Demanded} = 0.1 \times 5 = 0.5 \][/tex]
So, the quantity demanded of genetic testing will increase by 0.5 percent.
### Part b:
We need to determine the new number of procedures demanded after health insurance starts to cover 50 percent of the procedure's price.
1. Price Elasticity of Demand: This measures how sensitive the quantity demanded of a good is to a change in its price. We are given that the price elasticity of demand is 0.4.
2. Initial Price and Quantity: The initial price of the procedure is $5,000, and initially, there are 25,000 procedures done each year.
3. Change in Price: Health insurance will cover 50 percent of the procedure's price, so the new price consumers pay is:
[tex]\[ \text{New Price} = 5000 \times 0.5 = 2500 \][/tex]
4. Percentage Change in Price: The percentage change in the price is:
[tex]\[ \text{Percentage Change in Price} = \frac{\text{New Price} - \text{Initial Price}}{\text{Initial Price}} \times 100 \][/tex]
[tex]\[ \text{Percentage Change in Price} = \frac{2500 - 5000}{5000} \times 100 = -50 \][/tex]
The price decreased by 50 percent.
To find the percentage change in quantity demanded, we use the formula:
[tex]\[ \text{Percentage Change in Quantity Demanded} = \text{Price Elasticity of Demand} \times \text{Percentage Change in Price} \][/tex]
[tex]\[ \text{Percentage Change in Quantity Demanded} = 0.4 \times (-50) = -20 \][/tex]
To find the new quantity demanded, we apply this percentage change to the initial quantity:
[tex]\[ \text{New Quantity Demanded} = \text{Initial Quantity} \times \left(1 + \frac{\text{Percentage Change in Quantity Demanded}}{100}\right) \][/tex]
[tex]\[ \text{New Quantity Demanded} = 25000 \times \left(1 + \frac{-20}{100}\right) = 25000 \times 0.8 = 20000 \][/tex]
So, following the change in health insurance coverage, the total number of procedures demanded will be 20,000 procedures.
