Answer and Explanation:
To find the Hicksian substitution and income effects of the price change on the first good, we need to follow these steps:
1. Find the optimal allocation at $1 for both goods:
- Use the utility function u(x1, x2) = √(x1 * x2) to maximize utility, subject to the budget constraint p1x1 + p2x2 = I.
- At $1 per unit for both goods, the budget constraint becomes x1 + x2 = 80 (where I is $80).
- Solve for the optimal quantities x1 and x2 using the Lagrangian method, taking partial derivatives and setting them equal to zero.
- This will give us the optimal allocation at $1 for both goods.
2. Find the optimal allocation when good 1 costs $4:
- Adjust the budget constraint to reflect the new price, which becomes 4x1 + x2 = 80.
- Repeat the steps from above to find the optimal quantities x1 and x2 at the new prices.
3. Calculate the Hicksian substitution and income effects:
- The Hicksian substitution effect measures the change in the quantity demanded of a good due to a change in its relative price, while keeping utility constant.
- To calculate the Hicksian substitution effect for good 1, subtract the initial quantity demanded from the new quantity demanded at the new price, while keeping utility constant.
- The income effect measures the change in the quantity demanded of a good due to a change in income, while keeping prices constant.
- To calculate the income effect for good 1, subtract the initial quantity demanded from the new quantity demanded at the new price, while keeping income constant.
By following these steps, you will be able to find the Hicksian substitution and income effects of the price change on the first good.