I have a problem solving 6c (last picture). I've attached all my work till this point.
Can you help me to caluclate substitution effect of the increase in the price of good X, and income that I will need to attain orginal level of utility? Thank you
6. Assume you have $100 to spend on two goods, X and Y. You also know that your utility is given by the equation U = XY + 10X.
a. Assume that the prices of X and Y are each $1. Set up a Lagrangian and find your optimal consumption bundle.
b. If the price of good X rises to $5, what is the new optimal bundle? What is the total effect of the price change on the consumption of each good?
c. Use a Lagrangian to find the substitution effect of the increase in the price of good X on the consumption in each good. What income would you need in order to attain your original level of utility when the price of X rises to $5?
d. What is the income effect of the price increase? Are X and Y normal or inferior? Explain your answer.
e. Show that the total effect you found in part b is equal to the sum of the substitution and income effects you found in in part c and d.