Suppose that a consumer has utility given by U(X, Y) = XY + 10Y and income of $100 to spend on goods X and Y.
a. The prices of X and Y are both $1 per unit. Use a Lagrangian to solve for the optimal basket of goods.
b. Suppose that the price of X increases to $5 per unit. Use a Lagrangian to solve for the new optimal basket of goods. Find 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 of each good. What income would the consumer need to attain the original level of utility when the price of X increases to $5 per unit?
d. Find the income effect of the increase in the price of good X on the consumption of each good. Are the goods normal or inferior? Explain.
e. Show that the total effect of the increase in the price of X is equal to the sum of the substitution effect and the income effect.